STABILITY OF SWITCHED SYSTEMS

1. STABILITY OF SWITCHED SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

2. SWITCHED vs. HYBRID SYSTEMS Switched system: • is a family of systems • is a switching signal • state-dependent or time-dependent • autonomous or controlled Switching: Details of discrete behavior are “abstracted away” Hybrid systems give rise to classes of switching signals : stability Properties of the continuous state

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

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

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

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

7. COMMUTING STABLE MATRICES => GUES … t … quadratic common Lyap fcn [Narendra & Balakrishnan ’94]:

8. COMMUTATION RELATIONS and STABILITY Lie algebra: Lie bracket: GUES and quadratic common Lyap fcn guaranteed for: • 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) Further extension based only on Lie algebra is not possible [Agrachev & L ’01]

9. SWITCHED NONLINEAR SYSTEMS [Mancilla-Aguilar, Shim et al., Vu & L] => GUAS • Linearization (Lyapunov’s indirect method) • Commuting systems • Global results beyond commuting case – ??? [Unsolved Problems in Math. Systems and Control Theory]

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

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

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

13. SYSTEMS with SPECIAL STRUCTURE • Triangular systems • Feedback systems • passivity conditions • small-gain conditions • 2-D systems

14. TRIANGULAR SYSTEMS For linear systems, triangular form GUES quadratic common Lyap fcn exponentially fast exp fast For nonlinear systems, not true in general 0 Need to know (ISS) diagonal [Angeli & L ’00]

15. 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

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

17. TWO-DIMENSIONAL SYSTEMS Necessary and sufficient conditions for GUES known since 1970s worst-case switching quadratic common Lyap fcn <=> convex combinations of Hurwitz

18. (observability with respect to ) Example: => GAS observable WEAK LYAPUNOV FUNCTION Barbashin-Krasovskii-LaSalle theorem: is GAS if s.t. • (weak Lyapunov function) • is not identically zero along any nonzero solution

19. Theorem: is GAS if • . • observable for each • s.t. there are infinitely many • switching intervals of length COMMON WEAK LYAPUNOV FUNCTION To extend this to nonlinear switched systems and nonquadratic common weak Lyapunov functions, we need a suitable nonlinear observability notion

20. NONLINEAR VERSION Theorem: is GAS if s.t. • Each system is small-time norm-observable: pos. def. incr. : • s.t. there are infinitely many • switching intervals of length

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

22. MULTIPLE LYAPUNOV FUNCTIONS GAS respective Lyapunov functions => is GAS t Useful for analysis of state-dependent switching

23. MULTIPLE LYAPUNOV FUNCTIONS decreasing sequence decreasing sequence t => GAS [DeCarlo, Branicky]

24. DWELL TIME GES The switching times satisfy respective Lyapunov functions dwell time

25. DWELL TIME Need: t The switching times satisfy GES

26. DWELL TIME Need: must be The switching times satisfy GES

27. AVERAGE DWELL TIME average dwell time # of switches on dwell time: cannot switch twice if no switching: cannot switch if

28. AVERAGE DWELL TIME Theorem: [Hespanha] => is GAS if GAS is uniform over in this class Useful for analysis of hysteresis-based switching logics

29. MULTIPLE WEAK LYAPUNOV FUNCTIONS Theorem: is GAS if • . • observable for each • s.t. there are infinitely many • switching intervals of length • For every pair of switching times • s.t. • have – milder than a.d.t. Extends to nonlinear switched systems as before

30. APPLICATION: FEEDBACK SYSTEMS observable positive real Weak Lyapunov functions: Theorem: switched system is GAS if • s.t. infinitely many switching intervals of length • For every pair of switching times at • which we have (e.g., switch on levels of equal “potential energy”)

31. RELATED TOPICS NOT COVERED • Computational aspects (LMIs, Tempo & L) • Formal methods (work with Mitra & Lynch) • Stochastic stability (Chatterjee & L) • Switched systems with external signals • Applications to switching control design

32. REFERENCES Lie-algebras and nonlinear switched systems: [Margaliot & L ’04] Nonlinear observability, LaSalle: [Hespanha, L, Angeli & Sontag ’03] (http://decision.csl.uiuc.edu/~liberzon)