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THE ROLE OF LIE BRACKETS IN STABILITY OF LINEAR AND NONLINEAR SWITCHED SYSTEMS

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

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

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

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

  4. 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

  5. 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

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

  7. 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 Properties of the continuous state : stability Details of discrete behavior are “abstracted away” Further abstraction/relaxation: differential inclusion, measurable switching 2 of 24

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

  9. TWO BASIC PROBLEMS • Stability for arbitrary switching • Stability for constrained switching 4 of 24

  10. TWO BASIC PROBLEMS • Stability for arbitrary switching • Stability for constrained switching

  11. GUAS is Lyapunov stability plus asymptotic convergence GUES: GLOBAL UNIFORM ASYMPTOTIC STABILITY 5 of 24

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

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

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

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

  16. MORE GENERAL LIE ALGEBRAS Levi decomposition: radical (max solvable ideal) • is compact (purely imaginary eigenvalues) GUES, quadratic common Lyap fcn • is not compact not enough info in Lie algebra: 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] 10 of 24

  17. SUMMARY: LINEAR CASE Assuming GES of all modes, GUES is guaranteed for: • commuting subsystems: • nilpotent Lie algebras (suff. high-order Lie brackets are 0) • solvable Lie algebras (triangular up to coord. transf.) e.g. • solvable + compact (purely imaginary eigenvalues) Lie algebra w.r.t. Quadratic common Lyapunov function exists in all these cases Extension based only on the Lie algebra is not possible 11 of 24

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

  19. GUAS • Linearization (Lyapunov’s indirect method) • Global results beyond commuting case – ? [Unsolved Problems in Math. Systems and Control Theory] SWITCHED NONLINEAR SYSTEMS • Commuting systems Can prove by trajectory analysis[Mancilla-Aguilar ’00] or common Lyapunov function[Shim et al. ’98, Vu–L ’05] 13 of 24

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

  21. (original switched system ) Worst-case control law[Pyatnitskiy, Rapoport, Boscain, Margaliot]: fix and small enough OPTIMAL CONTROL APPROACH Associated control system: where 15 of 24

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

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

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

  25. 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

  26. Checkable conditions • In terms of the original data • Independent of representation • Not robust to small perturbations In anyneighborhood of any pair of matrices there exists a pair of matrices generating the entire Lie algebra [Agrachev–L ’01] REMARKS on LIE-ALGEBRAIC CRITERIA How to measure closeness to a “nice” Lie algebra? 20 of 24

  27. EXAMPLE Switching between and (discrete time, or cont. time periodic switching: ) Schur Stable for dwell time : Fact 1 Suppose . If then always stable: Fact 2 When we have where Stable if where is small enough s.t. Fact 3 This generalizes Fact 1 (didn’t need ) and Fact 2 ( ) 21 of 24

  28. EXAMPLE Switching between and (discrete time, or cont. time periodic switching: ) Schur Suppose . Stable for dwell time : Fact 1 If then always stable: Fact 2 When we have where 21 of 24

  29. MORE GENERAL FORMULATION Assume switching period : 22 of 24

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

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

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

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

  34. SOME OPEN ISSUES • Relation between and Lie brackets of and Suppose but Elementary shuffling: , not nec. close to but commutes with and Example: • Smallness of higher-order Lie brackets 23 of 24

  35. SOME OPEN ISSUES • Relation between and Lie brackets of and Suppose but Elementary shuffling: , (Gurvits: true for any ) This shows stability for In general, for can define by • Smallness of higher-order Lie brackets not nec. close to but commutes with and Example: 23 of 24

  36. SOME OPEN ISSUES • Relation between and Lie brackets of and can still define but relation with Lie brackets less clear especially when is not a multiple of • Smallness of higher-order Lie brackets • More than 2 subsystems • Optimal way to shuffle Thank you for your attention! 24 of 24

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