Sparse Filtering Under Norm-Bounded Exogenous Disturbances Using Observers


  • Mikhail Khlebnikov V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences



linear system, filtering, sparsity, exogenous disturbances, linear matrix inequalities, invariant ellipsoids


The paper considers the sparse filtering problem under arbitrary norm-bounded exogenous disturbances. We propose a simple and universal observer-based approach to its solution, based on the LMI technique and the method of invariant ellipsoids; it allows the use of a reduced number of system outputs. From a technical point of view of application, we reduce the original problem to semi-definite programming, which is easily solved numerically. The proposed simple approach is easy to implement and can be equally extended to systems in continuous and discrete time.


A. Y. Carmi, L. Mihaylova, and S. J. Godsill (Eds.), Compressed Sensing & Sparse Filtering. Berlin: Springer, 2014.

F. M. Zennaro and K. Chen, “Towards understanding sparse filtering: A theoretical perspective,” Neural Networks, vol. 98, pp. 154–177, 2018.

S. Yang, M. Wang, Z. Feng, Z. Liu, and R. Li, “Deep sparse tensor filtering network for synthetic aperture radar images classification,” IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 8, pp. 3919–3924, 2018.

Z. Zhang, S. Li, J. Wang, Y. Xin, and Z. An, “General normalized sparse filtering: A novel unsupervised learning method for rotating machinery fault diagnosis,” Mechanical Systems and Signal Processing, vol. 124, pp. 596–612, 2019.

C. Han, Y. Lei, Y. Xie, D. Zhou, and M. Gong, “Visual domain adaptation based on modified A-distance and sparse filtering,” Pattern Recognition, vol. 104, art. 107254, 2020.

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

A. B. Kurzhanskii, Control and Observation under Uncertainty. Moscow: Nauka, 1977. [in Russian]

F. L. Chernousko, State Estimation for Dynamic Systems. Moscow: Nauka, 1988. [in Russian]

B. T. Polyak and M. V. Topunov, “Filtering under nonrandom disturbances: The method of invariant ellipsoids,” Doklady Mathematics, vol. 77, no. 1, pp. 158–162, 2008.

M. V. Khlebnikov, “Robust filtering under nonrandom disturbances: The invariant ellipsoid approach,” Automation and Remote Control, vol. 70, no. 1, pp. 133–146, 2009.

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

K. O. Zheleznov, M. V. Khlebnikov, “Tracking problem for dynamical systems with exogenous and system disturbances,” Proc. 20th International Conference on System Theory, Control and Computing (ICSTCC 2016). Sinaia, Romania, October 13–15, 2016, pp. 125–128.

M. V. Khlebnikov, K. O. Zheleznov, “Feedback design for linear control systems with exogenous and system disturbances: robust statement,” Proc. 21st International Conference on System Theory, Control and Computing (ICSTCC 2017). Sinaia, Romania, October 19–21, 2017, pp. 716–721.

B. T. Polyak, M. V. Khlebnikov, and P. S. Shcherbakov, Control of Linear Systems Subject to Exogenous Disturbances: The Linear Matrix Inequalitiy Technique. Moscow: LENAND, 2014. [in Russian]

D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol. 52, pp. 1289–1306, 2006.

S.-J. Kim, K. Koh, S. Boyd, and D. Gorinevsky, “ℓ1-Trend filtering,” SIAM Review, vol. 51, no. 2, pp. 339–360, 2009.

F. Lin, M. Fardad, and M. Jovanovi´c, “Sparse feedback synthesis via the alternating direction method of multipliers,” Proc. 2012 Amer. Control Conf., Montreal, Canada, June 27–29, 2012, pp. 4765–4770.

F. Lin, M. Fardad, and M. Jovanovi´c, “Augmented lagrangian approach to design of structured optimal state feedback gains,” IEEE Transactions on Automatic Control, vol. 56, no. 12, pp. 2923–2929, 2011.

B. T. Polyak, M. V. Khlebnikov, and P. S. Shcherbakov, “An LMI approach to structured sparse feedback design in linear control systems,” Proc. 12th European Control Conference (ECC’13), Z¨urich, Switzerland, July 17–19, 2013, pp. 833–838.

J. L¨ofberg, YALMIP: Software for Solving Sonvex (and Nonconvex) Optimization Problems. URL∼joloef/wiki/pmwiki.php

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.0 beta. URL

M. Grant and S. Boyd, “Graph implementations for nonsmooth convex programs,” Recent Advances in Learning and Control (a tribute to M. Vidyasagar), V. Blondel, S. Boyd, and H. Kimura, editors. Springer, 2008. pp. 95–110.

M. V. Khlebnikov, “Sparse filtering under non-random bounded exogenous disturbances,” Proc. 25th International Conference on System Theory, Control and Computing (ICSTCC 2021), Ias¸i, Romania, October 20–23, 2021, pp. 200–205.

F. Leibfritz and W. Lipinski, Description of the Benchmark Examples in COMPleib 1.0. Technical report. University of Trier, 2003, URL




How to Cite

“Sparse Filtering Under Norm-Bounded Exogenous Disturbances Using Observers”, Syst. Theor. Control Comput. J., vol. 2, no. 1, pp. 1–7, Jun. 2022, doi: 10.52846/stccj.2022.2.1.29.