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The Problem of Absolute Stability

The Problem of Absolute Stability

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The Problem of Absolute Stability

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  1. + _ G(s) Plant + compensator Actuator The Problem of Absolute Stability • Motivation Assume that the actuator is linear Hurwitz sector

  2. Aizerman conjecture Assume now that the actuator is nonlinear, for instance, a saturator This question was posed by M.A. Aizerman in 1940’s. Aizerman conjecture :

  3. Kalman conjecture Kalman conjecture : A answer was first proposed by A.I. Lurie. Popov, Kalman, Yakubovich and others contributed to the solution. Sometimes this problem is called the Lurie problem. N.B :

  4. Problem Formulation Plant Assumptions

  5. Problem (Continued) We can rewrite the sector condition  as

  6. Problem (Continued) Consider the decentralized feedback Define and Then p-dim sector condition is symmetric positive definite diagonal matrix

  7. Def : Problem (Continued) - Generalization for centralized case Introduce Then where again

  8. Notation Define : Problem (Continued) Consider again

  9. Remarks Remarks • Absolute stability  not another type of stability • Absolute stability gives a measure of robustness • No constructive necessary and sufficient conditions have beenfound as yet. The main tool is the Kalman-Yakubovich-Popov Lemma.

  10. Solution • Approach to the solution To find out conditions of absolute stability Find a Lyapunov function good for a continuum of systems – all with nonlinearities in the sector  Two types of Lyapunov functions are typically used. Here the conditions are less conservative (Popov Criterion).

  11. Solution (Continued)

  12. Kalman – Yakubovich-Popov Lemma

  13. Circle Criterion • Circle Criterion

  14. Circle Criterion (Continued) Then

  15. K – Y – P Lemma

  16. K – Y – P Lemma (Continued) Lemma :

  17. 0 G(s) Generalization To eliminate the restriction on A to be Hurwitz  loop transformation (pole shifting) G(s)

  18. Generalization (Continued) Obviously

  19. Generalization : Case 1

  20. Generalization : Case 1

  21. Generalization : Case 1   

  22. Generalization : Case 1 

  23. Generalization : Case 1

  24. Generalization : Case 2

  25. Generalization : Case 3

  26. Summarizing

  27. Summarizing

  28. Popov Criterion • Popov Criterion Popov

  29. Popov Criterion (Continued)

  30. Popov Criterion (Continued)

  31. Popov Criterion (Continued)

  32. Popov Criterion (Continued)

  33. Popov Criterion (Continued)

  34. Theorem Theorem :

  35. Theorem(Continued)

  36. K can be as large as possible Finite sector Theorem(Continued) Graphically

  37. Theorem(Continued)

  38. Theorem(Continued)