Stability and stabilization of discrete-time descriptor systems with several extensions

This paper is concerned with stability criterion of discrete-time descriptor systems in terms of LMIs. Though there have been proposed several LMIs for stability of discrete-time descriptor systems, their use is seemingly limited in comparison with LMIs for continuous-time descriptor systems. A remedy of those existing results is that they are not able to be linearized with respect to the coefficient matrices of the descriptor equation. This makes it difficult to apply them to synthesis problems, while LMIs for analysis of state-space systems or those for continuous-time descriptor systems enjoy change-of-variable methods to obtain LMIs for synthesis. In this paper, a new LMI is presented for admissibility of discrete-time descriptor systems that has such a useful feature of affineness of coefficient matrices in the inequality. After showing the main result with the new LMI, we apply it to several problems on discrete-time descriptor systems, namely, state-feedback synthesis, stability analysis of δ-operator systems and D-stability problems generalized to descriptor systems, and a generalization of quadratic stabilization. Some of these are illustrated via numerical examples.