On the optimal control of the $n$-fold integrator

Yu. N. Gorelov

Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 10 (2015), pp. 114-143

Résumé

The optimal control problem $n$-fold integrator with arbitrary boundary conditions and functionals of type norms in spaces of $L_q[t_0,t_f]$, $q=1, 2, \infty$ is considered. First, it is the problem of minimizing the total controling impulse, which boils down to $L_\infty$-problem of moments; secondly, the problem of minimizing the maximum values of the control parameter (represented as $L_1$-problem of moments), and, finally, it is the problem of minimizing "generalized work control" (as $L_2$-problem of moments). Solving problems is obtained by using the method of moments in the form of the maximum principle by N. N. Krasovsky. It is shown that optimal control in the first problem is approximated by a $\delta$-impulsive control. Conditions for the existence of regular and singular solutions to this problem depending on the boundary conditions are also specified. The general solution of the second problem, which is the conditions for existence of regular and singular solutions and not equivalence with the mutual problem of time-optimal control is obtained. Examples of solution for the considered control tasks are given. In case of a quadratic functional general relations required for constructing a program optimal control were obtained.

Détail

Keywords: the $n$-fold integrator, optimal control, problem of moments, maximum principle by N.N. Krasovsky, Chebyshev polynomials.

Bibliographie

[1] Abgaryan K. A., Matrix calculous with applications in the theory of dynamic systems, Fizmatlit, M., 1994, 544 pp. (in Russian)

[2] Krasovsky N. N., Motion control theory: linear systems, Nauka, M., 1965, 476 pp. (in Russian) | MR

[3] Moroz A. I., Course of systems theory, Vysshaia shkola, M., 1987, 304 pp. (in Russian)

[4] Voronov A. A., Kim D. P., Lokhin V. M. et al., Theory of automatic control, In 2 parts, v. II, Theory of nonlinear and special systems of automatic control, 2nd ed., revised and enlarged, ed. A. A. Voronov, Vysshaia shkola, M., 1986, 504 pp. (in Russian)

[5] Butkovskij A. G., Methods of control of systems with distributed parameters, Nauka, M., 1975, 568 pp. (in Russian)

[6] Sinyakov A. N., Control system of elastic moving objects, Izd-vo LGU, L., 1981, 200 pp. (in Russian)

[7] Korobov V. I., Sklyar G. M., “Optimality and the power moment problem”, Sbornik: Mathematics, 134(176):2(10) (1987), 186–206 (in Russian) | MR | Zbl

[8] Korobov V. I., Sklyar G. M., “Exact solution of an n-dimensional optimal control problem”, Proceedings of the USSR Academy of Sciences, 296:6 (1988), 1304–1308 (in Russian)

[9] Widnall W. S., “The optimal law for the thrust control vector in the autopilot of the lunar module spacecraft “Apollo””, Control in space, Proceedings of the III International Symposium IFAC on automatic control in the peaceful uses of outer space, v. 2, Nauka, M., 1972, 36–49 (in Russian) | Zbl

[10] Gorelov Yu. N., Morozova M. V., “Optimal control of the threefold integrator according to minimum consumption”, Vestnik of Samara State University, 2012, no. 9(100), 118–129 (in Russian)

[11] Gorelov Yu. N., Morozova M. V., “Synthesis of optimal control of spacecraft partial rotation by moments method”, Proceedings of the Samara Scientific Center of the Russian Academy of Sciences, 14:6 (2012), 166–176 (in Russian)

[12] Yu. N. Gorelov et al., “On Optimization of Attitude Control Programs for Earth Remote Sensing Satellite”, Gyroscopy and Navigation, 5:2 (2014), 90–97 (in English) | DOI

[13] Gorelov Yu. N., “On the solution of the optimal control synthesis problem of reorientation in sensing hardware redirection by one successive approxmations method”, Proceedings of the Samara Scientific Center of the Russian Academy of Sciences, 16:4 (2014), 127–131 (in Russian)

[14] Danilov Yu. A., Chebyshev Polynomials, Editorial URSS, M., 2003, 160 pp. (in Russian)

[15] Voevodin V. V., Kuznetsov Yu. A., Matrices and calculations, Nauka, M., 1984, 320 pp. (in Russian) | MR

[16] Korneichuk N. P., Motorny V. P., “The Least deviating from zero polynomial”, Encyclopedia of Mathematics, v. 3, ed. I. M. Vinogradov, Izd-vo “Sovetskaia entsiklopediia”, M., 1982, 874–875 (in Russian)

[17] Loskutov E. M., “On the optimal reorientation problem of the spacecraft”, Space researches, 11:2 (1973), 180–187 (in Russian)

[18] Gorelov Yu. N., Titov B. A., “On optimal reorientation of a rotating spacecraft”, Space researches, 16:2 (1978), 157–162 (in Russian)