Relations between full information and Kalman-based estimation

For nonlinear state space systems with additive noises, sometimes the number of process noise signals could be less than the dimension of the state space. In order to improve the accuracy and stability of nonlinear state estimation, this paper provides for the first time the derivation of the full information estimator (FIE) for such nonlinear systems. We verify our derivation of the FIE by firstly proving the unbiasedness and minimum-variance of the FIE for linear time varying (LTV) systems, then showing the equivalence between the Kalman filter/smoother and the FIE for LTV systems. Finally, we prove that the FIE will provide more accurate state estimates than the extended Kalman filter (EKF) and smoother (EKS) for nonlinear systems.