Abstract
Since the innovation of the ubiquitous Kalman filter more than five decades back it is well known that to obtain the best possible estimates the tuning of its statistics \({\mathbf{X}}_{\mathbf{0}}\), \({\mathbf{P}}_{\mathbf{0}}\), \(\Theta \), R and Q namely initial state and covariance, unknown parameters, and the measurement and state noise covariances is very crucial. The manual and other approaches have not matured to a routine approach applicable for any general problem. The present reference recursive recipe (RRR) utilizes the prior, posterior, and smoothed state estimates as well as their covariances to balance the state and measurement equations and thus form generalized cost functions. The filter covariance at the end of each pass is heuristically scaled up by the number of data points and further trimmed to provide the \({\mathbf{P}}_{\mathbf{0}}\) for subsequent passes. The importance of \({\mathbf{P}}_{\mathbf{0}}\) as the probability matching prior between the frequentist approach via optimization and the Bayesian approach of the Kalman filter is stressed. A simultaneous and proper choice for Q and R based on the filter sample statistics and other covariances leads to a stable filter operation after a few iterations. A typical simulation study of a spring, mass and damper system with a weak nonlinear spring constant by RRR shows it to be better than earlier techniques. Part-2 of the paper further consolidates the present approach based on an analysis of real flight test data.
References
Kalman R E 1960 A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. 82 (Series D): 35–45
Kalman R E, Bucy R S 1961 New results in linear filtering and prediction theory. J. Basic Eng. 83(1): 95–108
Klein V and Morelli E A 2006 Aircraft system identification: theory and practice. In: AIAA Edu. Series
Bar-Shalom Y, Rong Li X and Kirubarajan T 2000 Estimation with applications to tracking and navigation, theory, algorithm and software. John Wiley and Sons. Inc
Ananthasayanam M R, Anilkumar A K and Subba Rao P V 2006 New approach for the evolution and expansion of space debris Scenario. J. Spacecraft Rockets 43(6): 1271–1282
Grewal M S, Lawrence R W and Andrews A P 2007 Global positioning systems, inertial Navigation, and Integration, 2nd Edition, John Wiley & Sons, Inc.
Kleusberg A and Teunissen P J G 1996 GPS for geodesy, 1st edition, Springer
Fruhwirth R, Regier M, Bock R K, Grote H and Notz D 2000 Data analysis techniques for high-energy physics. In:Cambridge monographs on particle physics, nuclear physics and cosmology
Federer W T and Murthy B R 1998 Kalman filter bibliography : agriculture, biology, and medicine. Technical Report BU-1436-M, Department of Biometrics, Cornell University
Visser H and Molenaar J 1988 Kalman filter analysis in dendroclimatology. Biometrics 44: 929–940
Wells C 1996 The Kalman filter in finance. Springer-Science+Business Media, BV
Costagli M and Kuruoglu E E 2007 Image separation using particle filters. Digital Signal Process. 17: 935–946
Evensen G 2009 Data assimilation: the ensemble Kalman Filter. Springer Verlag .
Brown R and Hwang P 2012 Introduction to random signals and applied Kalman filtering, with MATLAB exercises. John Wiley and Sons, Inc 4th edition
Kailath T 1970 An innovation approach to detection and estimation theory. Proc. lEEE 58(5): 680–695
Mehrotra K Mahapatra P R 1997 A jerk model for tracking highly maneuvering targets. IEEE Trans. Aerospace Electron. Syst. 33(4)
Maybeck P S 1979 Stochastic models, estimation, and control. Volume 1, New York: Academic Press
Candy J V 1986 Signal processing the Model based approach. Mcgraw Hill
Gemson R M 0 1991 Estimation of aircraft aerodynamic derivatives accounting for measurement and process noise By EKF through adaptive filter tuning. PhD Thesis, Department of Aerospace Engineering, IISc, Bangalore
Bohlin T 1976 Four cases of identification of changing systems. In: System identification: advances and case studies, 1st edition. Academic Press
Gemson R M O and Ananthasayanam M R 1998 Importance of initial state covariance matrix for the parameter estimation using adaptive extended Kalman filter. AIAA-98-4153, pp. 94–104
Ljung L 1979 Asymptotic behaviour of the EKF as a parameter estimator for linear systems. IEEE Trans. Autom. Control AC 24: 36–50
Ljungquist D and Balchen J G 1994 Recursive prediction error methods for online estimation in nonlinear state space models. Model. Identifi. Control 15(2): 109–121
Alspach D 1974 A parallel filtering algorithm for linear systems with unknown time varying noise statistics. IEEE Trans. Autom. Control 19(5): 552–556
Kashyap R 1970 Maximum likelihood identification of stochastic linear systems. IEEE Trans. Auto. Control 15(1): 25–34
Shumway R H and Stoffer D S 2000 Time series analysis and its applications. Springer, Verlag, NY
Bavdekar V A, Deshpande A P and Patwardhan S C 2011 Identification of process and measurement noise covariance for state and parameter estimation using extended Kalman filter. J. Process Control 21: 585–601
Myers K A and Tapley B D 1976 Adaptive sequential estimation with unknown noise statistics. IEEE Trans. Autom. Control 21: 520–525
Mohamed A H and Schwarz K P 1999 Adaptive Kalman filtering for INS/GPS. J. Geodesy 73(4): 193–203
Carew B and Belanger P R 1973 Identification of optimum filter steady state gain for systems with unknown noise covariances. IEEE Trans. Auto. Control 18(6): 582–587
Mehra R K 1970 On the identification of variances and adaptive Kalman filtering. IEEE Trans. Automat. Control 15(2): 175–184
Mehra R 1972 Approaches to adaptive filtering. IEEE Trans. Autom. Control 17: 903–908
Belanger P R 1974 Estimation of noise covariances for a linear time-varying stochastic process. Automatica 10(3): 267–275
Neethling C and Young P 1974 Comments on Identification of optimum filter steady state gain for systems with unknown noise covariances. IEEE Trans. Autom. Control 19(5): 623–625
Odelson B J, Lutz A and Rawlings J B 2006 The autocovariance-least squares method for estimating covariances: application to model-based control of chemical reactors. IEEE Trans. Control Syst. Technol. 14(3): 532–540
Valappil J and Georgakis C 2000 Systematic estimation of state noise statistics for extended Kalman filters. AIChe J. 46(2): 292–308
Manika S, Bhaswati G and Ratna G 2014 Robustness and sensitivity metrics for tuning the extended Kalman filter. IEEE Trans. Instrum. Meas. 63(4): 964–971
Powell T D 2002 Automated tuning of an extended Kalman filter using the downhill simplex algorithm. J. Guidance Control Dyn. 25(5): 901–908
Oshman Y and Shaviv I 2000 Optimal tuning of a Kalman filter using genetic algorithm. AIAA Paper 2000-4558
Anilkumar A K 2000 Application of controlled random search optimisation technique in MMLE with process noise. MSc Thesis, Department of Aerospace Engineering, IISc, Bangalore
Lau T and Lin K 2011 Evolutionary tuning of sigma-point Kalman filters. IEEE International Conference on Robotics and Automation (ICRA), pp. 771–776
Shyam M M, Naren Naik, Gemson R M O and Ananthasayanam M R 2015 Introduction to the Kalman filter and tuning its statistics for near optimal estimates and Cramer Rao bound. TR/EE2015/401, IIT Kanpur, http://arxiv.org/abs/1503.04313
Rauch H E, Tung F and Striebel C T 1965 Maximum likelihood estimates of linear dynamic systems. AIAA J. 3(8): 1445–1450
Shyam M M 2014 An iterative tuning strategy for achieving Cramer Rao bound using extended Kalman filter for a parameter estimation problem. MTech Thesis, IIT Kanpur
Ananthasayanam M R, Suresh H S and Muralidharan M R 2001 GUI based software for teaching parameter estimation technique using MMLE. Report 2001 FM 1, IISc
Samaniego F J 2011 A Comparison of the Bayesian and frequentist approaches to estimation. Springer Science+Business Media, LLC
Acknowledgements
Our grateful thanks are due to Profs R M Vasu (Department of Instrumentation and Applied Physics), D Roy (Department of Civil Engineering) and M R Muralidharan (Supercomputer Education and Research Centre) for help in a number of ways without which this work would just not have been possible at all and also for providing computational facilities at the IISc, Bangalore.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ananthasayanam, M.R., Mohan, M.S., Naik, N. et al. A heuristic reference recursive recipe for adaptively tuning the Kalman filter statistics part-1: formulation and simulation studies. Sādhanā 41, 1473–1490 (2016). https://doi.org/10.1007/s12046-016-0562-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12046-016-0562-z