Fast nonlinear filtering via Galerkin's method

Abstract

The conditional probability density function of the state of a stochastic dynamic system represents the complete solution to the nonlinear filtering problem because, with the conditional density in hand, optimal estimates of the state can be computed. It is well known that, for systems with continuous dynamics, the conditional density evolves, between measurements, according to Kolmogorov's forward equation. At a measurement, it is updated according to Bayes formula. Therefore, these two equations can be viewed as the dynamic equations of the conditional density and, hence, the optimal filter. In this paper, Galerkin's method is used to approximate the nonlinear filter by solving for the entire conditional density. Using an FFT to approximate the projections required in Galerkin's method leads to a computationally efficient nonlinear filter. The implementation details are given and performance is assessed through simulations.