Kalman filter for continuous state-space system with continuous, multirate, randomly sampled and delayed measurements

This paper presents an optimal, in the Kalman sense, filter for linear, continuous, stochastic state-space system with continuous, multirate, randomly sampled and delayed measurements. A general theorem on optimal filter of Ito-Volterra system with discontinuous measure is presented and then specialized to standard state-space model with both continuous and discrete measurements. The discontinuity of the measurement vector leads to the optimal filter with continuous and impulsive inputs, causing the discontinuity of the filter equations. The size of the jumps in state and covariance matrix can be explicitly calculated using the theory of vibrosolutions. A previously unknown optimal filter for continuous state space systems with continuous and sampled measurements, including multirate, randomly sampled and delayed measurements, is obtained. Under additional assumption, it is shown that the derived optimal filter recovers several known results, including the Kalman-Bucy and Jazwinski filters (continuous process with discrete measurements). The developed and the previously reported filters are compared using Monte Carlo simulations, which show that the optimal result gives the least-mean-square-error estimates of the states, and correctly predicts the goodness of the obtained estimates; the alternative filters tend to be overly optimistic in calculating the quality of the generated state estimates. Numerical simulations demonstrate that the proposed approach is convenient in practice as it neither requires implementation of multirate filters, nor any approximations to handle measurements arriving with different and, possibly, random sampling rates, as often is the case with human-in-the-loop and networked measurements.