Polynomial filtering for stochastic systems with Markovian switching coefficients

In this paper the state estimation problem for discrete-time Markovian switching systems affected by additive noise (not necessarily Gaussian) is solved following a polynomial approach. The key point for the derivation of the optimal polynomial filter is the possibility to represent the Markov switching systems as bilinear systems (linear drift, multiplicative noise) by means of a suitable state augmentation. By construction, the optimal polynomial filter of a given degree ν provides the minimum error variance among all polynomial output transformations of the same degree. Obviously, for ν > 1 better performances are obtained with respect to linear filters. Simulation results are reported as a validation of the theory.