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摘要: 在统计流形空间中, 从信息几何角度考虑非线性状态后验分布近似的实质是后验分布与相应参数化变分分布之间的Kullback-Leibler散度最小化问题, 同时也可以转化为变分置信下界的最大化问题. 为了提升非线性系统状态估计的精度, 在高斯系统假设条件下结合变分贝叶斯推断和Fisher信息矩阵推导出置信下界的自然梯度, 并通过分析其信息几何意义, 阐述在统计流形空间中置信下界沿其方向不断迭代增大, 实现变分分布与后验分布的 “紧密” 近似; 在此基础上, 以状态估计及其误差协方差作为变分超参数, 结合最优估计理论给出一种基于自然梯度的非线性变分贝叶斯滤波算法; 最后, 通过天基光学传感器量测条件下近地轨道卫星跟踪定轨和纯角度被动传感器量测条件下运动目标跟踪仿真实验验证: 与对比算法相比, 所提算法具有更高的精度.
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关键词:
- 非线性滤波 /
- 信息几何 /
- 变分贝叶斯推断 /
- 自然梯度 /
- Fisher信息矩阵
Abstract: In statistical manifold space, the essence of nonlinear state posterior distribution approximation from the perspective of information geometry is minimizing Kullback-Leibler divergence between posterior distribution and the corresponding approximated distribution; Meanwhile, it is equivalent to maximizing evidence low bound. Aiming at the problem of improving the estimation accuracy of nonlinear system state, the natural gradient of evidence lower bound is derived under Gaussian system assumption by combining with Fisher information matrix and variational Bayesian inference, which produces a faster movement direction to the posterior distribution, and realizing a close approximation between variational distribution and the posterior. On this basis, a variational Bayesian Kalman filtering algorithm using natural gradient is proposed for updating the variational hyperparameters of state estimation and the associated error covariance. Simulations in low earth orbit target tracking system with space-based optical sensors and bearing-only target tracking system are presented verifying that the proposed algorithm has higher accuracy than the comparison algorithms. -
图 1 变量分布近似过程中的KL散度示意图
Fig. 1 The KL divergence of the distribution approximation of a variable
图 2 非线性动态系统状态转移和量测的示意图
Fig. 2 The state transition and measurement in a nonlinear dynamic system
图 3 单变量高斯分布的欧氏距离示意图
Fig. 3 The Euclidean distance for univariate Gaussian distributions
图 4 多变量高斯分布的欧氏距离示意图
Fig. 4 The Euclidean distance for multivariate Gaussian distributions
图 5 非线性状态估计在不同空间中的含义示意图
Fig. 5 The meaning of nonlinear state estimation in different spaces
图 7 变分迭代过程中置信下界自然梯度的示意图
Fig. 7 The natural gradient of ELBO in variation1al Bayesian iteration
图 8 天基量测条件下LEO跟踪定轨仿真场景
Fig. 8 Scenario of LEO orbit determination and tracking with space-based measurement
表 1 文中变量和符号含义
Table 1 The meaning of variables and symbols
$ {\boldsymbol x}_k $ $ k $时刻目标状态真实值 $ {\boldsymbol x}_{k|k} $ $ k $时刻目标状态估计值 $ {\boldsymbol P}_{k|k} $ $ k $时刻目标状态估计误差协方差 $ {\boldsymbol z}_k $ 传感器在$ k $时刻的量测值 $ {\boldsymbol\omega}_k $ $ k $时刻的系统噪声 $ {\boldsymbol\upsilon}_k $ $ k $时刻的量测噪声 $ {\boldsymbol Q}_k $ $ k $时刻系统噪声方差 $ {\boldsymbol R}_k $ $ k $时刻量测噪声方差 $ d_x $ 目标状态向量的维数 $ d_z $ 量测向量的维数 $ {\boldsymbol F}_{k|k-1} $ $ k-1 $时刻到$ k $时刻的状态转移矩阵 $ {\boldsymbol H}_{k} $ $ k $时刻量测矩阵 $ {\boldsymbol\psi}_{k} $ 变分分布参数 $ p\left({\boldsymbol x}_k|{\boldsymbol z}_{k}\right) $ $ k $时刻目标状态后验分布 $ q\left({\boldsymbol x}_k|{\boldsymbol\psi}_{k}\right) $ 以$ {\boldsymbol\psi}_{k} $为参数的变分分布 $ \mathcal{L}\left({\boldsymbol\psi}_{k}\right) $ 以$ {\boldsymbol\psi}_{k} $为变分分布参数的置信下界 $ \mathbb{D}\left(q\left({\boldsymbol x}_k|{\boldsymbol\psi}_{k}\right)|| p\left({\boldsymbol x}_k|{\boldsymbol z}_{k}\right)\right) $ 变分分布$ q\left({\boldsymbol x}_k|{\boldsymbol\psi}_{k}\right) $与状态后验分布$ p\left({\boldsymbol x}_k|{\boldsymbol z}_{k}\right) $的KL散度 $ {\boldsymbol J}_{{\boldsymbol\psi}_k} $ 以$ {{\boldsymbol\psi}_k} $为参数的 Fisher 信息矩阵 $ \mathcal{M} $ 流型空间 $ \mathcal{S} $ 流型空间中的概率分布集合 $ \mathcal{F} $ 流型空间中的平滑映射函数 $ {\overrightarrow{\boldsymbol v}}_{OP} $ 流型空间中的$ O $点处指向$ P $点的切向量 $ |{\overrightarrow{\boldsymbol v}}_{OP}| $ 切向量$ {\overrightarrow{\boldsymbol v}}_{OP} $的模 
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表 2 目标的轨道根数
Table 2 The orbital elements of target
半长轴 (km) 离心率 倾角 (deg) 近地点角 (deg) 升交点赤经 (deg) 7500 0.1 15 30 12
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表 3 算法平均估计误差均值的对比
Table 3 Comparison of the mean estimation error of the algorithm
算法 EKF UKF IEKF VBKF-NG x 轴位置 6.8731 6.8493 6.8290 6.6025 x 轴速度 − 0.0382 0.0418 − 0.0368 − 0.0241 y 轴位置 2.8793 2.8770 2.8688 2.7563 y 轴速度 − 0.0257 − 0.0243 − 0.0222 − 0.0103 z 轴位置 4.7665 4.6759 4.5546 4.3286 z 轴速度 0.1272 0.1097 0.1076 0.1062
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表 4 算法平均运行时间$( \times 10^{-4}\;{\rm{s}}) $的对比
Table 4 The comparison of the average run time $( \times 10^{-4}\;{\rm{s}}) $ of algorithms
算法 EKF UKF IEKF VBKF-NG 时间 0.2580 0.9542 1.0989 1.1205
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表 5 算法平均RMSE的对比
Table 5 Comparison of average RMSE between algorithms
算法 径向距(km) 径向速度(k/min) EKF 1.5052 0.0813 UKF 1.1833 0.0670 VBKF-COD 0.9181 0.0432 VBKF-NG 0.2990 0.0161
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