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INTRODUCTION to SWITCHED SYSTEMS ; STABILITY under ARBITRARY SWITCHING Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign IAAC Workshop, Herzliya, Israel, June 1, 2009

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electric circuit SWITCHED and HYBRID SYSTEMS Hybrid systems combine continuous and discrete dynamics Which practical systems are hybrid? Which practical systems are not hybrid? More tractable models of continuous phenomena thermostat stick shift walking cell division

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MODELS of HYBRID SYSTEMS … [ Van der Schaft–Schumacher ’00 ] [ Proceedings of HSCC ] continuous discrete [ Nešić–L ‘05 ]

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SWITCHED vs. HYBRID 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” : stability and beyondProperties of the continuous state Discrete dynamics classes of switching signals

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STABILITY ISSUE unstable Asymptotic stability of each subsystem is not sufficient for stability

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TWO BASIC PROBLEMS Stability for arbitrary switching Stability for constrained switching

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TWO BASIC PROBLEMS Stability for arbitrary switching Stability for constrained switching

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GLOBAL UNIFORM ASYMPTOTIC STABILITY GUAS is: Lyapunov stability plus asymptotic convergence GUES:

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quadratic is GUES where is positive definite is GUAS if (and only if) s.t. COMMON LYAPUNOV FUNCTION

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OUTLINE Stability criteria to be discussed: Commutation relations (Lie algebras) Feedback systems (absolute stability) Observability and LaSalle-like theorems Common Lyapunov functions will play a central role

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COMMUTING STABLE MATRICES => GUES For subsystems – similarly (commuting Hurwitz matrices)

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quadratic common Lyapunov function [ Narendra–Balakrishnan ’94 ] COMMUTING STABLE MATRICES => GUES Alternative proof: is a common Lyapunov function

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Nilpotent means sufficiently high-order Lie brackets are 0 NILPOTENT LIE ALGEBRA => GUES Lie algebra: Lie bracket: Hence still GUES [ Gurvits ’95 ] For example: (2nd-order nilpotent) In 2nd-order nilpotent case Recall: in commuting case

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SOLVABLE LIE ALGEBRA => GUES Example: quadratic common Lyap fcn diagonal exponentially fast [ Kutepov ’82, L–Hespanha–Morse ’99 ] Larger class containing all nilpotent Lie algebras Suff. high-order brackets with certain structure are 0 exp fast Lie’s Theorem: is solvable triangular form

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SUMMARY: LINEAR CASE Lie algebra w.r.t. Assuming GES of all modes, GUES is guaranteed for: commuting subsystems: nilpotent Lie algebras (suff. high-order Lie brackets are 0) e.g. solvable Lie algebras (triangular up to coord. transf.) solvable + compact (purely imaginary eigenvalues) No further extension based on Lie algebra only [ Agrachev–L ’01 ] Quadratic common Lyapunov function exists in all these cases

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SWITCHED NONLINEAR SYSTEMS Lie bracket of nonlinear vector fields: Reduces to earlier notion for linear vector fields (modulo the sign)

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SWITCHED 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 – ? [ Unsolved Problems in Math. Systems & Control Theory ’04 ] Commuting systems GUAS

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SPECIAL CASE globally asymptotically stable Want to show: is GUAS Will show: differential inclusion is GAS

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OPTIMAL CONTROL APPROACH Associated control system: where (original switched system ) Worst-case control law [Pyatnitskiy, Rapoport, Boscain, Margaliot] : fix and small enough

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MAXIMUM PRINCIPLE is linear in at most 1 switch (unless ) GAS Optimal control: (along optimal trajectory)

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GENERAL CASE GAS Want: polynomial of degree (proof – by induction on ) bang-bang with switches [ Margaliot–L ’06, Sharon–Margaliot ’07 ]

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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 measure closeness to a “nice” Lie algebra?

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FEEDBACK SYSTEMS: ABSOLUTE STABILITY Circle criterion: quadratic common Lyapunov function is strictly positive real (SPR): For this reduces to SPR (passivity) Popov criterion not suitable: depends on controllable

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FEEDBACK SYSTEMS: SMALL-GAIN THEOREM controllable Small-gain theorem: quadratic common Lyapunov function

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OBSERVABILITY and ASYMPTOTIC STABILITY Barbashin-Krasovskii-LaSalle theorem: (observability with respect to ) observable => GAS Example: is GAS if s.t. is not identically zero along any nonzero solution (weak Lyapunov function)

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SWITCHED LINEAR SYSTEMS [ Hespanha ’04 ] Theorem (common weak Lyapunov function): observable for each To handle nonlinear switched systems and non-quadratic weak Lyapunov functions, need a suitable nonlinear observability notion Switched linear system is GAS if infinitely many switching intervals of length s.t..

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SWITCHED NONLINEAR SYSTEMS Theorem (common weak Lyapunov function): s.t. Switched system is GAS if infinitely many switching intervals of length Each system is norm-observable: [Hespanha–L–Sontag–Angeli ’05]

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