Optimal filtering for systems with unknown inputs via the descriptor Kalman filtering method

This paper addresses globally optimal unbiased minimum-variance state estimation for systems with unknown inputs that affect both the system and the output with the descriptor Kalman filtering method. It is shown that directly applying the conventional descriptor Kalman filter (DKF) to the considered problem may not yield the globally optimal solution because the unknown input vector may not be estimable. To remedy this problem, three approaches are proposed to facilitate optimal filter design: the transformed approach uses some input and output transformations, the untrammeled approach does not require any transformations, and the augmented approach reconstructs the unknown input dynamics. Then, three “5-block” forms of the extended DKF (5-block EDKF) are derived as globally optimal state estimators in the sense that the first two filters are equivalent to the recently developed extended recursive three-step filter and the third is equivalent to the conventional augmented state Kalman filter. The relationship between the proposed EDKFs and the existing results in the literature is addressed. Simulation results are given to illustrate the usefulness of the proposed filters.