A New Adaptive Extended Kalman Filter for a Class of Nonlinear Systems

Document Type : Research Paper


Department of Mechanical Engineering, Palestine Polytechnic University, Hebron, Palestine


This paper proposes a new adaptive extended Kalman filter (AEKF) for a class of nonlinear systems perturbed by noise which is not necessarily additive. The proposed filter is adaptive against the uncertainty in the process and measurement noise covariances. This is accomplished by deriving two recursive updating rules for the noise covariances, these rules are easy to implement and reduce the number of noise parameters that need to be tuned in the extended Kalman filter (EKF). Furthermore, the AEKF updates the noise covariances to enhance filter stability. Most importantly, in the worst case, the AEKF converges to the conventional EKF. The AEKF performance is determined based on the mean square error (MSE) performance measure and the stability is proven. The results illustrate that the proposed AEKF has a dramatic improved performance over the conventional EKF, the estimates are more stable with less noise.


Main Subjects

[1] Z. Zhou, J. Wu, Y. Li, C. Fu, and H. Fourati, Critical issues on Kalman filter with colored and correlated system noises, Asian Journal of Control, 19(6), 2017, 1905-1919.
[2] C. Fraser and S. Ulrich, An Adaptive Kalman Filter for Spacecraft Formation Navigation using Maximum Likelihood Estimation with Intrinsic Smoothing, in 2018 Annual American Control Conference (ACC), 2018, 5843-5848.
[3] X. Tong, Z. Li, G. Han, N. Liu, Y. Su, J. Ning, et al., Adaptive EKF Based on HMM Recognizer for Attitude Estimation Using MEMS MARG Sensors, IEEE Sensors Journal, 18(8), 2018, 3299-3310.
[4] Y. Xi, X. Zhang, Z. Li, X. Zeng, X. Tang, Y. Cui, et al., Double-ended travelling-wave fault location based on residual analysis using an adaptive EKF, IET Signal Processing, 12(8), 2018, 1000-1008.
[5] M. S. Grewal and A. P. Andrews, Kalman Filtering: Theory and Practice Using MATLAB, 2nd ed., John Wiley & Sons. New York, USA, 2001.
[6] S. Ulrich and J. Z. Sasiadek, Extended Kalman filtering for flexible joint space robot control, in American Control Conference (ACC), 2011, 1021-1026.
[7] V. A. Bavdekar, A. P. Deshpande, and S. C. Patwardhan, Identification of process and measurement noise covariance for state and parameter estimation using extended Kalman filter, Journal of Process Control, 21(4), 2011, 585-601.
[8] R. Jassemi-Zargani and D. Necsulescu, Extended Kalman filter-based sensor fusion for operational space control of a robot arm, IEEE Transactions on Instrumentation and Measurement, 51(6), 2002, 1279-1282.
[9] E. Hedberg, J. Norén, M. Norrlöf, and S. Gunnarsson, Industrial Robot Tool Position Estimation using Inertial Measurements in a Complementary Filter and an EKF, IFAC-PapersOnLine, 50(1), 2017, 12748-12752.
[10] U. Bussi, V. Mazzone, and D. Oliva, Control strategies analysis using EKF applied to a mobile robot, in Workshop on Information Processing and Control (RPIC), 2017, 1-6.
[11] Y. Xu, Y. S. Shmaliy, C. K. Ahn, G. Tian, and X. Chen, Robust and accurate UWB-based indoor robot localisation using integrated EKF/EFIR filtering, IET Radar, Sonar & Navigation, 12(7), 2018, 750-756.
[12] D. Simon, Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches,Wiley & Sons. Hoboken, New Jersey, USA, 2006.
[13] P. S. Maybeck, Stochastic models, estimation and control,Academic Press. New York, USA, 1982.
[14] H. Heffes, The effect of erroneous models on the Kalman filter response, IEEE Transactions on Automatic Control, 11(3), 1966, 541-543.
[15] R. J. Fitzgerald, Divergence of the Kalman filter, IEEE Transactions on Automatic Control, 16(6), 1971, 736-747.
[16] A. Mohamed and K. Schwarz, Adaptive Kalman filtering for INS/GPS, Journal of Geodesy, 73(4), 1999, 193-203.
[17] K. Myers and B. D. Tapley, Adaptive sequential estimation with unknown noise statistics, IEEE Transactions on Automatic Control, 21(4), 1976, 520-523.
[18] W. Ding, J. Wang, C. Rizos, and D. Kinlyside, Improving adaptive Kalman estimation in GPS/INS integration, Journal of navigation, 60(3), 2007, 517.
[19] E. Shi, An improved real-time adaptive Kalman filter for low-cost integrated GPS/INS navigation, in IEEE 2012 International Conference on Measurement, Information and Control (MIC) Harbin, 2012, 1093-1098.
[20] C. Hide, T. Moore, and M. Smith, Adaptive Kalman filtering for low-cost INS/GPS, Journal of Navigation, 56(1), 2003, 143-152.
[21] R. Mehra, On the identification of variances and adaptive Kalman filtering, IEEE Transactions on Automatic Control, 15(2), 1970, 175-184.
[22] P. R. Bélanger, Estimation of noise covariance matrices for a linear time-varying stochastic process, Automatica, 10(3), 1974, 267-275.
[23] M. Oussalah and J. D. Schutter, Adaptive kalman filter for noise identification, in Proceedings of the international Seminar on Modal Analysis, Kissimmee, Florida, 2001, 1225-1232.
[24] B. J. Odelson, M. R. Rajamani, and J. B. Rawlings, A new autocovariance least-squares method for estimating noise covariances, Automatica, 42(2), 2006, 303-308.
[25] H. Raghavan, A. K. Tangirala, R. Bhushan Gopaluni, and S. L. Shah, Identification of chemical processes with irregular output sampling, Control Engineering Practice, 14(5), 2006, 467-480.
[26] B. J. Odelson, A. Lutz, and J. B. Rawlings, The autocovariance least-squares method for estimating covariances: application to model-based control of chemical reactors, IEEE Transactions on Control Systems Technology, 14(3), 2006, 532-540.
[27] X. Wang, Vehicle health monitoring system using multiple-model adaptive estimation, MSc Thesis, Electrical Engineering, University of Hawaii at Manoa, Manoa, 2003.
[28] X. Rong Li and Z. Youmin, Multiple-model estimation with variable structure part V: Likely-model set algorithm, IEEE Transactions on Aerospace and Electronic Systems, 36(2), 2000, 448-466.
[29] M. Karasalo and X. Hu, An optimization approach to adaptive Kalman filtering, Automatica, 8(47), 2011, 1785-1793.
[30] Y. Yang and W. Gao, An Optimal Adaptive Kalman Filter, Journal of Geodesy, 80(4), 2006, 177-183.
[31] I. Hashlamon and K. Erbatur, An improved real-time adaptive Kalman filter with recursive noise covariance updating rules, Turkish Journal of Electrical Engineering & Computer Sciences, 24(2), 2016, 524-540.
[32] C. Biçer, E. K. Babacan, and L. Özbek, Stability of the adaptive fading extended Kalman filter with the matrix forgetting factor, Turkish Journal of Electrical Engineering & Computer Sciences, 20(5), 2012, 819-833.
[33] E. K. Babacan, L. Ozbek, and M. Efe, Stability of the extended Kalman filter when the states are constrained, IEEE Transactions on Automatic Control, 53(11), 2008, 2707-2711.
[34] H. K. Khalil, Nonlinear Systems, 3rd ed.,Prentice Hall. New Jersy, 2000.
[35] R.M. Murray, Z. Li , and S. Satry, A mathematical introduction to robotic manipulation, CRC Press, 1994.
[36] L. Yinan, Research on Joint Orientation Algorithm of Multi Sensor and Distributed Localization based on Quaternion EKF, Revista de la Facultad de Ingeniería, 32(12), 2017, 341-347.
[37] K. Feng, J. Li, X. Zhang, C. Shen, Y. Bi, T. Zheng, et al., A New Quaternion-Based Kalman Filter for Real-Time Attitude Estimation Using the Two-Step Geometrically-Intuitive Correction Algorithm, Sensors, 17(9), 2017, 2146.
[38] I. Hashlamon and K. Erbatur, Experimental verification of an orientation estimation technique for autonomous robotic platforms, Master Thesis, Sabanci University, Istanbul, 2010.
[39] A. Kim and M. F. Golnaraghi, A quaternion-based orientation estimation algorithm using an inertial measurement unit, IEEE. New York, 2004.
[40] P. Bauer and J. Bokor, Development and hardware-in-the-loop testing of an Extended Kalman Filter for attitude estimation, in 11th International Symposium on Computational Intelligence and Informatics (CINTI) 2010, 57-62.
[41] D. Roetenberg, "Inertial and magnetic sensing of human motion," PhD, University of Twente, Enschede, NL, 2006.
[42] E. J. Lefferts, F. L. Markley, and M. D. Shuster, Kalman Filtering for Spacecraft Attitude Estimation, Journal of Guidance, Control, and Dynamics, 5(5), 1982, 417-429.
[43] D. Simon, Kalman filtering with state constraints: a survey of linear and nonlinear algorithms, Control Theory & Applications, IET, 8(4), 2009, 1303-1318.
[44] A. J. Calise, Enforcing an Algebraic Constraint in Extended Kalman Filter Design, Journal of Guidance, Control, and Dynamics, 40(9), 2017, 2229-2236.
[45] V. Bonnet, R. Dumas, A. Cappozzo, V. Joukov, G. Daune, D. Kulić, et al., A constrained extended Kalman filter for the optimal estimate of kinematics and kinetics of a sagittal symmetric exercise, Journal of Biomechanics, 62, 2017, 140-147.
[46] V. Mahboub and D. Mohammadi, A Constrained Total Extended Kalman Filter for Integrated Navigation, Journal of Navigation, 71(4), 2018, 971-988.
[47] I. Hashlamon, A constrained quaternion extended Kalman filter, in Sixth Palestinian Conference on Modern Trends in Mathematics and Physics, Palestine, 2018.