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

In this paper, we investigate an optimal state estimation problem for Markovian Jump Linear Systems. We consider that the state has two components: the first component of the state 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 deterministic control input and a zero mean, white and Gaussian process noise. The observable output has two components: the first is the mode delayed by a fixed amount and the second is a linear combination of the continuous state observed in zero mean white Gaussian noise. 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 of 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 conditional on the observed output. We show that the optimal estimator is nonlinear on the observed output and on the control input. In addition, we show that the computation complexity of our recursive schemes is polynomial in the number of modes and exponential in the mode observation delay.