TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign European Control Conference, Budapest, Aug 2009

SWITCHED SYSTEMS Switched 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 matrices there 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