Non-linear filtering using probability density evolutions

The solution of the Fokker-Planck-Kolmogorov (FPK) forward diffusion equation in conjunction with Bayes' conditional density lemma provides optimal (minimum variance) state estimates of any general stochastic dynamic system (SDS). It has been well documented in non-linear filtering Literature that the analytical solution for the FPK equation is extremely difficult to obtain except in a few special cases. In this paper we propose the use of numerical solution of FPK to obtain the optimal state estimates of a non-linear dynamic system. The proposed method provides the conditional densities from which the conditional means (the optimal state estimates) can be easily evaluated. The estimated conditional densities clearly violate the assumptions of Gaussianity implicitly required by Extended Kalman Filter (EKF) based approaches. The performance of the proposed method is compared with that of the EKF. Monte-Carlo simulation results are provided to show the superior performance of the density evolution method.