Abstract
It is shown how the synthesis of stabilizing, H∞-optimal, and robust controllers of the prescribed order in the output for linear continuous and discrete systems reduces to a solution of linear matrix inequalities relative to two reciprocal matrices. To verify the solvability of such inequalities and to find their solutions, an algorithm realizable in the MATLAB package is proposed. An example of stabilization of an inverted pendulum with the aid of a controller of the reduced order in the output is given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Boyd, S., Al Ghaouri, L., Feron, E., et al., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.
Iwasaki, T. and Skelton, R.E., All Controllers for the General H ∞ Control Problem: LMI Existence Conditions and State Space Formulas, Automatica, 1994, vol. 30, no. 88, pp. 1307–1317.
Gahinet, P. and Apkarian, P., A Linear Matrix Inequality Approach to H ∞ Control, Int. J. Robust Nonlinear Control, 1994, vol. 4, pp. 421–448.
Gahinet, P., Nemirovski, A., Laub, A., et al., LMI Control Toolbox. For Use with MATLAB, Natick: The Math. Works Inc., 1995.
Apkarian, P. and Gahinet, P., A Convex Characterization of Gain-Scheduled H ∞ Controllers, IEEE Trans. Automat. Control, 1995, vol. 40, no. 5, pp. 853–864.
Apkarian, P. and Tuan, H.D., Robust Control via Concave Minimization Local and Global Algorithms, IEEE Trans. Automat. Control, 2000, vol. 45, no. 2, pp. 299–305.
El Ghaoui, L., Balakrishnan, V., Feron, E., et al., On Maximizing a Robustness Measure for Structured Nonlinear Perturbations, Proc. Am. Control Conf., 1992, pp. 2923–2924.
Packard, A., Zhou, K., Pandey, P., et al., Optimal Constant I/O Similarity Scaling for Full-Information and State Feedback Control Problems, Syst. Control Lett., 1992, vol. 19, pp. 271–280.
Yamada, Y. and Hara, S., Global Optimization for H ∞ Control with Constant Diagonal Scaling, IEEE Trans. Automat. Control, 1998, vol. 43, no. 2, pp. 191–203.
Scherer, C.W., Gahinet, P., and Chilali, M., Multi-Objective Output-Feedback Control via LMI Optimization, IEEE Trans. Automat. Control, 1997, vol. 42, pp. 896–911.
Leibfritz, F., A LMI-Based Algorithm for Designing Suboptimal Static Output Feedback Controllers, SIAM J. Contr. Optimiz., 2001, vol. 39, no. 6, pp. 1711–1735.
Balandin, D.V. and Kogan, M.M., Synthesis of Optimal Robust H ∞-Control of Convex Optimization Methods, Avtom. Telemekh., 2004, no. 7, pp. 71–81.
Balandin, D.V. and Kogan, M.M., An Optimization Algorithm for Checking Feasibility of Robust H ∞-Control Problem for Linear Time-Varying Uncertain Systems, Int. J. Control, 2004, vol. 77, no. 5, pp. 498–502.
Horn, R. and Johnson, C.R., Matrix Analysis, Cambridge: Cambridge Univ. Press, 1986. Translated under the title Matrichtyi analiz, Moscow: Mir, 1989.
Author information
Authors and Affiliations
Additional information
Translated from Avtomatika i Telemekhanika, No. 1, 2005, pp. 82–99.
Original Russian Text Copyright © 2005 by Balandin, Kogan.
This work was supported by the Russian Foundation for Basic Research, project nos. 04-01-0022, 02-01-00230, the “Universities of Russia” Program, project no. UR.03.01.013, and INTAS, project no. 03-51-5547.
Rights and permissions
About this article
Cite this article
Balandin, D.V., Kogan, M.M. Synthesis of controllers on the basis of a solution of linear matrix inequalities and a search algorithm for reciprocal matrices. Autom Remote Control 66, 74–91 (2005). https://doi.org/10.1007/s10513-005-0008-2
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10513-005-0008-2