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TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign, USA Yuliy Baryshnikov Bell Labs, Murray Hill, NJ, USA Andrei A. Agrachev S.I.S.S.A., Trieste, Italy 1 of 11

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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 2 of 11

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STABILITY ISSUE unstable Asymptotic stability of each subsystem is not sufficient for stability under arbitrary switching 3 of 11

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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 Stability is preserved under arbitrary switching 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 ] 4 of 11

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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 5 of 11

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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 capture closeness to a “nice” Lie algebra? 6 of 11

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DISCRETE TIME: BOUNDS on COMMUTATORS 7 of 11 – Schur stable Let GUES Idea of proof: take,, consider GUES:

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DISCRETE TIME: BOUNDS on COMMUTATORS 7 of 11 – Schur stable GUES: Let GUES Idea of proof: take,, consider

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DISCRETE TIME: BOUNDS on COMMUTATORS 7 of 11 induction basis contraction small – Schur stable GUES: Let GUES Idea of proof: take,, consider

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DISCRETE TIME: BOUNDS on COMMUTATORS 7 of 11 Accounting for linear dependencies among – easy – Schur stable GUES: Let GUES Extending to higher-order commutators – hard

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DISCRETE TIME: BOUNDS on COMMUTATORS 7 of 11 For they commute GUES, common Lyap fcn – Schur stable GUES: Let GUES Example: Our approach still GUES for Perturbation analysis based on GUES for

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CONTINUOUS TIME: LIE ALGEBRA STRUCTURE 8 of 11 compact set of Hurwitz matrices GUES:Lie algebra: Levi decomposition: ( solvable, semisimple) Switched transition matrix splits as where and Let and GUES robust condition but not constructive

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CONTINUOUS TIME: LIE ALGEBRA STRUCTURE 8 of 11 more conservative but easier to verify GUES There are also intermediate conditions GUES: Levi decomposition: ( solvable, semisimple) Switched transition matrix splits as where and Let and compact set of Hurwitz matrices Lie algebra:

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CONTINUOUS TIME: LIE ALGEBRA STRUCTURE 9 of 11 Levi decomposition: Switched transition matrix splits as Previous slide: small GUES But we also know: compact Lie algebra (not nec. small) GUES Cartan decomposition: ( compact subalgebra) Transition matrix further splits: where and Let GUES

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CONTINUOUS TIME: LIE ALGEBRA STRUCTURE 10 of 11 Example: GUES Levi decomposition: Cartan decomposition:

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SUMMARY Stability is preserved under arbitrary switching for: commuting subsystems: solvable Lie algebras (triangular up to coord. transf.) solvable + compact (purely imaginary eigenvalues) 11 of 11 approximately

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