Abstract
for the linear dynamic plant that is subjected to exogenous action and uncertainty generated by the unknown initial plant conditions, the level of perturbation suppression was determined as the greatest value of the ratio of the L2-norm of the objective output to the quadratic root of the sum of the squared L2-norm of the exogenous disturbance and norm of the initial state taken with a weight coefficient. Determination of the worst exogenous disturbance and the initial state maximizing this index was demonstrated. The role of the weight coefficient in the trade-off between the H∞-norm, that is, the level of suppression of the exogenous disturbance under the zero initial conditions, and the level γ0 of suppression of the initial uncertainties, was clarified in the absence of exogenous disturbance. The generalized H∞-optimal laws of state and output control minimizing the chosen criterion were designed in terms of the linear matrix inequalities. The advantage of the generalized controller over the ordinary H∞-optimal controller was demonstrated by way of an example of the linear oscillator with unknown initial conditions.
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Doyle, J.C., Glover, K., Khargonekar, P.P., and Francis, B.A., State-space Solutions to Standard H 2 and H∞ Control Problems, IEEE Trans. Automat. Control, 1989, vol. 34, no. 8, pp. 831–847.
Kwakernaak, H., Robust Control and H∞-optimization—Tutorial Paper, Automatica, 1993, vol. 29, no. 2, pp. 255–273.
Balandin, D.V. and Kogan, M.M., Linear-Quadratic and γ-Optimal Output Control Laws, Autom. Remote Control, 2008, no. 6, pp. 911–919.
Balandin, D.V. and Kogan, M.M., LMI Based Output-Feedback Controllers: γ-Optimal Versus Linear Quadratic, in Proc. 17th World IFAC Congr., Seoul, Korea, 2008, pp. 9905–9909.
Khargonekar, P.P., Nagpal, K.M., and Poolla, K.R., H∞ Control with Transients, SIAM J. Control Optim., 1991, vol. 29, no. 6, pp. 1373–1393.
Boyd, S., El Ghaoui L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.
Balandin, D.V. and Kogan, M.M., Sintez zakonov upravleniya na osnove lineinykh matrichnykh ner-avenstv (Design of the Control Laws Based on Linear Matrix Inequalities), Moscow: Fizmatlit, 2007.
Gahinet, P., Nemirovski, A., Laub, A., and Chilali, M., LMI Control Toolbox. For Use with MATLAB, Philadelphia: The Math Works Inc., 1995.
Basar, T. and Bernhard, P., H∞-optimal Control and Related Minimax Design Problems: A Dynamic Game Approach, Boston: Birkhauser, 1995.
Kogan, M.M., The Worst Perturbation and Minimax Control for Continuous Linear Systems: Solution of the Inverse Problems, Autom. Remote Control, 1997, no. 4, part 1, pp. 535–541.
Balandin, D.V., Limit Possibilities of Linear System Control, Dokl. Ross. Akad. Nauk, 1994, vol. 334, no. 5, pp. 571–573.
Horn, R. and Johnson, C, Matrix Analysis, New York: Cambridge Univ. Press, 1985. Translated under the title Matrichnyi analiz, Moscow: Mir, 1989.
Gelig, A.Kh., Leonov, G.A., and Yakubovich, V.A., Ustoichivost’ nelineinykh sistem s needinstvennym sostoyaniem ravnovesiya (Stability of Nonlinear Systems with Nonunique Equilibrium State), Moscow: Nauka, 1978.
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Original Russian Text © D.V. Balandin, M.M. Kogan, 2010, published in Avtomatika i Telemekhanika, 2010, No. 6, pp. 20–38.
This work was supported by the Russian Foundation for Basic Research, projects nos. 07-01-00481, 08-01-00422, 08-01-97034-r-povolzh’e and FTsP “Scientific and Pedagogical Personnel of the Innovation Russia.”
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Balandin, D.V., Kogan, M.M. Generalized H∞-optimal control as a trade-off between the H∞-optimal and γ-optimal controls. Autom Remote Control 71, 993–1010 (2010). https://doi.org/10.1134/S0005117910060020
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DOI: https://doi.org/10.1134/S0005117910060020