Journal of Applied Mathematics

Weighted Fusion Robust Steady-State Kalman Filters for Multisensor System with Uncertain Noise Variances

Wen-Juan Qi, Peng Zhang, and Zi-Li Deng

Full-text: Open access


A direct approach of designing weighted fusion robust steady-state Kalman filters with uncertain noise variances is presented. Based on the steady-state Kalman filtering theory, using the minimax robust estimation principle and the unbiased linear minimum variance (ULMV) optimal estimation rule, the six robust weighted fusion steady-state Kalman filters are designed based on the worst-case conservative system with the conservative upper bounds of noise variances. The actual filtering error variances of each fuser are guaranteed to have a minimal upper bound for all admissible uncertainties of noise variances. A Lyapunov equation method for robustness analysis is proposed. Their robust accuracy relations are proved. A simulation example verifies their robustness and accuracy relations.

Article information

J. Appl. Math., Volume 2014 (2014), Article ID 369252, 11 pages.

First available in Project Euclid: 2 March 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Qi, Wen-Juan; Zhang, Peng; Deng, Zi-Li. Weighted Fusion Robust Steady-State Kalman Filters for Multisensor System with Uncertain Noise Variances. J. Appl. Math. 2014 (2014), Article ID 369252, 11 pages. doi:10.1155/2014/369252.

Export citation


  • D. L. Hall and J. Llinas, “An introduction to multisensor data fusion,” Proceedings of the IEEE, vol. 85, no. 1, pp. 6–23, 1997.
  • X. R. Li, Y. Zhu, J. Wang, and C. Han, “Optimal linear estimation fusion–-part I: unified fusion rules,” IEEE Transactions on Information Theory, vol. 49, no. 9, pp. 2192–2323, 2003.
  • Z.-L. Deng, Y. Gao, L. Mao, Y. Li, and G. Hao, “New approach to information fusion steady-state Kalman filtering,” Automatica, vol. 41, no. 10, pp. 1695–1707, 2005.
  • S.-L. Sun and Z.-L. Deng, “Multi-sensor optimal information fusion Kalman filter,” Automatica, vol. 40, no. 6, pp. 1017–1023, 2004.
  • Z. L. Deng, P. Zhang, W. J. Qi, Y. Gao, and J. F. Liu, “The accuracy comparison of multisensor covariance intersection fuser and three weighting fusers,” Information Fusion, vol. 14, pp. 177–185, 2013.
  • Q. Gan and C. J. Harris, “Comparison of two measurement fusion method for Kalman filter based multisensor data fusion,” IEEE Transactions on Aerospace and Electronic Systems, vol. 37, no. 1, pp. 273–280, 2001.
  • Y. Gao, C.-J. Ran, X.-J. Sun, and Z.-L. Deng, “Optimal and self-tuning weighted measurement fusion Kalman filters and their asymptotic global optimality,” International Journal of Adaptive Control and Signal Processing, vol. 24, no. 11, pp. 982–1004, 2010.
  • C. J. Ran, Y. S. Hui, L. Gu, and Z. L. Deng, “Correlated measurement fusion steady-state Kalman filtering algorithms and their optimality,” Acta Automatica Sinica, vol. 34, no. 3, pp. 233–239, 2008.
  • Y. Theodor and U. Sharked, “Robust discrete-time minimum-variance filtering,” IEEE Transactions on Signal Processing, vol. 44, no. 2, pp. 181–189, 1996.
  • X. Zhu, Y. C. Soh, and L. Xie, “Design and analysis of discrete-time robust Kalman filters,” Automatica, vol. 38, no. 6, pp. 1069–1077, 2002.
  • X. Lu, L. Xie, H. Zhang, and W. Wang, “Robust Kalman filtering for discrete-time systems with measurement delay,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 54, no. 6, pp. 522–526, 2007.
  • K. Xiong, C. L. Wei, and L. D. Liu, “Robust Kalman filtering for discrete-time nonlinear systems with parameter uncertainties,” Aerospace Science and Technology, vol. 18, no. 1, pp. 15–24, 2012.
  • F. Yang, Z. Wang, and Y. S. Hung, “Robust Kalman filtering for discrete time-varying uncertain systems with multiplicative noises,” IEEE Transactions on Automatic Control, vol. 47, no. 7, pp. 1179–1183, 2002.
  • M. S. Mahmoud, L. Xie, and Y. C. Soh, “Robust Kalman filtering for discrete state-delay systems,” IEE Proceedings: Control Theory and Applications, vol. 147, no. 6, pp. 613–618, 2000.
  • Y. K. Foo and Y. C. Soh, “Robust Kalman filtering for uncertain discrete-time systems with probabilistic parameters bounded within a polytope,” Systems & Control Letters, vol. 57, no. 6, pp. 482–488, 2008.
  • L. Xie, L. Lu, D. Zhang, and H. Zhang, “Improved robust ${H}_{2}$ and ${H}_{\infty }$ filtering for uncertain discrete-time systems,” Automatica, vol. 40, no. 5, pp. 873–880, 2004.
  • F. Wang and V. Balakrishnan, “Robust steady-state filtering for systems with deterministic and stochastic uncertainties,” IEEE Transactions on Signal Processing, vol. 51, no. 10, pp. 2550–2558, 2003.
  • F. Yang and Y. Li, “Robust set-membership filtering for systems with missing measurement: a linear matrix inequality approach,” IET Signal Processing, vol. 6, no. 4, pp. 341–347, 2012.
  • X.-B. Jin, J. Bao, and J.-L. Zhang, “Centralized fusion estimation for uncertain multisensor system based on LMI method,” in Proceedings of the IEEE International Conference on Mechatronics and Automation (ICMA '09), pp. 2383–2387, Changchun, China, August 2009.
  • X. Qu and J. Zhou, “The optimal robust finite-horizon Kalman filtering for multiple sensors with different stochastic failure rates,” Applied Mathematics Letters, vol. 26, no. 1, pp. 80–86, 2013.
  • H. Xi, “The guaranteed estimation performance filter for discrete-time descriptor systems with uncertain noise,” International Journal of Systems Science, vol. 28, no. 1, pp. 113–121, 1997.
  • Z. Dong and Z. You, “Finite-horizon robust Kalman filtering for uncertain discrete time-varying systems with uncertain-covariance white noises,” IEEE Signal Processing Letters, vol. 13, no. 8, pp. 493–496, 2006.
  • L. Shi, K. H. Johansson, and R. M. Murray, “Kalman filtering with uncertain process and measurement noise covariances with application to state estimation in sensor networks,” in Proceedings of the 16th IEEE International Conference on Control Applications (CCA '07), pp. 1031–1036, Singapore, October 2007.
  • W. J. Qi, P. Zhang, and Z. L. Deng, “Robust weighted fusion Kalman filters for multisensor time-varying systems with uncertain noise variances,” Signal Processing, vol. 99, pp. 185–200, 2014.
  • B. Chen, L. Yu, W.-A. Zhang, and A. Liu, “Robust information fusion estimator for multiple delay-tolerant sensors with different failure rates,” IEEE Transactions on Circuits and Systems. I. Regular Papers, vol. 60, no. 2, pp. 401–414, 2013.
  • J. Feng, Z. Wang, and M. Zeng, “Distributed weighted robust Kalman filter fusion for uncertain systems with autocorrelated and cross-correlated noises,” Information Fusion, vol. 14, pp. 78–86, 2013.
  • A. Ahmad, M. Gani, and F. Yang, “Decentralized robust Kalman filtering for uncertain stochastic systems over heterogeneous sensor networks,” Signal Processing, vol. 88, no. 8, pp. 1919–1928, 2008.
  • S. J. Julier and J. K. Uhlmann, “Non-divergent estimation algorithm in the presence of unknown correlations,” in Proceedings of the American Control Conference, vol. 4, pp. 2369–2373, June 1997.
  • S. J. Julier and J. K. Uhlmann, “Simultaneous localization and map building using split covariance intersection,” in Proceedings of the IEEE International Conference on Intelligent Robots and Systems, pp. 1257–1262, 2001.
  • S. J. Julier and J. K. Uhlmann, “General decentralized data fusion with covariance intersection,” in Handbook of Multisensor Data Fusion, M. E. Liggins, D. L. Hall, and J. Llinas, Eds., Theory and Practice, pp. 319–342, CRC Press, 2nd edition, 2009.
  • W. Niehsen and R. B. Gmbh, “Information fusion based on fast covariance intersection filtering,” in Proceedings of the 5th International Conference on Information Fusion, pp. 901–905, 2002.
  • Z. Deng, P. Zhang, W. Qi, J. Liu, and Y. Gao, “Sequential covariance intersection fusion Kalman filter,” Information Sciences, vol. 189, pp. 293–309, 2012.
  • J. Sijs and M. Lazar, “State fusion with unknown correlation: ellipsoidal intersection,” Automatica, vol. 48, no. 8, pp. 1874–1878, 2012.
  • Q. Guo, S. Chen, H. Leung, and S. Liu, “Covariance intersection based image fusion technique with application to pansharpening in remote sensing,” Information Sciences, vol. 180, no. 18, pp. 3434–3443, 2010.
  • S. J. Julier and J. K. Uhlmann, “Using covariance čommentComment on ref. [35?]: This reference is a repetition of [29?]. Please check.intersection for SLAM,” Robotics and Autonomous Systems, vol. 55, no. 1, pp. 3–20, 2007.
  • J. Cesar Bolzani de Campos Ferreira and J. Waldmann, “Covariance intersection-based sensor fusion for sounding rocket tracking and impact area prediction,” Control Engineering Practice, vol. 15, no. 4, pp. 389–409, 2007.
  • S. B. Lazarus, I. Ashokaraj, A. Tsourdos et al., “Vehicle localization using sensors data fusion via integration of covariance intersection and interval analysis,” IEEE Sensors Journal, vol. 7, no. 9, pp. 1302–1314, 2007.
  • T. Kailath, A. H. Sayed, and B. Hassibi, Linear Estimation, Prentice Hall, New York, NY, USA, 2000.
  • E. W. Kamen and J. K. Su, Introduction to Optimal Estimation, Springer, London, UK, 1999.
  • X. Qu, J. Zhou, E. Song, and Y. Zhu, “Minimax robust optimal estimation fusion in distributed multisensor systems with uncertainties,” IEEE Signal Processing Letters, vol. 17, no. 9, pp. 811–814, 2010. \endinput