Minimization of State Bounding for Perturbed Positive Systems with Delays

Abstract

This paper studies the minimum state bounding problem of linear positive differential systems with discrete and distributed delays and unknown-but-bounded disturbances. Specifically, two problems are addressed: (i) The first problem is to derive the smallest componentwise bound of the state vector when the time tends to infinity, and (ii) the second problem is to derive the smallest possible componentwise bound of the state vector when the time tends to a prespecified finite time. A new method which is based on state transformations, the Lyapunov method, and optimization techniques is presented for deriving the smallest bounds of the state vector which solves the two stated problems. The obtained results are extended to a class of nonlinear systems which are upper bounded by linear positive systems. An application of the results to the $L_{\infty}$-gain problem for positive time-delay systems is also presented. The feasibility and effectiveness of our derived results are illustrated through two numerical examples.