Studying the extended Kalman filter — adaptive digital filter system operation as the non-stationary signal filter
Authors: Bezmen P.A. | Published: 27.09.2024 |
Published in issue: #10(775)/2024 | |
Category: Mechanical Engineering and Machine Science | Chapter: Robots, Mechatronics and Robotic Systems | |
Keywords: non-stationary signal, extended Kalman filter, adaptive digital filter, robotic device |
Processing the non-stationary signals of various origin in the robotic device control systems appears to be a relevant task. Introduction of the extended Kalman filter (EKF) in operation with the non-stationary signals, including their filtering, is limited by the EKF precise setting, i.e. by the observed system model and the EFK algorithm covariance matrices. It requires the algorithm adaptivity to the input noisy non-stationary signal changing rapidly. The paper proposes to filter the non-stationary signals using the EKF — adaptive digital filter (ADF) system. The system appears to be an EKF supplemented by the ADF with the NLMS (Normalized Least Mean Squares) adaptation algorithm and is introduced to compensate for the EKF operation error in filtering the non-stationary signals. Studying the EKF-ADF system operation as the non-stationary signal filter showed that with the non-stationary signal exposed to a significant noise level the RFK-ATS system algorithm, compared to the RFK algorithm with settings similar to the RFK settings in the RFK-ATS system, has the signal-to-noise ratio highest values for the filtering results. The RFK-ATS system could be introduced in observing the robotic device state, it is able to operate in the non-stationary processes.
EDN: ONANWI, https://elibrary/onanwi
References
[1] Schmidt S.F. Application of state-space methods to navigation problems. Advances in Control Systems, 1966, vol. 3, pp. 293–340, doi: https://doi.org/10.1016/B978-1-4831-6716-9.50011-4
[2] Wishner R.P., Tabaczynski J.A., Athans M.A. A comparison of three non-linear filters. Automatica, 1969, vol. 5, no. 4, pp. 487–496, doi: https://doi.org/10.1016/0005-1098(69)90110-1
[3] Bass R.W., Norum V.D., Swartz L. Optimal multichannel nonlinear filtering. J. Math. Anal. Appl., 1966, vol. 16, no. 1, pp. 152–164, doi: https://doi.org/10.1016/0022-247X(66)90193-4
[4] Jazwinski A.H. Filtering for nonlinear dynamical systems. IEEE Trans. Autom. Control, 1966, vol. 11, no. 4, pp. 765–766, doi: https://doi.org/10.1109/TAC.1966.1098431
[5] Julier S.J., Uhlmann J.K., Durrant-Whyte H. A new approach for filtering nonlinear systems. Proc. ACC, 1995, vol. 3, pp. 1628–1632, doi: https://doi.org/10.1109/ACC.1995.529783
[6] Julier S.J., Uhlmann J.K. A new extension of the Kalman filter to nonlinear systems. Proc. SPIE, 1997, vol. 3068, pp. 182–193, doi: https://doi.org/10.1117/12.280797
[7] Julier S., Uhlmann J., Durrant-Whyte H. A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Trans. Autom. Control, 2000, vol. 45, no. 3, pp. 477–482, doi: https://doi.org/10.1109/9.847726
[8] Evensen G. Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 1994, vol. 99, no. C5, pp. 10143–10162, doi: https://doi.org/10.1029/94JC00572
[9] Evensen G. The Ensemble Kalman Filter: theoretical formulation and practical implementation. Ocean Dynamics, 2003, vol. 53, no. 4, pp. 343–367, doi: https://doi.org/10.1007/s10236-003-0036-9
[10] Arasaratnam I., Haykin S., Elliott R.J. Discrete-time nonlinear filtering algorithms using Gauss — Hermite quadrature. Proc. IEEE, 2007, vol. 95, no. 5, pp. 953–977, doi: https://doi.org/10.1109/JPROC.2007.894705
[11] Arasaratnam I., Haykin S. Cubature Kalman filters. IEEE Trans. Autom. Control, 2009, vol. 54, no. 6, pp. 1254–1269, doi: https://doi.org/10.1109/TAC.2009.2019800
[12] Nørgaard M., Poulsen N.K., Ravn O. New developments in state estimation for nonlinear systems. Automatica, 2000, vol. 36, no. 11, pp. 1627–1638, doi: https://doi.org/10.1016/S0005-1098(00)00089-3
[13] Ito K., Xiong K. Gaussian filters for nonlinear filtering problems. IEEE Trans. Autom. Control, 2000, vol. 45, no. 5, pp. 910–927, doi: https://doi.org/10.1109/9.855552
[14] Khodarahmi M., Maihami V. A review on Kalman filter models. Arch. Computat. Methods Eng., 2023, vol. 30, no. 1, pp. 727–747, doi: https://doi.org/10.1007/s11831-022-09815-7
[15] Madhukar P.S., Madhukar S. Kalman Filters in different biomedical signals-an overview. ICOSEC, 2020, pp. 1268–1272, doi: https://doi.org/10.1109/ICOSEC49089.2020.9215335
[16] Kumari N., Kulkarni R., Ahmed M.R. et al. Use of Kalman filter and its variants in state estimation: a review. In: Artificial intelligence for a sustainable industry 4.0. Springer, 2021, pp. 213–230, doi: https://doi.org/10.1007/978-3-030-77070-9_13
[17] Shilman S.V. Adaptive Kalman filters. Doklady AN, 1994, vol. 338, no. 6, pp. 742–744. (In Russ.). (Eng. version: Dokl. Math., 1994, vol. 39, no. 10, pp. 687–689.)
[18] Liu S. An adaptive Kalman filter for dynamic estimation of harmonic signals. Proc. 8th Int. Conf. on Harmonics and Quality of Power, 1998, vol. 2, pp. 636–640, doi: https://doi.org/10.1109/ICHQP.1998.760120
[19] Kovač U., Košir A. Fast estimation of the non-stationary amplitude of a harmonically distorted signal using a Kalman filter. Metrol. Meas. Syst., 2013, vol. 20, no. 1, pp. 27–42.
[20] Bendat J.S., Piersol A.G. Measurement and analysis of random data. New York, Wiley, 1966. 414 p. (Russ. ed.: Izmerenie i analiz sluchaynykh protsessov. Moscow, Mir Publ., 1971. 408 p.)
[21] Bozhokin S.V., Lykov S.N. Continuous wavelet transformation. Nauchno-tekhnicheskie vedomosti SPbGPU. Fiziko-matematicheskie nauki [St. Petersburg Polytechnic University Journal - Physics and Mathematics], 2012, no. 1, pp. 146–151. (In Russ.).
[22] Bozhokin S.V. Continuous wavelet transform and exactly solvable model of nonstationary signals. Zhurnal tekhnicheskoy fiziki, 2012, vol. 82, no. 7, pp. 8–13. (In Russ.). (Eng. version: Tech. Phys., 2012, vol. 57, no. 7, pp. 900–906, doi: https://doi.org/10.1134/S1063784212070067)
[23] Bozhokin S.V., Suslova I.M. Double wavelet transform of frequency-modulated nonstationary signal. Zhurnal tekhnicheskoy fiziki, 2013, vol. 83, no. 12, pp. 26–32. (In Russ.). (Eng. version: Tech. Phys., 2013, vol. 58, vol. 12, pp. 1730–1736, doi: https://doi.org/10.1134/S1063784213120074)
[24] Bezmen P.A. The extended Kalman filter augmented by an adaptive digital filter for data fusion of a mobile robot control system. Fundamentalnye i prikladnye problemy tekhniki i tekhnologii [Fundamental and Applied Problems of Engineering and Technology], 2020, no. 2, pp. 85–94, doi: https://doi.org/10.33979/2073-7408-2020-340-2-85-94 (in Russ.).
[25] Bezmen P.A. Investigation of the operation of the extended kalman filter supplemented by an adaptive digital filter for integrating data from a mobile robot control system. Izvestiya Yugo-Zapadnogo gosudarstvennogo universiteta [Proceedings of the Southwest State University], 2020, vol. 24, no. 1, pp. 68–89, doi: https://doi.org/10.21869/2223-1560-2020-24-1-68-89 (in Russ.).
[26] Bezmen P.A. Tsifrovoy filtr dlya nestatsionarnykh signalov [Digital filter for non-stationary signals]. Patent RU 2747199. Appl. 05.07.2020, publ. 29.04.2021. (In Russ.).
[27] Bezmen P.A. Nabor bibliotek «RFK-ATsF-ARS» realizatsii sistemy upravleniya sostoyaniem obekta [Set of "RFK-ACF-ARS" libraries for implementation of the object state control system]. Software reg. certificate 2022663792 RF. Appl. 01.07.2022, publ. 20.07.2022. (In Russ.).
[28] Bezmen P.A. Integration of mobile robot control system data using the extended Kalman filter. Izvestiya Yugo-Zapadnogo gosudarstvennogo universiteta [Proceedings of the Southwest State University], 2019, vol. 23, no. 2, pp. 53–64, doi: https://doi.org/10.21869/2223-1560-2019-23-2-53-64 (in Russ.).