Optimal filtering for systems with unknown inputs via descriptor Kalman filtering

In this paper, we consider the global unbiased minimum-variance state estimation for systems with unknown inputs which affect both the system and the output via the descriptor Kalman filtering method. It is shown that the conventional descriptor Kalman filter (DKF) may not yield the optimal filtering performance. Using unknown input transformations, a so-called “5-block” form of the extended DKF (5-block EDKF) is proposed as a globally optimal state estimator in the sense that it is equivalent to the recently developed extended recursive three-step filter (ERTSF). The relationship between the 5-block EDKF and the ERTSF is clearly addressed. To simplify computational complexity, a compact version of the 5-block EDKF, named as the 4-block EDKF, is derived through further considering a specific output transformation. Moreover, a 5-block refined EDKF that does not need any transformations is also proposed. Simulation results are given to illustrate the usefulness of the proposed results.