Two Second-Order Nonlinear Extended Kalman Particle Filter Algorithms
ABSTRACT
In algorithms of nonlinear Kalman filter, the so-called extended Kalman filter algorithm actually uses first-order Taylor expansion approach to transform a nonlinear system into a linear system. It is obvious that this algorithm will bring some systematic deviations because of ignoring nonlinearity of the system. This paper presents two extended Kalman filter algorithms for nonlinear systems, called second-order nonlinear Kalman particle filter algorithms, by means of second-order Taylor expansion and linearization approximation, and correspondingly two recursive formulas are derived. A simulation example is given to illustrate the effectiveness of two algorithms. It is shown that the extended Kalman particle filter algorithm based on second-order Taylor expansion has a more satisfactory performance in reducing systematic deviations and running time in comparison with the extended Kalman filter algorithm and the other second-order nonlinear Kalman particle filter algorithm.
In algorithms of nonlinear Kalman filter, the so-called extended Kalman filter algorithm actually uses first-order Taylor expansion approach to transform a nonlinear system into a linear system. It is obvious that this algorithm will bring some systematic deviations because of ignoring nonlinearity of the system. This paper presents two extended Kalman filter algorithms for nonlinear systems, called second-order nonlinear Kalman particle filter algorithms, by means of second-order Taylor expansion and linearization approximation, and correspondingly two recursive formulas are derived. A simulation example is given to illustrate the effectiveness of two algorithms. It is shown that the extended Kalman particle filter algorithm based on second-order Taylor expansion has a more satisfactory performance in reducing systematic deviations and running time in comparison with the extended Kalman filter algorithm and the other second-order nonlinear Kalman particle filter algorithm.
Cite this paper
Dai, H. and Zou, L. (2015) Two Second-Order Nonlinear Extended Kalman Particle Filter Algorithms. Open Journal of Statistics, 5, 254-261. doi: 10.4236/ojs.2015.54027.
Dai, H. and Zou, L. (2015) Two Second-Order Nonlinear Extended Kalman Particle Filter Algorithms. Open Journal of Statistics, 5, 254-261. doi: 10.4236/ojs.2015.54027.
References
[1] Daum. F. (2005) Nonlinear Filters: Beyond the Kalman Filter. IEEE A&E Systems Magazine, 20, 57-69. http://dx.doi.org/10.1109/MAES.2005.1499276 [2] Doucet, A. and Johansen, A. (2011) A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later. In: Crisan, D. and Rozovskii, B., Eds., The Oxford Handbook of Nonlinear Filtering, Oxford University Press, New York, 656-704. [3] Cappe, O., Godsill, S.J. and Moulines, E. (2007) An Overview of Existing Methods and Recent Advances in Sequential Monte Carlo. Proceedings of the IEEE, 95, 899-924.
http://dx.doi.org/10.1109/JPROC.2007.893250 [4] Gao, Y., Gao, S. and Wu, J. (2014) Fading-Memory Square-Root Unscented Particle Filter Algorithm and Its Application in Integrated Navigation System. Journal of Chinese Inertial Technology, 22, 777-781. [5] Qian, Z. and Qi, Y. (2012) A SLAM Algorithm Based on an Iterated Central Difference Particle Filter. Journal of Harbin Engineering University, 33, 355-360. [6] Zhu, Z. (2012) Particle Filter Algorithm and Application. Science Press, Beijing. [7] Wang, F. and Guo, Q. (2010) Neural Network Training Based on Extended Kalman Particle Filter. Computer Engineering & Science, 32, 48:50. [8] Zhao, L. (2012) Nonlinear Kalman Filter Theory. National Defense Industry Press, Beijing. [9] Julier, S.J., Uhlmann, J.K. and Durrant-Whyten, H.F. (2000) A New Approach for the Nonlinear Transformation of Means and Covariance in Filters and Estimators. IEEE Transactions on Automatic Control, 45, 477-482. http://dx.doi.org/10.1109/9.847726 [10] Wang, F. and Zhao, Q. (2008) A New Particles to Solve the Problem of Nonlinear Filter Algorithm. Chinese Journal of Computers, 31, 346-352. [11] Liu, X., Tao, Z., Jin, Y. and Yang, Y. (2010) A Novel Multiple Model Particle Filter Algorithm Based on Particle Optimization. Acta Electronica Sinica, 38, 301-305. [12] Kolas, S., Foss, B.A. and Schei, T.S. (2009) Constrained Nonlinear State Estimation-Based on the UKF Approach. Computers & Chemical Engineering, 33, 1386-1401.
http://dx.doi.org/10.1016/j.compchemeng.2009.01.012 [13] Wang, F., Lu, M., Zhao, Q. and Yuan, Z. (2014) Particle Filtering Algorithm. Chinese Journal of Computers, 37, 1679-1693. [14] Xia, N., Qiu, T. and Li, J. (2013) A Nonlinear Kalman Filtering Algorithm Combining the Kalman Filter and the Particle Filter. Acta Electronica Sinica, 41, 148-152.
[1] Daum. F. (2005) Nonlinear Filters: Beyond the Kalman Filter. IEEE A&E Systems Magazine, 20, 57-69. http://dx.doi.org/10.1109/MAES.2005.1499276 [2] Doucet, A. and Johansen, A. (2011) A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later. In: Crisan, D. and Rozovskii, B., Eds., The Oxford Handbook of Nonlinear Filtering, Oxford University Press, New York, 656-704. [3] Cappe, O., Godsill, S.J. and Moulines, E. (2007) An Overview of Existing Methods and Recent Advances in Sequential Monte Carlo. Proceedings of the IEEE, 95, 899-924.
http://dx.doi.org/10.1109/JPROC.2007.893250 [4] Gao, Y., Gao, S. and Wu, J. (2014) Fading-Memory Square-Root Unscented Particle Filter Algorithm and Its Application in Integrated Navigation System. Journal of Chinese Inertial Technology, 22, 777-781. [5] Qian, Z. and Qi, Y. (2012) A SLAM Algorithm Based on an Iterated Central Difference Particle Filter. Journal of Harbin Engineering University, 33, 355-360. [6] Zhu, Z. (2012) Particle Filter Algorithm and Application. Science Press, Beijing. [7] Wang, F. and Guo, Q. (2010) Neural Network Training Based on Extended Kalman Particle Filter. Computer Engineering & Science, 32, 48:50. [8] Zhao, L. (2012) Nonlinear Kalman Filter Theory. National Defense Industry Press, Beijing. [9] Julier, S.J., Uhlmann, J.K. and Durrant-Whyten, H.F. (2000) A New Approach for the Nonlinear Transformation of Means and Covariance in Filters and Estimators. IEEE Transactions on Automatic Control, 45, 477-482. http://dx.doi.org/10.1109/9.847726 [10] Wang, F. and Zhao, Q. (2008) A New Particles to Solve the Problem of Nonlinear Filter Algorithm. Chinese Journal of Computers, 31, 346-352. [11] Liu, X., Tao, Z., Jin, Y. and Yang, Y. (2010) A Novel Multiple Model Particle Filter Algorithm Based on Particle Optimization. Acta Electronica Sinica, 38, 301-305. [12] Kolas, S., Foss, B.A. and Schei, T.S. (2009) Constrained Nonlinear State Estimation-Based on the UKF Approach. Computers & Chemical Engineering, 33, 1386-1401.
http://dx.doi.org/10.1016/j.compchemeng.2009.01.012 [13] Wang, F., Lu, M., Zhao, Q. and Yuan, Z. (2014) Particle Filtering Algorithm. Chinese Journal of Computers, 37, 1679-1693. [14] Xia, N., Qiu, T. and Li, J. (2013) A Nonlinear Kalman Filtering Algorithm Combining the Kalman Filter and the Particle Filter. Acta Electronica Sinica, 41, 148-152.