New results on the solution of the positive consensus problem

This paper addresses the positive consensus problem for a homogeneous multi-agent system, by assuming that the agents' dynamics is described by a positive single-input and continuous-time system. We first provide a necessary condition for the problem solvability that refers to the Frobenius normal form of the Metzler matrix describing the agents' unforced evolution. This result imposes quite significant constraints on the agents' description, and it allows to reduce the general problem to the special case when the aforementioned matrix is irreducible. Under this assumption, equivalent sets of conditions that prove to be sufficient for the existence of a solution for the positive consensus problem are derived. Numerical examples illustrate the proposed results.