TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS Daniel Liberzon...

TOWARDS ROBUST LIE-ALGEBRAIC STABILITY

CONDITIONS for SWITCHED LINEAR SYSTEMS

Daniel Liberzon

Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign

European Control Conference, Budapest, Aug 2009

SWITCHED SYSTEMSSwitched system:

• is a family of systems

• is a switching signal

Switching can be:

• State-dependent or time-dependent• Autonomous or controlled

Details of discrete behavior are “abstracted away”

: stabilityProperties of the continuous state

Discrete dynamics classes of switching signals

STABILITY ISSUE

unstable

Asymptotic stability of each subsystem is

not sufficient for stability under arbitrary switching

KNOWN RESULTS: LINEAR SYSTEMS

Lie algebra w.r.t.

No further extension based on Lie algebra only

Quadratic common Lyapunov function exists in all these cases

Assuming GES of all modes, GUES is guaranteed for:

• commuting subsystems:

[Narendra–Balakrishnan,

• nilpotent Lie algebras (suff. high-order Lie brackets are 0)e.g.

Gurvits,

• solvable Lie algebras (triangular up to coord. transf.)

Kutepov, L–Hespanha–Morse,

• solvable + compact (purely imaginary eigenvalues)

Agrachev–L]

KNOWN RESULTS: NONLINEAR SYSTEMS

• Linearization (Lyapunov’s indirect method)

Can prove by trajectory analysis [Mancilla-Aguilar ’00]

or common Lyapunov function [Shim et al. ’98, Vu–L ’05]

• Global results beyond commuting case

Recently obtained using optimal control approach

[Margaliot–L ’06, Sharon–Margaliot ’07]

• Commuting systems

GUAS

REMARKS on LIE-ALGEBRAIC CRITERIA

• Checkable conditions

• In terms of the original data

• Independent of representation

• Not robust to small perturbations

In any neighborhood of any pair of matricesthere exists a pair of matrices generating the entire Lie algebra [Agrachev–L ’01]

How to measure closeness to a “nice” Lie algebra?

EXAMPLE

(discrete time, or cont. time periodic switching: )

Suppose .

Fact 1 Stable for dwell time :

Fact 2 If then always stable:

This generalizes Fact 1 (didn’t need ) and Fact 2 ( )

Switching between and

Schur

Fact 3 Stable if where is small enough s.t.

When we have where

EXAMPLE

(discrete time, or cont. time periodic switching: )

Suppose .

Fact 1 Stable for dwell time :

Fact 2 If then always stable:

Switching between and

Schur

When we have where

MORE GENERAL FORMULATION

Assume switching period :

MORE GENERAL FORMULATION

Find smallest s.t.

Assume switching period

MORE GENERAL FORMULATION

Assume switching period

Intuitively, captures:

• how far and are from commuting ( )

• how big is compared to ( )

, define byFind smallest s.t.

MORE GENERAL FORMULATION

Stability condition: where

already discussed: (“elementary shuffling”)

Assume switching period

, define byFind smallest s.t.

SOME OPEN ISSUES

• Relation between and Lie brackets of and

general formula seems to be lacking

SOME OPEN ISSUES

• Relation between and Lie brackets of and

Suppose but

Elementary shuffling: ,

not nec. close to but commutes with and

• Smallness of higher-order Lie brackets

Example:

SOME OPEN ISSUES

• Relation between and Lie brackets of and

Suppose but

Elementary shuffling: ,

not nec. close to but commutes with and

This shows stability for

Example:

In general, for can define by

• Smallness of higher-order Lie brackets

(Gurvits: true for any )

SOME OPEN ISSUES

• Relation between and Lie brackets of and

• Smallness of higher-order Lie brackets

• More than 2 subsystems

can still define but relation with Lie brackets less clear

• Optimal way to shuffle

especially when is not a multiple of

• Current work investigates alternative approaches

that go beyond periodic switching