Gain Function Tracking in the Feedback Particle Filter

The feedback particle filter (FPF) was formulated to approximate the nonlinear filter and is motivated by techniques from mean-field game theory. The critical component in the implementation of the FPF is the innovations gain function. The exact computation of the gain requires obtaining the gradient of the solution to a version of Poisson's equation that is difficult to obtain. This paper advances the reproducing kernel Hilbert space (RKHS) based differential TD (temporal-difference)-learning algorithm for gain function approximation in an on-line setting. Algorithms for tracking the FPF gain are proposed based on known structure of the gain function, and by exploiting the time-continuity of the gain. Performance and parameter sensitivity are tested in simulations.