Nonlinear Filtering Method in the Presence of Unknown Noise Intensity
Keywords:
nonlinear filtering, unknown noise intensity, linear stochastic differential system, Kalman-Bucy filterAbstract
The paper describes the method of inverse statistical linearization – the method of nonlinear filtering for state estimation in linear Gaussian differential systems with observation noise of unknown intensity. The proposed technique is based on finding a nonlinear perturbation of the innovation process while keeping the gain coefficient in the Kalman-Bucy filter. As a result, the nonlinear filter is defined by a system of differential equations of the same order as the state vector without using any additional equations for the error covariance matrix. The nonlinear filter is found in an analytical form for a one-dimensional motion model in which only the highest order derivative is affected by a white-noise stochastic disturbance, and only one output is available for observing the position with additive noise of unknown intensity. The recurrent nonlinear filtering scheme is examined to establish the unbiasedness of the estimates and to obtain the steady-state equation for their error variances and covariances. The theoretical results are confirmed on the basis of computer simulations carried out to compare the estimation accuracy of the optimal filter and the proposed nonlinear filtering scheme.
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2. Sinitsyn I.N. Normal suboptimal filtering methods in implicit observable Gaussian stochastic systems. Sistemy i sredstva informatiki – Systems and Means of Informatics. 2025. vol. 35. no. 1. pp. 41–58. (In Russ.).
3. Miller B., Kolosov K. [Robust estimation based on the least absolute deviations method and the Kalman filter]. Automation and Remote Control. 2020. vol. 81. no. 11. pp. 1994–2010.
4. Kulikova M.V., Kulikov G.Y. On derivative-free extended Kalman filtering and its Matlaboriented square-root implementations for state estimation in continuous-discrete nonlinear stochastic systems. European J. Control. 2023. vol. 73.
5. Bosov A.V., Uryupin I.V. [A modified extended Kalman filter by the linear pseudomeasurement method]. Inform. i ee primenenie – Informatics and Applications. 2025. vol. 19. no. 2. pp. 17–26. (In Russ.).
6. Ananyev B.I. [On estimation of dynamical systems under inexact constraints on
parameters]. Trudy Instituta Matematiki i Mekhaniki UrO RAN – Tr. In-ta matematiki i mehaniki UrO RAN. 2025. vol. 31. no. 2. pp. 15–29. (In Russ.).
7. Wang S. Distributionally robust state estimation for nonlinear systems. IEEE Trans. Signal Process. 2022. vol. 70. pp. 4408–4423.
8. Stepanov O.A., Isaev A.M. [A procedure of comparative analysis of recursive nonlinear filtering algorithms in navigation data processing based on predictive modeling]. Gyroscopy Navig. 2023. vol. 14. pp. 213–224.
9. Olfati-Saber R. Kalman-consensus filter: Optimality, stability, and performance. Proc. 48th IEEE Conf. Decision and Control (CDC’2009) held jointly with 28th Chinese Control Conf. 2009. pp. 7036–7042. DOI: 10.1109/CDC.2009.5399678.
10. Sharma R., Beard R.W., Taylor C.N., Quebe S. Graph-based observability analysis of bearing-only cooperative localization. IEEE Trans. Robotics. 2011. vol. 28. no. 2. pp. 522–529.
11. Wang L., Liu J., Morse A.S. A distributed observer for a continuous-time linear system with time-varying network. arXiv:2003.02134. 2020.
12. Talebi P.S., Mandic D. On the dynamics of multi agent nonlinear filtering and learning. arXiv:2309.03557v2. 2023.
13. Andrievsky B.R., Matveev A.S. Fradkov A.L. [Control and estimation under information constraints: Toward a unified theory of control, computation and communications]. Automation and Remote Control. 2010. vol. 71. no. 4. pp. 572–633.
14. Song W., Wang Z., Li Z., Wang J., Han Q.-L. Nonlinear filtering with sample-based approximation under constrained communication: Progress, insights and trends. IEEE/CAA J. Autom. Sinica. 2024. vol. 11. no. 7. pp. 1539–1556.
15. Voortman Q., Efimov D., Pogromsky A., Richard J.-P., Nijmeijer H. Remote state estimation of steered systems with limited communications: An event-triggered approach. IEEE Trans. Automat. Control. 2024. vol. 69. no. 7. pp. 4199–4214.
16. Tanwani A. Suboptimal filtering over sensor networks with random communication. IEEE Trans. Automat. Control. 2022. vol. 67. no. 10. pp. 5456–5463.
17. Anan’ev B.I. [Minimax mean-square estimates in statistically indeterminate systems]. Differenc. uravnenija – Differential equations. 1984. vol. 20. no. 8. pp. 1291–1297. (In Russ.).
18. Lebedev M.V., Semenikhin K.V. [Minimax filtering in a stochastic differential system with non-stationary perturbations of unknown intensity]. Journal of Computer and System Sciences International. 2007. vol. 46. no. 2. pp. 206–217.
19. Pankov A.R., Platonov E.N., Semenikhin K.V. Robust filtering of a process in the stationary difference stochastic system. Automation and Remote Control. 2011. vol. 72. no. 2. pp. 377–392.
20. Kogan M.M. [Robust estimation and filtering in uncertain linear systems under unknown covariations]. Automation and Remote Control. 2015. vol. 76. no. 10. pp. 1751–1764.
21. Shafieezadeh Abadeh S., Nguyen V.A., Kuhn D., Mohajerin Esfahani P.M. Wasserstein distributionally robust Kalman filtering. Advances in Neural Information Processing Systems. 2018. vol. 31.
22. Kargin T., Hajar J., Malik V., Hassibi B. Distributionally robust Kalman filtering over finite and infinite horizon. arXiv:2407.18837v1. 2024.
23. Barabanov A.E. [Linear filtering with adaptive adjustment of the disturbance covariation matrices in the plant and measurement noise]. Automation and Remote Control. 2016. vol. 77. no. 1. pp. 21–36.
24. Li K., Zhao S., Liu F. Joint state estimation for nonlinear state-space model with unknown time-variant noise statistics. Internat. J. Adaptive Control and Signal Processing. 2021. vol. 35. no. 4. pp. 498–512.
25. Huang Y., Zhang Y., Wu Z., Li N., Chambers J. A novel adaptive Kalman filter with inaccurate process and measurement noise covariance matrices. IEEE Trans. Automat. Control. 2018. vol. 63. no. 2. pp. 594–601.
26. Chen Y., Li W., Wang Y. Online adaptive Kalman filter for target tracking with unknown noise statistics. IEEE Sensors Letters. 2021. vol. 5. no. 3. pp. 1–4.
27. Kanouj M.M., Klokov A.V. [Adaptive unscented Kalman filter for tracking GPS signals in the case of an unknown and time-varying noise covariance]. Gyroscopy Navig. 2021. vol. 12. pp. 224–235.
28. Grigor’ev F.N., Kuznetsov N.A. [Quasi-optimal filtering in the presence of unknown noise intensity]. Conference Proceedings: Doklady II-go Vsesoyuznogo soveschaniya po statisticheskim metodam teorii upravleniya [Reports of the II All-Union Meeting on Statistical Methods of Management Theory]. Moscow: Nauka, 1970. pp. 112–118. (In Russ.).
29. Grigor’ev F.N. Metody upravleniya protsessom nablyudeniya v nepreryvnykh sistemakh [Methods for control of observation processes in continuous-time systems]: Ph.D. Diss. kand. tekhn. nauk: 05.13.02. Moscow: Moscow Institute of Physics and Technology, 1975. 104 p. (In Russ.).
30. Teoreticheskiye voprosy postroyeniya ASU krupnotonnazhnymi transportnymi sudami: Sbornik statey [Theoretical issues of constructing ACSs (Automated Control Systems) for large-capacity transport vessels: Collection of articles]. Moscow: Nauka, 1978. 211 p. (In Russ.).
31. Liptser R.S., Shiryayev A.N. Statistics of random processes. 3rd edition. New York: Springer, 2005.
32. Polyak B.T., Khlebnikov M.V., Rapoport L.B. Matematicheskaja teorija avtomaticheskogo upravlenija [Mathematical theory of automatic control]. М.: LENAND, 2019. 500 p. (In Russ.).
33. Kalatchev M.G. [A method of multiple differentiation of signal in automatic control systems]. Avtomatika i Telemekhanika – Automation and telemechanics. 1970. no. 6. pp. 29–36. (In Russ.).
34. Miller B.M., Pankov A.R. Teoriya sluchaynykh protsessov v primerakh i zadachakh [Theory of Random Processes: Examples and Problems]. Moscow: Fizmatlit, 2002. 317 p. (In Russ.).
Published
2025-12-04
How to Cite
Kuznetsov, N., & Semenikhin, K. (2025). Nonlinear Filtering Method in the Presence of Unknown Noise Intensity. Informatics and Automation, 24(6), 1544-1567. https://doi.org/10.15622/ia.24.6.1
Section
Mathematical Modeling and Applied Mathematics
Copyright (c) Константин Владимирович Семенихин, Николай Александрович Кузнецов
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