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Stability Analysis of Switched Systems: A Variational Approach

This article discusses stability analysis of switched systems using a control-theoretic and geometric approach, with a focus on the application of differential inclusions. The article also explores optimal control approaches and provides examples of both linear and nonlinear switched systems.

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Stability Analysis of Switched Systems: A Variational Approach

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  1. Stability Analysis of Switched Systems: A Variational Approach Michael Margaliot School of EE-Systems Tel Aviv University Joint work with Daniel Liberzon (UIUC)

  2. Overview • Switched systems • Stability • Stability analysis: • A control-theoretic approach • A geometric approach • An integrated approach • Conclusions

  3. Switched Systems Systems that can switch between several modes of operation. Mode 1 Mode 2

  4. Example 1 server

  5. linear filter Example 2 Switched power converter 50v 100v

  6. Example 3 A multi-controller scheme plant + controller1 controller2 switching logic Switched controllers are “stronger” than regular controllers.

  7. More Examples • Air traffic control • Biological switches • Turbo-decoding • ……

  8. Synthesis of Switched Systems Driving: use mode 1 (wheels) Braking: use mode 2 (legs) The advantage: no compromise

  9. stronger MODELING CAPABILITY weaker Mathematical Modeling with Differential Inclusions easier ANALYSIS harder

  10. The Gestalt Principle “Switched systems are more than the sum of their subsystems.“  theoretically interesting  practically promising

  11. Differential Inclusions A solution is an absolutely continuous function satisfying (DI) for all t. Example:

  12. Stability The differential inclusion is called GAS if for any solution (i) (ii)

  13. The Challenge • Why is stability analysis difficult? • A DI has an infinite number of solutions for each initial condition. • The gestalt principle.

  14. Absolute Stability

  15. Problem of Absolute Stability The closed-loop system: A is Hurwitz, so CL is asym. stable for any The Problem of Absolute Stability: Find For CL is asym. stable for any

  16. Absolute Stability and Switched Systems The Problem of Absolute Stability: Find

  17. Example

  18. Trajectory of the Switched System This implies that

  19. Although both and are stable, is not stable. Instability requires repeated switching. This presents a serious problem in multi-controller schemes.

  20. Optimal Control Approach Write as a control system: Fix Define Problem: Find the control that maximizes is the worst-case switching law (WCSL). Analyze the corresponding trajectory

  21. Optimal Control Approach Consider as

  22. Optimal Control Approach Thm. 1 (Pyatnitsky) If then: (1) The function is finite, convex, positive, and homogeneous (i.e., ). (2) For every initial condition there exists a solution such that

  23. Solving Optimal Control Problems is a functional: Two approaches: • The Hamilton-Jacobi-Bellman (HJB) equation. • The Maximum Principle.

  24. The HJB Equation Find such that Integrating: or An upper bound for , obtained for the maximizing Eq. (HJB).

  25. The HJB for a LDI: Hence, In general, finding is difficult.

  26. The Maximum Principle Let Then, Differentiating we get A differential equation for with a boundary condition at

  27. Summarizing, The WCSL is the maximizing that is, We can simulate the optimal solution backwards in time.

  28. The Case n=2 Margaliot & Langholz (2003) derived an explicit solution for when n=2. This yields an easily verifiable necessary and sufficient condition for stability of second-order switched linear systems.

  29. The Basic Idea The function is a first integral of if We know that so Thus, is a concatenation of two first integrals and

  30. Example: where and

  31. Thus, so we have an explicit expression for V (and an explicit solution of HJB).

  32. Nonlinear Switched Systems where are GAS. Problem: Find a sufficient condition guaranteeing GAS of (NLDI).

  33. Lie-Algebraic Approach For the sake of simplicity, consider the LDI so

  34. Commutation and GAS Suppose that A and B commute, AB=BA, then Definition: The Lie bracket of Ax and Bx is [Ax,Bx]:=ABx-BAx. Hence, [Ax,Bx]=0 implies GAS.

  35. Lie Brackets and Geometry Consider Then:

  36. Geometry of Car Parking This is why we can park our car. The term is the reason it takes so long.

  37. Nilpotency Definition: k’th order nilpotency - all Lie brackets involving k+1 terms vanish. 1st order nilpotency: [A,B]=0 2nd order nilpotency: [A,[A,B]]=[B,[A,B]]=0 Q: Does k’th order nilpotency imply GAS?

  38. Some Known Results Switched linear systems: • k = 2 implies GAS (Gurvits,1995). • k’th order nilpotency implies GAS (Liberzon, Hespanha, and Morse, 1999).(The proof is based on Lie’s Theorem) Switched nonlinear systems: • k = 1 implies GAS. • An open problem: higher orders of k? (Liberzon, 2003)

  39. A Partial Answer Thm. 1 (Margaliot & Liberzon, 2004) 2nd order nilpotency implies GAS. Proof: Consider the WCSL Define the switching function

  40. Differentiating m(t) yields 1st order nilpotency   no switching in the WCSL. Differentiating again, we get 2nd order nilpotency    up to a single switch in the WCSL.

  41. Handling Singularity If m(t)0, then the Maximum Principle does not necessarily provide enough information to characterize the WCSL. Singularity can be ruled out using thenotion of strong extermality (Sussmann, 1979).

  42. 3rd order Nilpotency In this case: further differentiation cannot be carried out.

  43. 3rd order Nilpotency Thm. 2 (Sharon & Margaliot, 2005) 3rd order nilpotency implies The proof is based on using: (1) the Hall-Sussmann canonical system; and (2) the second-order Agrachev-Gamkrelidze MP.

  44. Hall-Sussmann System Consider the case [A,B]=0. Guess the solution: Then so and (HS system)

  45. Hall-Sussmann System If two controls u, v yield the same values for then they yield the same value for does not depend on u, Since we conclude that any and measurable control can be replaced with a bang-bang control with a single switch:

  46. 3rd order Nilpotency In this case, The HS system:

  47. Conclusions • Switched systems and differential inclusions are important in various scientific fields, and pose interesting theoretical questions. • Stability analysis is difficult. A natural and useful idea is to consider the most unstable trajectory.

  48. For more information, see the survey paper: “Stability analysis of switched systems using variational principles: an introduction”, Automatica 42(12), 2059-2077, 2006. Available online: www.eng.tau.ac.il/~michaelm

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