An extension of the positive real lemma to descriptor systems

Abstract

The well-known positive real lemma characterizes positive realness of transfer functions of time-invariant linear systems via the solvability of certain linear matrix inequalities (LMIs). In this article, we propose an extension of the positive real lemma and the underlying linear matrix inequalities to descriptor systems. We prove that the solvability of these LMIs is sufficient and, under an additional assumption, also necessary for positive realness of transfer functions of time-invariant linear descriptor systems. Our results also show why known extensions of the bounded real lemma to descriptor systems cannot be used to obtain the extension of the positive real lemma presented in this article.