A family of polynomial filters for discrete-time nonlinear stochastic systems

This work presents a family of polynomial filters for discrete-time nonlinear stochastic systems. These filters can be considered the polynomial version of the well known Extended Kalman Filter. The standard EKF consists in the optimal linear filter applied to the linear approximation of systems. The filters presented in this paper are polynomial filters applied to polynomial approximations of nonlinear systems, and therefore each of them is characterized by a pair of integers: the degree of the system approximation and the degree of the filter. The first filter of the family, the one of order (1,1), coincides with the EKF in the standard form. The implementation of the proposed filters does not require the complete knowledge of the noise distributions, but only the moments up to suitable orders. Numerical simulations show the performances of the filters for some values of the degrees.