On the zero pattern properties and asymptotic behavior of continuous-time positive system trajectories

In this paper, the zero pattern properties and the asymptotic evolution of the trajectories of a continuous-time positive system are investigated. To this end, we need to introduce some new tools and to derive some new results, within the broad research area of nonnegative matrix theory, which enable use to explore the zero pattern and the elementary modes of the exponential of a Metzler matrix. Specifically, a normal form for the exponential of a Metzler matrix is provided, and the concept of echelon basis (consisting of eigenvectors and generalized eigenvectors of the Metzler matrix) is introduced. By making use of these two ingredients, a detailed result about the dominant mode of each single block appearing in the normal form of the exponential matrix is provided. This allows to obtain a “modal decomposition” of the exponential matrix, emphasizing the column dominant modes. As a result, the zero pattern as well as the asymptotic behavior of every free state evolution, depending on the zero pattern of the initial state, can be easily determined.