Stationary filter for linear minimum mean square error estimator of discrete-time Markovian jump systems

We derive in this paper a stationary filter for the linear minimum mean square error estimator (LMMSE) of discrete-time Markovian jump linear systems (MJLSs). We obtain the convergence of the error covariance matrix of the LMMSE to a stationary value under the assumption of mean square stability of the MJLS and ergodicity of the associated Markov chain. It is shown that there exists a unique solution for the stationary Riccati filter equation and, moreover, this solution is the limit of the error covariance matrix of the LMMSE. The advantage of this scheme is that it is very easy to implement and all calculations can be performed off-line, leading to a linear time-invariant filter.