nonlinear predictive control for fast constrained systems by ahmed youssef
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Nonlinear Predictive Controlfor Fast Constrained Systems
By
Ahmed Youssef
What’s MBPC
N=N2-N1
Nu
CV
MV
N1 N2t
Introduction
CV: controlled variable MV: manipulated variable
IntroductionShortcomings of current industrial nonlinear MBPC•Computing the MBPC control law demands significant on-line computation effort •Inability to deal explicitly with the plant model uncertainty.
Objective of research workReducing the computational complexity of nonlinear MBPC & adding the robustness property whilst preserving its good attributes to make it more effective practical tool for controlling systems of fast-constrained dynamics.
Given the nonlinear dynamic model
Reformulate into nonlinear state-dependent form
This is not a linearisation
Trivial example
tttttttt xxCyuxBxxAx )()()(1
)(),(1 ttkkk xgyuxfx
)cos()(;)sin(
)(
)cos()sin(1
kkkk
kk
kkkkk
xxxBx
xxA
uxxxx
State Dependent State-Space Models
Hamilton-Jacobi-Bellman
uxgxfx )()(
0
)()()(min)( dtuxRuxxQxxV TT
tu
0)()(4
1)( 1
xQx
x
VxgRxg
x
Vxf
x
V
t
V T
T
T
T
The NLQGPC quadratic infinite horizon cost function:
The optimal control vector in terms of the states of the system and reference model:
11
, ( , ) ( , ) ( ) ( ) ( , )
1 2( , ) ( ) ( ) ( , ) ,ˆ
T T Tt N u t t N e t t N t t t t N e
TT Tt t N t t t N t t N e t N
U Q S Q S H S Q
A H A x H S Q R
N
jjtu
Tjtjtjte
Tjtjtt
T
tt
T
uQuryQryJ
JT
J
01111
01
1lim
NLQGPC Control Law
The Coupled Algebraic Riccati Equations
1 1 11 1
11 1
1 1
T TT Tj N e N j N e N j
T TT Tu N e N j N e N j
H A Q H A A Q S H
Q S Q S H S Q H A
2 2 11 1
11 2
1 1
TT T T Tj N e j N e N j
T TT Tu N e N j j N N e
H A Q A H A Q S H
Q S Q S H H S Q
Control Lyapunov Function
A C1 function V(x): n is said to be a discrete CLF for the system:
if V(x) is positive definite, unbounded, and if
for all x 0
Dealing with Stability Issue
Satisficing is based on a point-wise cost / benefit comparison of an action.
The benefits are given by the “Selectability” function Ps(u,x), while the costs are given by the
“Rejectability” function Pr(u,x).
The “satisficing” set is those options for which selectability exceeds rejectability:
Stability via Satisficing
Satisficing generates the state dependent set of controls that render the closed-loop system stable with respect to a known CLF.
uaug = uNLQGPC - ()(BTPB)-1BTPf
Start
uS
f, B, P,
Calculate
uNLQGPC
No
uS
implies
impl
ies
fPBBPBBPf
fPBuTTT
TTNLQGPC
1)(
fPBBPBBPf
xPxfPfTTT
TT
1)(
CLF-Based Satisficing Technique
• Magnitude Saturation
• Rate-Limited Actuators
• Actuator Dead-Zone
Therefore the common term is the Saturation function
,uu,u
,uuu,u
,uu,u
)u(satu
mininmin
maxinminin
maxinmax
inout
,u,
,u,u
,u,
)u(satu
mininmin
maxinminin
maxinmax
inout
)()( usatuuD dd
Dealing with Input Constraints
Examples of Actuator Constraints
u0 u
the actuator range of operation is limited
Limiting functions that map the interval (-,) onto (0, 1)
Limiting functions that map the interval (-,) onto (-1, 1)
Approximation of Magnitude Saturation
Error function (Blue)
Tanh function (Green)
Sigmoid function (Red)11S
10SSigmoid function (Black)
Approximation of Magnitude Saturation
Case Studies
F-8 fighter aircraft
F-16 fighter aircraft Caltech Ducted Fan
Controlling of F-8 Fighter
0 2 4 6 8 10 12 14-0.1
0
0.1
0.2
0.3
0.4
0.5
Time [s]
Ang
le o
f A
ttac
k [r
ad]
Unconstrained NLQGPC Constrained NLQGPC CNLQGPC-Satisficing (Nominal) CNLQGPC-Satisficing (Wind Gust)
0 2 4 6 8 10 12 14-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
Time [s]
Pitc
h R
ate
[rad
/s]
0 2 4 6 8 10 12 14-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
Time [s]
u [r
ad]
Elevator Deflection
radu 05236.0
-0.500.5
-1
0
1-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
x2
x1
Trajectories of Controlled System
x3
CONCLUSIONS
Properties of NLQGPC controller:
1. High performance
2. Less computational burden
3. Dealing with input constraints
4. Guaranteeing asymptotic stability to the closed-loop system.
5. Possesses both performance robustness & stability robustness
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