1 / 27

Revision:

Revision:. What are Lyapunov conditions for stability? What are Lyapunov conditions for GAS? How can we estimate DOA via a Lyapunov function?. Lecture 6. Linear systems Lyapunov matrix equation LaSalle invariance principle. Recommended reading. Khalil Chapter 3 (2 nd edition).

farica
Download Presentation

Revision:

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Revision: What are Lyapunov conditions for stability? What are Lyapunov conditions for GAS? How can we estimate DOA via a Lyapunov function?

  2. Lecture 6 Linear systems Lyapunov matrix equation LaSalle invariance principle

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

  4. Outline: • Linear systems • Lyapunov matrix equation • LaSalle’s invariance principle • Summary

  5. Linear systems

  6. Solutions of linear systems • Consider • This system always has unique solutions (why?) defined for all time t  0 given by where eAt is the matrix exponential.

  7. Matrix exponential • Given a square matrix A, its matrix exponential is a family of square matrices satisfying: • NOTE: It is not true for A=(aij) that eAt=(eaijt).

  8. Example • Consider the double integrator: Its matrix exponential is:

  9. Stability conditions • The origin is stable iff and each i(A)=0 has an associated Jordan block of order one (in the Jordan canonical form). • The origin is attractive (hence GAS) iff A is a stability (or Hurwitz) matrix, i.e.

  10. Global exponential stability (GES) • There exist K,>0: • For linear systems GAS  GES! • GES is a strong stability property that nonlinear systems may not possess. • Local ES (LES): above holds for small x0

  11. Quadratic forms (Lyapunov functions) • Quadratic form is positive definite iff P is positive definite. • Sylvester’s conditions for P=PT>0: All leading principal minors of P are positive.

  12. Example • The following matrix is positive definite since

  13. Quadratic Lyapunov function • Taking derivative of a quadratic V, we have If Q=QT>0, then we have GES (A is Hurwitz). • Note: we assumed P>0 and if Q>0 then A is Hurwitz. • The opposite is true!!

  14. Lyapunov matrix theorem • Theorem: A is Hurwitz iff for symmetric Q and P: Moreover, if A is Hurwitz then P is unique.

  15. Comments • Necessary and sufficient conditions for GES. • Provides an algorithm for computing V(x)! • Quadratic functions always work for linear systems. • Theoretically very important. • Similar algorithms exist for classes of nonlinear systems (only sufficient conditions).

  16. LaSalle’s invariance principle

  17. Motivating example • Lyapunov function candidate is the energy: • We know (from experience) that the origin is AS but above conditions imply only stability!

  18. Motivation Fact 1: Lyapunov theorem on AS requires Fact 2: We often can only prove (although AS holds!) Fact 3: Finding another V satisfying (1) is often hard! Fact 4: LaSalle is simpler to use and allows us to use (2) with some extra conditions to conclude AS .

  19. Invariant sets • A set M is invariant if the following holds • It is stronger than positive invariance: • M is invariant iff it is positively invariant for

  20. LaSalle’s theorem • Consider a time-invariant system Let  D be a compact, positively invariant set. Let V: D  R satisfy Let E:={x : dV/dt=0 } and M be the largest invariant subset of E. Then, every solution in  approaches M as t .

  21. Graphical interpretation : positively invariant set on which dV/dt  0 E: set on which dV/dt=0 All trajectories converge to M! M may be a much smaller set than E! M: largest invariant subset of E

  22. Comments • Positive invariance of  is crucial!! • Pros: • V does not have to be sign definite! • Estimates of DOA via LaSalle may be better. • Can be applied to conclude stability of sets. • Applicable to electro-mechanical systems and many other systems. • Cons: - Applicable only to time-invariant systems.

  23. Special case (V>0) • Suppose that: • V>0 on a domain D containing the origin • dV/dt  0 on D • No solution other than the trivial solution can stay in E={x: dV/dt=0 } for all time. Then, the origin is AS. If, moreover, • D=Rn and V is radially unbounded then, the origin is GAS.

  24. Example – origin is AS: • is the largest positively invariant subset of D (e.g. via the level sets of V).

  25. Summary: • We can always construct (quadratic) V for linear systems. • Linear systems whenever AS, they are GES. • LaSalle invariance principle is used to conclude AS when we only have dV/dt  0. • LaSalle extends the classical Lyapunov theorems in several directions. • LaSalle is only applicable to time-invariant systems.

  26. Next lecture: Homework: Chapter 3 in Khalil

  27. Thank you for your attention!

More Related