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Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 1
Lecture Notes onMinimum Time Control
Prof. Guy BealeElectrical and Computer Engineering
George Mason UniversityFairfax, Virginia
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 2
Objective
Find the optimal control to transfer the system from a given initial condition x(t0) to a (possibly moving) target set S[x(T),T] in the shortest possible time.
00
T
tJ dt T t= = −∫
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 3
Constraints
System Equations (nonlinear system, but linear in the control):
Boundary Conditions:
Control Constraints:
( ) [ ( ), ] [ ( ), ] ( )x t f x t t B x t t u t= +
( )( ) ( )
0 0
* * *,
x t x
x T S x T T
=
∈
[ ] [ ]0( ) 1 , , 1,iu t t t T i m≤ ∈ ∈
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 4
Reachability
What states can be reached from the initial condition in a finite time?Definitions:• u[t0,t] – admissible control in the time interval
[t0,t].• At – those states that can be reached at time t from
x(t0) by an admissible control. For each t, set is closed and bounded, and At grows with time.
• St – set of target states at time t.
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 5
At1
At2
At3
At4
At5x(t0)
St0
St1
St2 St3
St4
St5
T* = t4 = minimum time = time of first intersectionof St and At
x*
( )1 20 0 1 2,t tx t A A t t t⊂ ⊂ < <
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 6
Defining Minimum Time
No solution exists if
IF
THEN ti = T* is the minimum time solution = time of first intersection of At and St.
0t tA S t t∩ =∅ ∀ ≥
i it t
t t i
A S
A S t t
∩ ≠∅
∩ =∅ ∀ <
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 7
Solution Procedure
Pontryagin’s Minimum Principle provides necessary conditions for u*(t).Same system equations, control constraints, and performance index as previously shown.Target Set:
[ ] ( ) ( )*, 0 ,g x T x T S x T= ⇒ ∈
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 8
Procedure
The solution procedure follows that for the general, continuous-time optimal control problem• define the Hamiltonian• find the necessary conditions• put control law in feedback form
Hamiltonian
[ ]( ) 1 ( ) ( , ) ( , )TH t t f x t B x t uλ= + +
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 9
Necessary Conditions
[ ] [ ]( ) ( ), ( ), ) ( )
( , ) ( , )( ) ( ) ( ) ( )T T
H x t f x t t B x t t u t
H f x t B x tt t u t tx x x
λ
λ λ λ
∂= = +
∂
∂ ∂ ∂ = − = + ∂ ∂ ∂
( ) ( )* * *0 0 , , 0x t x g x T T = =
{ }* * * * *, , , min , , ,admissible
u
H x u t H x u tλ λ =
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 10
Finding the Optimal Control
* * * * *, , , , , ,
admissible ( )
H x u t H x u t
u t
λ λ ≤ ∀
* * * * *
* * * *
1 , ,
1 , ,
admissible ( )
T T
T T
f x t B x t u
f x t B x t u
u t
λ λ
λ λ
+ + ≤
+ + ∀
* * * * *, ,
admissible ( )
T TB x t u B x t u
u t
λ λ ≤ ∀
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 11
The Optimal Control
{ }* * * *
( ) 1
* * *
min ( ), ( ) ( ),
( ) ( ),
Component-by-Component
i
T T
u t
T
B x t t u t B x t t
u t SGN B x t t
λ λ
λ
≤ = −
= −
[ ]1 2
* *( ) ( )m
Ti i
B b b b
u t SGN b tλ
=
= −
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 12
Bang-Bang Control
*( )Tib tλ
*( )iu t
t
t
1
1−
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 13
Minimum Time Control
Optimal control is Bang-Bang – control is always at its maximum value.“Normal” minimum time problem: is 0 only at isolated points in time. “Singular” minimum time problem: is 0 over a non-zero time interval. The optimal control may exist during this interval, but ui(t) is not defined by –SGN[ ].
*( )Tib tλ
*( )Tib tλ
*( )Tib tλ
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 14
Example for aLinear Time-Invariant System
Double Integrator system and target set:
Hamiltonian:
( ) [ ]* *
( ) ( ) ( ), 2, 1
, 0 0 T
x t Ax t Bu t n m
S x T T
= + = =
=
0 1 0,
0 0 1A B
= =
( ) 1 2 21 1TH Ax Bu x uλ λ λ= + + = + +
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 15
Necessary Conditions:
Optimal Control:
11
22
1
2 1
0 01 0
( ) 0
( ) ( )
TH Ax
t
t t
λλλ λ
λλ
λ
λ λ
∂= = − = − = − ∂
=
= −
1 1
2 2 1
( ) (0)( ) (0) (0)tt t
λ λλ λ λ
== −
[ ] [ ]1*2
2
( ) 0 1Tu t SGN B SGN SGNλ
λ λλ
= − = − = −
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 16
Possible Control Trajectories2λ
2λ
2λ
2λ
t
t
t
t
2 1*
(0) 0, (0) 0( ) 1u t
λ λ> ≤
= −
[ ]2 1
*
(0) 0, (0) 0( ) 1, 1u t
λ λ> >
= − +
[ ]2 1
*
(0) 0, (0) 0( ) 1, 1u t
λ λ< <
= + −
2 1*
(0) 0, (0) 0( ) 1u t
λ λ< ≥
= +
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 17
Feedback Control
Right now, the optimal control is given in terms of the costate vector λ(t).We want the control to be given in terms of the state vector x(t) so we will have a closed-loop control system.The control system must also be causal, so that u(t) does not depend on future values of x(t).
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 18
2
2 2
1 22
1 1 2
( ) 1( ) (0)( ) ( )( ) (0) (0) 0.5
x t ux t x utx t x t
x t x x t ut
= = ±= +=
= + +
Solving the state equations:
Solving for time t:
2 2( ) (0)x t xtu−
=
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 19
Substituting t into x1(t) equation:
Multiplying through by u:
This is a set of parabolas opening about the x1axis.
2 21 1 2
22 2
( ) (0)( ) (0) (0)
( ) (0)0.5
x t xx t x xu
x t xuu
− = +
− +
[ ] [ ][ ]
1 1 2 2 2
22 2
( ) (0) (0) ( ) (0)
0.5 ( ) (0)
u x t x x x t x
x t x
− = −
+ −
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 20
x2
x1
u=+1
u=-1
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 21
Two parabolas pass through each point in x1-x2 space: one for u = +1 and one for u = -1.For a given x(0), using only u = +1 or u = -1 may not transfer the system from x(0) to the target set x = 0.
u = +1u = -1
x(0) x2
x1
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 22
x2
x1u = +1
u = -1
21
22
2
2
2
11 (0) 0.5 (0), (0) 01 (0) 0.5 (0), (0) 0u x x
u xx
x x=
= − ⇒− =
⇒ =
≥
+ ≤
1 2 2( ) 0.5 ( ) ( )Equation for the Switching Curve
x t x t x t= −
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 23
Optimal Control Law
If x(0) is to the left of the switching curve, apply u = +1 until the curve is reached, then apply u = -1 until the origin is reached.If x(0) is to the right of the switching curve, apply u = -1 until the curve is reached, then apply u = +1 until the origin is reached.If x(0) is on the switching curve, apply u = -SGN[x2(0)] until the origin is reached.Apply u = 0 when the origin is reached.
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 24
Examples of u*(t)
x2
x1
u = +1
u = -1
x(0)
u = +1
x(0)
u = -1
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 25
Optimal Feedback Control Law
Apply u(t) = +1 if
Apply u(t) = -1 if
Apply u(t) = -SGN[x2(t)] if
u(t) is generated by a relay nonlinearity following a nonlinear operation on x2(t).
1 2 2( ) 0.5 ( ) ( ) 0x t x t x t+ <
1 2 2( ) 0.5 ( ) ( ) 0x t x t x t+ >
1 2 2( ) 0.5 ( ) ( ) 0x t x t x t+ =
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 26
Closed-Loop Block Diagram
∫ ∫1
-1
Out
In-1u x2 x1
Plant
++Logic
0.5x2|x2|
x1
x2
Logic determines if v = x1 + 0.5x2|x2| = 0
Out
In
v
v orx2
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 27
Linear, Time-Invariant Systems
A linear, time-invariant (LTI) system is Normal if and only if it is completely controllable from each input channel.
A Normal LTI system always produces a Normal minimum time control problem.
[ ]
1
1,i
i
nc i i i
c
H b Ab A b
Rank H n i m
− = = ∀ ∈
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 28
Properties of the Optimal Control
Existence – u*(t) exists for S[x*(T*),T*] = 0 if Ahas no eigenvalues with positive real parts.Uniqueness – If u*(t) exists and system is Normal, u*(t) is unique.Switchings – If all n eigenvalues of A are real, and u*(t) exists, each input signal can switch values at most (n-1) times. For complex eigenvalues, number of switchings is finite, but depends on distance from x(0) to origin.
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 29
A Different Target Set
This example uses the same LTI double integrator system as before.The same initial condition and control constraints apply.The target set is now given by:
( )( )
( )
*2* * *
*1
0,
0
x TS x T T
x T α
= = ≤ >
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 30
x2
x1
α−α
u = -1
u = +1
G+
G-
Q+
Q-
1 2 20.5x x x α= − −
1 2 20.5x x x α= − +
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 31
Defining the Subspaces
2 2 1 2 2
*2
2 2 1 2 2
*2
0.5 0.5
0, ( ) 10.5 0.5
0, ( ) 1
G x x x x x
x u tG x x x x x
x u t
α α
α α
+
−
⇒ − − ≤ ≤ − +
< = +
⇒ − − ≤ ≤ − +
> = −
[ ]
[ ]
1 2 2
*
1 2 2
*
0.5
( ) 1, 1
0.5
( ) 1, 1
Q x x x
u t
Q x x x
u t
α
α
+
−
⇒ < − −
= + −
⇒ > − +
= − +
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 32
Comments on the Target Set
The previous switching curve is replaced by two switching curves.If x(0) is in Q+ or Q-, then x1(T*) will always be on the boundary of the target set.If x(0) is in G+ or G-, then x1(T*) will depend on the initial condition.What can be said about u*(t) if the target set is an open set rather than a closed set?
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 33
Switching Time and Final Time
These expressions are for the LTI double integrator system.The target set is the origin of state space.The expressions are derived by assuming first that u*(t) = [-1, +1], solving for the time to hit the switching curve, and solving for the time to reach the origin along the curve. This is repeated for u*(t) = [+1, -1], and the results are generalized.
Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA Minimum Time Control - 34
If x(0) is on the switching curve:
If x(0) is not on the switching curve:
[ ]
*2
*2
(0)
( ) (0)finalt T x
u t SGN x
= =
= −
[ ] [ ]
1/ 221 2 0 1 0 2
*1 0 2
*
0
0.5 (0) (0) (0)
2 (0)
( ) 1, 1 or 1, 1 is the first value in the control sequence
switch
final
t t x u x u x
t T t u x
u tu
= = − − = = +
= + − − +