Contents I. Introduction
Given a square transfer matrix , the Kalman– Yakubovich–Popov (KYP) Lemma relates an analytic property of a square transfer matrix in the frequency domain to a set of algebraic equations involving parameters of a minimal realization in time domain. See the original references [7], [18], and [13], [20]. Further important developments were given in [3], [12]. The lemma was generalized to the multivariable case in [2]. Extensions and clarifications appeared on [5], [16], and [10]. Clear presentations and relationships with other related results appeared in [17] and [8]. A novel proof based on convexity properties and linear algebra appeared recently in [14]. Based on this classical result, the following question with respect to minimality arises: is the KYP lemma valid for nonminimal realizations? This note addresses this question and gives a positive answer, i.e., the KYP lemma is valid for realizations which are stabilizable and observable. This extension has important applications in control systems theory and in the stability analysis of adaptive output feedback systems [6]. Some comments have appeared in the literature with respect to this relaxation. Meyer [11] made early comments on the minimality issue. A method for construction of Lyapunov functions for a positive real nonminimal systems was proposed in [6]. In a recent survey paper, the authors stated that the KYP lemma is valid for stabilizable realizations. However, they did not provide details of the proof. The objective of this note is to clarify and establish that the KYP lemma holds also for stabilizable and observable realizations.
