H ∞ Control for Linear Positive Discrete-Time Systems

Abstract

This paper is concerned with $H_{\infty }$ control for linear positive discrete-time systems. Positive systems are characterized by nonnegative restriction on systems’ variables. This restriction results in some remarkable results which are available only for linear positive systems. One of them is the celebrated diagonal positive definite matrix solutions to some existed well-known results for linear systems without nonnegative restriction. We provide an alternative proof for criterion of $H_{\infty }$ norm by using separating hyperplane theorem and Perron-Frobenius theorem for nonnegative matrices. We also consider $H_{\infty }$ control problem for linear positive discrete-time systems via state feedback. Necessary and sufficient conditions for such problem are presented under controller gain with and without nonnegative restriction, and then the desired controller gains can be obtained from the feasible solutions.