Optimization of Invariant Ellipsoid Technique for Sparse Controllers Design
https://doi.org/10.17587/mau.23.3-12
Abstract
The paper deals with the method for the design of linear controllers with sparse state feedback matrices for control the plants under conditions of unknown and bounded disturbances. The importance of the sparsity property in feedback can be explained by two factors. First, by minimizing the columnar norm of the feedback matrix in the control it becomes pos- sible to use a minimum number of measuring devices. Secondly, by minimizing the row norm of the feedback matrix, the required number of executive (control) devices is minimized. Both properties, if they are achievable in the synthesis of the controller, reduce the cost of the system and improve the fault tolerance and quality of regulation by reducing the structural complexity. The search algorithm for sparse matrices is based on the method of invariant ellipsoids and is formulated as a solution to a system of linear matrix inequalities with additional constraints. A special set of optimization conditions is
proposed which for a disturbed system minimizes overshoot and overshoots in transient processes of the disturbed closed- loop system simultaneously with minimizing errors in the steady state. The proposed method also assumes the possibility of minimizing both the row norm of the feedback matrix and the column one, while preserving the robustness properties, which makes it possible to solve the sparse control problem (a sparse control is understood as a linear controller with a sparse feedback matrix). The efficiency of the proposed control scheme is confirmed by the results of computer modeling and comparison with some existing ones.
Supporting agencies: The main result was obtained at the expense of a grant from the President of the Russian Federation (project no. MD-1054.2020.8) at IPME RAS. The convex programming problem was supported by a grant from the Russian Foundation for Basic Re- search (project No. 20-08-00610) at IPME RAS. Numerical studies were carried out with a grant from the Russian Foundation for Basic Research (project no. 19-08-00246) at IPME RAS.
About the Authors
S. A. Vrazhevsky
Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences; ITMO University
Russian Federation
Russian Federation
St. Petersburg, 199178
St. Petersburg, 197101
J. V. Chugina
Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences
Russian Federation
Russian Federation
St. Petersburg, 199178
I. B. Furtat
Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences
Russian Federation
Russian Federation
St. Petersburg, 199178
D. E. Konovalov
ITMO University
Russian Federation
Russian Federation
St. Petersburg, 197101
References
1. Polyak B. T., Khlebnikov M. V., Shcherbakov P. S. Sparse feedback in linear control systems, Automation and Remote Control, 2014, vol. 75, pp. 2099—2111.
2. Grant M., Boyd S. CVX: Matlab software for disciplined convex programming, version 2.0 beta, available at: http://cvxr.com/cvx, September 2013.
3. Grant M., Boyd S. Graph implementations for nonsmooth convex programs, Recent Advances in Learning and Control (a tribute to M. Vidyasagar), Lecture Notes in Control and Information Sciences, Springer, 2008, pp. 95—110, available at: http:// stanford.edu/boyd/graph dcp.html.
4. Boyd S., Ghaoui L. Feron E., Balakrishnan V. Linear matrix inequalities in system and control theory, SIAM studies in applied mathematics, 1994, vol. 15.
5. Furtat I. B., Gushchin P. A., Peregudin A. A. Disturbance Attenuation with Minimization of Ellipsoids Restricting Phase Trajectories in Transition and Steady State, Mekhatronika, Avtomatizatsiya, Upravlenie, 2020, vol. 21, no. 4, pp. 195—199 (in Russian).
6. Barabanov A. E., Granichin O. N. An optimal controller of a linear plant subjected to constrained noise, Automation and Remote Control, 1984, vol. 45, no. 5, pp. 578—584.
7. Tibshirani R. Regression shrinkage and selection via the lasso, Journal of the Royal Statistical Society: Series B (Methodological), 1996, vol. 58, no. 1, pp. 267—288.
8. Donoho D. L. Compressed sensing, IEEE Transactions on information theory, 2006, vol. 52, no. 4, pp. 1289—1306.
9. Kim S. J. et al. l1-trend filtering, SIAM review, 2009, vol. 51, no. 2, pp. 339—360.
10. Lin F., Fardad M., Jovanovic M. Sparse feedback synthesis via the alternating direction, American Control Conference (ACC), 2012, pp. 4765—4770.
11. Lin F., Fardad M., Jovanovic M. Augmented Lagrangian approach to design of structured optimal state feedback gains, IEEE Transactions on Automatic Control, 2011, vol. 56, no. 12, pp. 2923—2929.
12. Nazin S. A., Polyak B. T., Topunov M. V. Rejection of bounded exogenous disturbances by the method of invariant ellipsoids, Automation and Remote Control, 2007, vol. 68, pp. 467—486.
13. Lin F., Fardad M., Jovanovic M. A linear matrix inequality approach to peak-to-peak gain, International Journal of Robust and Nonlinear Control, 1996, vol. 6, pp. 899—927.
14. Blanchini F., Sznaier M. Persistent disturbance rejection via static-state feedback, IEEE Transactions on Automatic Control, 1995, vol. 40, no. 6, pp. 1127—1131.
15. Furtat I. B. Robust Syncronization of the Structural Uncertainty Nonlinear Network with Delay and Disturbances, IFAC-PapersOnline, 2013 (Proc. of the 11th IFAC International Workshop on Adaptation and Learning in Control and Signal Processing, ALCOSP 2013), pp. 227—232.
16. Furtat I. B., Fradkov A. L. Robust Control of Multimachine Power Systems with Compensation of Disturbances, International Journal of Electrical Power & Energy Systems, 2015, vol. 73, pp. 584—590.
Review
For citations:
Vrazhevsky S.A., Chugina J.V., Furtat I.B., Konovalov D.E. Optimization of Invariant Ellipsoid Technique for Sparse Controllers Design. Mekhatronika, Avtomatizatsiya, Upravlenie. 2022;23(1):3-12. (In Russ.) https://doi.org/10.17587/mau.23.3-12
Views:
566
.png)
Email this article 