Nonlinear Bayesian filtering via holonomic gradient method with quasi moment generating function
Nonlinear Bayesian filtering via holonomic gradient method with quasi moment generating function
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DOI：
10.1002/asjc.2970
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发表时间：
2022-10
影响因子：
2.4
通讯作者：
Tomoyuki Iori;T. Ohtsuka
中科院分区：
文献类型：
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作者：
Tomoyuki Iori;T. Ohtsuka
针对一类离散时间非线性随机系统的贝叶斯滤波问题，提出了一种符号-数值方法。我们首先将后验概率密度函数近似为高斯分布。均值和方差的更新律被表示为依赖于某些参数的几个积分的求值。与已有的扩展卡尔曼滤波(EKF)、无迹卡尔曼滤波(UKF)和粒子滤波(PF)等方法不同，该方法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母函数(MGF)激励的积分变换，我们称之为拟MGF。此外，准MGF与微分算子的傅里叶变换是相容的。我们利用这种相容来减少非对易微分算子环上的Gröbner基的计算次数，从而减少了离线计算时间。数值算例表明，该方法与EKF、UKF、PF等已有方法相比，具有较高的计算效率。
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.