A Methodology for Determining the Non-existence of Common Quadratic Lyapunov Functions for Pairs of Stable Systems

The existence of a common quadratic Lyapunov function (CQLF) for a switched linear system guarantees its global asymptotic stability. Although the progress in finding conditions for existence/non-existence of a CQLF has been significant in the last years, especially in switched linear systems with N subsystems of second order or two non-arbitrary subsystems of order n, the general case of N systems of order n still remains open. In this paper, based on a sufficient condition for the nonexistence of a CQLF for a pair of general subsystems of order n obtained from a lemma by Shorten et al., a new method for determining the non-existence of a CQLF, using Particle Swarm Optimization, is designed. A example illustrating the proposed method is introduced towards the end of the paper.