Linear Minimum Mean Square Filters for Markov Jump Linear Systems

This paper studies optimal mean square error estimation for discrete-time linear systems with observed Markov jump parameters. New linear estimators are introduced by considering a cluster information structure in the filter design. The set of filters constructed in this way can be ordered in a lattice according to the refines of clusters of the Markov chain, including the linear Markovian estimator at one end (with only one cluster) and the Kalman filter at the other end (with as many clusters as Markov states). The higher is the number of clusters, the heavier are precomputations and smaller is the estimation error for embedded sequences of partitions so that the cardinality and choice of the clusters allows for a tradeoff between performance and computational requirements. In this paper, we propose the estimator, give the formulas for precomputation of gains, present some properties, and give an illustrative numerical example.