H ∞ model reduction for discrete‐time positive systems with inhomogeneous initial conditions

This paper addresses the H _∞ model reduction problem of discrete‐time positive linear systems with inhomogeneous initial conditions. For an asymptotically stable positive system with non‐zero initial condition, our goal is to approximate it by a reduced‐order initial‐valued positive system without introducing significant error. We establish a necessary and sufficient condition for the existence of a desired reduced‐order model such that the output error between the original system and the reduced‐order one is bounded by a weighted sum of the magnitude of the input and that of the initial condition. Moreover, based on congruent transformation and the dual form of bounded real lemma, several equivalent conditions are derived in terms of LMIs and an iterative convex optimization algorithm is developed accordingly. Finally, an illustrative example is presented to show the effectiveness of the proposed methods. Copyright © 2013 John Wiley & Sons, Ltd.