Nonlinear Bayesian filtering based on Fokker-Planck equation and tensor decomposition

A nonlinear Bayesian filter is proposed in this paper for a general nonlinear system of continuous time dynamics and discrete time measurements. In this filter, a transient Fokker-Planck equation solver based on tensor decomposition is used for propagating the conditional state probability density function (PDF) in conjunction with a measurement update via Bayes´ rule. This filter is not restricted by assumptions of linearity or Gaussianity since it relies on the exact state PDF which captures the entire information of the underlying uncertainty. Moreover, it is suitable for system of high-dimensional state space by virtue of the efficient tensor decomposition scheme, which enables the computational efforts for solving the state PDF grow benignly with dimensionality. This is possible because every dimension of the state space as well as the time domain is separated from each other in the solution process, as a result of which originally expensive high-dimensional operations are decoupled into a series of simple one-dimensional operations. Numerical examples are provided to demonstrate the advantages of the proposed filter over the extended Kalman filter for state estimation.