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Stability Analysis of Continuous-Time Switched Systems: A Variational Approach

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### Stability Analysis of Continuous-Time Switched Systems: A Variational Approach

Michael Margaliot

School of EE-Systems Tel Aviv University, Israel

Joint work with: Michael S. Branicky (CWRU) Daniel Liberzon (UIUC)

Overview

- Switched systems
- Stability
- Stability analysis:
- A control-theoretic approach
- A geometric approach
- An integrated approach
- Conclusions

Example 2

A multi-controller scheme

plant

+

controller1

controller2

switching logic

Switched controllers are “stronger” than regular controllers.

More Examples

- Air traffic control
- Biological switches
- Turbo-decoding
- ……

For more details, see:

- Introduction to hybrid systems, Branicky

- Basic problems in stability and design of

switched systems, Liberzon & Morse

Synthesis of Switched Systems

Driving: use mode 1 (wheels)

Braking: use mode 2 (legs)

The advantage: no compromise

Gestalt Principle

“Switched systems are more than the

sum of their subsystems.“

theoretically interesting

practically promising

Differential Inclusions

A solution is an absolutely continuous function satisfying (DI) for almost all t.

Example:

Global Asymptotic Stability (GAS)

Definition The differential inclusion

is called GAS if for any solution

(i)

(ii)

The Challenge

- Why is stability analysis difficult?
- A DI has an infinite number of solutions for each initial condition.
- The gestalt principle.

Absolute Stability

The closed-loop system:

A is Hurwitz, so CL is asym. stable for

any

Absolute Stability Problem

Find

For CL is asym. stable for any

Absolute Stability and Switched Systems

Absolute Stability ProblemFind

A Solution of the Switched System

This implies that

Fix Define:

T

0.

Problem Find a control maximizing

Optimal Control ApproachWrite as the bilinear control

system:

is the worst-case switching law (WCSL).

Analyze the corresponding trajectory

Optimal Control Approach

Theorem (Pyatnitsky) If then:

(1) The function

is finite, convex, positive, and homogeneous (i.e.).

(2) For every initial condition there exists a solution such that

Solving Optimal Control Problems

is a functional:

Two approaches:

1. Hamilton-Jacobi-Bellman (HJB)

equation.

2. Maximum Principle.

HJB Equation

Find such that

Integrating:

or

An upper bound for ,

obtained for the maximizing Eq. (HJB).

The Case n=2

Margaliot & Langholz (2003) derived an

explicit solution for when n=2.

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

The function is a first integral of if

We know that so

Thus, is a concatenation of two first integrals and

Example:

where and

→ an explicit expression for V (and an explicit solution of the HJB).

Theorem For a planar bilinear control system

[Margaliot & Branicky, 2009]

Corollary GAS of 2nd-order positive

linear switched systems.

Nonlinear Switched Systems

where are GAS.

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

Commutation Relations and GAS

Suppose that A and B commute, i.e.

AB=BA, then

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

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

Nilpotency

Definitionk’th order nilpotency:

all Lie brackets involving k+1 terms vanish.

1st order nilpotency: [A,B]=0

2nd order nilpotency: [A,[A,B]]=[B,[A,B]]=0

Q: Does k’th order nilpotency imply GAS?

Known Results

Linear switched systems:

- k = 2 implies GAS (Gurvits,1995).
- k’th order nilpotency implies GAS (Liberzon, Hespanha, & Morse, 1999)(Kutepov, 1982)

Nonlinear switched systems:

- k = 1 implies GAS (Mancilla-Aguilar, 2000).
- An open problem: higher orders of k? (Liberzon, 2003)

A Partial Answer

Theorem(Margaliot & Liberzon, 2004)

2nd order nilpotency implies GAS.

Proof By the PMP, the WCSL satisfies

Let

up to a single switch in the WCSL.

Then

1st order nilpotency

Differentiating again yields:

Handling Singularity

If m(t)0, the Maximum Principle

does not necessarily provide enough

information to characterize the WCSL.

Singularity can be ruled out using

thenotion ofstrong extremality

(Sussmann, 1979).

3rd Order Nilpotency

Theorem (Sharon & Margaliot, 2007)

3rd order nilpotency implies

Proof

(1) Hall-Sussmann canonical system;

(2) A second-order MP

(Agrachev&Gamkrelidze).

Conclusions

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

- Stability analysis is difficult.

A natural and powerful idea is to

consider the “most unstable” trajectory.

More info on the variational approach:

“Stability analysis of switched systems using variational principles: an introduction”, Automatica 42(12): 2059-2077, 2006.

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

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