Nonlinear filter design using Fokker-Planck-Kolmogorov probability density evolutions

The Fokker-Planck-Kolmogorov equation (FPKE) in conjunction with Bayes' conditional density update formula provides optimal estimates for a general continuous-discrete nonlinear filtering problem. It is well known that the analytical solution of FPKE and Bayes' formula are extremely difficult to obtain except in a few special cases. Hence, we address this problem using numerical approaches. The efficient numerical solution of FPKE presented relies on the key issue of adaptively calculating the domain over which the state probability density function (pdf) is to be evaluated, which is done using Chebyshev's inequality. Application to a passive tracking example shows that this approach can provide consistent estimators when measurement nonlinearities and noise levels are high.