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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