1. INPUT-TO-STATE STABILITY of SWITCHED SYSTEMS Debasish Chatterjee, Linh Vu, Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

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

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

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

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

6. INVERTIBILITY of SWITCHED SYSTEMS Aneel Tanwani, Linh Vu, Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

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

8. MOTIVATING EXAMPLE because Guess:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

25. CONCLUSIONS Showed how results on stability under slow switching extend in a natural way to external stability (ISS) Studied new invertibility problem: recovering both the input and the switching signal Both problems have applications in control design General motivation/application: analysis and design of complex interconnected systems