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

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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

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

Overview

  • Switched systems

  • Stability

  • Stability analysis:

    • A control-theoretic approach

    • A geometric approach

    • An integrated approach

  • Conclusions


Switched systems

Switched Systems

Systems that can switch between

several modes of operation.

Mode 1

Mode 2


Example 1

linear filter

Example 1

Switched power converter

50v

100v


Example 2

Example 2

A multi-controller scheme

plant

+

controller1

controller2

switching logic

Switched controllers are “stronger” than regular controllers.


More examples

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

Synthesis of Switched Systems

Driving: use mode 1 (wheels)

Braking: use mode 2 (legs)

The advantage: no compromise


Gestalt principle

Gestalt Principle

“Switched systems are more than the

sum of their subsystems.“

 theoretically interesting

 practically promising


Differential inclusions

Differential Inclusions

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

Example:


Global asymptotic stability gas

Global Asymptotic Stability (GAS)

Definition The differential inclusion

is called GAS if for any solution

(i)

(ii)


The challenge

The Challenge

  • Why is stability analysis difficult?

  • A DI has an infinite number of solutions for each initial condition.

  • The gestalt principle.


Absolute stability lure 1944

Absolute Stability [Lure, 1944]


Absolute stability

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 and Switched Systems

Absolute Stability ProblemFind


Example

Example


A solution of the switched system

A Solution of the Switched System

This implies that


Two remarks

Two Remarks

Although both and are

stable, is not stable.

Instability requires repeated switching.


Optimal control approach

>

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


Optimal control approach1

Optimal Control Approach

Consider

as


Optimal control approach2

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

Solving Optimal Control Problems

is a functional:

Two approaches:

1. Hamilton-Jacobi-Bellman (HJB)

equation.

2. Maximum Principle.


Hjb equation

HJB Equation

Find such that

Integrating:

or

An upper bound for ,

obtained for the maximizing Eq. (HJB).


The case n 2

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.


Stability analysis of continuous time switched systems a variational approach

Basic Idea

The function is a first integral of if

We know that so

Thus, is a concatenation of two first integrals and


Example1

Example:

where and


Stability analysis of continuous time switched systems a variational approach

Thus,

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


Stability analysis of continuous time switched systems a variational approach

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

Nonlinear Switched Systems

where are GAS.

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


Lie algebraic approach

Lie-Algebraic Approach

For simplicity, consider the linear

differential inclusion:

so


Commutation relations and gas

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.


Lie brackets and geometry

Lie Brackets and Geometry

Consider

Then:


Geometry of car parking

Geometry of Car Parking

This is why we can park our car.

The term is the reason this takes

so long.


Nilpotency

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

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

A Partial Answer

Theorem(Margaliot & Liberzon, 2004)

2nd order nilpotency implies GAS.

Proof By the PMP, the WCSL satisfies

Let


Stability analysis of continuous time switched systems a variational approach

2nd order nilpotency  

 up to a single switch in the WCSL.

Then

1st order nilpotency

Differentiating again yields:


Handling singularity

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).


Stability analysis of continuous time switched systems a variational approach

3rd Order Nilpotency

In this case:

further differentiation cannot be carried out.


3rd order nilpotency

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

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.


Stability analysis of continuous time switched systems a variational approach

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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