Fractional Kalman filters

The nonlinear Kalman filters usually suffer from degradation when the measurements appear in the tail of prior distribution. This article presents a new nonlinear filter named the fractional Kalman filter (FKF) for high robustness against linearization errors of both the time and the measurement updates. Based on the Bayesian theory and Gaussian assumption, an optimal fractional Gaussian filter (OFGF) is developed by maximizing the posterior PDFs, which consists of three basic steps namely time update, fractional measurement update and fractional state fusion. Specifically, the prior probability density function (PDF) and the likelihood function are factored into $N$ parts with fractional exponents, which increases the overlapping area of the prior and the likelihood distributions for linearization improvement. With the OFGF and the Kullback–Leibler divergence (KLD) analysis, the FKF is developed by a deterministic sampling-based linearization for approximation of means and covariances. From the performance analysis, it reveals that the risk of destroying the stabilities of both the time update and the measurement update is reduced by introducing the fractional exponents. Finally, the effectiveness and superiority of the proposed FKF method are verified by simulations of a target tracking example.