Volume 3, Issue 6, November 2018, Page: 113-128
Robust Time-Varying Kalman State Estimators with Uncertain Noise Variances
Wenjuan Qi, School of Mechnical and Electronical Engineering, Heilongjiang University, Harbin, China
Zunbing Sheng, School of Mechnical and Electronical Engineering, Heilongjiang University, Harbin, China
Received: Sep. 7, 2018;       Accepted: Sep. 19, 2018;       Published: Jan. 4, 2019
DOI: 10.11648/j.mcs.20180306.11      View  237      Downloads  74
Abstract
This paper addresses the design of robust Kalman estimators (filter, predictor and smoother) for the time-varying system with uncertain noise variances. According to the unbiased linear minimum variance (ULMV) optimal estimation rule, the robust time-varying Kalman estimators are presented. Specially, two robust Kalman smoothing algorithms are presented by the augmented and non-augmented state approaches, respectively. They have the robustness in the sense that their actual estimation error variances are guaranteed to have a minimal upper bound for all admissible uncertainties of noise variances. Their robustness is proved by the Lyapunov equation approach, and their robust accuracy relations are proved. The corresponding steady-state robust Kalman estimators are also presented for the time-invariant system, and the convergence in a realization between the time-varying and steady-state robust Kalman estimators is proved by the dynamic error system analysis (DESA) method and the dynamic variance error system analysis (DVESA) method. A simulation example is given to verify the robustness and robust accuracy relations.
Keywords
Uncertain System, Uncertain Noise Variance, Robust Kalman Filtering, Minimax Estimator, Robust Accuracy, Lyapunov Equation Approach, Convergence
To cite this article
Wenjuan Qi, Zunbing Sheng, Robust Time-Varying Kalman State Estimators with Uncertain Noise Variances, Mathematics and Computer Science. Vol. 3, No. 6, 2018, pp. 113-128. doi: 10.11648/j.mcs.20180306.11
Copyright
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
F. L. Lewis, L. H. Xie, D. Popa, Optimal and Robust Estimation, Second Edition, New York: CRC Press, 2008.
[2]
J. L. Liang, F. Wang, Z. D. Wang, X. H. Liu, Robust Kalman filtering for two-dimensional systems with multiplicative noise and measurement degradations: The finite-horizon case, Automatica 96 (2018) 166-177.
[3]
H. M. Qian, Z. B. Qiu, Y. H. Wu, Robust extended Kalman filtering for nonlinear stochastic systems with random sensor delays, packet dropouts and correlated noises, Aerospace Science and Technology 66 (2017) 249-261.
[4]
J. X. Feng, Z. D. Wang, M. Zeng, Distributed weighted robust Kalman filter fusion for uncertain systems with autocorrelated and cross-correlated noises, Information Fusion 14(1) (2013) 78-86.
[5]
C. S. Yang, Z. B. Yang, Z. L. Deng, Robust weighted state fusion Kalman estimators for networked systems with mixed uncertainties, Information Fusion 45 (2019) 246-265.
[6]
B. Chen, L. Yu, W. A. Zhang, Robust Kalman filtering for uncertain state delay systems with random observation delays and missing measurements, IET Control Theory 5 (17) (2012) 1945-1954.
[7]
Z. J. Shi, L. Y. Zhao, Robust model reference adaptive control based on linear matrix inequality, Aerospace Science and Technology. 66 (2017) 152-159.
[8]
F. Yang, Y. Li, Robust set-membership filtering for systems with missing measurement: a linear matrix inequality approach, IET Signal Process. 6 (4) (2012) 341-347.
[9]
Q. H. Zhang, Adaptive Kalman filter for actuator fault diagnosis, Automatica 93 (2018) 333-342.
[10]
Z. B. Qiu, H. M. Qian, G. Q. Wang, Adaptive robust cubature Kalman filtering for satellite attitude estimation, Chinese Journal of Aeronautics 31(4) (2018) 806-819.
[11]
H. R. Wang, An adaptive Kalman filter estimating process noise covariance, Neurocomputing 223 (2017) 12-17.
[12]
W. Q. Liu, X. M. Wang, Z. L. Deng, Robust Kalman estimators for systems with multiplicative and uncertain variance linearly correlated additive white noises, Aerospace Science and Technology 72 (2018) 230-247.
[13]
X. M. Wang, W. Q. Liu, Z. L. Deng, Robust weighted fusion Kalman estimators for systems with multiplicative noises, missing measurements and uncertain-variance linearly correlated white noises, Aerospace Science and Technology 68 (2017) 331-344.
[14]
W. J. Qi, P. Zhang, Z. L. Deng, Robust weighted fusion Kalman filters for multisensor time-varying systems with uncertain noise variances, Signal Processing 99 (2014) 185-200.
[15]
W. J. Qi, P. Zhang, Z. L. Deng, Robust weighted fusion Kalman predictors with uncertain noise variances, Digital Signal Processing 30(2014) 37-54.
[16]
W. J. Qi, P. Zhang, Z. L. Deng, Robust weighted fusion time-varying Kalman smoothers for multisensor time-varying systems with uncertain noise variances, Information Science 282 (2014) 15-37.
[17]
J. F. Liu, Z. L. Deng, Self-tuning weighted measurement fusion Kalman filter for ARMA signals with colored noise, Applied Mathematics and Information Sciences 6 (1) (2012) 1-7.
[18]
C. J. Ran, G. L. Tao, J. F. Liu, Z. L. Deng, Self-tuning decoupled fusion Kalman predictor and its convergence analysis, IEEE Sensor Journal 9 (12) (2009) 2024-2032.
[19]
S. S. Lan, T. Bullock, Analysis of discrete-time Kalman filtering under incorrect noise variances, IEEE Trans. Automatic Control 38 (12) (1990) 1304-1309.
[20]
T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation, New York, Prentice Hall, 2000.
[21]
C. J. Ran, Z. L. Deng, Self-tuning measurement fusion Kalman predictors and their convergence analysis, International Journal of Systems Science 42 (10) (2011) 1697-1708.
[22]
H. Sriyananda, A simple method for the control of divergence in Kalman filter algorithms, International Journal of Control 16 (6) (1972) 1101-1106.
[23]
E. W. Kamen, J. K. Su, Introduction to optimal estimation, Springer Verlag, London Berlin Heidelberg 1999.
[24]
R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985.
[25]
Z. L. Deng, P. Zhang, W. J. Qi, J. F. Liu, Y. Gao, Sequential covariance intersection fusion Kalman filter, Information Sciences 189 (2012) 293-309.
[26]
L. Ljung, System identification, Theory for the User, Second Edition. Practice Hall PTR, 1999.
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