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.