Applying Sliding Mode Technique to Filter and Controller Design for Nonlinear Polynomial Stochastic Systems

This chapter addresses the mean-square and mean-module filtering problems for stochastic polynomial systems 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 mean-square filter generates the mean-square estimate, which has the same minimum estimation error variance as the best estimate given by the conventional mean-square polynomial 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 conventional polynomial mean-square filter. The theoretical results are complemented with illustrative examples verifying performance of the designed filters. It is demonstrated that the estimates produced by the designed sliding mode mean-square filter and the conventional polynomial mean-square filter yield the same estimation error variance, and there is an advantage in favor of the designed sliding mode mean-module filter. The chapter then presents the solution to the optimal controller problems for a polynomial system over linear observations with respect to a Bolza-Meyer criterion, where the integral control and state energy terms are quadratic and the non-integral term is of the first degree. The simulation results confirm an advantage in favor of the designed sliding mode controller.