Optimal state estimation for stochastic systems: An information theoretic approach

In this paper, we examine the problem of optimal state estimation or filtering in stochastic systems using an approach based on information theoretic measures. In this setting, the traditional minimum mean-square measure is compared with information theoretic measures, Kalman filtering theory is re-examined, and some new interpretations are offered. We show that for a linear Gaussian system, the Kalman filter is the optimal filter not only for the mean-square error measure, but for several information theoretic measures which are introduced in this work. For nonlinear systems, these same measures generally are in conflict with each other, and the feedback control policy has a dual role with regard to regulation and estimation. For linear stochastic systems with general noise processes, a lower bound on the achievable mutual information between the estimation error and the observation are derived. The properties of an optimal (probing) control law and the associated optimal filter, which achieve this lower bound, and their relationships are investigated. It is shown that for a linear stochastic system with an affine linear filter for the homogeneous system, under some reachability and observability conditions, zero mutual information between estimation error and observations can be achieved only when the system is Gaussian.