INTRODUCTION to SWITCHED SYSTEMS ; STABILITY under ARBITRARY SWITCHING

1. 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

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

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

4. 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 Discrete dynamics classes of switching signals Properties of the continuous state Details of discrete behavior are “abstracted away” : stability and beyond

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

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

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

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

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

10. 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

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

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

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

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

15. 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 No further extension based on Lie algebra only [Agrachev–L ’01]

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

17. GUAS • Linearization (Lyapunov’s indirect method) • Global results beyond commuting case – ? [Unsolved Problems in Math. Systems & Control Theory ’04] SWITCHED NONLINEAR SYSTEMS • Commuting systems Can prove by trajectory analysis[Mancilla-Aguilar ’00] or common Lyapunov function[Shim et al. ’98, Vu–L ’05]

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

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

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

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

22. 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?

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

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

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

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

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