Applying Sliding Mode Technique to Optimal Filter and Controller Design

Abstract

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. Then, the paper addresses the optimal controller problem for a linear system over linear observations with respect to different Bolza-Meyer criteria, where 1) the integral control and state energy terms are quadratic and the non-integral term is of the first degree or 2) the control energy term is quadratic and the state energy terms are of the first degree. The optimal solutions are obtained as sliding mode controllers, each consisting of a sliding mode filter and a sliding mode regulator, whereas the conventional feedback LQG controller fails to provide a causal solution. Performance of the obtained optimal controllers is verified in the illustrative example against the conventional LQG controller that is optimal for the quadratic Bolza-Meyer criterion. The simulation results confirm an advantage in favor of the designed sliding mode controllers.