1

Stability Analysis of LinearSwitched Systems:

An Optimal Control ApproachMichael Margaliot

School of Elec. Eng.

Tel Aviv University, Israel

Joint work with: Gideon Langholz (TAU), Daniel Liberzon (UIUC), Michael S. Branicky (CWRU), Joao Hespanha (UCSB).

Part 1

2

Overview Switched systems Global asymptotic stability The edge of stability Stability analysis:

An optimal control approach A geometric approach An integrated approach

Conclusions

3

Switched Systems

Systems that can switch between

several possible modes of operation.

Mode 1

Mode 2

4

Example 1

serve

r

1x 2x

C

)(2 ta)(1 ta

1 1

2 2

( )

( )

x a t

x a t C

1 1

2 2

( )

( )

x a t C

x a t

5

Example 2

Switched power converter

100v 50vlinear filter

6

Example 3

A multi-controller scheme

plant

controller1

+

switching logiccontroller

2

Switched controllers are stronger than “regular” controllers.

7

More Examples

Air traffic controlBiological switchesTurbo-decoding……

8

Synthesis of Switched Systems

Driving: use mode 1 (wheels)

Braking: use mode 2 (legs)

The advantage: no compromise

Linear Systems

Solution:

9

.x Ax

( ) exp( ) (0).x t At x

Theorem:

Definition: The system is globally asymptotically stable if lim ( ) 0, (0).

tx t x

Re( ) 0, eig( ).λ λ A

A is called a Hurwitz matrix.

stability

10

Linear Switched Systems

A system that can switch between them:

: {1,2}.σ R

Two (or more) linear systems:

( )( ) ( ),σ tx t A x t

1( ) ( ),x t A x t2( ) ( ).x t A x t

10

t

( )σ t

21

...

4 1 3 2 2 1 1 2 0( ) ...exp( )exp( )exp( )exp( ) .x T t A t A t A t A x1t 1 2t t

11

StabilityLinear switched system:

: {1,2}.σ R

Definition: Globally uniformly asymptotically stable (GUAS): ( ) 0 0 ., ( ),x t x σ

AKA, “stability under arbitrary switching”.

( )( ) ( ),σ tx t A x t

11

3 2 2 1 1 2 0lim ( ) lim(...exp( )exp( )exp( ) ) 0t tx t t A t A t A x

for any 1 2, ,.... 0t t

In other words,

12

A Necessary Condition for GUASThe switching law yields

Thus, a necessary condition for GUAS

is that both are Hurwitz.

Then instability can only arise due to

repeated switching.

12

( ) 1σ t 1( ) ( ).x t A x t

1 2,A A

13

Why is the GUAS problem difficult?

Answer 1:

The number of possible switching laws

is huge.

13

14

Why is the GUAS problem difficult?

0 1

2 1x x

0 1

12 1x x

14

Answer 2: Even if each linear subsystem is

stable, the switched system may not be GUAS.

15

Why is the GUAS problem difficult?Answer 2: Even if each linear subsystem is

stable, the switched system may not be GUAS.

15

16

Stability of Each Subsystem is Not Enough

A multi-controller scheme

plant

controller1

+

switching logiccontroller

2

Even when each closed-loop is stable,the switched system may not be GUAS.

17

Easy Case #1

A trajectory of the switched system:

Suppose that the matrices commute:

17

1 2 2 1.A A A A

4 1 3 2 2 1 1 2 0( ) ...exp( )exp( )exp( )exp( ) .x T t A t A t A t A x

Then

and since both matrices are Hurwitz, the

switched system is GUAS.

1 2 0( ) exp(( ) ) exp( ) ,x T T s A sA x

18

Easy Case #2Suppose that both matrices are upper

triangular:

Then

18

1 1,

0 2x x

3 7

.0 2.5

x x

2 , 2.5 ,2 2 2 2x x x x so

| ( ) | exp( 2 ) | (0) | .2 2x t t x

Now , 3 71 1 2 1 1 2x x x x x x so ( ) 0.1x t

This proves GUAS.

19

Optimal Control Approach

Basic idea:

(1) A relaxation: linear switched

system → bilinear control system

(2) characterize the “most

destabilizing” control

(3) the switched system is GUAS iff *( ) 0x t

*u

Pioneered by E. S. Pyatnitsky (1970s).

19

Optimal Control Approach

Relaxation: the switched system:

,σx A x : {1,2},σ R

1 2 1( ( ) ) ,x A A A u x

→ a bilinear control system:

,u U

where is the set of measurable functions taking values in [0,1].

U

1u 0u 1x A x 2x A x

1/ 2u 1 2(1/ 2)( )x A A x

20

The bilinear control system (BCS)

1 2 1( ( ) ) ,x A A A u x ,u U

( ) 0, (0), .ux t Ux

is globally asymptotically stable (GAS) if:

Theorem The BCS is GAS if and only if the linear switched system is GUAS.

21

Optimal Control Approach

Optimal Control Approach

The most destabilizing control:

( ) | ( ; ) | .fJ u x t u

1 2 1( ( ) ) ,x A A A u x ,u U

Fix a final time . Let

Optimal control problem: find a control that maximizes

0(0) .x x

( ).J u

*u U

Intuition: maximize the distance to the origin.22

0.ft

Optimal Control Approach and Stability

Theorem The BCS is GAS iff *( ) 0.fx t

23

Edge of Stability

GAS

24

1 2 1( ) , .x A Bu x B A A The BCS: Consider 1 2 1( ) , ( ) .k kx A B u x B A A k

1k 0k

1( 0 )x A u x 0k ε

1( ) ,εx A B u x

GAS original BCS

1 1( ) ,x A Bu x

Edge of Stability

GAS

25

1 2 1( ) , .x A Bu x B A A The BCS: Consider 1 2 1( ) , ( ) .k kx A B u x B A A k

1k 0k

1( 0 )x A u x 0k ε

1( ) ,εx A B u x

GAS original BCS

1 1( ) ,x A Bu x

Definition: k* is the minimal value of k>0

such that GAS is lost.

Edge of Stability

26

1 2 1( ) , .x A Bu x B A A The BCS: Consider 1 2 1( ) , ( ) .k kx A B u x B A A k

Definition: k* is the minimal value of k>0

such that GAS is lost.

The system is said to be on

the edge of stability. 1 *( )kx A B u x

Edge of Stability

27

1 2 1( ) , .x A Bu x B A A The BCS: Consider 1 2 1( ) , ( ) .k kx A B u x B A A k

Definition: k* is the minimal value of k>0

such that GAS is lost.

Proposition: our original BCS is GAS iff

k*>1.

0 1 k 0 1 kk* k*

Edge of Stability

28

1 2 1( ) , .x A Bu x B A A The BCS: Consider 1 2 1( ) , ( ) .k kx A B u x B A A k

Proposition: our original BCS is GAS iff

k*>1.

→ we can always reduce the problem of analyzing GUAS to the problem of determining the edge of stability.

Edge of Stability When n=2

29

Consider 1( ) .kx A B u x

*k k *k k*k k0x

0x

The trajectory x* corresponding to u*:

A closed

periodic trajectory

30

Solving Optimal Control Problems

is a functional:

Two approaches:

1. The Hamilton-Jacobi-Bellman (HJB) equation.

2. The Maximum Principle.

2| ( ; ) |fx t u

( ; ) ( , [0, ])f fx t u F u(t) t t

31

Solving Optimal Control Problems

1. The HJB equation.

Intuition: there exists a function

and V can only decrease on any other

trajectory of the system.

( , *( )) const,V t x t ( , ) : nV R R R

32

The HJB Equation

Find such that

Integrating:

or

An upper bound for ,

obtained for the maximizing (HJB).

( , ( )) 0. (HJB)[0,1]

dMAX V t x t

dtu

( , ( )) (0, (0)) 0f fV t x t V x

2| ( ) | / 2fx t

( , ) : nV R R R 2( , ) || || / 2,fV t y y

*u

2| ( ) | / 2 (0, (0)).fx t V x

33

The HJB for a BCS:

Hence,

In general, finding is difficult.

1, ( ) 0,

* 0, ( ) 0,

?, ( ) 0.

x

x

x

V A B x

u V A B x

V A B x

V

})({max

)})1(({max

}{max0

xx

x

x

xBAuVBxVV

BxuuAxVV

xVV

tu

tu

tu

Note: u* depends on only. xV

34

The Maximum Principle

Let Then,

Differentiating we get

A differential equation for with a boundary condition at

0 ,t xV V x

0 ( (1- ) )

( (1- ) )

( (1- ) )

tx xx x

ddt x x

V V x V uA u B

V V uA u B

uA u B

( ) : ( , *( )). xt V t x t 2( ) ( ) / 2 ( ). xf f ft x t x t

.ft( ),t

35

Summarizing,

The WCSL is the maximizing

that is,

We can simulate the optimal solution backwards in time.

0

( (1- ) ) , ( ) ( )

( (1- ) ) , (0)

Tf fuA u B t x t

x uA u B x x x

( (1 ) )T

t x tV V x V uA u B x

1, ( )( ) ( ) 0,*( )

0, ( )( ) ( ) 0.

T

T

t A B x tu t

t A B x t

*u

36

Result #1 (Margaliot & Langholz, 2003)

An explicit solution for the HJB equation, when n=2, and {A,B} is on the “edge of stability”.

This yields an easily verifiable necessary and sufficient condition for stability of second-order switched linear systems.

37

Basic Idea

The HJB eq. is:

Thus,

Let be a first integral of

that is,

Then is a concatenation of two first integrals and

x x 0= max{ ( ) }.uV Bx uV A B x

x

x

0 0=

1 0=

u V Bx

u V Ax

0 ( ( )) .A Ax

dH x t H Ax

dt ( ) ( ),x t Ax t

( ).BH x( )AH x

V

2:AH R R

38

Example:

12

10A

12

10

kB

10

1 2

72( ) exp( arctan( ))

27A T x

H x x P xx x

Bxx

Axx

1

1 2

7 42( ) exp( arctan( ))

27 4B T

k

k xH x x P x

x xk

1

1

12/1

2/12 kPkwhere and ...985.6* k

39

Nonlinear Switched Systems

with GAS.

Problem: Find a sufficient condition guaranteeing GAS of (NLDI).

1 2 { ( ), ( )} (NLDI)x f x f x

( )ix f x

40

Lie-Algebraic Approach

For the sake of simplicity, we present

the approach for LDIs, that is,

and

},{ BxAxx

2 1( ) ...exp( )exp( ) (0).x t Bt At x

41

Commutation and GAS

Suppose that A and B commute,AB=BA, then

Definition: The Lie bracket of Ax and Bx is [Ax,Bx]:=ABx-BAx.

Hence, [Ax,Bx]=0 implies GAS.

3 2 1

3 1 4 2

( ) ...exp( )exp( )exp( ) (0)

exp( (... )) exp( (... )) (0)

x t At Bt At x

A t t B t t x

42

Lie Brackets and Geometry

Consider

A calculation yields:

{ , , , }.x Ax Ax Bx Bx

x Ax

x Axx Bx

x Bx

)0(x

)4( x

2(4 ) (0) [ , ]( (0)).x ε x ε A B x

43

Geometry of Car Parking

This is why we can park our car.

The term is the reason it takes

so long.

Bx

2

Ax

Bx

[ , ]( )A B x

Ax

44

NilpotencyWe saw that [A,B]=0 implies GAS.What if [A,[A,B]]=[B,[A,B]]=0?

Definition: k’th order nilpotency -all Lie brackets involving k terms vanish.

[A,B]=0 → 1st order nil.[A,[A,B]]=[B,[A,B]]=0 → 2nd order nil.

45

Nilpotency and Stability

We saw that 1st order nilpotency

Implies GAS.

A natural question:

Does k’th order nilpotency imply GAS?

46

Some Known ResultsSwitched linear systems:k=2 implies GAS (Gurvits,1995).k order nilpotency implies GAS

(Liberzon, Hespanha, and Morse, 1999).(The proof is based on Lie’s Theorem)

Switched nonlinear systems:k=1 implies GAS.An open problem: higher orders of k?

(Liberzon, 2003)

47

A Partial Answer

Result #2 (Margaliot & Liberzon, 2004)

3rd order nilpotency implies GAS.

Proof: Consider the WCSL

Define the switching function

1, ( )( ) ( ) 0*( )

0, ( )( ) ( ) 0

T

T

t A B x tu t

t A B x t

BACtCxttm T ),()(:)(

48

Differentiating m(t) yields

2nd order nilpotency no switching in the WCSL!Differentiating again, we get

3rd order nilpotency up to a single switching in the WCSL.

( ) ( ) ( ) ( ) ( )

( )[ , ] ( ).

T T

T

m t t Cx t t Cx t

t C A x t

xBACuxAAC

xACxACm

TT

TT

]],,[[]],,[[

],[],[

0m ( ) constm t

battm )(0m

49

Singular Arcs

If m(t)0, then the Maximum Principle

provides no direct information.

Singularity can be ruled out using

the auxiliary system.

50

Summary Parking cars is an underpaid job.

Stability analysis is difficult. A natural and useful idea is to consider the worst-case trajectory.

Switched systems and differential inclusions are important in various scientific fields, and pose interesting theoretical questions.

Summary: Optimal Control Approach

*u Advantages: reduction to a single control leads to necessary and sufficient conditions

for GUAS allows the application of powerful tools (high-order MPs, HJB equation, Lie-

algebraic ideas,….) applicable to nonlinear switched systemsDisadvantages: requires characterizing explicit results for particular cases only

*u

51

5252

1. Margaliot. “Stability analysis of switched systems using variational principles: an introduction”, Automatica, 42: 2059-2077, 2006.

2. Sharon & Margaliot. “Third-order nilpotency, nice reachability and asymptotic stability”, J. Diff. Eqns., 233: 136-150, 2007.

3. Margaliot & Branicky. “Nice reachability for planar bilinear control systems with applications to planar linear switched systems”, IEEE Trans. Automatic Control, 54: 1430-1435, 2009.

Available online: www.eng.tau.ac.il/~michaelm

More Information