Control for stability and positivity: equivalent conditions and computation

This paper investigates the stabilizability of linear systems with closed-loop positivity. A necessary and sufficient condition for the existence of desired state-feedback controllers guaranteeing the resultant closed-loop system to be asymptotically stable and positive is obtained. Both continuous and discrete-time cases are considered, and all of the conditions are expressed as linear matrix inequalities which can be easily verified by using standard numerical software. Numerical examples are provided to illustrate the proposed conditions.

I. Introduction

In Many practical systems, variables are constrained to be nonnegative. Such constraints abound in physical systems where variables are used to represent levels of heat, population, and storage. For instance, age-structured populations described by certain Leslie models [6], compartmental models used in hydrology and biology applications, can be described by positive systems [13], [18], whose states and outputs are nonnegative whenever the initial condition and input signal are nonnegative. Since positive systems are defined on cones, not on linear spaces, many well-established results of general linear systems cannot be simply applied to positive systems. Therefore, in recent years, many researchers have shown their interests in positive systems and many fundamental results have been reported (see, for instance, [1]–[3], [7], [11], [12], [16], [17], [19], and [20] and the references therein).