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

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

Switched Systems

Systems that can switch between

several modes of operation.

Mode 1

Mode 2

linear filter

Example 1

Switched power converter



Example 2

A multi-controller scheme





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.


Global Asymptotic Stability (GAS)

Definition The differential inclusion

is called GAS if for any solution



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

The closed-loop system:

A is Hurwitz, so CL is asym. stable for


Absolute Stability Problem


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:



Problem Find a control maximizing

Optimal Control Approach

Write as the bilinear control


is the worst-case switching law (WCSL).

Analyze the corresponding trajectory

Optimal Control Approach



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)


2. Maximum Principle.

HJB Equation

Find such that



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


where and


→ 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:


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



Geometry of Car Parking

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?

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


2nd order nilpotency  

 up to a single switch in the WCSL.


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


(1) Hall-Sussmann canonical system;

(2) A second-order MP



  • 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.


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