Galerkin Approximations of the Kushner Equationin Nonlinear Estimation

If a continuous-time nonlinear system with continuous measurements is subjected to process and measurement noise, the evolution of the probability density function associated with the state is given by the solution to a nonlinear stochastic partial diierential equation called the Kushner equation. If the Kushner equation can be solved then various lters based on diierent optimality criteria could be designed, including the minimum variance lter which corresponds to the Kalman lter for linear systems. Unfortunately, the Kushner equation cannot be solved in general. As a result, nonlinear lters are usually constructed via heuristic methods such as the extended Kalman lter. In this paper, we show how the Galerkin spectral method can be used to approximate the solution to the Kushner equation. The result is a lter that is based on the nonlinear model of the system and not its linearization. The lter is practically implemented for low order systems and out-performs the EKF for both large variations in the state and for model mismatches. An application where the extended Kalman lter completely fails is also presented.