A general Bayesian nonlinear estimation method using resampled Smooth Particle Hydrodynamics solutions of the underlying Fokker–Planck Equation

Abstract

The effectiveness of nonlinear filters depends on many factors, but one of the most important is how accurately the filter is able to predict the state dynamics of the underlying system between measurements. For a wide class of Gaussian white noise driven nonlinear systems the Bayesian optimal prior can be obtained by solving the system's corresponding Fokker–Planck Equation (FPE). Unfortunately the Fokker–Planck Equation is a partial differential equation with dimension equal to the number of states in the underlying dynamical system, making it extremely difficult to solve for realistic systems due to Curse of Dimensionality scaling issues. As a result it has been and still largely remains computationally impractical to simulate higher dimensional Fokker–Planck equations, at least while obtaining very high accuracy across the entire transient probability density function. This paper presents a general nonlinear filter based on solving the transient Fokker–Planck equation via Smooth Particle Hydrodynamics (SPH) at lower resolution, which turns out to still allow for accurate state estimation. The filter is enabled by an efficient heuristic resampling scheme of the SPH solution also presented here. The FPE-SPH Filter is able to replicate the accuracy of the Particle Filter and Extended Kalman filter (EKF) for lower-dimensional systems, while also being more robust than the EKF on certain classes of system.

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