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Stability Analysis of Linear Switched Systems: An Optimal Control Approach

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  1. Part 1 Stability Analysis of LinearSwitched Systems:An Optimal Control Approach Michael Margaliot School of Elec. Eng. Tel Aviv University, Israel Joint work with: Gideon Langholz (TAU), Daniel Liberzon (UIUC), Michael S. Branicky (CWRU), Joao Hespanha (UCSB).

  2. Overview • Switched systems • Global asymptotic stability • The edge of stability • Stability analysis: • An optimal control approach • A geometric approach • An integrated approach • Conclusions

  3. Switched Systems Systems that can switch between several possible 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. Linear Systems Solution: Definition: The system is globallyasymptotically stable if Theorem: stability A is called a Hurwitzmatrix.

  10. Linear Switched Systems A system that can switch between them: Two (or more) linear systems: 10

  11. Stability Linear switched system: Definition: Globally uniformly asymptotically stable (GUAS): In other words, for any AKA, “stability under arbitrary switching”. 11

  12. A Necessary Condition for GUAS The switching law yields Thus, a necessary condition for GUAS is that both are Hurwitz. Then instability can only arise due to repeated switching. 12

  13. Why is the GUAS problem difficult? Answer 1: The number of possible switching laws is huge. 13

  14. Why is the GUAS problem difficult? Answer 2: Even if each linear subsystem is stable, the switched system may not be GUAS. 14

  15. Why is the GUAS problem difficult? Answer 2: Even if each linear subsystem is stable, the switched system may not be GUAS. 15

  16. Stability of Each Subsystem is Not Enough A multi-controller scheme plant + controller1 controller2 switching logic Even when each closed-loop is stable, the switched system may not be GUAS.

  17. Easy Case #1 A trajectory of the switched system: Suppose that the matrices commute: Then and since both matrices are Hurwitz, the switched system is GUAS. 17

  18. Easy Case #2 Suppose that both matrices are upper triangular: Then so Now so This proves GUAS. 18

  19. Optimal Control Approach Basic idea: (1) A relaxation: linear switched system → bilinear control system (2) characterize the “most destabilizing” control (3) the switched system is GUAS iff Pioneered by E. S. Pyatnitsky (1970s). 19

  20. where is the set of measurable functions taking values in [0,1]. Optimal Control Approach Relaxation: the switched system: → a bilinear control system:

  21. Optimal Control Approach The bilinear control system (BCS) is globally asymptotically stable (GAS) if: TheoremThe BCS is GAS if and only if the linear switched system is GUAS.

  22. Optimal Control Approach The most destabilizing control: Fix a final time . Let Optimal control problem: find a control that maximizes Intuition: maximize the distance to the origin.

  23. Optimal Control Approach and Stability Theorem The BCS is GAS iff

  24. Edge of Stability GAS The BCS: Consider original BCS GAS 24

  25. Edge of Stability GAS The BCS: Consider original BCS GAS Definition: k* is the minimal value of k>0 such that GAS is lost. 25

  26. Edge of Stability The BCS: Consider Definition: k* is the minimal value of k>0 such that GAS is lost. The system is said to be on the edge of stability. 26

  27. Edge of Stability The BCS: Consider Definition: k* is the minimal value of k>0 such that GAS is lost. k k 0 1 k* 0 1 k* Proposition: our original BCS is GAS iff k*>1. 27

  28. Edge of Stability The BCS: Consider Proposition: our original BCS is GAS iff k*>1. → we can always reduce the problem of analyzing GUAS to the problem of determining the edge of stability. 28

  29. Edge of Stability When n=2 Consider The trajectory x* corresponding to u*: A closed periodic trajectory 29

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

  31. Solving Optimal Control Problems 1. The HJB equation. Intuition: there exists a function and V can only decrease on any other trajectory of the system.

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

  33. The HJB for a BCS: Hence, In general, finding is difficult. Note: u* depends on only.

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

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

  36. Result #1 (Margaliot & Langholz, 2003) An explicit solution for the HJB equation, when n=2, and {A,B} is on the “edge of stability”. This yields an easily verifiable necessary and sufficient condition for stability of second-order switched linear systems.

  37. Basic Idea The HJB eq. is: Thus, Let be a first integral of that is, Then is a concatenation of two first integrals and

  38. Example: where and

  39. Nonlinear Switched Systems with GAS. Problem: Find a sufficient condition guaranteeing GAS of (NLDI).

  40. Lie-Algebraic Approach For the sake of simplicity, we present the approach for LDIs, that is, and

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

  42. Lie Brackets and Geometry Consider A calculation yields:

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

  44. Nilpotency We saw that [A,B]=0 implies GAS. What if [A,[A,B]]=[B,[A,B]]=0? Definition: k’th order nilpotency - all Lie brackets involving k terms vanish. [A,B]=0 → 1st order nil. [A,[A,B]]=[B,[A,B]]=0 → 2nd order nil.

  45. Nilpotency and Stability We saw that 1st order nilpotency Implies GAS. A natural question: Does k’th order nilpotency imply GAS?

  46. Some Known Results Switched linear systems: • k=2 implies GAS (Gurvits,1995). • k 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)

  47. A Partial Answer Result #2 (Margaliot & Liberzon, 2004) 3rd order nilpotency implies GAS. Proof: Consider the WCSL Define the switching function

  48. Differentiating m(t) yields 2nd order nilpotency   no switching in the WCSL! Differentiating again, we get 3rd order nilpotency    up to a single switching in the WCSL.

  49. Singular Arcs If m(t)0, then the Maximum Principle provides no direct information. Singularity can be ruled out using the auxiliary system.

  50. Summary • Parking cars is an underpaid job. • 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 worst-case trajectory.