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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).

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  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!

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