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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)

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

- Conclusions

Systems that can switch between

several modes of operation.

Mode 1

Mode 2

linear filter

Switched power converter

50v

100v

A multi-controller scheme

plant

+

controller1

controller2

switching logic

Switched controllers are “stronger” than regular controllers.

- 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

Driving: use mode 1 (wheels)

Braking: use mode 2 (legs)

The advantage: no compromise

“Switched systems are more than the

sum of their subsystems.“

theoretically interesting

practically promising

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

Example:

Definition The differential inclusion

is called GAS if for any solution

(i)

(ii)

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

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 ProblemFind

This implies that

Although both and are

stable, is not stable.

Instability requires repeated switching.

>

Fix Define:

T

0.

Problem Find a control maximizing

Write as the bilinear control

system:

is the worst-case switching law (WCSL).

Analyze the corresponding trajectory

Consider

as

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

is a functional:

Two approaches:

1. Hamilton-Jacobi-Bellman (HJB)

equation.

2. Maximum Principle.

Find such that

Integrating:

or

An upper bound for ,

obtained for the maximizing Eq. (HJB).

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.

Basic Idea

The function is a first integral of if

We know that so

Thus, is a concatenation of two first integrals and

where and

Thus,

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

More on the Planar Case

Theorem For a planar bilinear control system

[Margaliot & Branicky, 2009]

Corollary GAS of 2nd-order positive

linear switched systems.

where are GAS.

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

For simplicity, consider the linear

differential inclusion:

so

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.

Consider

Then:

This is why we can park our car.

The term is the reason this takes

so long.

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?

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)

Theorem(Margaliot & Liberzon, 2004)

2nd order nilpotency implies GAS.

Proof By the PMP, the WCSL satisfies

Let

2nd order nilpotency

up to a single switch in the WCSL.

Then

1st order nilpotency

Differentiating again yields:

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

In this case:

further differentiation cannot be carried out.

Theorem (Sharon & Margaliot, 2007)

3rd order nilpotency implies

Proof

(1) Hall-Sussmann canonical system;

(2) A second-order MP

(Agrachev&Gamkrelidze).

- 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