Part 2
Stability Analysis of Linear
Switched Systems:
An Optimal Control Approach
Michael Margaliot
School of Elec. Eng.
Tel Aviv University, Israel
Joint work with Lior Fainshil
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Outline
• 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
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Linear Switched Systems
Two (or more) linear systems:
x(t )  A1 x(t ),
x(t )  A2 x(t ).
A system that can switch between them:
x(t )  Aσ (t ) x(t ),
σ : R  {1,2}.
Global Uniform Asymptotic Stability (GUAS):
x(t )  0,
x(0), σ .
AKA, “stability under arbitrary switching”.
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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
x
x
 2 1
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 0 1
x
x
 12 1
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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Variational Approach
Pioneered by E. S. Pyatnitsky (1970s).
Basic idea:
(1) relaxation: linear switched system →
bilinear control system
(2) characterize the “most destabilizing
control” u *
(3) the switched system is GUAS iff x * (t )  0
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Variational Approach for Positive
Linear Switched Systems
Basic idea:
(1) positive linear switched system →
positive bilinear control system (PBCS)
(2) characterize the “most destabilizing
control” u *
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Positive Linear Systems
Motivation: suppose that the state
variables can never attain negative values.
x(0)  0 
x(t )  0, t  0.
In a linear system x  Ax, this holds if
aij  0, i  j.
i.e., off-diagonal entries are non-negative.
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Such a matrix is called a Metzler matrix.
Positive Linear Systems
x  Ax,
with aij  0, i  j.
Theorem x(0)  0  x(t )  0, t  0.
An example:
 1 3 
x
x

 5 2 
x1   x1  a non-negative number
x1  0   0, x2  0   0  x1 (t )  0, t  0.
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Positive Linear Systems
The solution of x  Ax is x(t )  exp( At ) x(0).
transition matrix
If A is Metzler then for any t  0
exp( At ) : Rn  Rn
so exp( At )  0.
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The transition matrix is a non-negative
matrix.
Perron-Frobenius Theory
Definition Spectral radius of a matrix C
ρ(C )  max{| λ |: λ  eig(C )}.
Example Let
 0 1
C 
.

 1 0 
The eigenvalues are λ1  j, λ2   j,
so
ρ(C )  max{| λ1 |,| λ2 |}  1.
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Perron-Frobenius Theorem
Theorem Suppose that C  0.
• C has a real eigenvalue λmax such that:
λmax  ρ(C ) : max{| λ |: λ  eig(C )}
 ρ(C ').
• The corresponding eigenvectors of
C, C ' , denoted v, w, satisfy v  0, w  0.
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Positive Linear Switched Systems:
A Variational Approach
Relaxation: x   A  Bu  x, u U .
“Most destabilizing control”: maximize the
spectral radius of the transition matrix.
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Positive Linear Switched Systems:
A variational Approach
x   A  Bu  x.
Theorem For any T>0, x(T ; u )  C (T ; u ) x(0)
where C (T ; u ) is the solution at time T of
C  t    A  Bu  t   C  t  ,
C 0  I .
C is called the transition matrix
corresponding to u.
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Transition Matrix of a Positive System
x(T ; u )  C (T ; u ) x(0)
C  t    A  Bu  t   C  t  ,
C 0  I .
If A1 , A2 are Metzler, then C (t )  0, t  0.
C (T ) and C '(T ) admit a real and
eigenvalue λ(T ) such that:
λ(T )  ρ(C (T ))  ρ(C '(T )).
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The corresponding eigenvectors
satisfy v  0, w  0.
Optimal Control Problem
C  t    A  Bu  t   C  t  ,
C 0  I .
Fix an arbitrary T>0.
Problem: find a control u* U that
maximizes ρ(C (T , u )).
We refer to u * as the “most destabilizing”
control.
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Relation to Stability
C  t    A  Bu  t   C  t  ,
C 0  I .
Define:
ρT ( A, B)  max uU (C (T , u )) .
1/T
ρ( A, B)  limsupT  ρT ( A, B).
Theorem: the PBCS is GAS if and only if
ρ( A, B )  1.
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Main Result: A Maximum Principle
Theorem Fix T>0. Consider
C  t    A  Bu  t   C  t  , C  0   I .
Let u * be optimal. Let C*  C (T , u*),
and let v*, w * denote the factors of C *.
Define
p   A  Bu * p,
p  0   v*,
q    A  Bu * ' q, q  0   w*,
and let m(t )  q '  t  Bp  t .
Then
1, m(t )  0,
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
u *(t )  
0, m(t )  0.
Comments on the Main Result
p   A  Bu * p,
p  0   v*,
q    A  Bu * ' q, q  0   w*,
m(t )  q '  t  Bp  t .
1, m(t )  0,
u *(t )  
0, m(t )  0.
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1. Similar to the Pontryagin MP, but with
one-point boundary conditions;
2. The unknown v*, w * play an important
role.
Comments on the Main Result
p   A  Bu * p,
p  0   v*,
q    A  Bu * ' q, q  0   w*,
m(t )  q '  t  Bp  t .
3. The switching function satisfies:
m(T )  q ' T  Bp T 
 q '(T ) BC *(T ) p  0 
 (q '  0  / λ ) Bλ p  0 
max
 q '  0  Bp  0 
 m(0).
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max
Comments on the Main Result
m(T )  m(0).
m (t )
0 t1
t2
t3
t4 T
t
The number of switching points in a bangbang control must be even.
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Main Result: Sketch of Proof
Let u* U be optimal. Introduce a needle
variation u with perturbation width ε  0.
Let C denote the corresponding transition
matrix.
u *(t )
1
0
u (t )
1
0
T
t
0
0
ε
By optimality, ρ(C (T ))  ρ(C * (T )).
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T
t
Sketch of Proof
Let γ  ε   ρ  C T  . Then γ  0   ρ  C * T    ρ *.
We know that
γ  ε   γ  0   εγ  0   ...
with
d

γ  0    w * '  C T   v *.
 dε

ε 0
Since u * is optimal, γ  0   ρ*  γ  ε  , so
d

 w * ' C T   v*  0
ε 0 
 dε
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Sketch of Proof
Since u * is optimal, γ  0   ρ*  γ  ε  , so
d

 w * '  C T   v*  0.
ε 0 
 dε
We can obtain an expression for
C (T )  C * (T )
to first order in ε, as u is a needle variation.
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Applications of Main Result
Assumptions: A1 , A2  R
nn
are Metzler
kA1  (1  k ) A2 is Hurwitz k  [0,1].
Proposition 1 If there exist α, β  R such that
αA1  βA2  0, the switched system is GUAS.
Proposition 2 If A2  A1  bc ' and either b  0
or c  0, the switched system is GUAS.
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Applications of Main Result
Assumptions: A1 , A2  R
nn
are Metzler
kA1  (1  k ) A2 is Hurwitz k  [0,1].
Proposition 3 If A2  A1  bc ' then any
bang-bang control with more than one
switch includes at least 4 switches.
Conjecture If A2  A1  bc ' then the
switched system is GUAS.
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Conclusions
We considered the stability of positive
switched linear systems using a variational
approach.
The main result is a new MP for the control
maximizing the spectral radius of the
transition matrix.
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Further research: numerical algorithms for
calculating the optimal control; consensus
problems; switched monotone control
systems,…
More Information
Margaliot. “Stability analysis of switched systems
using variational principles: an introduction”,
Automatica, 42: 2059-2077, 2006.
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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