Nonlinear projection filter based on Galerkin approximation

The conditional probability density function of the state of a stochastic d ynamic system represents the complete solution to the nonlinear filtering p roblem because, with the conditional density in hand, all estimates of the state, optimal or otherwise, can be computed. It is well known that, for sy stems with continuous dynamics, the conditional density evolves, between me asurements, according to Kolmogorov's forward equation, At a measurement, i t is updated according to Bayes formula. Therefore, these two equations can be viewed as the dynamic equations of the conditional density and, hence, the exact nonlinear filter. In this paper, Galerkin's method is used to app roximate the nonlinear filter by solving for the entire conditional density . Using a discrete cosine transform to approximate the projections required in Galerkin's method leads to a computationally realizable nonlinear filte r, The implementation details are given and performance is assessed through simulations.