Optimal Linear Mean Square Filter for Markov Jump Linear Systems with Hidden Markov Chain and Partial Observations of the State

In this paper we make further foray on the filtering problem for Markov jump linear systems (MJLS) in the worst case scenario in which both, the state signal and the operation mode (Markov chain) are partially observables. Since the optimal nonlinear filter for the state signal in this setting is infinite dimensional (in the sense of the optimal filtering theory), an optimal linear filter has been derived in [1]. A peculiar feature of the approach used in [1] is the fact that the optimal linear filter for the state variable is derived by circumventing the need to deal with the unobservable Markov chain. Having the result derived in [1] and [2] as the linchpin for our approach, the main contribution of this paper is to derive an optimal linear filter for the operation mode in the worst case scenario. As far as the authors are aware, this is the first optimal filter for the operation mode, in this scenario. In addition, relying on Murayama's stochastic numerical method and the results in [3], we carry out a simulation to evaluate the filter performance.