Abstract
This paper studies two optimal control problems for a fractional-order pendulum in the case when admissible control actions belong to the class of square integrable functions on a segment. The first problem is to find control actions transferring a system to a given state with the minimum control norm under a fixed control time. The second problem is to find control actions transferring the system to a given state within the minimum time under a given constraint on the control norm. The authors demonstrate that the problem can be reduced to the problem of moments, as well as derive the feasible statement and solvability conditions for the latter. Solution of the problem is obtained analytically in the form of quadratures. A series of computing experiments are conducted and the qualitative features of system dynamics are discussed.
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Tarasov, V.E., Modeli teoreticheskoi fiziki s integro-differentsirovaniem drobnogo poryadka (Models ofTheoretical Physics with Fractional-order Integro-Differentiation), Izhevsk: RKhD, 2011.
Uchaikin, V.V., Metod drobnykh proizvodnykh (Method of Fractional Derivatives), Ul’yanovsk: Artishok, 2008.
Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J., Theory and Applications of Fractional DifferentialEquations, Amsterdam: Elsevier, 2006.
Butkovskii, A.G., Postnov, S.S., and Postnova, E.A., Fractional Integro-Differential Calculus and ItsControl-Theoretical Applications. I. Mathematical Fundamentals and the Problem of Interpretation, Autom. Remote Control, 2013, vol. 74, no. 4, pp. 543–574.
Butkovskii, A.G., Postnov, S.S., and Postnova, E.A., Fractional Integro-Differential Calculus and ItsControl-Theoretical Applications. II. Fractional Dynamic Systems: Modeling and Hardware Implementation, Autom. Remote Control, 2013, vol. 74, no. 5, pp. 725–749.
Dulf, E.-H., Pop, C.-I., and Dulf, F.-V., Fractional Calculus in 13C Separation Column Control, Signal, Image Video Proces., 2012, vol. 6, pp. 479–485.
Bruzzone, L. and Fanghella, P., Fractional-Order Control of a Micrometric Linear Axis, J. Control Sci.Eng., 2013, vol. 2013, article ID 947428.
Monje, C.A., et al., Fractional-order Systems and Controls: Fundamentals and Applications, London: Springer-Verlag, 2010.
Caponetto, R., et al., Fractional Order Systems. Modeling and Control Applications, Singapore: World Scientific, 2010.
Agrawal, O.P., A General Formulation and Solution Scheme for Fractional Optimal Control Problems, Nonlin. Dynam., 2004, vol. 38, pp. 323–337.
Agrawal, O.P., A Formulation and Numerical Scheme for Fractional Optimal Control Problems, J. Vibr.Control, 2008, vol. 14, no. 9–10, pp. 1291–1299.
Frederico, G.S.F. and Torres, D.F.M., Fractional Optimal Control in the Sense of Caputo and theFractional Noether’s Theorem, Int. Math. Forum, 2008, vol. 3, no. 10, pp. 479–493.
Postnov, S.S., A Study of the Optimal Control Problem for Single and Double Fractional Order Integratorswith the Method of Moments, Probl. Upravlen., 2012, no. 5, pp. 9–17.
Kubyshkin, V.A. and Postnov, S.S., Investigation of Optimal Control Problem for Single and Double Integrators of Fractional Order Using the Method of Moments in CaseWhen Admissible Control Belongs to Lp[0, T] Space Probl. Upravlen. 2013, no. 3, pp. 9–17.
Krasovskii, N.N., Teoriya upravleniya dvizheniem (Theory of Motion Control), Moscow: Nauka, 1968.
Butkovskii, A.G., Fazovye portrety upravlyaemykh dinamicheskikh sistem (Phase Portraits of ControllableDynamic Systems), Moscow: Nauka, 1985.
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Original Russian Text © V.A. Kubyshkin, S.S. Postnov, 2014, published in Problemy Upravleniya, 2014, No. 3, pp. 14–22.
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Kubyshkin, V.A., Postnov, S.S. Analysis of two optimal control problems for a fractional-order pendulum by the method of moments. Autom Remote Control 76, 1302–1314 (2015). https://doi.org/10.1134/S0005117915070152
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DOI: https://doi.org/10.1134/S0005117915070152