A method for determining the non-existence of a common quadratic Lyapunov function for switched linear systems based on particle swarm optimisation

Abstract

The existence of a common quadratic Lyapunov function (CQLF) for a switched linear system guarantees its global asymptotic stability. Even if progress in finding the conditions for the existence/non-existence of a CQLF is significant, especially in switched linear systems consisting of N second-order systems or two systems of order n, the general case of N systems of order n still remains open. In this article, a sufficient condition for the non-existence of a CQLF for N systems of order n is derived. Based on the condition, a new method for determining the non-existence of a CQLF, using particle swarm optimisation, was designed and is described. Examples illustrating the proposed method are introduced at the end of this article.

Keywords:

• switched linear systems
• stability
• common quadratic Lyapunov function
• particle swarm optimisation

Acknowledgements

The work reported in this article has been supported by CONICYT-CHILE through grant FONDECYT No. 1090208, Programa de Financiamiento Basal ‘Centro de Tecnología para la Minería’ FB0809, and the Grant Agency of the Czech Republic under contract No. 103/12/2431 (P. Zagalak).