Optimal filtering for partially measured polynomial system states

In this paper, the optimal filtering problem for polynomial systems with partially measured linear part 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 over linear observations with delay is then established, which yields the explicit closed form of the filtering equations in the particular case of a bilinear system state. In the example, performance of the designed optimal filter is verified for a quadratic-linear state with unmeasured linear part over linear observations against the conventionally designed extended Kalman-Bucy filter.