Nonlinear Kalman Filter Based on Gaussian-generalized-hyperbolic Mixing Distribution
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摘要: 本文研究带非平稳厚尾非高斯量测噪声的非线性系统状态估计问题. 考虑到广义双曲分布包含多种常见厚尾分布特例, 且其混合分布为共轭的广义逆高斯分布, 选用广义双曲分布建模厚尾噪声; 进而引入伯努利变量构建高斯–广义双曲混合分布来建模非平稳厚尾噪声, 并利用该分布的高斯分层结构得到系统的概率模型. 随后采用变分贝叶斯方法实现对系统状态以及噪声参数的后验估计, 得到针对此类噪声系统的卡尔曼滤波 (Kalman filter, KF) 框架, 现有的几种鲁棒滤波算法均是本文算法的特例. 机器人跟踪仿真实验表明, 所提算法与同类算法相比具有更好的估计精度和数值稳定性, 且对于初始参数具有较好的鲁棒性.Abstract: In this paper, we consider the state estimation problem of nonlinear systems with non-stationary heavy-tailed non-Gaussian noise. Considering that many often encountered heavy-tailed noises are the special cases of generalized hyperbolic distribution, and its mixing distribution is the conjugate generalized inverse Gaussian distribution, we adopt the generalized hyperbolic distribution to model the heavy-tailed noise. After that, we employ the Bernoulli variable to construct the Gaussian-generalized-hyperbolic mixing distribution to model the non-stationary heavy-tailed noise, and then the probability model of the target system is constructed based on the hierarchical Gaussian structure of this distribution. The Kalman filter (KF) framework for systems with this type of noises is achieved by applying the variational Bayes method to estimate the posteriori distribution of system states and the parameters of the noise, and several existing robust filtering algorithms are the special cases of the proposed algorithm. Simulation results of robot tracking experiments demonstrate that the proposed algorithm has better estimation accuracy and numerical stability than the existing ones, and is robust to its initial parameters.
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图 2 基于传感器网络的机器人跟踪示意图
Fig. 2 The illustration of tracking a robot with the sensor network
图 3 仿真中产生一维噪声的幅值以及概率密度函数
Fig. 3 The amplitude and probability density functions of the one-dimensional noise used in the simulation
图 4 本文所提算法与同类方法的
${{\rm{RMSE}}}_{\rm pos}$ Fig. 4 The
${{\rm{RMSE}}}_{\rm pos}$ of the proposed algorithms and related ones
图 5 本文所提算法与同类方法的
${{\rm{RMSE}}}_{\rm vel}$ Fig. 5 The
${{\rm{RMSE}}}_{\rm vel}$ of the proposed algorithms and related ones
图 6 本文算法在迭代次数
$ N$ 不同时的$ {{\rm{RMSE}}}_{\rm pos}$ Fig. 6 The
$ {{\rm{RMSE}}}_{ \rm pos}$ of the proposed algorithms with different iteration number$ N$ 
图 7 本文算法在迭代次数
$N$ 不同时的$ {{\rm{RMSE}}}_{\rm vel}$ Fig. 7 The
$ {{\rm{RMSE}}}_{\rm vel}$ of the proposed algorithms with different iteration number$N$ 
图 8 本文算法在
$\omega_0$ 和$\eta_0$ 不同时的${{\rm{RMSE}}}_{\rm {pos}}$ Fig. 8 The
${{\rm{RMSE}}}_{\rm {pos}}$ of the proposed algorithms with different$\omega_0$ and$\eta_0$ 
图 9 本文算法在
$\omega_0$ 和$\eta_0$ 不同时的$ {{\rm{RMSE}}}_{\rm vel}$ Fig. 9 The
$ {{\rm{RMSE}}}_{\rm vel}$ of the proposed algorithms with different$\omega_0$ and$\eta_0$ 
图 10 本文算法在
$\delta_0$ 不同时的$ {{\rm{RMSE}}}_{\rm pos}$ Fig. 10 The
$ {{\rm{RMSE}}}_{\rm pos}$ of the proposed algorithms with different$\delta_0$ 
图 11 本文算法在
$\delta_0$ 不同时的$ {{\rm{RMSE}}}_{\rm vel}$ Fig. 11 The
${{\rm{RMSE}}}_{\rm vel}$ of the proposed algorithms with different$\delta_0$ 表 1 广义双曲分布的几种特殊分布
Table 1 Several special cases of generalized hyperbolic distribution
分布名称 关键参数 高斯分布 $\delta\rightarrow+\infty$或者$\delta\rightarrow-\infty$ 正态逆高斯分布 $\delta=-0.5$ 双曲分布 $\delta=1$ K 分布 $\omega=0$ 广义双曲学生 t 分布 $\eta=0$ 
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表 2 不同算法总运行时间
Table 2 The total simulation time of different algorithms
算法名称 仿真时间 (s) UKF 15.28 STUKF 139.66 MCUKF 22.51 RSE-ML 151.75 SEUKF 146.92 OUKF 14.67 G-GHUKF1 167.36 G-GHUKF2 165.32 G-GHUKF3 166.65 G-GHUKF4 145.77
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