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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

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
Switched Systems

Systems that can switch between

several modes of operation.

Mode 1

Mode 2

Example 1

linear filter

Example 1

Switched power converter



Example 2
Example 2

A multi-controller scheme





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.


Global asymptotic stability gas
Global Asymptotic Stability (GAS)

Definition The differential inclusion

is called GAS if for any solution



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
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 and Switched Systems

Absolute Stability ProblemFind

Two remarks
Two Remarks

Although both and are

stable, is not stable.

Instability requires repeated switching.

Optimal control approach


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 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)


2. Maximum Principle.

Hjb equation
HJB Equation

Find such that



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.

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
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:


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.

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.


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


2nd order nilpotency  

 up to a single switch in the WCSL.


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).

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


(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.