Sliding mode filtering for polynomial systems

This paper addresses the mean-square and mean-module filtering problems for a nonlinear polynomial stochastic 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 estimate given by the conventional mean-square polynomial filter [24], 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 conventional mean-square polynomial 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 conventional mean-square polynomial filter yield the same estimation error variance, and there is an advantage in favor of the designed sliding mode mean-module filter.