Stability Analysis of Positive Linear Switched Systems: A Variational Approach

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Stability Analysis of Positive LinearSwitched
SystemsA Variational Approach
Michael Margaliot School of Elec. Eng. -Systems
Tel Aviv University, Israel
Joint work with Lior Fainshil
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Outline

• Stability of linear switched systems
• Variational approach to stability analysis
• Relaxation a bilinear control system
• The most destabilizing control

• 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 Systems
Solution
Definition The system is stable if
Theorem
stability
A is called a Hurwitz matrix.
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Linear Switched Systems
Two (or more) linear systems

• A system that can switch between them

Global Uniform Asymptotic Stability (GUAS)
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.

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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
switching logic
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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
• (3) the switched system is GUAS iff

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Variational Approach

• Relaxation the switched system

? a bilinear control system
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Variational Approach

• The bilinear control system (BCS)

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 Approach

• The most destabilizing control

Fix Tgt0. Let
Optimal control problem find a control that
maximizes
Intuition maximize the distance to the origin.
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Variational Approach and Stability

• Theorem The BCS is GAS iff

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Variational Approach
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 systems
Disadvantages ? requires characterizing
? explicit results for particular cases only
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Part 2 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
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Positive Linear Systems
Motivation suppose that the state variables can
never attain negative values.
In a linear system
this holds if
i.e., off-diagonal entries are non-negative.
Such a matrix is called a Metzler matrix.
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Positive Linear Systems
with
Theorem
An example
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Positive Linear Systems
The solution of
is
transition matrix
The transition matrix is non-negative.
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Perron-Frobenius Theorem
Definition Spectral radius of a matrix
has a real eigenvalue such that
The corresponding eigenvectors of ,
denoted , satisfy
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Some Perturbation Theory
Let be a smooth
parameter-dependent non-negative matrix. Denote
dominant eigenvalue of
dominant eigenvectors of
Then,
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Sketch of Proof
Differentiate with respect to
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Positive Linear Switched Systems A Variational
Approach
Relaxation
Most destabilizing control maximize the
spectral radius of the transition matrix.
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Positive Linear Switched Systems A variational
Approach
Theorem For any Tgt0,
where is the solution at time T of

is called the transition matrix corresponding
to u.
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Transition Matrix of a Positive System
If are Metzler, then
admit a real and
eigenvalue
such that
The corresponding eigenvectors satisfy
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Optimal Control Problem
Fix an arbitrary Tgt0.
Problem find a control that
maximizes
We refer to as the most destabilizing
control.
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Relation to Stability
Define
Theorem the PBCS is GAS if and only if
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Main Result A Maximum Principle
Theorem Fix Tgt0. Consider
Let be optimal. Let and let
denote the factors of Define
and let
Then
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Comments on the Main Result
1. Similar to the Pontryagin MP, but with
one-point boundary conditions 2. The unknown
play an important role.

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Comments on the Main Result
3. The switching function satisfies

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Comments on the Main Result
The number of switching points in a bang-bang
control must be even.
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Main Result Sketch of Proof
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Sketch of Proof
Let Then We know that
with
Since is optimal,
so
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Sketch of Proof
We can obtain an expression for
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Applications of Main Result
are Metzler
Assumptions
is Hurwitz
Proposition 2 If
and either
or
the switched system is GUAS.
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Applications of Main Result
Assumptions
are Metzler
is Hurwitz
Conjecture If
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.
Further research numerical algorithms for
calculating the optimal control consensus
problems switched monotone control systems.
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More Information

• 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/michael
  m