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Automation and Remote Control

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01-04-2023 | STOCHASTIC SYSTEMS

A Comparison of Guaranteeing and Kalman Filters

Author: M. V. Khlebnikov

Published in: Automation and Remote Control | Issue 4/2023

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Abstract

We propose a new approach to filtering under arbitrary bounded exogenous disturbances based on reducing this problem to an optimization problem. The approach has a low computational complexity since only Lyapunov equations are solved at each iteration. At the same time, it possesses advantages essential from an engineering-practical point of view, namely, the possibilities to limit the filter matrix and to construct optimal filter matrices separately for each coordinate of the system’s state vector. A gradient method for finding the filter matrix is presented. According to the examples, the proposed recurrence procedure is rather effective and yields quite satisfactory results. This paper continues the series of research works devoted to feedback control design from an optimization perspective.

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It actually takes 3–4 iterations to obtain a solution with high accuracy if the starting point is not too close to the boundaries of the interval (ρ²(A – L_j _+ 1C), 1).

Kalman, R.E., A New Approach to Linear Filtering and Prediction Problems, J. Basic Engineer., 1960, vol. 82, no. 1, pp. 35–45.MathSciNetCrossRef

Kailath, T., Sayed, A.H., and Hassibi, B., Linear Estimation, New Jersey: Prentice Hall, 2000.MATH

Matasov, A.I., Osnovy teorii fil’tra Kalmana (Foundations of Kalman Filter Theory), Moscow: Mosk. Gos. Univ., 2021.

Schweppe, F.C., Uncertain Dynamic Systems, New Jersey: Prentice Hall, 1973.

Kurzhanskii, A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti (Control and Observation under Uncertainty), Moscow: Nauka, 1977.

Chernous’ko, State Estimation for Dynamic Systems, Boca Raton: CRC Press, 1994.

Polyak, B.T. and Topunov, M.V., Filtering under Nonrandom Disturbances: The Method of Invariant Ellipsoids, Dokl. Math., 2008, vol. 77, no. 1, pp. 158–162.MathSciNetCrossRefMATH

Khlebnikov, M.V. and Polyak, B.T., Filtering under Arbitrary Bounded Exogenous Disturbances: The Technique of Linear Matrix Inequalities, The 13th Multiconference on Control Problems (MCCP 2020), Proceedings of the 32nd Conference in Memory of Nikolay Ostryakov, St. Petersburg, October 6–8, 2020, Concern CSRI Elektropribor, pp. 291–294.

Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.CrossRefMATH

10.

Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Upravlenie lineinymi sistemami pri vneshnikh vozmu-shcheniyakh: tekhnika lineinykh matrichnykh neravenstv (Control of Linear Systems under Exogenous Disturbances: The Technique of Linear Matrix Inequalities), Moscow: LENAND, 2014.

11.

Fazel, M., Ge, R., Kakade, S., and Mesbahi, M., Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator, Proc. 35th Int. Conf. Machine Learning, Stockholm, July 10–15, 2018, vol. 80, pp. 1467–1476.

12.

Mohammadi, H., Zare, A., Soltanolkotabi, M., and Jovanović, M.R., Global Exponential Convergence of Gradient Methods over the Nonconvex Landscape of the Linear Quadratic Regulator, Proc. 2019 IEEE 58th Conf. Decision Control, Nice, December 11–13, 2019, pp. 7474–7479.

13.

Zhang, K., Hu, B., and Başar, T., Policy Optimization for \({{\mathcal{H}}_{2}}\) Linear Control with \({{\mathcal{H}}_{\infty }}\) Robustness Guarantee: Implicit Regularization and Global Convergence, Proc. 2nd Conference on Learning for Dynamics and Control (2nd L4DC), Zürich, June 11–12, 2020, pp. 179–190.

14.

Bu, J., Mesbahi, A., Fazel, M., and Mesbahi, M., LQR through the Lens of First Order Methods: Discrete-Time Case, arXiv:1907.08921, 2019.

15.

Fatkhullin, I. and Polyak, B., Optimizing Static Linear Feedback: Gradient Method, SIAM J. Control Optim., 2021, vol. 59, no. 5, pp. 3887–3911.MathSciNetCrossRefMATH

16.

Polyak, B.T. and Khlebnikov, M.V., Static Controller Synthesis for Peak-to-Peak Gain Minimization as an Optimization Problem, Autom. Remote Control, 2021, vol. 82, no. 9, pp. 1530–1553.MathSciNetCrossRefMATH

17.

Polyak, B.T. and Khlebnikov, M.V., Observer-Aided Output Feedback Synthesis as an Optimization Problem, Autom. Remote Control, 2022, vol. 83, no. 3, pp. 303–324.MathSciNetCrossRefMATH

18.

Polyak, B.T. and Khlebnikov, M.V., New Criteria for Tuning PID Controllers, Autom. Remote Control, 2022, vol. 83, no. 11, pp. 1724–1741.MathSciNetCrossRefMATH

19.

Luenberger, D.G., Observing the State of a Linear System, IEEE Transactions on Military Electronics, 1964, vol. 8, pp. 74–80.CrossRef

20.

Luenberger, D.G., An Introduction to Observers, IEEE Trans. Autom. Control, 1971, vol. 35, pp. 596–602.CrossRef

21.

Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Linear Matrix Inequalities in Control Systems with Uncertainty, Autom. Remote Control, 2021, vol. 82, no. 1, pp. 1–40.MathSciNetCrossRefMATH

22.

Nazin, S.A., Polyak, B.T., and Topunov, M.V., Rejection of Bounded Exogenous Disturbances by the Method of Invariant Ellipsoids, Autom. Remote Control, 2007, vol. 68, no. 3, pp. 467–486.MathSciNetCrossRefMATH

23.

en.wikipedia.org/wiki/Kalman\_filter.

24.

Humpherys, J., Redd, P., and West, J., A Fresh Look at the Kalman Filter, SIAM Rev., 2012, vol. 54, no. 4, pp. 801–823.MathSciNetCrossRefMATH

25.

Tang, W., Zhang, Q., Wang, Z., and Shen, Y., Ellipsoid Bundle and Its Application to Set-Membership Estimation, IFAC-PapersOnLine, 2020, vol. 53, no. 2, pp. 13688–13693.CrossRef

26.

Tang, W., Zhang, Q., Wang, Z., and Shen, Y., Set-Membership Filtering with Incomplete Observations, Inform. Sci., 2020, vol. 517, pp. 37–51.MathSciNetCrossRef

27.

Polyak, B.T., Nazin, S.A., Durieu, C., and Walter, E., Ellipsoidal Parameter or State Estimation under Model Uncertainty, Automatica, 2004, vol. 40, no. 7, pp. 1171–1179.MathSciNetCrossRefMATH

28.

Durieu, C., Walter, E., and Polyak, B., Multi-Input Multi-Output Ellipsoidal State Bounding, J. Optim. Theory Appl., 2001, vol. 111, no. 2, pp. 273–303.MathSciNetCrossRefMATH

29.

Kwon, W.H., Moon, Y.S., and Ahn, S.C., Bounds in Algebraic Riccati and Lyapunov Equations: A Survey and Some New Results, Int. J. Control, 1996, vol. 64, pp. 377–389.MathSciNetCrossRefMATH

Title: A Comparison of Guaranteeing and Kalman Filters
Author: M. V. Khlebnikov
Publication date: 01-04-2023
Publisher: Pleiades Publishing
Published in: Automation and Remote Control / Issue 4/2023
Print ISSN: 0005-1179
Electronic ISSN: 1608-3032
DOI: https://doi.org/10.1134/S0005117923040094

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