Sliding mode filter design for linear systems with unmeasured states

This paper addresses the mean-square and mean-module filtering problems for a linear system with Gaussian white noises. The obtained solutions contain a sliding mode term, signum of the innovations process. It is shown that the designed sliding mode mean-square filter generates the mean-square estimate, which has the same minimum estimation error variance as the best estimate given by the classical Kalman-Bucy filter, although the gain matrices of both filters are different. The designed sliding mode mean-module filter generates the mean-module estimate, which yields a better value of the mean-module criterion in comparison to the mean-square Kalman-Bucy filter. The theoretical result is complemented with an illustrative example verifying performance of the designed filters. It is demonstrated that the estimates produced by the designed sliding mode mean-square filter and the Kalman-Bucy filter yield the same estimation error variance, and there is an advantage in favor of the designed sliding mode mean-module filter.