Abstract
The classical problem of analytical design of the optimal controllers which was stated and solved by A.M. Letov for continuous linear systems was considered for the class of logicdynamic (hybrid) systems. Equations to determine the optimal feedback control were derived on basis of the sufficient optimality conditions. In distinction to classical case, the optimal positional control is realized by the piecewise-linear controller with the piecewise-quadratic Bellman function. In contrast to the continuous, discrete or discrete-continuous systems, the optimal processes of logic-dynamic systems, can have multiple switchings in the logic block, thus hindering the optimal control design. Therefore it was suggested to consider a simplified problem where the logic-dynamic system is replaced by a discrete-continuous system admitting instantaneous multiple switchings of the discrete part. The resulting control for the logic-dynamic system becomes suboptimal. Application of the optimality and suboptimality conditions was demonstrated by way of several examples.
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References
Letov, A.M., Dinamika poleta i upravlenie (Flight Dynamics and Control), Moscow: Nauka, 1973.
Propoi, A.I., Elementy teorii optimal’nykh diskretnykh sistem (Elements of the Theory of Optimal Discrete Systems), Moscow: Nauka, 1973.
Bortakovskii, A.S. and Panteleev, A.V., Sufficient Conditions of Optimal Control of Continuous-Discrete Systems, Autom. Remote Control, 1987, vol. 48, no. 7, part 1, pp. 880–887.
Roitenberg, Ya.N., Avtomaticheskoe upravlenie (Automatic Control), Moscow: Nauka, 1977.
Butkovskii, A.G., Teoriya optimal’nogo upravleniya sistemami s raspredelennymi parametrami (Theory of Optimal Control of the Distributed-parameter Systems), Moscow: Nauka, 1985.
Zhuk, K.D. and Timchenko, A.A., Avtomatizirovannoe proektirovanie logiko-dinamicheskikh sistem (Computer-aided Design of Logic-Dynamic Systems), Kiev: Naukova Dumka, 1981.
Semenov, V.V., Dynamic Programming in the Design of Logic-Dynamic Systems, Priborostroenie, 1984, no. 9, pp. 71–77.
Vassilyev, S.N., Zherlov, A.K., Fedosov, E.A., et al., Intellektnoe upravlenie dinamicheskimi sistemami (Intelligent Control of Dynamic Systems), Moscow: Fizmatlit, 2000.
IFAC Workshop: Modelling and Analysis of Logic Controlled Dynamic Systems, Irkutsk: Inst. Syst. Dyn. Control Theory, Sib. Branch, Russ. Acad. Sci., 2003.
Bortakovskii, A.S., Suboptimal Control of Logical-Dynamic Systems under Parametric Uncertainty Conditions, Autom. Remote Control, 2007, vol. 68, no. 11, pp. 1986–2001.
Hybrid Systems, Grossmann, R.L., Nerode, A., Ravn, A.P., and Rischel, H., Eds., Berlin: Springer, 1993 (Lect. Notes Comput. Sci.; vol. 736).
Hybrid Systems. III, Alur, R., Henzinger, T.A., and Sontag, E.D., Eds., Berlin: Springer, 1996 (Lect. Notes Comput. Sci.; vol. 1066).
Hybrid Systems. V, Ahtsaklis, P., Kohn, W., Lemmon, M., Nerode, A., and Sastry, S., Eds., Berlin: Springer, 1999. (Lect. Notes Comput. Sci.; vol. 1567).
Matveev, A.S. and Savkin, A.V., Qualitative Theory of Hybrid Dynamical Systems, Boston: Birkhäuser, 2000.
Cassandras, C.G., Peryne, D.L., and Wardi, Y., Optimal Control of a Class of Hybrid Systems, IEEE Trans. Automat. Control, 2001, vol. 46, no. 3, pp. 398–415.
Savkin, A.V. and Evans, R.J., Hybrid Dynamical Systems: Controller and Sensor Switching Problems, Boston: Birkhäuser, 2002.
Hedlund, S. and Rantzer, A., Optimal Control of Hybrid Systems, in Proc. 38 IEEE Conf. Decision Control, Phoenix, 1999, pp. 3972–3977.
Gurman, V.I., Vyrozhdennye zadachi optimal’nogo upravleniya (Degenerate Problems of Optimal Control), Moscow: Nauka, 1973.
Bortakovskii, A.S. and Pegachkova, E.A., Design of Control of Satellite Active Stabilization on the Basis of the Necessary Optimality Conditions of the Logic-Dynamic Systems, Vestn. Mosk. Aviats. Inst., 2008, vol. 15, no. 2, pp. 28–35.
Emel’yanov, S.V., Sistemy avtomaticheskogo upravleniya s peremennoi strukturoi (Variable-structure Systems of Automatic Control), Moscow: Nauka, 1967.
Liberson, D., Switching in Systems and Control, Berlin: Springer, 2003.
Li, Z., Soh, Y., and Wen, C., Switched and Impulsive System: Analysis, Design and Applications, Berlin: Springer, 2005.
Miller, B.M. and Rubinovich, E.Ya., Optimizatsiya dinamicheskikh sistem s impul’snymi upravleniyami (Optimization of Dynamic Systems with Pulse Controls), Moscow: Nauka, 2004.
Kurzhanskii, A.B. and Tochilin, P.A., Pulse Control in Models of Hybrid Systems, Diff. Uravn., 2009, vol. 45, no. 3, pp. 716–727.
Bortakovskii, A.S., Design of Logic-Dynamic Systems on the Basis of Sufficient Optimality Conditions, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2010, no. 2, pp. 54–68.
Krotov, V.F. and Gurman, V.I., Metody i zadachi optimal’nogo upravleniya (Methods and Problems of Optimal Control), Moscow: Nauka, 1973.
Bortakovskii, A.S., Necessary Conditions for Optimality of the Automatic Part of the Logic-Dynamic System, Tr. Mat. Inst. Akad. Nauk, 2008, vol. 262, pp. 50–63.
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Original Russian Text © A.S. Bortakovskii, 2011, published in Avtomatika i Telemekhanika, 2011, No. 12, pp. 3–23.
Articles published on pp. 2425–2457, are the termination of thematic issue devoted to centenary of the birthday of A.M. Letov (see Autom. Remote Control, 2011, no. 11).
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Bortakovskii, A.S. Analytical design of optimal controllers in the class of logic-dynamic (hybrid) systems. Autom Remote Control 72, 2425–2444 (2011). https://doi.org/10.1134/S0005117911120010
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DOI: https://doi.org/10.1134/S0005117911120010