Equilibrium Point(Examples)

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!!