Optimal Filtering for Incompletely Measured Polynomial States over Linear Observations

In this paper, the optimal filtering problem for incompletely measured polynomial system states over linear observations is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. In contrast to the previous works, the nonlinear polynomial states are allowed to be unmeasured in this problem. The procedure for obtaining a closed system of the filtering equations for any polynomial state over linear observations is then established, which yields the explicit closed form of the filtering equations in the particular case of a bilinear state equation. In the example, performance of the designed optimal filter is verified against a conventional extended Kalman-Bucy filter.