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 innovation 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 with the mean-square Kalman-Bucy filter. The theoretical result is complemented with an illustrative example verifying the 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.