Stability and Stabilization of Positive Switching Linear Discrete-Time Systems

Introduction

In the literature, systems whose states are nonnegative, whenever the initial conditions are nonnegative, are referred to as positive [84]. The design of controllers for these positive systems has been studied (see, for example, [1], [99] and references therein). However, to the best of our knowledge, few works have directly considered positive switching linear systems as in [29]. We can also cite [146], which studies the stability of positive continuous-time switching systems composed of two subsystems. It must be pointed out that in [96], it is shown that the conjecture of [146], which says that the “Hurwitz stability of the convex hull of a set of Metzler matrices is a necessary and sufficient condition for the asymptotic stability of the associated switched linear system under arbitrary switching”, is not true in general. Another interesting problem that has been studied in the literature is the reachability problem for positive discrete-time switching systems [143]. In particular, in [78], systems whose state variables remain nonnegative independently of the control and the external events are studied; then, control strategies are studied to maintain the nonnegative state evolution, against external events acting on the system.