Optimal filtering for polynomial systems with partially measured states and multiplicative noises

In this paper, the optimal filtering problem for polynomial systems with partially measured linear part and polynomial multiplicative noise over linear 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 any polynomial state with partially measured linear part and polynomial multiplicative noise over linear observations is then established, which yields the explicit closed form of the filtering equations in the particular case of a bilinear system state with bilinear multiplicative noise. In the example, the designed optimal filter is applied to solution of the optimal cubic sensor filtering problem, assuming a Gaussian initial condition for the cubic state. The resulting filter yields a reliable and rapidly converging estimate.