Minimax Regret filter for uncertainty Single-Input Single-Output systems: simulation study
DOI:
https://doi.org/10.17533/udea.redin.20240412Keywords:
Minimax regret approach, Unknown-but-bounded, Unknown distribution error, Grid hyperparameter optimizationAbstract
The Kalman filter, widely used since its introduction in 1960, assumes Gaussian random disturbances. However, this assumption can be inappropriate in non-Gaussian contexts, leading to suboptimal performance. Researchers have proposed robust filters like minimax filters to address this limitation, but these filters can overly conservative estimates. This research introduces a novel approach that combines unknown-but-bounded dynamics for the state process and stochastic processes for the measurement equation along with a Minimax Regret framework to improve state estimation in one-dimensional linear dynamic models. We evaluate the proposed method through two simulation studies. The first study optimizes the hyperparameter value using Grid Search. In contrast, the second compares the performance of the proposed method with conventional methods, including the Kalman filter and a robust version of the RobKF filter implemented in R software, using a suitable performance metric such as mean squared error. The results demonstrate the superiority of the proposed algorithm.
Downloads
References
R. E. Kalman, “A new approach to linear filtering and prediction problems,” Journal of Basic Engineering, vol. 82, no. 1, 1960. [Online]. Available: https://doi.org/10.1115/1.3662552
J. Iqbal, M. Rehman-Saad, A. Mahmood-Tahir, and A. Malik, “State estimation technique for a planetary robotic rover,” Revista Facultad De Ingenieria Universidad De Antioquia, no. 73, Oct-Dec 2014. [Online]. Available: https://doi.org/10.17533/udea.redin.17275
Q. Ge, J. Ma, H. He, H. Li, and G. Zhang, “A basic smart linear kalman filter with online performance evaluation based on observable degree,” Applied Mathematics and Computation, vol. 367, 2020. [Online]. Available: https://doi.org/10.1016/j.amc.2019.124603
Y. Li, X. Song, Z. Zhang, D. Zhao, and Z. Wang, “h∞ deconvolution filter design for uncertain linear discrete time-variant systems: A krein space approach,” Applied Mathematics and Computation, vol. 361, 2019. [Online]. Available: https://doi.org/10.1016/j.amc.2019.05.015
T. S. Badings, A. Abate, N. Jansen, D. Parker, H. A. Poonawala, and M. Stoelinga, “Sampling-based robust control of autonomous systems with non-gaussian noise,” Proceedings of the AAAI Conference on Artificial Intelligence, vol. 36, no. 9, Jun. 2022. [Online]. Available: https://doi.org/10.1609/aaai.v36i9.21201
M. Mintz, “A note on minimax estimation and kalman filtering,” IEEE Transactions on Automatic Control, vol. 23, no. 4, 1978. [Online]. Available: https://doi.org/10.1109/TAC.1969.1099239
M. Mintz, “A kalman filter as a minimax estimator,” Journal of Optimization Theory and Applications, vol. 9, no. 1, 1972. [Online]. Available: https://doi.org/10.1007/BF00932347
N. Castaño, J. P. Fernández-Gutiérrez, V. Azhmyakov, and P. Graczyk, “Non-parametric identification of upper bound covariance matrices for min-sup robust kalman filter: Application to the ar case,” in IFAC-PapersOnLine, vol. 55, no. 40, 2022. [Online]. Available: https://doi.org/10.1016/j.ifacol.2023.01.068
M. J. Morris, “The kalman filter: A robust estimator for some classes of linear quadratic problems,” IEEE Transactions on Information Theory, vol. 22, no. 5, 1976. [Online]. Available: https://doi.org/10.1109/TIT.1976.1055611
Y. Sholeh and K. Pelckmans, “Worst-case prediction performance analysis of the kalman filter,” IEEE Transactions on Automatic Control, vol. 63, no. 6, 2018. [Online]. Available: https://doi.org/10.1109/TAC.2017.2757908
A. Pankov and G. Miller, “Minimax filter for statistically indeterminate stochastic differential system,” IFAC Proceedings Volumes, vol. 38, no. 1, 2005. [Online]. Available: https://doi.org/10.3182/20050703-6-CZ-1902.00389
E. Gyurkovics and T. Takács, “Robust energy-to-peak filter design for a class of unstable polytopic systems with a macroeconomic application,” Applied Mathematics and Computation, vol. 420, 2022. [Online]. Available: https://doi.org/10.1016/j.amc.2021.126729
Y. C. Eldar, A. Ben-Tal, and A. Nemirovski, “Minimax regret estimation in linear models,” in 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 2, 2004. [Online]. Available: https://doi.org/10.1109/ICASSP.2004.1326219
B. Hassibi, A. H. Sayed, and T. Kailath, “Linear estimation in krein spaces-theory,” IEEE Transactions on Automatic Control, vol. 41, no. 1, pp. 18–33, 1996. [Online]. Available: https://doi.org/10.1109/9.481605
B. Hassibi and T. Kailath, “H/sup /spl infin// bounds for least-squares estimators,” IEEE Transactions on Automatic Control, vol. 46, no. 2, 2001. [Online]. Available: https://doi.org/10.1109/9.905700
M. M. J. Mirza, “A modified kalman filter for non-gaussian measurement noise,” in Communication Systems and Information Technology, M. Ma, Ed. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. [Online]. Available: https://doi.org/10.1007/978-3-642-21762-3_52
X. Sun, J. Duan, X. Li, and X. Wang, “State estimation under non-gaussian levy noise: A modified kalman filtering method,” arXiv e-prints, 2013. [Online]. Available: https://ui.adsabs.harvard.edu/abs/2013arXiv1303.2395S
V. Azhmyakov, “Optimal control of a well-stirred bioreactor in the presence of stochastic perturbations,” Informatica, vol. 13, no. 2, 2002. [Online]. Available: https://doi.org/10.3233/INF-2002-13201
J. C. Spall, “The kantorovich inequality for error analysis of the kalman filter with unknown noise distributions,” Technical Communique, vol. 31, no. 10, 1995. [Online]. Available: https://doi.org/10.1016/0005-1098(95)00069-9
J. L. Maryak, J. C. Spall, and B. D. Heydon, “Use of the Kalman filter for inference in state-space models with unknown noise distributions,” IEEE Transactions on Automatic Control, vol. 46, no. 2, 2001. [Online]. Available: https://doi.org/10.1109/TAC.2003.821415
H.-S. Ahn, Y.-S. Kim, and Y. Chen, “An interval kalman filtering with minimal conservatism,” Applied Mathematics and Computation, vol. 218, no. 18, 2012. [Online]. Available: https://doi.org/10.1016/j.amc.2012.02.050
Y. Liu, Y. Zhao, and F. Wu, “Ellipsoidal state-bounding-based set-membership estimation for linear system with unknown-but-bounded disturbances,” IET Control Theory & Applications, vol. 10, no. 4, 2016. [Online]. Available: https://doi.org/10.1049/iet-cta.2015.0654
Z. Wang, G. Xu, Z. Liu, Y. Wang, and Z. Ji, “Orthotopic-filtering-based fault diagnosis algorithms for nonlinear systems with slowly varying faults,” Journal of the Franklin Institute, vol. 357, 2020. [Online]. Available: https://doi.org/10.1016/j.jfranklin.2020.03.033
L. Savage, “The theory of statistical decision,” Journal of the American Statistical Association, vol. 46, 1951. [Online]. Available: https://doi.org/10.1080/01621459.1951.10500768
N. Oliver, F. Perez-Cruz, S. Kramer, J. Read, and J. Lozano, Machine Learning and Knowledge Discovery in Databases. Research Track European Conference, ECML PKDD 2021, Bilbao, Spain, September 13–17, 2021, Proceedings, Part III: European Conference, ECML PKDD 2021, Bilbao, Spain, September 13–17, 2021, Proceedings, Part III, 2021. [Online]. Available: https://doi.org/10.1007/978-3-030-86486-6
A. No and T. Weissman, “Minimax filtering regret via relations between information and estimation,” IEEE Transactions on Information Theory, vol. 60, no. 8, 2014. [Online]. Available: https://doi.org/10.1109/TIT.2014.2323069
J. Y. Halpern and S. Leung, “Minimizing regret in dynamic decision problems,” Theory and Decision, vol. 81, 2016. [Online]. Available: https://doi.org/10.1007/s11238-015-9526-8
Y. Eldar, A. Ben-Tal, and A. Nemirovski, “Linear minimax regret estimation of deterministic parameters with bounded data uncertainties,” IEEE Transactions on Signal Processing, vol. 52, no. 8, 2004. [Online]. Available: https://doi.org/10.1109/TSP.2004.831144
L. Chang, B. Hu, G. Chang, and A. Li, “Robust derivative-free kalman filter based on huber’s m-estimation methodology,” Journal of Process Control, vol. 23, no. 10, 2013. [Online]. Available: https://doi.org/10.1016/j.jprocont.2013.05.004
A. T. M. Fisch, D. Grose, I. A. Eckley, P. Fearnhead, and L. Bardwell, RobKF: Innovative and/or Additive Outlier Robust Kalman Filtering, 2022, r package version 1.0.2. [Online]. Available: https://doi.org/10.48550/arXiv.2007.03238
J. Fan, R. Li, C.-H. Zhang, and H. Zou, “Statistical foundations of data science,” in Statistical Foundations of Data Science. Chapman and Hall/CRC, 2021.
D. Akkaya and M. Pınar, “Minimizers of sparsity regularized huber loss function,” Journal of Optimization Theory and Applications, vol. 187, 2020. [Online]. Available: https://doi.org/10.1007/s10957-020-01745-3
A. Courand and M. Metz, “Evaluation of a robust regression method (roboost-plsr) to predict biochemical variables for agronomic applications: Case study of grape berry maturity monitoring,” Chemometrics and Intelligent Laboratory Systems, vol. 221, 2022. [Online]. Available: https://doi.org/10.1016/j.chemolab.2021.104485
C. Körner, “Mastering azure machine learning,” in Mastering Azure Machine Learning. Packt Publishing, 2022.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Revista Facultad de Ingeniería Universidad de Antioquia
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Revista Facultad de Ingeniería, Universidad de Antioquia is licensed under the Creative Commons Attribution BY-NC-SA 4.0 license. https://creativecommons.org/licenses/by-nc-sa/4.0/deed.en
You are free to:
Share — copy and redistribute the material in any medium or format
Adapt — remix, transform, and build upon the material
Under the following terms:
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
NonCommercial — You may not use the material for commercial purposes.
ShareAlike — If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original.
The material published in the journal can be distributed, copied and exhibited by third parties if the respective credits are given to the journal. No commercial benefit can be obtained and derivative works must be under the same license terms as the original work.
Twitter