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Stability with respect to periodic switching laws does not imply global stability under arbitrary switching

Ian D. Morris

Abstract

R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King have asked whether a linear switched system is guaranteed to be globally uniformly stable under arbitrary switching if it is known that every trajectory induced by a periodic switching law converges exponentially to the origin. Positive answers to this question have previously been announced for linear switched systems of order two and three. We answer this question negatively in all higher orders by constructing a fourth-order linear switched system with four switching states which is not uniformly exponentially stable but which has the property that every trajectory defined by a periodic switching law converges exponentially to the origin. We argue informally that positive linear systems with this combination of properties are likely to exist in sufficiently high dimensions.

Stability with respect to periodic switching laws does not imply global stability under arbitrary switching

Abstract

R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King have asked whether a linear switched system is guaranteed to be globally uniformly stable under arbitrary switching if it is known that every trajectory induced by a periodic switching law converges exponentially to the origin. Positive answers to this question have previously been announced for linear switched systems of order two and three. We answer this question negatively in all higher orders by constructing a fourth-order linear switched system with four switching states which is not uniformly exponentially stable but which has the property that every trajectory defined by a periodic switching law converges exponentially to the origin. We argue informally that positive linear systems with this combination of properties are likely to exist in sufficiently high dimensions.

Paper Structure

This paper contains 6 sections, 6 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical background and statements of theorems
  3. Proof of Theorem \ref{['th:not-periodic']}
  4. An explicit example
  5. Conclusions
  6. Proof of Theorem \ref{['th:example']}

Key Result

Theorem 1

There exists a set $\mathsf{A}$ of four $4\times 4$ real matrices with the property that every solution of eq:dimple corresponding to a periodic Lebesgue measurable switching law $A(t)$ converges exponentially to zero, but such that the linear switched system defined by $\mathsf{A}$ is not GUAS.

Figures (1)

  • Figure 1: The black line shows a periodic trajectory of the linear switched system defined by the matrices $A_0$, $A_1$ considered in section \ref{['se:explicit']}. Blue and brown flow lines follow the vector fields defined by $A_0$ and $A_1$ respectively. The periodic trajectory describes a level curve of the non-strict Lyapunov function $f$, and switches between the two vector fields upon crossing the horizontal or vertical axis.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • proof