Abstract
The conditional probability density function of the state of a stochastic dynamic system represents the complete solution to the nonlinear ltering problem because, with the conditional density in hand, all estimates of the state, optimal or otherwise, 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 exact nonlinear lter. In this paper, Galerkin’s method is used to approximate the nonlinear lter 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 lter. The implementation details are given and performance is assessed through simulations.
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