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 : stability Switching: state-dependent or time-dependent autonomous or controlled Properties of the continuous state Switched system: is a family of systems is a switching signal Details of discrete behavior are abstracted away Hybrid systems give rise to classes of switching signals

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

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

7 COMMUTING STABLE MATRICES => GUES quadratic common Lyap fcn [Narendra & Balakrishnan 94]: … t …

8 COMMUTATION RELATIONS and STABILITY 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) Lie algebra: Lie bracket: Further extension based only on Lie algebra is not possible [Agrachev & L 01]

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

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

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

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

13 GENERAL CASE Theorem: suppose GAS, backward complete, analytic s.t. and Then switched system is GUAS

14 SYSTEMS with SPECIAL STRUCTURE Triangular systems Feedback systems passivity conditions small-gain conditions 2-D systems

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

16 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

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

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

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

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

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

22 TWO BASIC PROBLEMS Stability for arbitrary switching Stability for constrained switching

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

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

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

26 DWELL TIME The switching times satisfy GES t Need:

27 DWELL TIME The switching times satisfy GES must be Need:

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

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

30 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

31 APPLICATION: FEEDBACK SYSTEMS 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) observable positive real Weak Lyapunov functions:

32 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

33 REFERENCES Lie-algebras and nonlinear switched systems: [Margaliot & L 04] Nonlinear observability, LaSalle: [Hespanha, L, Angeli & Sontag 03] (