Optimal filtering for polynomial states over polynomial observations

In this paper, the optimal filtering problem for polynomial system states over polynomial observations is studied proceeding from the general expression for the stochastic Ito differentials of the optimal estimate and the error variance. In contrast to the previously obtained results, the paper deals with the general case of nonlinear polynomial states and observations. 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 any polynomial state over observations with any polynomial drift is then established. In the example, the obtained optimal filter is applied to solve the optimal third order sensor filtering problem for a quadratic state, assuming a Gaussian initial condition for the extended third order state vector. The simulation results show that the designed filter yields a reliable and rapidly converging estimate.