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TopA significant amount of research efforts have been dedicated to the problem of nonlinear filtering. The extended Kalman filter (EKF) is an approximation method, in which nonlinear system equations are linearized by the Taylor series and the linear states are assumed to obey the Gaussian distribution. The linearization stage of the state equations may lead to the problem of divergence or instability (Cappe, Godsill, & Moulines, 2007). The EKF also requires the calculation of Jacobian matrix, which is a difficult and time-consuming process. Jacobian matrix may not even exist in some cases.
The unscented Kalman filter (UKF) and central difference Kalman filter (CDKF) are belonged to a single family of non-derivative Kalman filters. Both are based on statistical approximations of system equations without requiring the calculation of Jacobian matrix. Therefore, they have the higher accuracy and convergent speed than the EKF. The UKF combines the concept of unscented transform with the linear update structure of the Kalman filtering. However, it considers the second-order system momentum only, leading to the limited accuracy (Cappe, Godsill, & Moulines, 2007; Sarkka, 2007; Del Moral, Doucet, & Jasra, 2006; van der Merwe, Doucet, de Freitas, & Wan, 2000; Tenne, & Singh, 2003; Gordon, Salmond, & Smith, 1993). The CDKF approximates the state estimation and covariances of stochastic variables through central difference transformation. Its approximation accuracy is at least in the second order, leading to the higher filtering accuracy than the UKF. The CDKF also has a much simple structure than the UKF, as only one single scaling parameter is required for the CDKF while three parameters for the UKF. However, both CDKF and UKF require the system state obey the Gaussian distribution. Further, if the nonlinearity of a dynamic system is strong, both will lead to the biased or even divergent filtering solution (Simon, 2006).