Stabilizability of Discrete-Time Controlled Switched Linear Systems

Abstract

The goal of this paper is to study the stabilizability of controlled discrete-time switched linear systems (CSLSs), where the discrete switching control input and the continuous control input coexist. For a given positive integer h, we propose a quantitative metric of stabilizability, called h-contraction rate (h-CR), and prove that it serves as an indicator of the stabilizability in the following sense: the h-CR is less than one for sufficiently large h if and only if the switched linear system is stabilizable. In addition, we develop computational tools to estimate upper and lower bounds of the h-CR. In particular, the upper bound estimation is expressed as a semidefinite programming problem. We derive its Lagrangian dual problem and prove that the dual problem with an additional rank constraint gives an exact characterization of the h-CR. An algorithm is developed to solve the dual problem with the rank constraint.