Abstract
This paper considers a control problem for a linear singularly perturbed system with minimum energy. The terminal state of the system and the transition time are given. We construct asymptotic approximations of optimal programmed control and optimal feedback control in the problem. A main advantage of the proposed algorithms is decomposing the initial optimal control problem into two unperturbed problems of smaller dimension.
Similar content being viewed by others
References
Dmitriev, M.G. and Kurina, G.A., Singular Perturbation in Control Problems, Autom. Remote Control, 2006, vol. 67, no. 1, pp. 1–43.
Kalinin, A.I., Asymptotics of the Solutions of Perturbed Optimal Control Problems, J. Comput. Syst. Sci. Int., 1995, vol. 33, no. 6, pp. 75–84.
Kokotovic, P.V. and Khalil, H.K., Singular Perturbations in Systems and Control, New York: IEEE Press, 1986.
Rakitskii, Yu.V., Ustinov, S.M., and Chernorutskii, I.G., Chislennye metody resheniya zhestkikh sistem (Numerical Methods for Solving Stiff Systems), Moscow: Nauka, 1979.
Kokotovic, P.V. and Jackel, R.A., Singular Perturbation of Linear Regulators: Basic Theorems, IEEE Trans. Automat. Control, 1972, vol. 17, no. 1, pp. 29–37.
Wilde, R.R. and Kokotovic, P.V., Optimal Open- and Closed Loop Control of Singularly Perturbed Linear Systems, IEEE Trans. Automat. Control, 1973, vol. 18, no. 6, pp. 616–626.
Glizer, V.Ya. and Dmitriev, M.G., Singular Perturbations in the Linear Optimal Control Problem with a Quadratic Functional, Dokl. Akad. Nauk SSSR, 1975, vol. 225, no. 5, pp. 997–1000.
O’Malley, R.E., Jr., Singular Perturbation and Optimal Control, Lecture Notes. Math., 1978, vol. 680, pp. 171–218.
Kalinin, A.I. and Lavrinovich, L.I., Asymptotics of the Solution to a Singularly Perturbed Linear- Quadratic Optimal Control Problem, Comput. Math. Math. Phys., 2015, vol. 55, no. 2, pp. 194–205.
Krasovskii, N.N., Teoriya upravleniya dvizheniem (Theory of Motion Control), Moscow: Nauka, 1968.
Mordukhovich, B.Sh., The Existence of Optimal Controls, in Sovremennye problemy matematiki (Itogi nauki i tekhniki) (Modern Problems of Mathematics (The Outcomes of Science and Technology)), Moscow: VINITI, 1976, vol. 6, pp. 207–271.
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F., Matematicheskaya teoriya optimal’nykh protsessov, Moscow: Nauka, 1983. Translated under the title L.S. Pontryagin Selected Works, vol 4: The Mathematical Theory of Optimal Processes, New York: Gordon and Breach, 1986.
Gabasov, R. and Kirillova, F.M., Optimizatsiya lineinykh sistem (Optimization of Linear Systems), Minsk: BGU, 1973.
Donchev, A.L. and Gichev, T.R., The Correctness of Singular Perturbation of Optimal Control Problems with a Convex Performance Criterion, Dokl. Bolgarsk. Akad. Nauk, 1977, vol. 30, no 10, pp. 1393–1394.
Vasil’eva, A.B. and Butuzov, V.F., Asimptoticheskie razlozheniya reshenii singulyarno vozmushchennykh uravnenii (Asymptotic Expansions of Solutions of Singularly Perturbed Equations), Moscow: Nauka, 1973.
Gabasov, R. and Kirillova, F.M., Konstruktivnye metody optimizatsii. Chast’ 2 (Constructive Optimization Methods. Part 2), Minsk: Universitetskoe, 1984.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.I. Kalinin, L.I. Lavrinovich, 2016, published in Avtomatika i Telemekhanika, 2016, No. 5, pp. 3–18.
Rights and permissions
About this article
Cite this article
Kalinin, A.I., Lavrinovich, L.I. Application of the small parameter method to the singularly perturbed linear-quadratic optimal control problem. Autom Remote Control 77, 751–763 (2016). https://doi.org/10.1134/S0005117916050015
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117916050015