Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: a set-theory approach

Mirko Fiacchini (1) , Marc Jungers (2)

1 GIPSA-SYSCO - GIPSA - Systèmes non linéaires et complexité
2 CRAN - Centre de Recherche en Automatique de Nancy

In this paper, the stabilizability of discrete-time linear switched systems is considered. Several sufficient conditions for stabilizability are proposed in the literature, but no necessary and sufficient. The main contributions are the necessary and sufficient conditions for stabilizability based on set-theory and the characterization of a universal class of Lyapunov functions. An algorithm for computing the Lyapunov functions and a procedure to design the stabilizing switching control law are provided, based on such conditions. Moreover a sufficient condition for non-stabilizability for switched system is presented. Several academic examples are given to illustrate the efficiency of the proposed results. In particular, a Lyapunov function is obtained for a system for which the Lyapunov-Metzler condition for stabilizability does not hold.

Mots clés: Switched linear systems; Set-theory; Stabilizability; Invariance

Domaines: Systèmes dynamiques [math.DS]; Automatique / Robotique

Mirko Fiacchini, Marc Jungers. Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: a set-theory approach. Automatica, 2014, 50 (1), pp.75-83. ⟨10.1016/j.automatica.2013.09.038⟩. ⟨hal-00875424⟩