Stability Analysis of Positive LinearSwitched Systems:

A Variational Approach

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Michael MargaliotSchool of Elec. Eng. -Systems

Tel Aviv University, Israel

Joint work with Lior Fainshil

Outline• Stability of linear switched systems• Variational approach to stability analysis Relaxation: a bilinear control system The “most destabilizing” control *u

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• Positive linear switched systems• Variational approach Relaxation: a positive bilinear control

system Maximizing the spectral radius of the

transition matrix Main result: A maximum principle Applications

Linear Switched Systems

A system that can switch between them:

Global Uniform Asymptotic Stability (GUAS):

: {1,2}.σ R

( ) 0 0 ., ( ),x t x σ

AKA, “stability under arbitrary switching”.

Two (or more) linear systems:

( )( ) ( ),σ tx t A x t

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

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Why is the GUAS problem difficult?

1. The number of possible switching laws is huge.

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Why is the GUAS problem difficult?2. Even if each linear subsystem is stable, the

switched system may not be GUAS.

0 1

2 1x x

0 1

12 1x x

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Why is the GUAS problem difficult?

2. Even if each linear subsystem is stable, the switched system may not be GUAS.

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Switched Systems: An Example

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plant

controller1

+

controller2switching logic

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Variational Approach

Basic idea: (1) 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).

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Variational ApproachRelaxation: 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 x1/ 2u

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

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Variational ApproachThe 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.

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Variational ApproachThe most destabilizing control:

( ) | ( ; ) | .J u x T u

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

Fix T>0. Let

Optimal control problem: find a control that maximizes

0(0) .x x

( ).J u

*u U

Intuition: maximize the distance to the origin.12

Variational Approach and Stability

Theorem The BCS is GAS iff *( ) 0.x T

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Variational 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

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Part 2: Variational Approach for Positive Linear Switched Systems

*u

Basic idea: (1) positive linear switched system → positive bilinear control system (PBCS) (2) characterize the “most destabilizing control”

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Positive Linear Systems

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,x Ax

0, .ija i j

Motivation: suppose that the state variables can never attain negative values.

(0) 0 ( ) 0, 0.x x t t

In a linear system this holds if

Such a matrix is called a Metzler matrix.

i.e., off-diagonal entries are non-negative.

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Positive Linear Systems

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,x Ax

with 0, .ija i j

Theorem (0) 0 ( ) 0, 0.x x t t

An example: 1 3

5 2x x

1 1 a non-negative numberx x

1( ) 0, 0.x t t 1 20 0, 0 0x x

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Positive Linear Systems

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If A is Metzler then for any

exp( ) 0.At

0t

exp( ) : n nAt R R

so

transition matrix

The solution of x Ax is ( ) exp( ) (0).x t At x

The transition matrix is non-negative. 18

Perron-Frobenius Theorem

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has a real eigenvalue such that:

The corresponding eigenvectors of , denoted , satisfy

Theorem Suppose that

max ( ) : max{| |: eig( )}

( ').

λ ρ C λ λ C

ρ C

• C maxλ •

•, 'C C ,v w 0, 0.v w

0.C

( ) max{| |: eig( )}.ρ C λ λ C Definition Spectral radius of a matrix C

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Positive Linear Switched Systems: A Variational Approach

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,x A Bu x Relaxation:

“Most destabilizing control”: maximize the spectral radius of the transition matrix.

.u U

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Positive Linear Switched Systems: A variational Approach

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.x A Bu x

Theorem For any T>0,

,

0 .

C t A Bu t C t

C I

is called the transition matrix corresponding to u.

( ; ) ( ; ) (0)x T u C T u x

where is the solution at time T of ( ; )C T u

C

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Transition Matrix of a Positive System

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If are Metzler, then

,

0 .

C t A Bu t C t

C I

( ; ) ( ; ) (0)x T u C T u x

( ) 0, 0.C t t

eigenvalue such that: ( ) and '( )C T C T admit a real and

( )λ T

( ) ( ( )) ( '( )).λ T ρ C T ρ C T

The corresponding eigenvectors satisfy 0, 0.v w

1 2,A A

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Optimal Control Problem

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,

0 .

C t A Bu t C t

C I

Fix an arbitrary T>0. Problem: find a control that maximizes

*u U( ( , )).ρ C T u

We refer to as the “most destabilizing” control.

*u

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Relation to Stability

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,

0 .

C t A Bu t C t

C I

Define:

Theorem: the PBCS is GAS if and only if( , ) 1.ρ A B

1/( , ) max ( ( , )) .

( , ) limsup ( , ).

TT u U

T T

ρ A B C T u

ρ A B ρ A B

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Main Result: A Maximum Principle

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, 0 .C t A Bu t C t C I Theorem Fix T>0. Consider

Let be optimal. Let and let denote the factors of Define

* , 0 *,

* ' , 0 *,

p A Bu p p v

q A Bu q q w

* ( , *),C C T u

and let

1, ( ) 0,*( )

0, ( ) 0.

m tu t

m t

Then ( ) ' .m t q t Bp t

*, *v w

*u

*.C

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Comments on the Main Result

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1. Similar to the Pontryagin MP, but with one-point boundary conditions; 2. The unknown play an important role.

* , 0 *,

* ' , 0 *,

p A Bu p p v

q A Bu q q w

1, ( ) 0,*( )

0, ( ) 0.

m tu t

m t

( ) ' .m t q t Bp t

*, *v w

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Comments on the Main Result

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3. The switching function satisfies:

* , 0 *,

* ' , 0 *,

p A Bu p p v

q A Bu q q w

( ) ' .m t q t Bp t

max max

( ) '

'( ) *( ) 0

( ' 0 / ) 0

' 0 0

(0).

m T q T Bp T

q T BC T p

q λ Bλ p

q Bp

m

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Comments on the Main Result

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( ) (0).m T m

t

( )m t

1t 2t 3t 4t T

The number of switching points in a bang-bang control must be even.

0

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Main Result: Sketch of Proof

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Let be optimal. Introduce a needle variation with perturbation width Let denote the corresponding transition matrix.

*u U0.ε u

C

εT

*( )u t

t0

1

0 T

( )u t

t0

1

0

By optimality, ( ( )) ( *( )).ρ C T ρ C T

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Sketch of Proof

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Let ThenWe know that

Since is optimal, so

.γ ε ρ C T 0 * *.γ ρ C T ρ

0 0 ...γ ε γ εγ

*u 0 * ,γ ρ γ ε

with

0

0 * ' *.ε

dγ w C T v

dε

0

* ' * 0ε

dw C T v

dε

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Sketch of Proof

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We can obtain an expression for

Since is optimal, so 0 * ,γ ρ γ ε

0

* ' * 0.ε

dw C T v

dε

( ) *( )C T C T

*u

to first order in as is a needle variation.,ε u

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1 2(1 )kA k A

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Applications of Main Result Assumptions: are Metzler

is Hurwitz [0,1].k

1 2 0,αA βA Proposition 1 If there exist ,α βR such that

the switched system is GUAS.

Proposition 2 If 2 1 'A A bc and either 0bor 0,c the switched system is GUAS.

1 2, n nA A R

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1 2(1 )kA k A

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Applications of Main Result Assumptions: are Metzler

is Hurwitz [0,1].k

Proposition 3 If 2 1 'A A bc then any bang-bang control with more than one switch includes at least 4 switches.

1 2, n nA A R

Conjecture If 2 1 'A A bc switched system is GUAS.

then the

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ConclusionsWe considered the stability of positive switched linear systems using a variational approach.

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The main result is a new MP for the control maximizing the spectral radius of the transition matrix.

Further research: numerical algorithms for calculating the optimal control; consensus problems; switched monotone control systems.

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

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

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

Fainshil & Margaliot. “Stability analysis of positive linear switched systems: a variational approach”, submitted.

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

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