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Switched System Stability Analysis and Invertibility Challenges

Explore the input-to-state stability (ISS) and invertibility challenges of switched systems. Discover the impact of functions and dwell time on ISS and how stability margin plays a role. Delve into the invertibility problem and its applications in various systems. Learn about switch-singular pairs and the theorem on invertibility in the presence of these pairs. Get insights into output generation and the complexities of the invertibility problem in different scenarios.

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Switched System Stability Analysis and Invertibility Challenges

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  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 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]

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

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

  5. VARIANTS • Stability margin • Integral ISS (with stability margin) 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]

  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) • Undesirable: security (in multi-agent networked systems) Related work:[Sundaram–Hadjicostis, Millerioux–Daafouz]; [Vidal et al., Babaali et al., De Santis et al.]

  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 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:

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

  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 degreeat : 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 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

  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 no switch-singular pairs can recover subsystems are invertible can recover Idea of proof: The devil is in the details

  17. 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:

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

  19. 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:

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  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

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