Causal H∞ Control of Descrete-time Descriptor Systems: an Lmi Approach in Two Steps

The H∞ control problem for linear discrete-time descriptor systems which are of high index and possibly non-regular is considered. The goal is to compute a descriptor feedback such that the closed loop system is stable, causal, and H∞ norm bounded by a prescibed real number. In a recent solution of this problem, where causality is established by a preliminary feedback and where stability and H∞ norm bound are guaranteed in a second step, the pratically important question remained open whether the achievalbe H∞ performance of the closed loop is determined by the preliminary feedback. We show that this is not the case. Actually, the achievalbe H∞ performance is invariant with respect to any preliminary feedback achieving causality of the closed loop system. A numerically appealing LMI based solution to the problem is presented that explicitly shows all degrees of freedom in guaranteeing causality on the one hand and H∞ performance on the other hand.