Optimal state estimation with continuous, multirate and randomly sampled measurements

This paper presents an optimal filter for a continuous dynamic system with continuous, multirate and randomly sampled measurements. Using the optimal filtering theory for the Ito-Volterra systems with discontinuous measure, the optimal filter for linear state space model with continuous and discrete measurements is rigorously derived, and several known results are recovered, including the Kalman-Bucy and Jazwinski filters. A previously unknown optimal filter for the continuous systems with continuous and sampled measurements, including the case of multirate and random sampling, is obtained. Using the Monte Carlo simulations, the derived filter is compared with the previously reported alternatives. The comparison shows that the developed filter gives the least-mean-square estimates of the states and the correct estimation error covariance. The alternative filters produce less than optimal estimates, and, at the same time, tend to overestimate the quality of the obtained estimations. Numerical simulations demonstrate that the proposed approach is more convenient in practice: it allows one to simultaneously handle analog and sampled measurements without approximations, and is particularly convenient in the case of the multirate and randomly sampled measurements, often present with a human-in-the-loop and networked data acquisition.