Optimal filtering for linear states over polynomial observations

In this paper, the optimal filtering problem for linear system states over polynomial observations is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. As a result, the Ito differentials for the optimal estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining a closed system of the filtering equations for a linear state over observations with any polynomial drift is then established. In the example, the obtained optimal filter is applied to solution of the optimal third order sensor filtering problem, assuming a conditionally Gaussian initial condition for the third degree state. This assumption is quite admissible in the filtering framework, since the real distributions of the first and third degree states are actually unknown. The resulting filter yields a reliable and rapidly converging estimate.