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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

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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.

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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.

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

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

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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

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

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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.

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Asymptotic stability of the origin >0

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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:.

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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.

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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

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A geometric observation A local observation (at a point)

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Theorem 1: An instability result

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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.

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Proof: A sufficient condition for instability (Floquet theory)

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Proof: A sufficient condition for instability

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

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

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Proof: A theorem by Kato and Lancaster

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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].

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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.

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Pair-wise triangular switching systems

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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?

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Robustness of triangular systems

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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.

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Pairwise triangularisability

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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.

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