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Revision:. Statement of converse UGES theorem?. Lecture 10. Vanishing and non-vanishing perturbations. Recommended reading. Khalil Chapter 5 (2 nd edition). Outline:. Stability of perturbed systems Vanishing perturbations (UGES) Non-vanishing perturbations (ultimate boundedness).

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  1. Revision: Statement of converse UGES theorem?

  2. Lecture 10 Vanishing and non-vanishing perturbations

  3. Recommended reading • Khalil Chapter 5 (2nd edition)

  4. Outline: • Stability of perturbed systems • Vanishing perturbations (UGES) • Non-vanishing perturbations (ultimate boundedness). • Summary

  5. Motivation • Often we prove stability of a model of interest via an auxiliary model which is easier to analyze. • We saw a special case of this approach in linearization. • While ideas are quite simple this approach is quite useful and appears in averaging, singular perturbations, slowly varying systems, and so on.

  6. Problem set-up • Check stability of the nominal model via an auxiliary model that is UGAS/UGES • We rewrite the nominal model: • g(t,x) is not known but a bound on |g(t,x)| is.

  7. Vanishing perturbation

  8. Vanishing perturbation • n is UGES if the following holds • a is UGES: • g is “sufficiently small”:

  9. Sketch of proof

  10. Comments • This is a qualitative result – it shows that UGES is a robust property. • Local version of the result can be stated. • If g(t,0)=0 and g is globally Lipschitz, then there exists L so that |g(t,x)|  L|x| holds. • An application of this result to linearization yields a result from Lecture 8: is ULES L is ULES

  11. Non-vanishing perturbation

  12. Important remark • Often we do not have that • In such cases, it is not possible to prove UGES of n. • We can still prove another useful stability property for n that we refer to as ultimate boundedness.

  13. Motivating example • Consider the system where g(t)=g=const.  0. • The origin of this system can never be UGES since the origin is NOT an equilibrium for any g  0. • We can write

  14. Motivating example • Consider the same system with g(t)=a sin(t) • The solution of this system is: which implies:

  15. Uniform ultimate boundedness (UUB) • The solutions of are uniformly ultimately bounded (UUB) if

  16. Graphical interpretation (time-invariant) |x(t)| c  |x0| b T t

  17. Graphical interpretation x0 All trajectories starting in Bc eventually enter and stay forever in Bb c b x=0

  18. ULES with non-vanishing perturbation • Suppose that: • a is ULES: • g satisfies: • Then, n solutions are UUB with

  19. Comments • If the auxiliary system is UGES, then for any fixed  (0,1) we have that • Hence, the perturbed system is UUB for any perturbation that is uniformly bounded! UGES has very good robustness properties. • This is not true for UGAS!

  20. ULAS with non-vanishing perturbation • Suppose that: • a is ULAS: • g satisfies: • Then, n solutions are UUB with

  21. Summary: • Lyapunov conditions guarantee a certain degree of nominal robustness (total stability). • We can quantify the amount of perturbation that the system can cope with without becoming unstable. • UGES/UGAS + vanishing  UGES/UGAS • UGES/UGAS + non-vanishing  UUB • Class K functions are a tool in estimating all the bounds.

  22. Next lecture: Homework: read Chapter 5 in Khalil

  23. Thank you for your attention!

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