1. IFAC AIRTC, Budapest, October 2000 On the Dynamic Instability of a Class of Switching System Robert Noel Shorten Department of Computer Science National University of Ireland Maynooth, Ireland Fiacre Ó Cairbre Department of Mathematics National University of Ireland Maynooth, Ireland Paul Curran Department of Electrical Engineering National University of Ireland Dublin, Ireland

2. The switching system: Motivation We are interested in the asymptotic stability of linear switching systems: where x(t)  R n, and where A(t)  R n  n belongs to the finite set {A 1,A 2,…A M }. Switching is now common place in control engineering practice. –Gain scheduling. –Fuzzy and Hybrid control. –Multiple models, switching and tuning. Recent work by Douglas Leith (VB families) has shown a dynamic equivalence between classes of linear switching systems and non-linear systems.

3. So what is the problem: Asymptotic stability The dynamic system, A  R n  n, is asymptotically stable if the matrix A is Hurwitz ( i (A), i  {1,…,n}  C - ). If A is Hurwitz, the solution of the Lyapunov equation, is P = P T >0, for all Q=Q T >0.

4. Switching systems: The issue of stability The car in the desert scenario!

5. Periodic oscillations and instability Instability due to switching Periodic orbit due to switching

6. Traditional approach to analysis Stability: Most results in the literature pertain to stability –Lyapunov –Input-output –Slowly varying systems …. –Conservatism is well documented Instability: Few results concern instability –Describing functions –Chattering (sliding modes) –Routes to instability (chaos) : Potentially much tighter conditions

7. Overview of talk Some background discussion and definitions. Some geometric observations Main theorem and proof. Consequences of main theorem. Extensions Concluding remarks

8. Hurwitz matrices Hurwitz matrices: The matrix A, is said to be Hurwitz if its eigenvalues lie in the open left-half of the complex plane. A matrix A is said to be not-Hurwitz if some of its eigenvalues lie in the open right-half of the complex plane.

9. Asymptotic stability of the origin >0

10. Instability The switching system [1] is unstable if some switching sequence exists such that as time increases the magnitude of the solution to [1], x(t) is unbounded:.

11. Matrix pencils A matrix pencil is defined as: If the eigenvalues of  [A i,M] are in C - for all non-negative  i, then  [A i,M] is referred to as a Hurwitz pencil.

12. Common quadratic Lyapunov function (CQLF) V(x) = x T Px is said to be a common quadratic Lyapunov function (CQLF) for the dynamic systems, if and

13. A geometric observation A local observation (at a point)

14. Theorem 1: An instability result

15. Outline of the proof We consider a periodic switching sequence for [1]. We use known instability conditions for periodic systems using Floquet theory. We show that Theorem 1 implies instability for such systems.

16. Proof: A sufficient condition for instability (Floquet theory)

17. Proof: A sufficient condition for instability

18. Proof: an approximation So, for T small enough, the effect of the higher order terms become negligible, and we have,

19. Proof: an approximation So, for T small enough, the effect of the higher order terms become negligible, and we have,

20. Proof: A theorem by Kato and Lancaster

21. General switching systems: The existence of CQLF It has long been known that a necessary condition for the existence of a CQLF is that the matrix pencil: is Hurwitz. In general, this is a very conservative condition. Now we know that this conditions is necessary for stability of the system [1].

22. Equivalence of stability and CQLF for low order systems Necessary and sufficient conditions for the existence of a CQLF for two second order systems is that the matrix pencils are both Hurwitz. Non-existence of CQLF implies that one of the dual switching systems is unstable.

23. Pair-wise triangular switching systems

24. Pair-wise triangular switching systems: Comments A single T implies the existence of a CQLF for each of the component systems. Is this a robust result? Pair-wise triangularisability and some extra conditions imply global attractivity. Are general pairwise triangularisable systems stable?

25. Robustness of triangular systems

27. Consider the periodic switching system with duty cycle 0.5 with: As L increases A 1 and A 2 become more and more triangularisable. However, for K>4,L>2, an unstable switching sequence always exists.

28. Pairwise triangularisability

30. Conclusions Looked at a local stability theorem. Presented a formal proof. Used theorem to answer some open questions. Presented some extensions to the work. Gained insights into conservatism (or non- conservatism) of the CQLF.