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Equilibrium Point(Examples)

Equilibrium Point(Examples). Ex:.  Find equilibrium point. (i). This implies that. (ii).  Analyze the stability of the equilibrium point. (i). Equilibrium Point(Examples). (ii). Thus the system is (globally) asymptotically stable. Instability Theorem. Instability Theorem.

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Equilibrium Point(Examples)

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  1. Equilibrium Point(Examples) Ex:  Find equilibrium point. (i) This implies that (ii)  Analyze the stability of the equilibrium point. (i)

  2. Equilibrium Point(Examples) (ii) Thus the system is (globally) asymptotically stable.

  3. Instability Theorem • Instability Theorem Motivation : Formulation :

  4. Instability Theorem(Continued) If these conditions are met, the following can be constructed (1)

  5. Theorem Theorem : Proof :

  6. LaSalle’s Theorem • LaSalle’s Theorem (invariance principle) Motivation :

  7. Theorem Theorem :

  8. Theorem Theorem : continuously differentiable p.d. radially unbounded Proof : Using the idea of limit set & invariant set, it can be proved. Ex:

  9. Examples (Continued)

  10. Linear System • Linear System leading principal minors are all positive

  11. Theorem Theorem : Proof : Sufficiency follows from the Lyapunov stability theorem.

  12. Proof (Continued) Then

  13. A-Hurwitz Routh-Hurwitz test Proof (Continued) Thus

  14. Examples Ex:

  15. Stability Analysis through Linearization • Stability Analysis through Linearization : The first(indirect) Lyapunov method • Motivation : • Formulation : Theorem

  16. Proof

  17. Proof (Continued)

  18. Advantage & Disadvantage of the indirect method • Advantage of the indirect method : Easy to apply • Disadvantage • Only asymptotic stability can be investigated • : continuously differentiable. • Critical case : • Domain of attraction is unknown

  19. Application to Control • Application to Control Theorem:

  20. Theorem Proof:

  21. Selection of Lyapunov function Candidates • Remarks on the selection of Lyapunov function Candidates • Quadratic form : Works for linear system. • Quadratic form plus integral of nonlinearity

  22. Krasovskii’s method • Krasovskii’s method Premise :

  23. Proof Proof:

  24. Examples Ex:

  25. Variable gradient method • Variable gradient method Idea :

  26. Variable gradient method (Continued)

  27. Examples Ex:

  28. Computer generation of Lyapunov function.This approach is used to construct Lyapunov function so that the estimate of the domain of attraction is easy to obtain • When nothing works : THINK!!

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