On the Stability of Switched Positive Linear Systems

It was recently conjectured 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. In this note, we show that (1) this conjecture is true for systems constructed from a pair of second-order Metzler matrices; (2) the conjecture is true for systems constructed from an arbitrary finite number of second-order Metzler matrices; and (3) the conjecture is in general false for higher order systems. The implications of our results, both for the design of switched positive linear systems, and for research directions that arise as a result of our work, are discussed toward the end of the note.

I. Introduction

Positive dynamical systems are of fundamental importance to numerous applications in areas such as Economics, Biology, Sociology and Communications. Historically, the theory of positive linear time-invariant (LTI) systems has assumed a position of great importance in systems theory and has been applied in the study of a wide variety of dynamic systems [1]–[4]. Recently, new studies in communication systems [5], formation flying [6], and other areas, have highlighted the importance of switched (hybrid) positive linear systems (PLS). In the last number of years, a considerable effort has been expended on gaining an understanding of the properties of general switched linear systems [7], [8]. As is the case for general switched systems, even though the main properties of positive LTI systems are well understood, many basic questions relating to switched PLS remain unanswered. The most important of these concerns their stability, and in this note, we present some initial results on the stability of switched PLS.