Optimal filtering for incompletely measured polynomial states with multiplicative noise

In this paper, the optimal filtering problem for polynomial system states with polynomial multiplicative noise over linear observations with an arbitrary, not necessarily invertible, observation matrix 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. A transformation of the observation equation is introduced to reduce the original problem to the previously solved one with an invertible observation matrix. The procedure for obtaining a closed system of the filtering equations for any polynomial state with polynomial multiplicative noise over linear observations is then established, which yields the explicit closed form of the filtering equations in the particular cases of linear and bilinear state equations. In the example, performance of the designed optimal filter is verified against the optimal filter for a quadratic state with a state-independent noise and a conventional extended Kalman-Bucy filter.