Abstract
A simple, universal approach to solving the sparse filtering problem—problem using a reduced number of outputs—with arbitrary bounded exogenous disturbances by employing an observer is presented. The approach is based on the invariant ellipsoid method and the technique of linear matrix inequalities. An application of this concept has made it possible to reduce the original problem to a semidefinite programming problem that is easy to solve numerically. The approach is distinguished by its simplicity and ease of implementation and covers both continuous- and discrete-time statements of the problem. The efficiency of the procedure proposed is demonstrated by a test example.
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REFERENCES
Zennaro, F.M. and Chen, K., Towards understanding sparse filtering: a theoretical perspective, Neural Networks, 2018, vol. 98, pp. 154–177.
Zhang, Z., Li, S., Wang, J., Xin, Y., and An, Z., General normalized sparse filtering: a novel unsupervised learning method for rotating machinery fault diagnosis, Mech. Syst. Signal Process., 2019, vol. 124, pp. 596–612.
Han, C., Lei, Y., Xie, Y., Zhou, D., and Gong, M., Visual domain adaptation based on modified \( \mathcal A \)-distance and sparse filtering, Pattern Recognit., 2020, vol. 104, article ID 107254.
Schweppe, F.C., Uncertain Dynamic Systems., Hoboken: Prentice Hall, 1973.
Kurzhanskii, A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti (Control and Observation under Uncertain Conditions), Moscow: Nauka, 1977.
Chernous’ko, F.L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem. (Estimating Phase State of Dynamic Systems), Moscow: Nauka, 1988.
Furasov, V.D., Zadachi garantirovannoi identifikatsii (Problems of Guaranteed Identification), Moscow: Binom, 2005.
Chernousko, F. and Polyak, B., Special issue on set-membership modelling of uncertainties in dynamical systems, Math. Comput. Model. Dyn. Syst., 2005, vol. 11, no. 2, pp. 123–124.
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.
Khlebnikov, M.V., Robust filtering under nonrandom disturbances: the invariant ellipsoid approach, Autom. Remote Control, 2009, vol. 70, no. 1, pp. 133–146.
Khlebnikov, M.V. and Polyak, B.T., Filtering under arbitrary bounded exogenous disturbances: technique of linear matrix inequalities, 13-ya Mul’tikonf. po probl. upr. (MKPU-2020). Mater. XXXII konf. pamyati vydayushchegosya konstruktora giroskopicheskikh priborov N.N. Ostryakova (13th MultiConf. Probl. Control (MCPC-2020). Mater. XXXII Conf. in Memory of the Outstanding Designer of Gyroscopic Instruments N.N. Ostryakov) (St. Petersburg, October 6–8, 2020), pp. 291–294.
Boyd, S., El, GhaouiL., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory., Philadelphia: SIAM, 1994.
Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Upravlenie lineinymi sistemami pri vneshnikh vozmushcheniyakh. Tekhnika lineinykh matrichnykh neravenstv (Control of Linear Systems under Exogenous Disturbances. Technique of Linear Matrix Inequalities), Moscow: LENAND, 2014.
Donoho, D.L., Compressed sensing, IEEE Trans. Inform. Theory, 2006, vol. 52, pp. 1289–1306.
Kim, S.-J., Koh, K., Boyd, S., and Gorinevsky, D., \( \ell _1 \)-Trend filtering, SIAM Rev., 2009, vol. 51, no. 2, pp. 339–360.
Lin, F., Fardad, M., and Jovanovic, M., Sparse feedback synthesis via the alternating direction method of multipliers, Proc. 2012 Am. Control Conf. (Montreal, June 27–29, 2012), pp. 4765–4770.
Lin, F., Fardad, M., and Jovanovic, M., Augmented Lagrangian approach to design of structured optimal state feedback gains, IEEE Trans. Autom. Control, 2011, vol. 56, no. 12, pp. 2923–2929.
Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., An LMI approach to structured sparse feedback design in linear control systems, Proc. 12th Eur. Control Conf. (ECC’13) (Zurich, July 17–19, 2013), pp. 833–838.
Quattoni, A., Carreras, X., Collins, M., and Darrell, T., An efficient projection for \( \ell _{1,\infty } \) regularization, Proc. 26th Annual Int. Conf. Mach. Learn. (Montreal, June 14–18, 2009), pp. 857–864.
Kvinto, Ya.I., Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Some experiments on obtaining sparse controllers, Tr. 11-i Vseross. shkoly-konf. molodykh uchenykh “Upravlenie bol’shimi sistemami” (UBS’2014) (Proc. 11th Russ. Conf. Young Sci. “Control of Large Systems”) (Arzamas, September 9–12, 2014), Moscow: Inst. Probl. Upr., 2014, pp. 227–238.
Leibfritz, F. and Lipinski, W., Description of the benchmark examples in COMPleib 1.0, Tech. Rep., Univ. Trier, 2003.
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This work was supported by the Russian Science Foundation, project no. 21-71-30005.
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Translated by V. Potapchouck
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Khlebnikov, M.V. Sparse Filtering under Bounded Exogenous Disturbances. Autom Remote Control 83, 191–203 (2022). https://doi.org/10.1134/S0005117922020035
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DOI: https://doi.org/10.1134/S0005117922020035