New positive realness conditions for uncertain discrete descriptor systems: analysis and synthesis

This paper deals with the problems of positive real (PR) analysis and PR control for uncertain discrete-time descriptor systems. The parameter uncertainties are assumed to be time-invariant norm bounded and appear in both the state and input matrices. A new necessary and sufficient condition for a discrete-time descriptor system to be regular, causal, stable and extended strictly PR (ESPR) is proposed in terms of a strict linear matrix inequality. Based on this, the concepts of strong robust admissibility with ESPR and strong robust admissibilizability with ESPR were introduced. Without any additional assumptions on the system matrices, necessary and sufficient conditions for strong robust admissibility with ESPR and strong robust admissibilizability with ESPR are obtained. Through these results, the problems of PR analysis and PR control are solved. Furthermore, an explicit expression of a desired state feedback controller is also given, which involves no decomposition of the system matrices.

I. Introduction

Positive real (PR) control is a research problem of recurring interest in the past years [1], [3], [10]. The study of this problem is motivated by the robust and nonlinear control where a well-known fact is that the positive realness of a certain loop transfer function will guarantee the overall stability of a feedback system if uncertainty or nonlinearity can be characterized by a PR system [12]. By solving a pair of Riccati inequalities, a solution to this problem was given in [10]. When parameter uncertainties appear in a system model, the PR control problem was studied in [14], where state feedback controllers were designed and a Riccati equation approach was developed, while in [7] observer-based dynamic output-feedback controllers were constructed in terms of solutions to certain linear matrix inequalities (LMIs). The corresponding results for discrete-time systems can be found in [8]. It is worth mentioning that the results in [8] were further extended to two-dimensional (2-D) systems in [16].