Abstract
To improve the estimation accuracy of the Kalman filter in the scenario of random measurement delay and non-Gaussian process and measurement noises, a new variational Bayesian (VB)-based Kalman filter is proposed in this paper. First, the state expansion method and Bernoulli random variable (BRV) are utilized to characterize random measurement delay. Second, the one-step predicted probability density function (PDF) and measurement noise vectors are modeled as Student’s t (ST) distributions. Third, the likelihood function of two ST distributions is converted from a weighted sum to an exponential product to establish a hierarchical Gaussian state space model (HGSSM). Finally, the system state, BRV and intermediate random variables (IRV) are simultaneously estimated using the variational Bayesian (VB) method. Simulation experiment results indicate that the proposed filter has superior estimation performance to current filters to address the filtering problem of random measurement delay and non-Gaussian process and measurement noises.
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References
D. Simon, Optimal State Estimation: Kaiman, H∞, and Nonlinear Approaches, John Wiley&Sons, 2006.
A. Ferrero, H. V. Jetti, and S. Salicone, “The possibilistic Kalman filter: Definition and comparison with the available methods,” IEEE Transactions on Instrumentation and Measurement, vol. 70, pp. 1–11, July 2021.
C. Hajiyev and D. C. Guler, “Review on gyroless attitude determination methods for small satellites,” Progress in Aerospace Sciences, vol. 90, pp. 54–66, April 2017.
G. Y. Kulikov and M. V. Kulikova, “The accurate continuous-discrete extended Kalman filter for radar tracking,” IEEE Transactions on Signal Processing, vol. 64, no. 4, pp. 948–958, February 2016.
F. Zhao, C. Chen, W. He, and S. S. Ge, “A filtering approach based on MMAE for a SINS/CNS integrated navigation system,” IEEE/CAA Journal of Automatica Sinica, vol. 5, no. 6, pp. 1113–1120, November 2018.
T. Cui, F. Ding, X. B. Jin, A. Alsaedi, and T. Hayat, “Joint multi-innovation recursive extended least squares parameter and state estimation for a class of state-space systems,” International Journal of Control, Automation, and Systems, vol. 18, no. 6, pp. 1412–1424, June 2020.
L. Xu, W. L. Xiong, A. Alsaedi, and T. Hayat, “Hierarchical parameter estimation for the frequency response based on the dynamical window data,” International Journal of Control, Automation, and Systems, vol. 16, no. 4, pp. 1756–1764, August 2018.
J. Pan, X. Jiang, X. K. Wan, and W. F. Ding, “A filtering based multi-innovation extended stochastic gradient algorithm for multivariable control systems,” International Journal of Control, Automation, and Systems, vol. 15, no. 3, pp. 1189–1197, June 2017.
J. Pan, W. Li, and H. P. Zhang, “Control algorithms of magnetic suspension systems based on the improved double exponential reaching law of sliding mode control,” International Journal of Control, Automation, and Systems, vol. 16, no. 6, pp. 2878–2887, December 2018.
M. H. Li and X. M. Liu, “Maximum likelihood least squares based iterative estimation for a class of bilinear systems using the data filtering technique,” International Journal of Control, Automation, and Systems, vol. 18, no. 6, pp. 1581–1592, June 2020.
F. Ding, L. Lv, J. Pan, X. K. Wan, and X. B. Jin, “Two-stage gradient-based iterative estimation methods for controlled autoregressive systems using the measurement data,” International Journal of Control, Automation, and Systems, vol. 18, no. 4, pp. 886–896, April 2020.
L. J. Wan, F. Ding, X. M. Liu, and C. P. Chen, “A new iterative least squares parameter estimation approach for equation-error autoregressive systems,” International Journal of Control, Automation, and Systems, vol. 18, no. 3, pp. 780–790, March 2020.
D. Wang, Z. D. Wang, B. Shen, and F. E. Alsaadi, “Security-guaranteed filtering for discrete-time stochastic delayed systems with randomly occurring sensor saturations and deception attacks,” International Journal of Robust and Nonlinear Control, vol. 27, no. 7, pp. 1194–1208, May 2017.
J. Y. Mao, D. R. Ding, Y. Song, Y. R. Liu, and F. E. Alsaadi, “Event-based recursive filtering for time-delayed stochastic nonlinear systems with missing measurements,” Signal Processing, vol. 134, pp. 158–165, May 2017.
X. X. Wang, Y. Liang, Q. Pan, and C. H. Zhao, “Gaussian filter for nonlinear systems with one-step randomly delayed measurements,” Automatica, vol. 49, no. 4, pp. 976–986, April 2013.
X. Wang, Q. Pan, Y. Liang, and F. Yang, “Gaussian smoothers for nonlinear systems with one-step randomly delayed measurements,” IEEE Transactions on Automatic Control, vol. 58, no. 7, pp. 1828–1835, July 2013.
Z. Wang, Y. Huang, Y. Zhang, G. Jia, and J. Chambers, “An improved Kalman filter with adaptive estimate of latency probability,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 67, no. 10, pp. 2259–2263, November 2020.
A. Vasilijevic, B. Borovic, and Z. Vukic, “Underwater vehicle localization with complementary filter: Performance analysis in the shallow water environment,” Journal of Intelligent & Robotic Systems, vol. 68, no. 3, pp. 373–386, December 2012.
M. Roth, E. Özkan, and F. Gustafsson, “A Student’s t filter for heavy tailed process and measurement noise,” Proc. of IEEE International Conference on Acoustics, Speech and Signal Processing, Vancouver, pp. 5770–5774, May 2013.
Y. Huang, Y. Zhang, Y. Zhao, and J. A. Chambers, “A novel robust Gaussian-Student’s t mixture distribution based Kalman filter,” IEEE Transactions on Signal Processing, vol. 67, no. 13, pp. 3606–3620, May 2019.
B. Chen and J. C. Principe, “Maximum correntropy estimation is a smoothed MAP estimation,” IEEE Signal Processing Letters, vol. 19, no. 8, pp. 491–494, June 2012.
M. V. Kulikova, “Square-root approach for Chandrasekhar-based maximum correntropy Kalman filtering,” IEEE Signal Processing Letters, vol. 26, no. 12, pp. 1803–1807, October 2019.
X. Liu, Z. G. Ren, H. Q. Lyu, Z. H. Jiang, P. J. Ren, and B. D. Chen, “Linear and nonlinear regression-based maximum correntropy extended Kalman filtering,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 51, no. 5, pp. 3093–3102, May 2021.
Z. Li and S. Guan, “Diffusion normalized Huber adaptive filtering algorithm,” Journal of the Franklin Institute-Engineering and Applied Mathematics, vol. 355, no. 8, pp. 3812–3825, May 2018.
Z. B. Qiu, Y. L. Huang, and H. M. Qian, “Adaptive robust nonlinear filtering for spacecraft attitude estimation based on additive quaternion,” IEEE Transactions on Instrumentation and Measurement, vol. 69, no. 1, pp. 100–108, January 2020.
Y. Huang, Y. Zhang, N. Li, Z. Wu, and J. A. Chambers, “A novel robust Student’s t-based Kalman filter,” IEEE Transactions on Aerospace and Electronic Systems, vol. 53, no. 3, pp. 1545–1554, January 2017.
Y. Huang, Y. Zhang, B. Xu, Z. Wu, and J. A. Chambers, “A new adaptive extended Kalman filter for cooperative localization,” IEEE Transactions on Aerospace and Electronic Systems, vol. 54, no. 1, pp. 353–368, September 2018.
Y. Huang, G. Jia, B. Chen, and Y. Zhang, “A new robust Kalman filter with adaptive estimate of time-varying measurement bias,” IEEE Signal Processing Letters, vol. 27, pp. 700–704, March 2020.
Y. Huang, Y. Zhang, Z. Wu, N. Li, and J. Chambers, “A novel adaptive Kalman filter with inaccurate process and measurement noise covariance matrices,” IEEE Transactions on Automatic Control, vol. 63, no. 2, pp. 594–601, September 2018.
D. G. Tzikas, A. C. Likas, and N. P. Galatsanos, “The variational approximation for Bayesian inference,” IEEE Signal Processing Magazine, vol. 25, no. 6, pp. 131–146, December 2008.
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This work was supported by the China Scholarship Council (CSC, No. 202006680080), and the National Natural Science Foundation of China (61573113).
Chenghao Shan received his B.E. degree in automation from Harbin University of Science and Technology, Harbin, China, in 2017. He is currently pursuing the Ph.D. degree in control science and engineering with Harbin Engineering University, Harbin, China. Since 2021, he has also been a joint Ph.D. student funded by the China Scholarship Council, at Department of Electrical and Computer Engineering, National University of Singapore (NUS), His research interests include statistical signal processing, variational Bayesian filtering methods, information fusion and their applications in target tracking and integrated navigation.
Weidong Zhou received his B.E. and M.E. degrees in automatic control from Harbin Institute of Technology, in 1988, and Harbin University of Science and Technology, in 1991, respectively. He received a Ph.D. degree in navigation from Harbin Engineering University, China, in 2006. Since 2005, he has been a full professor at Department of Intelligent Systems Science and Engineering, Harbin Engineering University, China. He is an editorial board member of National Marine Boat Standardization Technical Committee, and a member of the Northeast Regional Branch of the Chinese Society of Inertial Technology. His research interests include automatic control theory, integrated navigation, estimation theory, and multi-sensor data fusion.
Hanyu Shan received her B.E. degree in electronic information engineering from Harbin Engineering University, Harbin, China, in 2020. She is currently pursuing a Ph.D. degree in information and communication engineering with Harbin Engineering University, Harbin, China. Her research interests include digital beamforming, array signal processing, and spatial filtering theory.
Lu Liu received her B.E. degree in automation from Henan University, Henan, China, in 2012, and her M.E. and Ph.D. degrees in control science and engineering from Harbin Engineering University, Harbin, China, in 2015 and 2020, respectively. She currently works at Beijing Institute of Control and Electronic Technology. Her research interests include nonlinear state estimation and information fusion and their applications in target tracking.
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Shan, C., Zhou, W., Shan, H. et al. A New Variational Bayesian-based Kalman Filter with Random Measurement Delay and Non-Gaussian Noises. Int. J. Control Autom. Syst. 20, 2594–2605 (2022). https://doi.org/10.1007/s12555-021-0467-4
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DOI: https://doi.org/10.1007/s12555-021-0467-4