To improve the filtering effect of the sparse grid quadrature filter (SGQF) under non-Gaussian conditions, the Gaussian sum technique is introduced, and the Gaussian sum sparse grid quadrature filter (GSSGQF) is developed. We present a systematic formulation of the SGQF and extend it to the discrete-time nonlinear system with the non-Gaussian noise. The proposed algorithm approximates the non-Gaussian probability densities by a finite number of weighted sums of Gaussian densities, and takes the SGQF as the Gaussian sub-filter to conduct the time and measurement update for each Gaussian component. An application in the discrete-time nonlinear system with the non-Gaussian noise has been shown to demonstrate the accuracy of the GSSGQF. It outperforms the unscented Kalman filter (UKF), the cubature Kalman filter (CKF) and the SGQF. Theoretical analysis and simulation results prove that the GSSGQF provides significant performance improvement in the calculation accuracy for nonlinear non-Gaussian filtering problems.

为了提高稀疏网格正交滤波器（SGQF）在非高斯条件下的滤波效果，引入了高斯求和技术，开发了高斯求和稀疏网格正交滤波器（GSSGQF）。我们提出了 SGQF 的系统公式，并将其扩展到具有非高斯噪声的离散时间非线性系统。该算法通过有限数量的高斯密度加权和来近似非高斯概率密度，并以SGQF作为高斯子滤波器对每个高斯分量进行时间和测量更新。已经证明在具有非高斯噪声的离散时间非线性系统中的应用证明了 GSSGQF 的准确性。它优于无迹卡尔曼滤波器 (UKF)、体积卡尔曼滤波器 (CKF) 和 SGQF。