Abstract
A new approach to realization of the time-optimal control in real time for linear systems under control with a constraint is proposed. It is based on dividing the computer costs into those made in advance of the control process and those carried out as it proceeds. The preliminary computations do not depend on a particular initial condition and rely on approximation of attainability sets in different periods of time by a union of hyperplanes. Methods of their construction and identification of the support hyperplane are given. Methods of approximate finding and subsequent refining of the normalized vector of the initial conditions of the adjoint system as well as switching times and instants of switching of time-optimal control are proposed. Results of modeling and numerical calculations are presented.
Similar content being viewed by others
References
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.
Boltyanskii, V.G., Matematicheskie metody optimal’nogo upravleniya (Mathematical Methods of the Optimal Control), Moscow: Nauka, 1969.
Fedorenko, R.P., Priblizhennoe reshenie zadach optimal’nogo upravleniya (Approximate Solution to Optimal Control Problems), Moscow: Nauka, 1978.
Lyubushin, A.A., Application of Modifications of the Method of Consecutive Approximations for Solving Optimal Control Problems, Zh. Vych. Mat. Mat. Fiz., 1982, vol. 22, no. 1, pp. 30–35.
Grachev, N.I. and Evtushenko, Yu.G., Library of Programs for Solving Optimal Control Problems, Zh. Vych. Mat. Mat. Fiz., 1979, vol. 19, no. 2, pp. 367–387.
Gabasov, R. and Kirillova, F.M., Constructing Consecutive Approximations for Some Optimal Control Problems, Avtom. Telemekh., 1966, vol. 27, no. 2, pp. 5–17.
Pshenichnyi, B.N. and Sobolenko, L.A., Accelerated Method for Solving a Linear Time-Optimal Control Problem, Zh. Vych. Mat. Mat. Fiz., 1968, vol. 8, no. 6, pp. 1343–1351.
Belolipetskii, A.A., Numerical Method for Solving a Linear Problem of the Optimal Control by Reducing It to the Cauchy Problem, Zh. Vych. Mat. Mat. Fiz., 1977, vol. 17, no. 6, pp. 1380–1386.
Kiselev, Yu.N. and Orlov, M.V., Numerical Algorithms of Linear Time-Optimal Control, Zh. Vych. Mat. Mat. Fiz., 1991, vol. 31, no. 12, pp. 1763–1771.
Srochko, V.A., Iteratsionnye metody resheniya zadach optimal’nogo upravleniya (IterativeMethods for Solving Optimal Control Problems), Moscow: Fizmatlit, 2000.
Aleksandrov, V.M., Numerical Method for Solving a Linear Time-Optimal Control Problem, Zh. Vych. Mat. Mat. Fiz., 1998, vol. 38, no. 6, 918–931.
Chernous’ko, F.L., Ellipsoidal Estimates of the Attainability Domain of a Controlled System, Prikl. Mat. Mekh., 1981, vol. 45, no. 1, pp. 11–19.
Aleksandrov, V.M., Time-Optimal Position-Program Control of Linear Dynamic Systems, Sib. Elektron. Mat. Izv., 2009, vol. 6, pp. 385–439.
Gantmakher, F.R., Teoriya matrits (Theory ofMatrices), Moscow: Nauka, 1966.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.M. Aleksandrov, 2012, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2012, Vol. 15, No. 1, pp. 1–19.
Rights and permissions
About this article
Cite this article
Aleksandrov, V.M. Forming an approximating construction for calculation and implementation of optimal control in real time. Numer. Analys. Appl. 5, 1–16 (2012). https://doi.org/10.1134/S1995423912010016
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995423912010016