Generalized $H_\infty$-optimal filtering for discrete-time plants over finite and infinite time horizons

We consider linear discrete-time plants subjected to exogenous disturbances and initial perturbations induced by the unknown initial conditions. A performance index for the accuracy of filtering is introduced which accounts for the effects of exogenous and initial perturbations on the filtering error over finite and infinite time horizons. Linear generalized $H_\infty$-optimal discrete-time filters in the form of observer are designed, which minimize the criterion adopted. Design is performed by using the minimax approach and the apparatus of linear matrix inequalities.