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.