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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

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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

### Switched Systems

Systems that can switch between

several modes of operation.

Mode 1

Mode 2

linear filter

### Example 1

Switched power converter

50v

100v

### 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)

### 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

### Two Remarks

Although both and are

stable, is not stable.

Instability requires repeated switching.

>

Fix Define:

T

0.

Problem Find a control maximizing

### Optimal Control Approach

Write as the bilinear control

system:

is the worst-case switching law (WCSL).

Analyze the corresponding trajectory

Consider

as

### 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.

Basic Idea

The function is a first integral of if

We know that so

Thus, is a concatenation of two first integrals and

### Example:

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.

### Nonlinear Switched Systems

where are GAS.

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

### Lie-Algebraic Approach

For simplicity, consider the linear

differential inclusion:

so

### 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.

Consider

Then:

### Geometry of Car Parking

This is why we can park our car.

The term is the reason this takes

so long.

### 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)

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:

### 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

In this case:

further differentiation cannot be carried out.

### 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.