An LMI approach towards H∞ control of discrete-time descriptor systems

In this paper the H_∞ descriptor feedback control of high index or even non-regular linear discrete-time descriptor systems is considered. The approach is based on a linear matrix inequality (LMI) characterization (with a Lyapunov-type matrix as matrix variable) of causal, stable and H_∞ norm-bounded descriptor systems. Applying this result to the problem to stabilize and additionally to establish an H_∞ norm bound for a controlled discrete-time descriptor system by descriptor feedback such that the closed loop is causal, renders a nonlinear matrix inequality with the controller gain matrix and a Lyapunov-type matrix as variables. For the corresponding non-descriptor problem this matrix inequality can be reduced to an LMI by applying Schur´s lemma and a subsequent linearizing change of variables. In the descriptor setup this procedure is not applicable due to the indefiniteness of the occurring Lyapunov-type matrix. Instead, a two step controller computation is presented, which first renders the closed loop system causality. In the second step an LMI based procedure is used to guarantee the stability and H_∞ norm-boundedness without affecting causality of the closed loop.