Abstract
The method of computation of control in real time of a linear system with disturbance is suggested. The system of linear algebraic equations is obtained, which links the deviations of phase coordinates to the deviations of initial conditions of the normalized conjugate system and to the deviation of the finite moment. The calculations reduce to the sequence of the solutions of systems of linear algebraic equations and the integration of a matrix differential equation over transfer intervals of the control switching moments and the finite moment of time. The correction of switching moments and the finite moment of control in the accompaniment of the phase trajectory of motion of a controllable object is considered. Simple constructive conditions of the origin of the sliding mode, motions of the representative point over manifolds of switchings, and changes of the control structure in accompanying the phase trajectory of the system motion are obtained. The convergence of the computational method is proved.
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Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F., Matematicheskaya teoriya optimal’nykh protsessov (Mathematical Theory of Optimal Processes), Moscow: Nauka, 1976.
Fedorenko, R.P., Priblizhennoe reshenie zadach optimal’nogo upravleniya (Approximate Solution of Optimal Control Problems), Moscow: Nauka, 1978.
Dikusar, V.V. and Milyutin, A.A., Kachestvennye i chislennye metody v printsipe maksimuma (Qualitative and Numerical Methods in the Maximum Pricniple), Moscow: Nauka, 1989.
Srochko, V.A., Iteratsionnye metody resheniya zadach optimal’nogo upravleniya (Iterative Methods of Solution of Optimal Control Problems), Moscow: Fizmatlit, 2000.
Dyurkovich, E., Numerical Method of Solution of Linear Speed Problems with Accuracy Estimation, Dokl. Akad. Nauk SSSR, 1982, vol. 265, no. 4, pp. 793–797.
Kiselev, Yu.N., Fast-Convergent Algorithms for Linear Optimal Speed, Kibernetika, 1990, vol. 62, no. 6, pp. 47–57.
Shevchenko, G.V., Numerical Algorithm of Solution of a Linear Optimal Speed Problem, Zh. Vychisl. Mat. Mat. Fiz., 2002, vol. 42, no. 8, pp. 1184–1196.
Hartl, R.E., Sethi, S.P., and Vickson, R.G., A Servey of the Maximum Principle for Optimal Control Problems with State Constraints, SIAM Rev., 1995, vol. 37, pp. 181–218.
Aleksandrov, V.M., Numerical Method of Solution of a Linear Speed Problem, Zh. Vychisl. Mat. Mat. Fiz., 1998, vol. 38, no. 6, pp. 918–931.
Aleksandrov, V.M., Sequential Synthesis of Optimal Speed Control, Zh. Vychisl. Mat. Mat. Fiz., 1999, vol. 33, no. 9, pp. 1464–1478.
Gabasov, R., Kirillova, F.M., and Kostyukova, O.I., Optimization of the Linear Control System in Real Time Mode, Izv. Ross. Akad. Nauk, Tekhn. Kibern., 1992, no. 4, pp. 3–19.
Blashevich, N.V., Gavasov, R., and Kirillova, F.M., Numerical Methods of Programmed and Positional Optimization of Linear Control Systems, Zh. Vychisl. Mat. Mat. Fiz., 2000, vol. 40, no. 6, pp. 838–859.
Gabasov, R. and Kirillova, F.M., Optimal Control in Real Time Mode, Second Int. Conf. Probl. Control, Plenary Reports, Moscow: Inst. Probl. Upravlen., 2003, pp. 20–47.
Aleksandrov, V.M., Optimal Speed Control in Real time Mode, Proc. XIII Baikal. Int. School-Seminar, Optim. Control, 2005, vol. 2, pp. 57–63.
Aleksandrov, V.M., Optimal Speed Control in Real Time of Linear Systems, III Int. Conf. Probl. Control, Plenary Reports and Selec. Works, Moscow: Inst. Probl. Upravlen., 2006, pp. 156–163.
Aleksandrov, V.M., Iterative Method of Calculation in Real Time of Optimal Speed Control, Sib. Zh. Vychisl. Mat., 2007, vol. 10, no. 1, pp. 1–28.
Aleksandrov, V.M., Sequential Synthesis of Optimal Speed Control of Linear Systems with Disturbances, Sib. Zh. Vychisl. Mat., 2008, vol. 11, no. 3, pp. 251–270.
Aleksandrov, V.M., Sequential Synthesis of the Time-optimal Control in Real Time, Autom. Remote Control, 2008, no. 8, pp. 1271–1288.
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Original Russian Text © V.M. Aleksandrov, 2009, published in Avtomatika i Telemekhanika, 2009, No. 4, pp. 58–77.
This work was supported by the Russian Foundation for Basic Research, project no. 09-01-00155 and Ural Branch, Russian Academy of Sciences, project no. 85.
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Aleksandrov, V.M. Features of motion of dynamic systems with disturbances in the neighborhood of manifolds of switchings. Autom Remote Control 70, 615–632 (2009). https://doi.org/10.1134/S0005117909040080
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DOI: https://doi.org/10.1134/S0005117909040080