Nonlinear Bayesian filtering via holonomic gradient method with quasi moment generating function

A symbolic‐numeric method is proposed for addressing the Bayesian filtering problems of a class of discrete‐time nonlinear stochastic systems. We first approximate the posterior probability density function to be Gaussian. The update law of the mean and variance is formulated as the evaluation of several integrals depending on certain parameters. Unlike existing methods, such as the extended Kalman filter (EKF), unscented Kalman filter (UKF), and particle filter (PF), this formulation considers the nonlinearity of system dynamics exactly. To evaluate the integrals efficiently, we introduce an integral transform motivated by the moment generating function (MGF), which we call a quasi MGF. Furthermore, the quasi MGF is compatible with the Fourier transform of differential operators. We utilize this compatibility to decrease the number of computations of Gröbner bases in the noncommutative rings of differential operators, which reduces the offline computational time. A numerical example is presented to show the efficiency of the proposed method compared to that of other existing methods such as the EKF, UKF, and PF.

提出了一种符号 - 数值方法来解决一类离散时间非线性随机系统的贝叶斯滤波问题。我们首先将后验概率密度函数近似为高斯函数。均值和方差的更新定律被表述为对几个依赖于某些参数的积分的求值。与现有的方法，如扩展卡尔曼滤波器（EKF）、无迹卡尔曼滤波器（UKF）和粒子滤波器（PF）不同，这种表述精确地考虑了系统动力学的非线性。为了有效地计算积分，我们引入了一种由矩生成函数（MGF）激发的积分变换，我们称之为准矩生成函数。此外，准矩生成函数与微分算子的傅里叶变换是兼容的。我们利用这种兼容性来减少微分算子非交换环中格罗比纳基的计算次数，从而减少离线计算时间。给出了一个数值例子，以表明与其他现有方法（如EKF、UKF和PF）相比，所提出方法的有效性。