Filtering and model reduction for positive linear systems using the H∞ norm in finite frequency ranges

Abstract

This paper investigates the problems of model-order reduction and reduced-order filtering for continuous and discrete-time positive linear systems, utilizing the H∞ norm as performance criterion within finite frequency ranges. Both problems are formulated in terms of bilinear matrix inequalities, employing additional slack variables derived from the application of Finsler’s lemma, which exhibits the closed-loop dynamic matrix isolated from other variables. A solution based on linear matrix inequalities is proposed through an iterative algorithm, exploring relaxations and a systematic initialization procedure. An advantage over the existing methods is a less conservative treatment of positivity constraints, along with the absence of the need to provide initial reduced models (or filters) for the algorithm. Numerical examples and comparisons with techniques from the literature are provided to illustrate the results.