Optimal discrete-time unbiased filtering for systems with unknown inputs

This paper concerns the H_∞ functional filtering design problem for discrete-time systems with unknown inputs. Our interest is focused on the unbiased filtering satisfying the H_∞ performance. First the unbiasedness conditions of the estimation error are presented. The dynamic of this error is dependent on the present and future values of the noise. Different cases are presented, the first one is based on the introduction of an algebraic constraint leading to the dynamic error dependent only on the present noise, the second case is more general and can be treated by modifying the performance index or by using the descriptor systems formulation. The conditions of the existence of an unbiased filter are presented and the designs of the different filters, based on the bounded real lemma, are given in the form of the linear matrix inequalities (LMI). The presented approach unifies the design of functional, reduced-order and full-order filters for discrete-time systems. Three numerical examples are presented to illustrate the approach described in the paper and shows the performance of the proposed functional filter.