A Unified Approach to Nonlinear Filtering by Statistical Equivalent Linearization

从统计等价线性化的角度，分析了等效线性化卡尔曼滤波(EqKF)、扩展卡尔曼滤波(EKF)、高斯滤波(GF)和无迹卡尔曼滤波(UKF)算法。对于立方传感器问题，我们证明了EqKF接近于GF，而UKF介于EqKF和GF之间，但EKF与其他滤波器有很大的不同。此外，针对GPS应用，在假设状态的条件概率密度函数为高斯和的假设下，应用计算条件期望的Gauss-Hermite积分公式，导出了一种新的高斯和等价线性化滤波器。此外，我们还得到了两个连续-离散(CD)滤波器，即CD-EqKF和CD-GF，表明这两个滤波器具有相同的时间更新方程，而不同之处在于观测更新方程。基于Heun方案的仿真结果表明，CD-EqKF和CD-GF优于经典的CD-EKF。

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.