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Designs and Theory for State-ConstrainedNonlinear Feedback Controls forDelay Systems An Infomercial
Michael Malisoff Roy P Daniels Professor of MathematicsLouisiana State University
JOINT WITH ENGINEERING AND MATHEMATICS COLLEAGUES AND STUDENTS
SPONSORED BY NSFECCSEPAS AND NSFCMMISDC PROGRAMS
Applied Analysis SeminarLSU Department of Mathematics
November 17 2014
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
113
Collaborators Fumin Zhang from Georgia Tech and Miroslav Krstic (MK) from UCSDCurrent PhD Students Ruzhou Yang Ningshi Yao
Total 3-Year Budget $871000 (LSU Portion $380928)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn
We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function u
The functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty
D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ)
Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
113
Collaborators Fumin Zhang from Georgia Tech and Miroslav Krstic (MK) from UCSDCurrent PhD Students Ruzhou Yang Ningshi Yao
Total 3-Year Budget $871000 (LSU Portion $380928)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn
We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function u
The functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty
D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ)
Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn
We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function u
The functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty
D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ)
Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn
We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function u
The functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty
D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ)
Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn
We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function u
The functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty
D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ)
Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function u
The functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty
D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ)
Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty
D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ)
Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ)
Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ)
Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
213
What Do We Mean By Delayed Control Systems
These are doubly parameterized families of ODEs of the form
Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)
) Y (t) isin Y (1)
Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm
Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family
Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)
where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)
Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded
γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
313
What is One Possible Control Objective
Input-to-state stability generalizes global asymptotic stability
Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)(UGAS)
Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin
Y prime(t) = G(t Yt δ(t)
) Y (t) isin Y (Σpert)
|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])
)+ γ2(|δ|[t0t]) (ISS)
Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn)
and2 d
dt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example
The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D
Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
413
What is a Lyapunov-Krasovskii Functional (LKF)
Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that
1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0+infin)times C([minusτ0]Rn) and
2 ddt
[V ](t Yt )
]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)
along all trajectories of the system
Example The function V (Y ) = 12 |Y |
2 is an ISS-LKF forY prime(t) = minusY (t) + 1
4Y (t) + δ(t) for any D Fix τ gt 0
V ](Yt ) = V (Y (t)) + 14
int ttminusτ |Y (`)|2d`+ 1
8τ
int ttminusτ
[int ts |Y (r)|2dr
]ds
is an ISS-LKF for Y prime(t) = minusY (t) + 14Y (t minus τ) + δ(t)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
Background on NMES Rehabilitation
It artificially stimulates skeletal muscles to restore functionalityin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)
It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion
Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes
Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)
Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
(Loading Video)
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
513
NMES on Leg Extension Machine
Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delay
A =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i
where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi
The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller (continued)
Euler iterations used for control
zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where
z0 =
(tan (q(Ti))minus tan (qd (Ti))
q(Ti )cos2(q(Ti ))
minus qd (Ti )cos2(qd (Ti ))
) hi = τ
Ni
and Ω [0+infin)2 times R2 rarr R2 is defined by
Ω(T h x v) =
[Ω1(T h x v)Ω2(T h x v)
](7)
and the formulas
Ω1(T h x v) = x1 + hx2 and
Ω2(T h x v) = x2 + ζ2d (T ) +int T+h
T g1(ζd (s)+x)ds
+int T+h
T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
813
NMES Theorem (IK MM MK Ruzhou IJRNC)
For all positive constants τ and r there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with
Ni = N(∣∣∣(tan(q(Ti)) q(Ti )
cos2(q(Ti ))
)minus ζd (Ti)
∣∣∣+||v minus vd ||[Timinusτ Ti ]
) (8)
satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]
le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|
cos2(q(0)) + ||v0 minus vd ||[minusτ 0])
for all t ge 0
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delay
A =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i
where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi
The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller (continued)
Euler iterations used for control
zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where
z0 =
(tan (q(Ti))minus tan (qd (Ti))
q(Ti )cos2(q(Ti ))
minus qd (Ti )cos2(qd (Ti ))
) hi = τ
Ni
and Ω [0+infin)2 times R2 rarr R2 is defined by
Ω(T h x v) =
[Ω1(T h x v)Ω2(T h x v)
](7)
and the formulas
Ω1(T h x v) = x1 + hx2 and
Ω2(T h x v) = x2 + ζ2d (T ) +int T+h
T g1(ζd (s)+x)ds
+int T+h
T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
813
NMES Theorem (IK MM MK Ruzhou IJRNC)
For all positive constants τ and r there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with
Ni = N(∣∣∣(tan(q(Ti)) q(Ti )
cos2(q(Ti ))
)minus ζd (Ti)
∣∣∣+||v minus vd ||[Timinusτ Ti ]
) (8)
satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]
le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|
cos2(q(0)) + ||v0 minus vd ||[minusτ 0])
for all t ge 0
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delay
A =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i
where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi
The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller (continued)
Euler iterations used for control
zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where
z0 =
(tan (q(Ti))minus tan (qd (Ti))
q(Ti )cos2(q(Ti ))
minus qd (Ti )cos2(qd (Ti ))
) hi = τ
Ni
and Ω [0+infin)2 times R2 rarr R2 is defined by
Ω(T h x v) =
[Ω1(T h x v)Ω2(T h x v)
](7)
and the formulas
Ω1(T h x v) = x1 + hx2 and
Ω2(T h x v) = x2 + ζ2d (T ) +int T+h
T g1(ζd (s)+x)ds
+int T+h
T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
813
NMES Theorem (IK MM MK Ruzhou IJRNC)
For all positive constants τ and r there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with
Ni = N(∣∣∣(tan(q(Ti)) q(Ti )
cos2(q(Ti ))
)minus ζd (Ti)
∣∣∣+||v minus vd ||[Timinusτ Ti ]
) (8)
satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]
le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|
cos2(q(0)) + ||v0 minus vd ||[minusτ 0])
for all t ge 0
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i
where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi
The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller (continued)
Euler iterations used for control
zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where
z0 =
(tan (q(Ti))minus tan (qd (Ti))
q(Ti )cos2(q(Ti ))
minus qd (Ti )cos2(qd (Ti ))
) hi = τ
Ni
and Ω [0+infin)2 times R2 rarr R2 is defined by
Ω(T h x v) =
[Ω1(T h x v)Ω2(T h x v)
](7)
and the formulas
Ω1(T h x v) = x1 + hx2 and
Ω2(T h x v) = x2 + ζ2d (T ) +int T+h
T g1(ζd (s)+x)ds
+int T+h
T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
813
NMES Theorem (IK MM MK Ruzhou IJRNC)
For all positive constants τ and r there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with
Ni = N(∣∣∣(tan(q(Ti)) q(Ti )
cos2(q(Ti ))
)minus ζd (Ti)
∣∣∣+||v minus vd ||[Timinusτ Ti ]
) (8)
satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]
le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|
cos2(q(0)) + ||v0 minus vd ||[minusτ 0])
for all t ge 0
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i
where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi
The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller (continued)
Euler iterations used for control
zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where
z0 =
(tan (q(Ti))minus tan (qd (Ti))
q(Ti )cos2(q(Ti ))
minus qd (Ti )cos2(qd (Ti ))
) hi = τ
Ni
and Ω [0+infin)2 times R2 rarr R2 is defined by
Ω(T h x v) =
[Ω1(T h x v)Ω2(T h x v)
](7)
and the formulas
Ω1(T h x v) = x1 + hx2 and
Ω2(T h x v) = x2 + ζ2d (T ) +int T+h
T g1(ζd (s)+x)ds
+int T+h
T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
813
NMES Theorem (IK MM MK Ruzhou IJRNC)
For all positive constants τ and r there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with
Ni = N(∣∣∣(tan(q(Ti)) q(Ti )
cos2(q(Ti ))
)minus ζd (Ti)
∣∣∣+||v minus vd ||[Timinusτ Ti ]
) (8)
satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]
le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|
cos2(q(0)) + ||v0 minus vd ||[minusτ 0])
for all t ge 0
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i
where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi
The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller (continued)
Euler iterations used for control
zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where
z0 =
(tan (q(Ti))minus tan (qd (Ti))
q(Ti )cos2(q(Ti ))
minus qd (Ti )cos2(qd (Ti ))
) hi = τ
Ni
and Ω [0+infin)2 times R2 rarr R2 is defined by
Ω(T h x v) =
[Ω1(T h x v)Ω2(T h x v)
](7)
and the formulas
Ω1(T h x v) = x1 + hx2 and
Ω2(T h x v) = x2 + ζ2d (T ) +int T+h
T g1(ζd (s)+x)ds
+int T+h
T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
813
NMES Theorem (IK MM MK Ruzhou IJRNC)
For all positive constants τ and r there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with
Ni = N(∣∣∣(tan(q(Ti)) q(Ti )
cos2(q(Ti ))
)minus ζd (Ti)
∣∣∣+||v minus vd ||[Timinusτ Ti ]
) (8)
satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]
le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|
cos2(q(0)) + ||v0 minus vd ||[minusτ 0])
for all t ge 0
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
613
Mathematical ModelMI (q)︷︸︸︷Jq +
Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +
Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)
+Mgl sin (q)︸ ︷︷ ︸Mg (q)
= A (q q) v (t minus τ) q isin (minusπ2
π2 )
(3)
J andM are inertia and mass of the lower limbmachine thebi rsquos and ki rsquos are positive damping and elastic constantsrespectively l is the distance between the knee joint and thecenter of the mass of the lower limbmachine τ gt 0 is a delayA =scaled moment arm v = voltage potential control
q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)
qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (5)
max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (6)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i
where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi
The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller (continued)
Euler iterations used for control
zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where
z0 =
(tan (q(Ti))minus tan (qd (Ti))
q(Ti )cos2(q(Ti ))
minus qd (Ti )cos2(qd (Ti ))
) hi = τ
Ni
and Ω [0+infin)2 times R2 rarr R2 is defined by
Ω(T h x v) =
[Ω1(T h x v)Ω2(T h x v)
](7)
and the formulas
Ω1(T h x v) = x1 + hx2 and
Ω2(T h x v) = x2 + ζ2d (T ) +int T+h
T g1(ζd (s)+x)ds
+int T+h
T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
813
NMES Theorem (IK MM MK Ruzhou IJRNC)
For all positive constants τ and r there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with
Ni = N(∣∣∣(tan(q(Ti)) q(Ti )
cos2(q(Ti ))
)minus ζd (Ti)
∣∣∣+||v minus vd ||[Timinusτ Ti ]
) (8)
satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]
le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|
cos2(q(0)) + ||v0 minus vd ||[minusτ 0])
for all t ge 0
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i
where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi
The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller (continued)
Euler iterations used for control
zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where
z0 =
(tan (q(Ti))minus tan (qd (Ti))
q(Ti )cos2(q(Ti ))
minus qd (Ti )cos2(qd (Ti ))
) hi = τ
Ni
and Ω [0+infin)2 times R2 rarr R2 is defined by
Ω(T h x v) =
[Ω1(T h x v)Ω2(T h x v)
](7)
and the formulas
Ω1(T h x v) = x1 + hx2 and
Ω2(T h x v) = x2 + ζ2d (T ) +int T+h
T g1(ζd (s)+x)ds
+int T+h
T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
813
NMES Theorem (IK MM MK Ruzhou IJRNC)
For all positive constants τ and r there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with
Ni = N(∣∣∣(tan(q(Ti)) q(Ti )
cos2(q(Ti ))
)minus ζd (Ti)
∣∣∣+||v minus vd ||[Timinusτ Ti ]
) (8)
satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]
le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|
cos2(q(0)) + ||v0 minus vd ||[minusτ 0])
for all t ge 0
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi
The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller (continued)
Euler iterations used for control
zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where
z0 =
(tan (q(Ti))minus tan (qd (Ti))
q(Ti )cos2(q(Ti ))
minus qd (Ti )cos2(qd (Ti ))
) hi = τ
Ni
and Ω [0+infin)2 times R2 rarr R2 is defined by
Ω(T h x v) =
[Ω1(T h x v)Ω2(T h x v)
](7)
and the formulas
Ω1(T h x v) = x1 + hx2 and
Ω2(T h x v) = x2 + ζ2d (T ) +int T+h
T g1(ζd (s)+x)ds
+int T+h
T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
813
NMES Theorem (IK MM MK Ruzhou IJRNC)
For all positive constants τ and r there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with
Ni = N(∣∣∣(tan(q(Ti)) q(Ti )
cos2(q(Ti ))
)minus ζd (Ti)
∣∣∣+||v minus vd ||[Timinusτ Ti ]
) (8)
satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]
le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|
cos2(q(0)) + ||v0 minus vd ||[minusτ 0])
for all t ge 0
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller
v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))
for all t isin [Ti Ti+1] and each i where
g1(x) = minus(1 + x21 )dF
dq (tanminus1(x1)) +2x1x2
21+x2
1minus (1 + x2
1 )H(
x21+x2
1
)
g2(x) = (1 + x21 )G
(tanminus1(x1) x2
1+x21
)
ζd (t) = (ζ1d (t) ζ2d (t)) =(
tan(qd (t)) qd (t)cos2(qd (t))
)
ξ1(t) = eminusmicro(tminusTi )
(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)
+ ξ1(Ti) cos(t minus Ti)
ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)
)sin(t minus Ti)
+ ξ2(Ti) cos(t minus Ti)
and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller (continued)
Euler iterations used for control
zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where
z0 =
(tan (q(Ti))minus tan (qd (Ti))
q(Ti )cos2(q(Ti ))
minus qd (Ti )cos2(qd (Ti ))
) hi = τ
Ni
and Ω [0+infin)2 times R2 rarr R2 is defined by
Ω(T h x v) =
[Ω1(T h x v)Ω2(T h x v)
](7)
and the formulas
Ω1(T h x v) = x1 + hx2 and
Ω2(T h x v) = x2 + ζ2d (T ) +int T+h
T g1(ζd (s)+x)ds
+int T+h
T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
813
NMES Theorem (IK MM MK Ruzhou IJRNC)
For all positive constants τ and r there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with
Ni = N(∣∣∣(tan(q(Ti)) q(Ti )
cos2(q(Ti ))
)minus ζd (Ti)
∣∣∣+||v minus vd ||[Timinusτ Ti ]
) (8)
satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]
le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|
cos2(q(0)) + ||v0 minus vd ||[minusτ 0])
for all t ge 0
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
713
Voltage Potential Controller (continued)
Euler iterations used for control
zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where
z0 =
(tan (q(Ti))minus tan (qd (Ti))
q(Ti )cos2(q(Ti ))
minus qd (Ti )cos2(qd (Ti ))
) hi = τ
Ni
and Ω [0+infin)2 times R2 rarr R2 is defined by
Ω(T h x v) =
[Ω1(T h x v)Ω2(T h x v)
](7)
and the formulas
Ω1(T h x v) = x1 + hx2 and
Ω2(T h x v) = x2 + ζ2d (T ) +int T+h
T g1(ζd (s)+x)ds
+int T+h
T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
813
NMES Theorem (IK MM MK Ruzhou IJRNC)
For all positive constants τ and r there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with
Ni = N(∣∣∣(tan(q(Ti)) q(Ti )
cos2(q(Ti ))
)minus ζd (Ti)
∣∣∣+||v minus vd ||[Timinusτ Ti ]
) (8)
satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]
le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|
cos2(q(0)) + ||v0 minus vd ||[minusτ 0])
for all t ge 0
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
813
NMES Theorem (IK MM MK Ruzhou IJRNC)
For all positive constants τ and r there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with
Ni = N(∣∣∣(tan(q(Ti)) q(Ti )
cos2(q(Ti ))
)minus ζd (Ti)
∣∣∣+||v minus vd ||[Timinusτ Ti ]
) (8)
satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]
le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|
cos2(q(0)) + ||v0 minus vd ||[minusτ 0])
for all t ge 0
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)
+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2
π2 )
(9)
τ = 007s A(q q) = aeminus2q2sin(q) + b
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(10)
qd (t) =π
8sin(t) (1minus exp (minus8t)) rad (11)
q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
913
First Simulation
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
We took τ = 007s and the same model parameters
J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m
(12)
qd (t) =π
3(1minus exp (minus3t)) rad (13)
q(0) = π18 q(0) = v0(t) = 0 Ni = N = 10 Ti+1 minus Ti = 0014
We used these mismatched parameters in the control
J prime = 125J bprime1 = 12b1 bprime2 = 09b2 aprime = 1185abprime3 = 085b3 k prime1 = 11k1 k prime2 = 0912k2 bprime = 098bk prime3 = 09k3 Mprime = 097M and l prime = 1013l
(14)
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1013
Simulated Robustness Test
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1113
Summary of NMES Research
NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders
It produces difficult tracking control problems that containdelays state constraints and uncertainties
Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories
By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value
Our control used a new numerical solution approximationmethod that covers many other time-varying models
In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1213
Openings for PhD Students
I have openings for PhD students that each pay up to $22000per school year plus a tuition remission plus certain travel
One would be co-advised by Miroslav Krstic Daniel L AlspachEndowed Chair in Dynamic Systems and Control at UCSD
Miroslav and I are studying time delayed state constrainedordinary and hyperbolic PDEs using predictive controls
Potential engineering applications include laser pulse shapingNMES and slugging in oil production
Even though my projects are sponsored by NSF Engineeringyou do not need any engineering background to be considered
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
I also have at least one opening on my joint project with FuminZhang from Georgia Tech ECE on a totally different topic
Fumin is a rising star who won an ONR YIP Award a LockheedInspirational Young Faculty Award and an NSF CAREER grant
Our project is on model predictive control which is an excitingmix of controls and optimization on a receding time horizon
Fumin and I are developing a new timing model to help resolvecontentions on computer networks eg in cars
It uses event-triggered controls and robust forward invarianceto help determine how much uncertainty a system can tolerate
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays
1313
Openings for PhD Students (contrsquod)
malisofflsuedu 225-578-6714
Malisoff (LSU) and his Colleagues and Students State-Constrained Nonlinear Feedback Controls with Delays