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THE ROLE OF LIE BRACKETS IN STABILITY OF LINEAR AND NONLINEAR SWITCHED SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign Semi-plenary lecture, MTNS, Blacksburg, VA, 7/31/08 1 of 24

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THE ROLE OF LIE BRACKETS IN STABILITY OF LINEAR AND NONLINEAR SWITCHED SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

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THE ROLE OF LIE BRACKETS IN STABILITY OF LINEAR AND NONLINEAR SWITCHED SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

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THE ROLE OF LIE BRACKETS IN STABILITY OF LINEAR AND NONLINEAR SWITCHED SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

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THE ROLE OF LIE BRACKETS IN STABILITY OF LINEAR AND NONLINEAR SWITCHED SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

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THE ROLE OF LIE BRACKETS IN STABILITY OF LINEAR AND NONLINEAR SWITCHED SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

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SWITCHED vs. HYBRID SYSTEMS Switching can be: State-dependent or time-dependent Autonomous or controlled Further abstraction/relaxation: differential inclusion, measurable switching Details of discrete behavior are “abstracted away” : stabilityProperties of the continuous state Switched system: is a family of systems is a switching signal 2 of 24

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STABILITY ISSUE unstable Asymptotic stability of each subsystem is not sufficient for stability 3 of 24

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TWO BASIC PROBLEMS Stability for arbitrary switching Stability for constrained switching 4 of 24

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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: 5 of 24

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COMMUTING STABLE MATRICES => GUES For subsystems – similarly (commuting Hurwitz matrices) 6 of 24

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...... quadratic common Lyapunov function [ Narendra–Balakrishnan ’94 ] COMMUTING STABLE MATRICES => GUES Alternative proof: is a common Lyapunov function 7 of 24

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LIE ALGEBRAS and STABILITY Nilpotent means suff. high-order Lie brackets are 0 e.g. is nilpotent if s.t. is solvable if s.t. Lie algebra: Lie bracket: Nilpotent GUES [ Gurvits ’95 ] 8 of 24

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SOLVABLE LIE ALGEBRA => GUES Example: quadratic common Lyap fcn diagonal exponentially fast 0 exp fast [ L–Hespanha–Morse ’99 ] Lie’s Theorem: is solvable triangular form 9 of 24

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MORE GENERAL LIE ALGEBRAS Levi decomposition: radical (max solvable ideal) There exists one set of stable generators for which gives rise to a GUES switched system, and another which gives an unstable one [ Agrachev–L ’01 ] is compact (purely imaginary eigenvalues) GUES, quadratic common Lyap fcn is not compact not enough info in Lie algebra: 10 of 24

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SUMMARY: LINEAR CASE Extension based only on the Lie algebra is not possible Lie algebra w.r.t. Quadratic common Lyapunov function exists in all these cases 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) 11 of 24

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SWITCHED NONLINEAR SYSTEMS Lie bracket of nonlinear vector fields: Reduces to earlier notion for linear vector fields (modulo the sign) 12 of 24

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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 and Control Theory] Commuting systems GUAS 13 of 24

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SPECIAL CASE globally asymptotically stable Want to show: is GUAS Will show: differential inclusion is GAS 14 of 24

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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 15 of 24

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MAXIMUM PRINCIPLE is linear in at most 1 switch (unless ) GAS Optimal control: (along optimal trajectory) 16 of 24

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SINGULARITY Need: nonzero on ideal generated by (strong extremality) At most 2 switches GAS Know: nonzero on strongly extremal (time-optimal control for auxiliary system in ) constant control (e.g., ) Sussmann ’79: 17 of 24

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GENERAL CASE GAS Want: polynomial of degree (proof – by induction on ) bang-bang with switches 18 of 24

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THEOREM Suppose: GAS, backward complete, analytic s.t. and Then differential inclusion is GAS, and switched system is GUAS [ Margaliot–L ’06 ] Further work in [ Sharon–Margaliot ’07 ] 19 of 24

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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 20 of 24 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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EXAMPLE 21 of 24 ( 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

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EXAMPLE 21 of 24 ( 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

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MORE GENERAL FORMULATION 22 of 24 Assume switching period :

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MORE GENERAL FORMULATION 22 of 24 Find smallest s.t. Assume switching period

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MORE GENERAL FORMULATION 22 of 24 Assume switching period Intuitively, captures: how far and are from commuting ( ) how big is compared to ( ), define byFind smallest s.t.

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MORE GENERAL FORMULATION 22 of 24 Stability condition: where already discussed: (“elementary shuffling”) Assume switching period, define byFind smallest s.t.

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SOME OPEN ISSUES 23 of 24 Relation between and Lie brackets of and general formula seems to be lacking

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SOME OPEN ISSUES 23 of 24 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:

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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 23 of 24 Smallness of higher-order Lie brackets ( Gurvits: true for any )

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SOME OPEN ISSUES Relation between and Lie brackets of and 24 of 24 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 Thank you for your attention!

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Lecture #11 Stability of switched system: Arbitrary switching João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched Systems.

Lecture #11 Stability of switched system: Arbitrary switching João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched Systems.

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