STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS

1. STABILITY under CONSTRAINED SWITCHING;SWITCHED SYSTEMS with INPUTS and OUTPUTS 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. TWO BASIC PROBLEMS • Stability for arbitrary switching • Stability for constrained switching

3. MULTIPLE LYAPUNOV FUNCTIONS – GAS – respective Lyapunov functions is GAS Useful for analysis of state-dependent switching

4. MULTIPLE LYAPUNOV FUNCTIONS decreasing sequence GAS decreasing sequence [DeCarlo, Branicky]

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

6. DWELL TIME The switching times satisfy Need: – GES

7. DWELL TIME The switching times satisfy – GES Need:

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

9. average dwell time – dwell time: cannot switch twice if # of switches on AVERAGE DWELL TIME

10. average dwell time Theorem: [Hespanha ‘99]Switched system is GAS if Lyapunov functions s.t. • . # of switches on AVERAGE DWELL TIME Useful for analysis of hysteresis-based switching logics

11. 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 ADT Extends to nonlinear switched systems as before

12. APPLICATION: FEEDBACK SYSTEMS (Popov criterion) linear system observable positive real Weak Lyapunov functions: See also invariance principles for switched systems in: [Lygeros et al., Bacciotti–Mazzi, Mancilla-Aguilar, Goebel–Sanfelice–Teel] Corollary: switched system is GAS if • s.t. infinitely many switching intervals of length • For every pair of switching times at • which we have

13. STATE-DEPENDENT SWITCHING Switched system unstable for some no common But switched system is stable for (many) other switch on the axes is a Lyapunov function

14. STATE-DEPENDENT SWITCHING Switch on y-axis level sets of level sets of GAS Switched system unstable for some no common But switched system is stable for (many) other

15. STABILIZATION by SWITCHING – both unstable Assume: stable for some

16. STABILIZATION by SWITCHING – both unstable Assume: stable for some So for each either or [Wicks et al. ’98]

17. UNSTABLE CONVEX COMBINATIONS Can also use multiple Lyapunov functions Linear matrix inequalities

18. SWITCHED SYSTEMS with INPUTS and OUTPUTS Outline: • Background • Input-to-state stability (ISS) • Main results • ISS under ADT switching • Invertibility of switched systems

19. INPUT-TO-STATE STABILITY (ISS) Nonlinear gain functions: ISS[Sontag ’89]: (means: pos.def., rad.unbdd.) class class without loss of generality, can replace by class , e.g. Equivalent Lyapunov characterization [Sontag–Wang]:

20. ISS under ADT SWITCHING Suppose functions class functions and constants such that : • . each subsystem is ISS If has average dwell time then switched system is ISS [Vu–Chatterjee–L, Automatica, Apr 2007]

21. SKETCH of PROOF Let be switching times on Consider Recall ADT definition: 1 1 2 3

22. SKETCH of PROOF – ISS 1 2 3 2 1 3 • GAS when • ISS under arbitrary switching if (common ) • ISS without switching (single ) Special cases:

23. Integral ISS: VARIANTS finds application in switching adaptive control • Output-to-state stability (OSS) [M. Müller] • Stochastic versions of ISS for randomly switched • systems [D. Chatterjee] • Some subsystems not ISS [Müller, Chatterjee]

24. SWITCHED SYSTEMS with INPUTS and OUTPUTS Outline: • Background • Input-to-state stability (ISS) • Main results • ISS under ADT switching • Invertibility of switched systems [Vu–L, Automatica, Apr 2008; Tanwani–L, CDC 2008]

25. PROBLEM FORMULATION Invertibility problem: recover uniquely from for given • Desirable: fault detection (in power systems) • Undesirable: security (in multi-agent networked systems) Related work:[Sundaram–Hadjicostis, Millerioux–Daafouz]; [Vidal et al., Babaali et al., De Santis et al.]

26. MOTIVATING EXAMPLE because Guess:

27. INVERTIBILITY of NON-SWITCHED SYSTEMS Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham]

28. INVERTIBILITY of NON-SWITCHED SYSTEMS Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham] Nonlinear: [Hirschorn, Isidori–Moog, Nijmeijer, Respondek, Singh]

29. INVERTIBILITY of NON-SWITCHED SYSTEMS Suppose it has relative degree at : Then we can solve for : Inverse system Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham] Nonlinear: [Hirschorn, Isidori–Moog, Nijmeijer, Respondek, Singh] SISO nonlinear system affine in control:

30. BACK to the EXAMPLE – similar We can check that each subsystem is invertible For MIMO systems, can use nonlinear structure algorithm

31. SWITCH-SINGULAR PAIRS Consider two subsystems and is a switch-singular pair if such that |||

32. FUNCTIONAL REPRODUCIBILITY SISO system affine in control with relative degreeat : For given and , that produces this output if and only if

33. CHECKING for SWITCH-SINGULAR PAIRS is a switch-singular pair for SISO subsystems with relative degrees if and only if For linear systems, this can be characterized by a matrix rank condition MIMO systems – via nonlinear structure algorithm Existence of switch-singular pairs is difficult to check in general

34. MAIN RESULT Theorem: Switched system is invertible at over output set if and only if each subsystem is invertible at and there are no switched-singular pairs no switch-singular pairs can recover subsystems are invertible can recover Idea of proof: The devil is in the details

35. BACK to the EXAMPLE Stop here because relative degree For every , and with form a switch-singular pair Switched system is not invertible on the diagonal Let us check for switched singular pairs:

36. OUTPUT GENERATION Given and , find (if exist) s.t. may be unique for some but not all Recall our example again:

37. OUTPUT GENERATION Given and , find (if exist) s.t. may be unique for some but not all Solution from : switch-singular pair Recall our example again:

38. OUTPUT GENERATION Given and , find (if exist) s.t. may be unique for some but not all Solution from : Recall our example again: switch-singular pair

39. OUTPUT GENERATION Given and , find (if exist) s.t. may be unique for some but not all Case 1: no switch at Then up to At , must switch to 2 But then Recall our example again: won’t match the given output

40. OUTPUT GENERATION Given and , find (if exist) s.t. may be unique for some but not all Recall our example again: Case 2: switch at No more switch-singular pairs

41. OUTPUT GENERATION Given and , find (if exist) s.t. may be unique for some but not all Recall our example again: Case 2: switch at No more switch-singular pairs

42. OUTPUT GENERATION Given and , find (if exist) s.t. may be unique for some but not all We also obtain from Recall our example again: Case 2: switch at No more switch-singular pairs We see how one switch can help recover an earlier “hidden” switch