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Optimization of parameters of asymptotically stable systems. (English) Zbl 1214.93097

Summary: This work deals with numerical methods of parameter optimization for asymptotically stable systems. We formulate a special mathematical programming problem that allows us to determine optimal parameters of a stabilizer. This problem involves solutions to a differential equation. We show how to chose the mesh in order to obtain discrete problem guaranteeing the necessary accuracy. The developed methodology is illustrated by an example concerning optimization of parameters for a satellite stabilization system.

MSC:

93D21 Adaptive or robust stabilization
93D20 Asymptotic stability in control theory
49M25 Discrete approximations in optimal control
49N90 Applications of optimal control and differential games
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References:

[1] R. N. Ismailov, “The peack effect in stationary linear systems with scalar inputs and outputs,” Automation and Remote Control, vol. 48, no. 8, part 1, pp. 1018-1024, 1987. · Zbl 0711.93040
[2] H. J. Sussmann and P. V. Kokotović, “The peaking phenomenon and the global stabilization of nonlinear systems,” IEEE Transactions on Automatic Control, vol. 36, no. 4, pp. 424-440, 1991. · Zbl 0749.93070 · doi:10.1109/9.75101
[3] Ya. Z. Cypkin and P. V. Bromberg, “On the degree of stability of linear systems,” USSR Academy of Sciences, Branch of Technical Sciences, vol. 1945, no. 12, pp. 1163-1168, 1945. · Zbl 0060.05705
[4] F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1983. · Zbl 0582.49001
[5] G. Smirnov and F. Miranda, “Filippov-Gronwall inequality for discontinuous differential inclusions,” International Journal of Mathematics and Statistics, vol. 5, no. A09, pp. 110-120, 2009.
[6] V. A. Sarychev, “Investigation of the dynamics of a gravitational stabilization system,” in XIIIth International Astronautical Congress, pp. 658-690, Varna, Bulgaria, 1964.
[7] R. L. Borrelli and I. P. Leliakov, “An optimization technique for the transient response of passively stable satellites,” Journal of Optimization Theory and Applications, vol. 10, pp. 344-361, 1972. · Zbl 0233.70018 · doi:10.1007/BF00935399
[8] V. A. Sarychev and V. V. Sazonov, “Optimal parameters of passive systems for satellite orientation,” Cosmic Research, vol. 14, no. 2, pp. 183-193, 1976.
[9] R. Fletcher, Practical Methods of Optimization, John Wiley & Sons, Chichester, UK, 2nd edition, 1987. · Zbl 0905.65002
[10] A. M. Liapunov, Stability of Motion, Academic Press, New York, NY, USA, 1966.
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