Sliding mode mean-square filtering for linear stochastic systems

This paper addresses the mean-square filtering problem for a linear system with Gaussian white noises. The obtained solution contains a sliding mode term, signum of the innovations process. It is shown that the designed sliding mode 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 theoretical result is complemented with an illustrative example verifying performance of the designed filter. It is demonstrated that the estimates produced by the designed filter and the Kalman-Bucy filter yield the same estimation error variance.