Optimal state estimation for discrete-time Markovian Jump Linear Systems, in the presence of delayed output observations

In this paper, we investigate the design of optimal state estimators for Markovian jump linear systems. We consider that the state has two components: the first component is finite valued and is denoted as mode, while the second (continuous) component is in a finite dimensional Euclidean space. The continuous state is driven by a zero mean, white and Gaussian process noise. The observation output has two components: the first is the mode and the second is a linear combination of the continuous state observed and zero mean, white Gaussian noise. Both output components are affected by delays, not necessarily equal. Our paradigm is to design optimal estimators for the current state, given the current output observation. We provide a solution to this paradigm by giving a recursive estimator for the continuous state, in the minimum mean square sense, and a finitely parameterized recursive scheme for computing the probability mass function of the current mode conditioned on the observed output. We show that when the mode is observed with a greater delay then the continuous output component, the optimal estimator nonlinear in the observed outputs.