Discussion on: 'Positive Real Control for Uncertain Singular Time-delay Systems via Output Feedback Controllers'

摘要

The purpose of this discussion is to bring about some relevant points that will help the readers and researchers to better appreciate the significance of the paper by Xu and Lam in the main stream of singular time-delay systems. The fundamental concept of positive realness plays a central role in stability theory of dynamical systems and has its roots in circuit theory. Historically, the properties of a positive real (PR) function of a complex variable s have been studied extensively in passive network theory and related definitions and measures of strictly positive real (SPR) functions have been developed. An important link in relating SPR functions to the existence of Lyapunov functions, and hence the stability of a corresponding dynamical system, is established by the Kalman-Yakubovich lemma. Moreover, the concepts of positive real-ness, passivity and dissipativity have found significant applications in systems theoretic investigations and stability of feedback systems. In this regard, it has been proved in [16] that the stability behavior of a feedback configuration composed of PR plant in the forward path and a SPR system in the negative feedback path is guaranteed in face of the plant variations so long as the plant remains PR. This brings about PR control theory for which a solution of a class of linear time-invariant plants has been established in [15] in terms of algebraic Riccati matrix inequalities.