Adaptive Kalman Filter for Noise Estimation and Identification with Bayesian Approach

Authors: Farhad Asadi, S. Hossein Sadati

Abstract:

Bayesian approach can be used for parameter identification and extraction in state space models and its ability for analyzing sequence of data in dynamical system is proved in different literatures. In this paper, adaptive Kalman filter with Bayesian approach for identification of variances in measurement parameter noise is developed. Next, it is applied for estimation of the dynamical state and measurement data in discrete linear dynamical system. This algorithm at each step time estimates noise variance in measurement noise and state of system with Kalman filter. Next, approximation is designed at each step separately and consequently sufficient statistics of the state and noise variances are computed with a fixed-point iteration of an adaptive Kalman filter. Different simulations are applied for showing the influence of noise variance in measurement data on algorithm. Firstly, the effect of noise variance and its distribution on detection and identification performance is simulated in Kalman filter without Bayesian formulation. Then, simulation is applied to adaptive Kalman filter with the ability of noise variance tracking in measurement data. In these simulations, the influence of noise distribution of measurement data in each step is estimated, and true variance of data is obtained by algorithm and is compared in different scenarios. Afterwards, one typical modeling of nonlinear state space model with inducing noise measurement is simulated by this approach. Finally, the performance and the important limitations of this algorithm in these simulations are explained. 

Keywords: adaptive filtering, Bayesian approach Kalman filtering approach, variance tracking

References:

[1] Kim Y., Bang H. “Introduction to Kalman filter and its applications. Introduction and Implementations of the Kalman Filter”, F. Govaers, Ed. IntechOpen. 2019 May 22.
[2] Ribeiro M. I. “Kalman and extended kalman filters: Concept, derivation and properties”. Institute for Systems and Robotics. 2004 Feb; 43:46.
[3] Li X. R., Bar-Shalom Y. “A recursive multiple model approach to noise identification”. IEEE Transactions on Aerospace and Electronic Systems. 1994 Jul; 30(3):671-84.
[4] Kim T., Park T. H. “Extended Kalman filter (EKF) design for vehicle position tracking using reliability function of radar and lidar”. Sensors. 2020 Jan; 20(15):4126.
[5] Asadi F., Mollakazemi M. J., Atyabi S. A., Uzelac I. L., Ghaffari A. Cardiac arrhythmia recognition with robust discrete wavelet-based and geometrical feature extraction via classifiers of SVM and MLP-BP and PNN neural networks. In2015 Computing in Cardiology Conference (CinC) 2015 Sep 6 (pp. 933-936). IEEE.
[6] Asadi F., Mollakazemi M. J., Ghiasi S., Sadati S. H. Enhancement of life-threatening arrhythmia discrimination in the intensive care unit with morphological features and interval feature extraction via random forest classifier. In2016 Computing in Cardiology Conference (CinC) 2016 Sep 11 (pp. 57-60). IEEE.
[7] Asadi F., Mollakazemi M. J., Uzelac I. L., Moosavian S. A. A novel method for arterial blood pressure pulse detection based on a new coupling strategy and discrete wavelet transform. In2015 Computing in Cardiology Conference (CinC) 2015 Sep 6 (pp. 1081-1084). IEEE.
[8] Almagbile A., Wang J., Ding W. “evaluating the performances of adaptive Kalman filter methods in GPS/INS integration”. Journal of Global Positioning Systems. 2010 Jun; 9(1):33-40.
[9] Honarmand-Shazilehei F., Pariz N., Sistani M. B. “Sensor fault detection in a class of nonlinear systems using modal Kalman filter”. ISA transactions. 2020 Dec 1; 107:214-23.
[10] Arikan O., Belge M., Cetin A. E., Erzin E. “Adaptive filtering approaches for non-Gaussian stable processes”. In1995 International Conference on Acoustics, Speech, and Signal Processing 1995 May 9 (Vol. 2, pp. 1400-1403). IEEE.
[11] Oussalah M., De Schutter J. “Adaptive Kalman filter for noise identification”. In Proceedings of the international Seminar on Modal Analysis 2001 (Vol. 3, pp. 1225-1232). KU Leuven; 1998.
[12] Mehra R. “Approaches to adaptive filtering”. IEEE Transactions on automatic control. 1972 Oct; 17(5):693-8.
[13] Zhang L., Sidoti D., Bienkowski A., Pattipati K. R., Bar-Shalom Y., Kleinman D. L. “On the identification of noise covariances and adaptive Kalman Filtering: A new look at a 50 year-old problem”. IEEE Access. 2020 Mar 23; 8:59362-88.
[14] Storvik G. “Particle filters for state-space models with the presence of unknown static parameters”. IEEE Transactions on signal Processing. 2002 Aug 7; 50(2):281-9.
[15] Djurić P. M., Miguez J. “Sequential particle filtering in the presence of additive Gaussian noise with unknown parameters”. In2002 IEEE International Conference on Acoustics, Speech, and Signal Processing 2002 May 13 (Vol. 2, pp. II-1621). IEEE.
[16] Arnold A. “Using Monte Carlo Particle Methods to Estimate and Quantify Uncertainty in Periodic Parameters”. Advances in Mathematical Sciences. 2020:213-26.
[17] Ma Z. S., Li L., Ding Q. “Multivariate recursive Bayesian linear regression and its applications to output-only identification of time-varying mechanical systems”. Journal of Vibration and Control. 2021 Jun; 27(11-12):1395-406.
[18] Asadi F., Khorram M., Moosavian S. A. CPG-based gait transition of a quadruped robot. In2015 3rd RSI International Conference on Robotics and Mechatronics (ICROM) 2015 Oct 7 (pp. 210-215). IEEE.
[19] Asadi F., Khorram M., Moosavian S. A. CPG-based gait planning of a quadruped robot for crossing obstacles. In2015 3rd RSI International Conference on Robotics and Mechatronics (ICROM) 2015 Oct 7 (pp. 216-222). IEEE.
[20] Beal M. J., Ghahramani Z. “The variational Kalman smoother”. Gatsby Computational Neuroscience Unit, University College London, Tech. Rep. GCNU TR. 2001 Apr 6; 3:2001.
[21] Zhu H., Zhang G., Li Y., Leung H. “An adaptive Kalman filter with inaccurate noise covariances in the presence of outliers”. IEEE Transactions on Automatic Control. 2021 Feb 2.
[22] Valpola H., Harva M., Karhunen J. “Hierarchical models of variance sources”. Signal Processing. 2004 Feb 1; 84(2):267-82.
[23] Sarkka S., Nummenmaa A. “Recursive noise adaptive Kalman filtering by variational Bayesian approximations”. IEEE Transactions on Automatic control. 2009 Feb 27; 54(3):596-600.