Abstract
We propose a new approach for solving the filtering problem in linear systems based on incomplete measurements, where the characteristics of the dynamic noise are not known exactly, and measurements may contain anomalous non-Gaussian errors. The proposed algorithm is based on the idea of using the adaptive Kalman filter and the generalized least absolute deviations method jointly. With numerical modeling, we show that, compared to the classical optimal linear filtering method, our solution has lower sensitivity to short-term outliers in measurements and provides a quick adjustment of the parameters of the system dynamics. The proposed algorithm can be used to solve onboard navigation and tracking problems on aircrafts. To implement the method of least absolute deviations, we use an efficient L1-optimization algorithm.
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References
Liptser, R. Sh & Shiryaev, A. N. Statistika sluchainykh protsessov (Statistics of Random Processes). (Nauka, Moscow, 1974).
Sinitsyn, I. N. Filatry Kalmana i Pugacheva (Kalman and Pugachev Filters). (Logos, Moscow, 2006).
Miller, A.B. and Miller, B.M.Tracking of the UAV Trajectory on the Basis of Bearing-Only Observations, 53rd IEEE Conf. on Decision and Control, Los Angeles, 2014, pp. 4178–4184.
Salychev, O. S. Mems-based Inertial Navigation: Expectations and Reality. (Bauman Moscow State Technical University, Moscow, 2012).
Vremeenko, K. K., Zheltov, S. Yu, Kim, N. V., Kozorez, D. A., Krasil’shchikov, M. N., Sebryakov, G. G., Sypalo, K. I. & Chernomorskii, A. I. Sovremennye informatsionnye tekhnologii v zadachakh navigatsii i navedeniya bespilotnykh manevrennykh letatelanykh apparatov (Modern Information Technologies in Navigation and Tracking Problems for Unmanned Maneuverable Aerial Vehicles). (Fizmatlit, Moscow, 2009).
Lainiotis, D. Optimal Adaptive Estimation: Structure and Parameter Adaption. IEEE Trans. Autom. Control no. 16, 160–170 (1971).
Yang, Y. & Gao, W. An Optimal Adaptive Kalman Filter. J. Geodesy no. 80, 177–183 (2006).
Mohamed, A. H. & Schwarz, K. P. Adaptive Kalman Filtering for INS/GPS. J. Geodesy no. 73, 193–203 (1999).
Reina, G., Vargas, A., Nagatani, K., and Yoshida, K.Adaptive Kalman Filtering for GPS-based Mobile Robot Localization, Int. Workshop on Safety, Security and Rescue Robotics, Rome, Italy, September 2007.
Bosov, A. V. & Pankov, A. R. Robust Recurrent Estimations of Processes in Stochastic Systems. Autom. Remote Control 53(no. 9), 1395–1402 (1992).
Koch, K. R. & Yang, Y. Robust Kalman Filter for Rank Deficient Observation Models. J. Geodesy no. 72, 436–441 (1998).
Chang, Guobin & Liu, Ming M-estimator-based Robust Kalman Filter for Systems with Process Modeling Errors and Rank Deficient Measurement Models. Nonlin. Dynam. no. 80, 1431–1449 (2015).
Cao, L., Qiao, D., and Chen, X.Laplace L1 Huber Based Cubature Kalman Filter for Attitude Estimation of Small Satellite, Acta Astronaut., 2018, vol. 148. https://doi.org/10.1016/j.actaastro.2018.04.020
Huber, P. J. Robust Statistics. (Wiley, New York, 1981).
Armstrong, R. D. & Frome, E. L. A Comparison of Two Algorithms for Absolute Deviation Curve Fitting. J. Am. Statist. Association 71(354), 328–330 (1976).
Maybeck, P.S.Stochastic Models, Estimation, and Control, New York: Academic, 1982, vol. 2, pp. 70–129.
Kotz, S., Kozubowski, T. J. & Podgorski, K. The Laplace Distribution and Generalizations. (Springer, New York, 2001).
Barrodale, I. & Roberts, F. An Improved Algorithm for Discrete L1 Linear Approximation. SIAM J. Numer. Anal. no. 10, 839–848 (1973).
Abdelmalek, NabihN. On the Discrete Linear L1 Approximation and L1 Solutions of Overdetermined Linear Equations. J. Approx. Theory no. 11, 38–53 (1974).
Abdelmalek, NabihN. An Efficient Method for the Discrete Linear L1 Approximation Problem. Math. Comput. 29(no. 131), 844–850 (1975).
Hogben, L. Handbook of Linear Algebra. 2nd ed (CRC Press, Boca Raton, 2013).
Teunissen, P. J. G. Distributional Theory for the DIA Method. J. Geodesy 92, 59–80 (2018).
Xu, Changhui, Rui, Xiaoping, Song, Xianfeng & Gao, Jingxiang Generalized Reliability Measures of Kalman Filtering for Precise Point Positioning. J. Syst. Eng. Electron. 24(no. 4), 699–705 (2013).
Seo, Seong-Hun, Jee, Gyu-In, Byung-Hyun, Lee, Sung-Hyuck, Im and Kwan-Sung, KimHypothesis Test for Spoofing Signal Identification using Variance of Tangent Angle of Baseline Vector Components, Proc. 30th Int. Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2017), Portland, Oregon, September 2017, pp. 1229–1240.
Julier, S.J., Uhlmann, J.K., and Durrant-Whyte, H.F.A New Approach for Filtering Nonlinear Systems, Proc. IEEE Am. Control Conf., 1995, pp. 1628–1632.
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Miller, B., Kolosov, K. Robust Estimation Based on the Least Absolute Deviations Method and the Kalman Filter. Autom Remote Control 81, 1994–2010 (2020). https://doi.org/10.1134/S0005117920110041
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DOI: https://doi.org/10.1134/S0005117920110041