Stability of Discrete-Time Switched Homogeneous Systems on Cones and Conewise Homogeneous Inclusions

Abstract

This paper presents a stability analysis of switched homogeneous systems on cones under arbitrary and optimal switching rules with extensions to conewise homogeneous or linear inclusions. Several interrelated approaches, such as the joint spectral radius approach and the generating function approach, are exploited to derive necessary and sufficient stability conditions and to develop suitable algorithms for stability tests. Specifically, the generalized joint spectral radius and the generalized joint lower spectral radius are introduced to characterize the radii of domains of strong and weak attraction. Furthermore, strong and weak generating functions and their radii of convergence are employed to derive stability conditions; their analytic properties, numerical approximations, and convergence analysis are established. Extensions to conewise homogeneous or linear inclusions are made to address state-dependent switching dynamics. Relations between different stability notions in the strong or weak sense are studied; Lyapunov techniques are used for stability analysis of the conewise linear inclusions.