A Unified Approach to Nonlinear Filtering by Statistical Equivalent Linearization

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From the point of view of statistical equivalent linearization, we analyze algorithms of the equivalent linealization Kalman filter (EqKF), the extended Kalman filter (EKF), the Gaussian filter (GF) and the unscented Kalman filter (UKF). For a cubic sensor problem, we show that EqKF is close to GF, and UKF is in-between EqKF and GF, but the EKF is quite different from other filters. Also, with the aim to GPS applications, a new Gaussian sum equivalent linearization filter is derived by applying the Gauss-Hermite integral formula for evaluating the conditional expectations, under the assumption that the conditional probability density function of the state is a Gaussian sum. Moreover, we obtain two continuous-discrete (CD) filters, i.e. CD-EqKF and CD-GF, showing that both filters have the same time update equations, and a difference is in the observation update equations. Heun scheme-based simulations show that CD-EqKF and CD-GF are superior to the classical CD-EKF.