Kalman Filter Reinforced by Least Mean Square for Systems with Unknown Inputs

This paper addresses the state estimation problem of linear discrete-time time-varying stochastic systems with unknown inputs (UIs). It is shown that the globally optimal unbiased minimum-variance filters may not satisfy the minimum-variance property, and hence they cannot eliminate noises appropriately. If this is the case, the well-known Kalman filter may give a better solution, which however may also not be the best one due to that the imbedded unknown input model may not be practical. To remedy the filtering degradation problem, a robust filter named as the KFLMS, which has good noise rejection property for such systems, is developed in this paper, where the UI estimates are obtained by using least mean square algorithm and the state estimation is achieved via the previous proposed two-stage Kalman filtering approach. Numerical examples are provided to show the effectiveness of the proposed results. Specifically, simulation results illustrate the goodness of the new method in the sense of lower root mean square error and better noise rejection property.