Polynomial Extended Kalman Filtering for discrete-time nonlinear stochastic systems

This paper deals with the state estimation problem for a discrete-time nonlinear system driven by additive noise (not necessarily Gaussian). The solution here proposed is a filtering algorithm which is a polynomial transformation of the measurements. The first step for the filter derivation is the embedding of the nonlinear system into an infinite-dimensional bilinear system (linear drift and multiplicative noise), following the Carleman approach. Then, the infinite dimensional system is approximated by neglecting all the powers of the state up to a chosen degree μ, and the minimum variance estimate among all the μ-degree polynomial transformations of the measurements is computed. The proposed filter can be considered a Polynomial Extended Kalman Filter (PEKF), because when μ=1 the classical EKF algorithm is recovered. Numerical simulations support the theoretical results and show the improvements of a quadratic filter with respect to the classical EKF.