Abstract
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.
The author thanks the Mexican National Science and Technology Council (CONACyT) for financial support under Grants 55584, 129081 and joint Mexico-EU FONCICyT Grant 93302.
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Basin, M., Rodriguez-Ramirez, P. (2013). Applying Sliding Mode Technique to Filter and Controller Design for Nonlinear Polynomial Stochastic Systems. In: Bandyopadhyay, B., Janardhanan, S., Spurgeon, S. (eds) Advances in Sliding Mode Control. Lecture Notes in Control and Information Sciences, vol 440. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36986-5_8
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