Abstract
To improve the filtering effect of the sparse grid quadrature filter (SGQF) under non-Gaussian conditions, the Gaussian sum technique is introduced, and the Gaussian sum sparse grid quadrature filter (GSSGQF) is developed. We present a systematic formulation of the SGQF and extend it to the discrete-time nonlinear system with the non-Gaussian noise. The proposed algorithm approximates the non-Gaussian probability densities by a finite number of weighted sums of Gaussian densities, and takes the SGQF as the Gaussian sub-filter to conduct the time and measurement update for each Gaussian component. An application in the discrete-time nonlinear system with the non-Gaussian noise has been shown to demonstrate the accuracy of the GSSGQF. It outperforms the unscented Kalman filter (UKF), the cubature Kalman filter (CKF) and the SGQF. Theoretical analysis and simulation results prove that the GSSGQF provides significant performance improvement in the calculation accuracy for nonlinear non-Gaussian filtering problems.
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References
L. Wang and X. H. Cheng, “Algorithm of Gaussian sum filter based on high-order UKF for dynamic state estimation,” International Journal of Control, Automation and System, vol. 13, no. 3, pp. 652–661, March 2015.
R. Radharkrishnam, S. Bhaumik, and N. K. Tomar, “Gaussian sum shifted Rayleigh filter for underwater bearings-only target tracking problems,” IEEE Journal of Oceanic Engineering, vol. 44, no. 2, pp. 492–501, April 2019.
L. Zhang, C. Yang, Q. W. Chen, and F. Yan, “Robust H-infinity CKF/KF hybrid filtering method for SINS alignment,” IET Science, Measurement & Technology, vol. 10, no. 8, pp. 916–925, November 2016.
W. H. Li, Y. Fan, F. Ringbeck, D. Jst, and D. U. Sauer, “Electrochemical model-based state estimation for lithium-ion batteries with adaptive unscented Kalman filter,” Journal of Power Sources, vol. 476, no. 228534, November 2020.
H. M. Wang, Y. P. Pan, S. H. Li, and H. Y. Yu, “Robust sliding mode control for robots driven by compliant actuators,” IEEE Transactions on Control Systems Technology, vol. 27, no. 3, pp. 1259–1266, May 2019.
H. M. Wang, Q. Y. Zhang, J. Xian, and I. Chen, “Robust finite-time output feedback control for systems with unpredictable time-varying disturbances,” IEEE Access, vol. 8, pp. 52268–52277, March 2020.
Z. Gao, “Kalman filters for continuous-time fractional-order systems involving fractional-order colored noises using tustin generating function,” International Journal of Control, Automation, and Systems, vol. 16, no. 3, pp. 1049–1059, March 2018.
H. J. Kushner, “Approximations to optimal nonlinear filters,” IEEE Transactions on Automatic Control, vol. 12, no. 5, pp. 546–556, October 1967.
X. X. Wang, Z. S. Xu, X. J. Gou, and L. Trajkovi, “Tracking a maneuvering target by multiple sensors using extended Kalman filter with nested probabilistic-numerical linguistic information,” IEEE Transactions on Fuzzy Systems, vol. 28, no. 2, pp. 346–360, Feburary 2020.
L. Zhong and S. C. Chan, “Adaptive fading bayesian unscented Kalman filter and smoother for state estimation of unmanned aircraft systems,” IEEE Access, vol. 8, pp. 119470–119486, June 2020.
F. Deng, H. L. Yang, and L. J. Wang, “Adaptive unscented Kalman filter based estimation and filtering for dynamic positioning with model uncertainties,” International Journal of Control, Automation, and Systems, vol. 17, no. 3, pp. 667–678, March 2019.
I. Kazufumi and K. Q. Xiong, “Gaussian filters for nonlinear filtering problems,” IEEE Transactions on Automatic Control, vol. 45, no. 5, pp. 910–927, May 2009.
B. Jia, M. Xin, and Y. Cheng, “Anisotropoc sparse Gauss-Hermite quadrature filter,” Proceedings of the IEEE, vol. 35, no. 3, pp. 1014–1022, August 2012.
B. Jia, M. Xin, and Y. Cheng, “Sparse-grid quadrature nonlinear filtering,” Automatica, vol. 48, no. 2, pp. 327–341, February 2012.
W. M. Shao, Z. Q. Ge, and Z. H. Song, “Semisupervised Bayesian Gaussian mixture models for non-Gaussian soft sensor,” IEEE Transactions on Cybernetics, vol. 99, pp. 1–14, November 2019.
J. M. Pak, “Gaussian sum FIR filtering for 2D target tracking,” International Journal of Control, Automation, and Systems, vol. 18, no. 3, pp. 643–649, March 2020.
M. Lyu, C. Lambelet, D. Woolley, X. Zhang, W. Chen, X. Ding, R. Gassert, and N. Wenderoth, “Comparison of particle filter to established filtering methods in electromyography biofeedback,” Biomedical Signal Processing and Control, vol. 60, 101949, July 2020.
H. Shariati, H. Moosavi, and M. Danesh, “Application of particle filter combined with extended Kalman filter in model identification of an autonomous underwater vehicle based on experimental data,” Applied Ocean Research, vol. 82, pp. 32–40, January 2019.
R. Havangi, “Intelligent adaptive unscented particle filter with application in target tracking,” Signal Image and Video Processing, vol. 14, pp. 1487–1495, May 2020.
H. L. Feng and Z. W. Cai, “Target tracking based on improved cubature particle filter in UWSNs,” IET Radar, Sonar & Navigation, vol. 13, no. 4, pp. 638–645, April 2019.
K. K. Kottakki, M. Bhushan, and S. Bhartiya, “Monte carlo Gaussian sum filter for state estimation of nonlinear dynamical systems,” IFAC-PapersOnline, vol. 49, no. 1, pp. 65–70, April 2016.
X. Hong, W. C. Xie, H. D. Yuan, K. Q. Duan, W. J. Liu, and Y. L. Wang, “Fixed-point iteration Gaussian sum filtering estimator with unknown time-varying non-Gaussian measurement noise,” Signal Processing, vol. 153, pp. 132–142, December 2018.
J. H. Kotecha and P. M. Djuric, “Gaussian sum particle filtering,” IEEE Transactions on Signal Processing, vol. 51, no. 10, pp. 2602–2612, October 2012.
I. Arasaratnam, S. Haykin, and R. J. Elliott, “Discrete-time nonlinear filtering algorithm using Gauss-Hermite quadrature,” Proceedings of the IEEE, vol. 95, no. 5, pp. 953–977, July 2007.
L. Wang and X. H. Cheng, “Algoritm of Gaussian sum filter based on high-order UKF for dynamic state estimation,” International Journal of Control, Automation, and Systems, vol. 13, no. 3, pp. 652–661, March 2015.
F. Heiss and V. Winschel, “Likelihood approximation by numerical integration on sparse grids,” Journal of Econometrics, vol. 144, no. 1, pp. 62–80, May 2008.
K. Zhang and G. L. Shan, “Nonlinear non-Gaussian filtering algorithm based on Gaussian sum filter and SCKF,” Chinese Journal of Scientific Instrument, vol. 35, no. 11, pp. 2524–2530, November 2014.
L. Zhang, S. Li, E. Z. Zhang, and Q. W. Chen, “Robust measure of non-linearity-based cubature Kalman filter,” IET Science, Measurement & Technology, vol. 11, no. 7, pp. 929–938, September 2017.
I. Arasaratnam and S. Haykin, “Cubature Kalman filters,” IEEE Transactions on Automatic Control, vol. 54, no. 6, pp. 1254–1269, June 2009.
C. Qian, S. Li, Q. W. Chen, J. Guo, and F. Yan, “MASGQF with application to SINS alignment,” IET science, Measurement & Technology, vol. 14, no. 5, pp. 525–535, July 2020.
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This work is supported by the Natural Science Foundation of China (No. 61673217, No. 61333008, and No. 61673219), Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX19_0300) and National defense basic scientific research program(JCKY2019606D001).
Chen Qian is a Ph.D. student in the School of Automation, Nanjing University of Science & Technology. His research interests include adaptive control, Kalman filter, integrated navigation, and nonlinear system filtering.
Chengying Song is a Ph.D. student in the School of Automation, Nanjing University of Science & Technology. Her research interests include integrated navigation, starlight navigation, nonlinear system control, and multi-sensor fusion.
Sheng Li is an Assocciate Professor in School of Automation, Nanjing University of Science & Technology. His research interests include nonlinear system control, robot control, and process control.
Qingwei Chen is a Professor in the School of Automation, Nanjing University of Science & Technology. His research interests include servo system control, fuzzy control, integrated navigation, and nonlinear system control.
Jian Guo is a Professor in the School of Automation, Nanjing University of Science & Technology. His research interests include intelligent control and intelligent system, robot system, and high precision motor control.
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Qian, C., Song, C., Li, S. et al. Algorithm of Gaussian Sum Filter Based on SGQF for Nonlinear Non-Gaussian Models. Int. J. Control Autom. Syst. 19, 2830–2841 (2021). https://doi.org/10.1007/s12555-020-0490-x
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DOI: https://doi.org/10.1007/s12555-020-0490-x