分维自适应稀疏网格积分非线性滤波器

徐嵩; 孙秀霞; 刘树光; 刘希; 蔡鸣

doi:10.3724/SP.J.1004.2014.01249

分维自适应稀疏网格积分非线性滤波器

doi: 10.3724/SP.J.1004.2014.01249

1.
空军工程大学航空航天工程学院西安 710038

基金项目:

航空科学基金（20121396008），国家大学生创新训练计划（201390052011）资助

详细信息

作者简介:
徐嵩空军工程大学航空航天工程学院博士研究生. 主要研究方向为最优估计，信息融合和视觉导航.E-mail：xusongpla@163.com

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出版历程
- 收稿日期: 2013-07-25
- 修回日期: 2013-09-18
- 刊出日期: 2014-06-20

Dimension-wise Adaptive Spare Grid Quadrature Nonlinear Filter

1.
School of Aeronautics and Astronautics Engineering, Air Force Engineering University, Xi'an 710038

Funds:

Supported by Aviation Science Foundation of China (20121396008),and Chinese National Innovation Plan for Undergraduate (201390052011)

摘要

摘要: 针对含加性高斯噪声的非线性离散系统，提出了可分别根据各维状态及量测方程的非线性函数特性来确定采样点及其权重的积分滤波器.设计了基于嵌入式高斯采样积分和稀疏网格法则的自适应多变量采样积分方法，可在匹配函数高阶泰勒展开项时，利用低阶采样点，提出了高效的数据结构和遍历算法，便于采用该积分方法分别估计系统状态/量测的预测均值和协方差矩阵.该滤波器既能根据各维非线性函数的特性确定采样点，又实现了对采样值和权重的完全复用，保证了算法效率.理论分析和仿真表明，该滤波算法中自适应调整的运算量小于计算非线性函数采样值.该滤波器与无迹卡尔曼滤波相比，提高了滤波精度，与固定形式的稀疏网格滤波器相比，提高了采样效率，且该方法为两者的广义形式.仿真实验也验证了状态估计的精确性和函数采样的高效性.
- 高斯滤波 /
- 稀疏网格 /
- 自适应采样 /
- 高斯-厄米特积分 /
- 快速遍历
Abstract: For nonlinear discrete systems with addictive Gaussian noises, a new quadrature filter is proposed, which can fix sample points according to each dimension0s nonlinear function, respectively. In order to match higher-order terms of the nonlinear function0s Taylor expanding with reusing the sample points matching lower-order ones, an adaptive sampled multi variable quadrature rule is designed based on the embedded Gaussian sampled quadrature and the spare grid quadrature (SGQ) formula. A group of effective data structures and traversal algorithms are proposed for the sampled quadrature rule to be used for calculating the predict expectations of the states and measurements with their covariances. This filter could not only fix sampled points for different dimensions separately, but also reuse these points and their weights completely, thus enhancing the efficiency of the filter. This filter achieves a higher accuracy than the unscented Kalman filter (UKF), more efficiency than the fixed SGQ filter, as well as generalized form of these two filters. The calculating cost of adaptive steps is much less than computing the function sampled values. Simulations also validates the accuracy and efficiency of this filter.
- Gaussian filter /
- spare grid /
- adaptive sampling /
- Gauss-Hermite quadrature (GHQ) /
- fast traversal

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参考文献(28)

[1]	Sun Yao, Zhang Qiang, Wan Lei. Small autonomous underwater vehicle navigation system based on adaptive UKF algorithm. Acta Automatica Sinica, 2011, 37(3): 343-354(孙尧, 张强, 万磊. 基于自适应UKF算法的小型水下机器人导航系统. 自动化学报, 2011, 37(3): 343-354)
[2]	Bernardo J M, Smith A F M. Bayesian Theory. Berlin: Springer-Verlag, 1994. 339-353
[3]	Wang Xiao-Xu, Pan Quan, Huang He, Gao Ang. Overview of deterministic sampling filtering algorithms for nonlinear system. Control and Decision, 2012, 27(6): 801-812 (王小旭, 潘泉, 黄鹤, 高昂. 非线性系统确定采样型滤波算法综述. 控制与决策, 2012, 27(6): 801-812)
[4]	Budhiraja A, Chen L J, Lee C H. A survey of numerical methods for nonlinear filtering problems. Physica D: Nonlinear Phenomena, 2007, 230(1): 27-36
[5]	Nie P Y, Fan J Y. A derivative-free filter method for solving nonlinear complementarilty problems. Applied Mathematics and Computation, 2005, 161(3): 787-797
[6]	Julier S J, Uhlmann J K. Unscented filtering and nonlinear estimation. Proceedings of the IEEE, 2004, 92(3): 401-422
[7]	Arasaratnam I, Haykin S. Cubature Kalman filters. IEEE Transactions on Automatic Control, 2009, 54(6): 1254-1269
[8]	Chang L B, Hu B Q, Li A, Qin F J. Transformed unscented Kalman filter. IEEE Transactions on Automatic Control, 2013, 58(1): 252-257
[9]	Genz A, Keister B D. Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight. Journal of Computational and Applied Mathematics, 1996, 71(2): 299-309
[10]	Ito K, Xiong K Q. Gaussian filters for nonlinear filtering problems. IEEE Transactions on Automatic Control, 2000, 45(5): 910-927
[11]	Arasaratnam I, Haykin S, Elliott R J. Discrete-time nonlinear filtering algorithms using Gauss-Hermite quadrature. Proceedings of the IEEE, 2007, 95(5): 953-977
[12]	Heiss F, Winschel V. Likelihood approximation by numerical integration on sparse grids. Journal of Econometrics, 2008, 144(1): 62-80
[13]	Smolyak S A. Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Mathematics Doklady, 1963, 4: 240-243
[14]	Gerstner T, Griebel M. Dimension-adaptive tensor-product quadrature. Computing, 2003, 71(1): 65-87
[15]	Bungartz H J, Dirnstorfer S. Multivariate quadrature on adaptive sparse grids. Computing, 2003, 71(1): 88-114
[16]	Jia B, Xin M, Cheng Y. Sparse Gaussian-Hermite quadrature filter with application to spacecraft attitude estimation. Journal of Guidance, Control and Dynamics, 2011, 27(2): 367-379
[17]	Bin J, Ming X, Yang C. Sparse-grid quadrature nonlinear filtering. Automatica, 2012, 48(2): 327-341
[18]	Wu Zong-Wei, Yao Min-Li, Ma Hong-Guang, Jia Wei-Min, Tian Fang-Hao. Sparse-grid square-root quadrature nonlinear filter. Acta Electronica Sinica, 2012, 40(7): 1298-1303(伍宗伟, 姚敏立, 马红光, 贾维敏, 田方浩. 稀疏网格平方根求积分非线性滤波器. 电子学报, 2012, 40(7): 1298-1303)
[19]	Baek K, Bang H. Adaptive sparse grid quadrature filter for spacecraft relative navigation. Acta Astronautica, 2013, 87(3): 96-106
[20]	Griebel M, Holtz M. Dimension-wise integration of high-dimensional functions with applications to finance. Journal of Complexity, 2010, 26(5): 455-489
[21]	Wang X Q, Fang K T. The effective dimension and quasi-Monte Carlo integration. Journal of Complexity, 2003, 19(2): 101-124
[22]	Dellaportas P, Wright D. Positive embedded integration Bayesian analysis. Statistics and Computing, 1991, 1(1): 1-12
[23]	Imai J, Tan K S. A general dimension reduction technique for derivative pricing. Journal of Computing Finance, 2006, 10(2): 129-155
[24]	Glasserman P. Monte Carlo Methods in Financial Engineering. Berlin: Springer, 2003. 226-228
[25]	Zhao Y Z. On the use of dimension reduction techniques in quasi-Monte Carlo methods. Mathematical and Computer Modeling, 2008, 48(11-12): 1925-1937
[26]	Cheng Yun-Peng, Zhang Kai-Yuan, Xu Zhong. Matrix Theory. Xi'an: Press of Northwestern Polytechnical University, 2006, 94-106(程云鹏, 张凯院, 徐仲. 矩阵论. 西安: 西北工业大学出版社, 2006, 94-106)
[27]	Sarkka S. On unscented Kalman filtering for state estimation of continuos-time nonlinear systems. IEEE Transactions on Automatic Control, 2007, 52(9): 1631-1641
[28]	Arasaratnam I, Haykin S, Thomas R H. Cubature Kalman filtering for continuous-discrete systems: theory and simulations. IEEE Transaction on Signal Processing, 2010, 58(10): 2207-2218