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摘要: 提出了一种改进的高斯近似(Gaussian approximate, GA)滤波方法, 推导了它的一般解和特殊解, 并证明了现有的高斯近似滤波方法是所提出的方法的一种特例.在提出的方法中, 不需要基于高斯假设重复地产生求积点, 而是直接地更新求积点.与现有的高斯近似滤波方法相比, 提出的方法利用了量测求积点修正状态求积点, 从而可以更好地捕获状态一步预测密度和状态后验密度的非高斯信息和高阶矩信息.此外, 提出的方法不仅适用于确定的系统模型而且还适用于随机的系统模型.单变量非平稳增长模型、垂直落体模型、再入飞行器目标跟踪的仿真验证了提出的高斯近似滤波方法的有效性和与现有方法相比的优越性.Abstract: In this paper, an improved Gaussian approximate (GA) filtering method is proposed. Its general solution and special solution are derived, and the existing GA filtering method is proved to be its special case of the proposed method. In the proposed method, the quadrature points are no longer generated repeatedly based on Gaussian assumption, but updated directly. As compared with the existing GA filtering method, the proposed method can better capture the non-Gaussian information and high-order moment information of the one-step predicted density and posterior density of state, since the measurement quadrature points in the proposed method are used to correct the state quadrature points. Moreover, the proposed method is suitable for not only deterministic process model but also random process model. The efficiency and superiority of the proposed method are illustrated by simulations of univariate non-stationary growth model, vertically falling body model, and target tracking of re-entry vehicle.
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图 1 提出的改进高斯近似滤波算法的流程图
Fig. 1 Diagram of the proposed improved Gaussian approximate filtering algorithm
图 4 不同滤波方法的高度估计平均绝对值误差
Fig. 4 Averaged absolute error of the altitude of different filtering methods
图 5 不同滤波方法的速度估计平均绝对值误差
Fig. 5 Averaged absolute error of the velocity of different filtering methods
图 6 不同滤波方法的弹道系数估计平均绝对值误差
Fig. 6 Averaged absolute error of the ballistic coefficient of different filtering methods
图 9 不同滤波方法的飞行器系数RMSE
Fig. 9 RMSEs of the vehicle coefficient of different filtering methods
表 1 不同滤波方法的平均RMSE
Table 1 The averaged RMSEs of different filtering methods
滤波器 标准的CKF 标准的DDF2和UKF (n=1时等价) 现有的改进UKF 提出的UKF 平均RMSE 7.417 7.787 7.271 7.009 
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表 2 不同滤波方法的单步运行时间
Table 2 The run time of different filtering methods at single step
滤波器 标准的CKF 标准的DDF2和UKF (n=1时等价) 现有的改进UKF 提出的UKF 单步运行时间(秒) 0.21 × 10-3 0.22 × 10-3 0.28 × 10-3 0.32 × 10-3
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表 3 不同滤波方法在最后30秒内的平均绝对值误差的均值
Table 3 The means of averaged absolute errors of different filtering methods over the last 30 s
滤波器 高度估计的平均绝对值
误差的均值（ft)速度估计的平均绝对值
误差的均值（ft/s)弹道系数估计的平均
绝对值误差的均值标准的DDF2 143.701 9.636 5.174 ×10-5 标准的CKF和UKF(n=3时等价） 144.003 9.659 5.182 ×10-5 现有的改进UKF 92.060 5.723 3.087 ×10-5 提出的UKF 73.210 3.499 1.886 ×10-5
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表 4 不同滤波方法的单步运行时间
Table 4 The run time of different filtering methods at single step
滤波器 标准的DDF2 标准的CKF和UKF (n=3时等价） 现有的改进UKF 提出的UKF 单步运行时间（秒） 0.28 × 10-3 0.25 × 10-3 0.35 × 10-3 0.50 × 10-3
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表 5 不同滤波方法在最后25秒内的平均RMSE
Table 5 The averaged RMSEs of different filtering methods over the last 25 s
滤波器 位置的平均RMSE(m) 速度的平均RMSE(m/s) 飞行器系数的平均RMSE 标准的DDF2 12.142 17.023 6.212 × 10-2 标准的CKF 14.577 20.747 8.323 × 10-2 标准的UKF 12.143 17.025 6.213 × 10-2 现有的改进UKF 11.505 16.026 5.601 × 10-2 提出的UKF 9.844 13.332 3.726 × 10-2
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表 6 不同滤波方法的单步运行时间
Table 6 The run time of different filtering methods at single step
滤波器 标准的DDF2 标准的CKF 标准的UKF 现有的改进UKF 提出的UKF 单步运行时间（秒） 1.08 × 10-3 0.97 × 10-3 1.02 × 10-3 1.16 × 10-3 1.20 × 10-3
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