notes 620 minimum time - George Mason Universitygbeale/ece_620/notes_620_minimum_time.pdf · Dr....

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


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= = −∫


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≤ ∈ ∈


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.


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⊂ ⊂ < <


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

∩ ≠∅

∩ =∅ ∀ <


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= ⇒ ∈


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= + +


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 =


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 ≤ ∀


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

=

= −


Bang-Bang Control

*( )Tib tλ

*( )iu t

t

t

1

1−


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


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= + + = + +


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

= − = − = −


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< ≥

= +


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).


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−

=


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

− = −

+ −


x2

x1

u=+1

u=-1


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


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= −


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.


Examples of u*(t)

x2

x1

u = +1

u = -1

x(0)

u = +1

x(0)

u = -1


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+ =


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


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

− = = ∀ ∈


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.


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

= = ≤ >


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= − +


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

+

−

⇒ < − −

= + −

⇒ > − +

= − +


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?


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.


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

= = − − = = +

= + − − +

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