Eigenvalue right-outer bounds for polytopes of Metzler matrices and systems applications

Abstract

Metzler matrices are algebraically characterized by having nonnegative off-diagonal entries and, from the dynamical point of view, define continuous-time positive systems. Our work studies polytopes of Metzler matrices and linear dynamics generated by such sets. The first part of the paper explores the algebraic properties of matrix polytopes, by focusing on the estimation of a right outer bound for all eigenvalues. The second part analyzes the dynamical properties of positive polytopic systems, by revealing connections between the estimated right outer bound and the evolution of trajectories (related to copositive Lyapunov functions and invariant sets).