Methodology for monitoring the technical condition of onboard launch vehicle systems based on the processing of rapidly changing

DOI: 10.34759/trd-2021-121-19

Аuthors

Uryupin I. P.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: uryupin93@yandex.ru

Abstract

The purpose of the research consists in developing an optimization algorithm for continuous systems in the class of piecewise constant controls. In classical problems of optimal control of continuous systems, the admissible controls are, as a rule, bounded measurable ones (in applied problems, they are piecewise continuous). In some complex continuous systems, physical implementation of such admissible control seems impossible Then one can narrow down the set of admissible controls, for example, to the class of piecewise constant controls with a fixed number of switchings and search for a solution in this narrow class. It is clear, the suboptimal controls herewith will be obtained, which, however, with an unlimited increase in the number of switchings, will tend to the optimal one. Thus, the problem of minimizing the number of switchings becomes actual.

To solve the set problem, a numerical-analytical algorithm has been developed based on the necessary conditions for optimality of switched systems. Boundary problems and formulas expressing the optimal values of piecewise constant control are obtained analytically, according to the necessary conditions. The resulting system of equations, as a rule, is transcendental and its solution is rather difficult. Thus, the author proposes to use the numerical minimization of the functional, followed by fulfillment verification of the necessary conditions to search for the optimal switching moments. The proposed algorithm is realized in MATLAB.

The problem of a linear oscillator controlling with quadratic functional is being considered as an example. The optimal solution to this problem in the class of continuous controls was obtained using the maximum principle. In this work, the optimal solution to the problem in a narrower class of piecewise constant controls is being searched for. This solution is being found using necessary conditions for the switched systems optimality. It may be regarded as a solution close to optimal continuous control. Besides the problem of an optimal piecewise constant control synthesizing, the problem of finding the minimum number of switchings, at which the difference between the approximate solution and the exact one does not exceed a given error is also solved.

The main result of the work is a numerical-analytical algorithm for minimizing the number of switchings of optimal piecewise-constant controls for an approximate solution of the optimal control problem for continuous systems.

Keywords:

theory of control, switched systems, switching minimization, optimization, algorithm, hybrid systems

References

1. Savkin A.V., Evans R.J. Hybrid dynamical systems: Controller and sensor switching problems, Boston, Birkhäuser, 2002. DOI:10.1007/978-1-4612-0107-6
2. Hedlund S., Rantzer A. Optimal Control of Hybrid Systems, Proceedings of the 38th IEEE Conference on Decision and Control, 1999, pp. 3972-3977. DOI:10.1109/CDC.1999.827981
3. Gurman V.I. Izvestiya RAN. Teoriya i sistemy upravleniya, 2004, no. 4, pp. 70-75.
4. Branicky M.S., Borkar V.S., Mitter S.K. A unified framework for hybrid control: Model and optimal control theory, IEEE Transactions on Automatic Control, 1998, vol. 43, no. 1, pp. 31-45.
5. Liberzon D. Switching in Systems and Control, Berlin, Springer, 2003, 243 p.
6. Kezling G.B. Tekhnicheskie sredstva ASU (Тechnical means automatic control system), Leningrad, Mashinostroenie, 1986. Vol. 2, 780 p.
7. Bortakovskii A.S. Izvestiya RAN. Teoriya i sistemy upravleniya, 2010, no. 2, pp. 41-55.
8. Zhuk K.D., Timchenko A.A., Dalenko T.I. Issledovanie struktur i modelirovanie logiko-dinamicheskikh system (Research of structures and modeling of logical-dynamic systems), Kiev, Naukova dumka, 1975, 199 p.
9. Bortakovskii A.S., Pegachkova E.A. Trudy MAI, 2007, no. 27. URL: http://www.mai.ru/eng/publications/index.php?ID=33013
10. Bortakovskii A.S. Trudy MIAN, 2008, vol. 262, pp. 50–63.
11. Nemychenkov G.I. Trudy MAI, 2016, no. 89. URL: http://trudy.mai.ru/eng/published.php?ID=73376
12. Pegachkova E.A. Trudy MAI, 2011, no. 49. URL: http://trudymai.ru/eng/published.php?ID=27614&PAGEN_2=2
13. Sun Z., Ge S. Analysis and Synthesis of Switched Linear Control Systems, Automatica, 2005, vol. 41 (2), pp. 181-195. DOI:10.1016/j.automatica.2004.09.015
14. Xu X., Antsaklis P.J. Optimal Control of Switched Systems Based on Parameterization of the Switching Instants, IEEE Transactions on Automatic Control, 2004, vol. 49 (1), pp. 2-16. DOI:10.1109/TAC.2003.821417
15. Bortakovskii A.S., Uryupin I.V. Izvestiya RAN. Teoriya i sistemy upravleniya, 2019, no. 4, pp. 29-46.
16. Bortakovsky A.S., Uryupin I.V. Optimization of Switchable Systems’ Trajectories, Journal of Computer and Systems Sciences International, 2021, vol. 60 (5), pp.701—718. DOI: 10.1134/s1064230721050051
17. Bortakovskii A.S., Uryupin I.V. Vestnik komp’yuternykh i informatsionnykh tekhnologii, 2019, no. 11, pp. 13 — 22.
18. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. Matematicheskaya teoriya optimal’nykh protsessov (Mathematical theory of optimal processes), Moscow, Fizmatgiz, 1961, 391 p.
19. Uryupin I.V. Trudy MAI, 2018, no. 100. URL: http://trudymai.ru/eng/published.php?ID=93440
20. Zhuravin Yu. Razgonnyi blok «Briz-M», Novosti kosmonavtiki, 2000, vol. 10, no. 8(211), pp. 52-55.
21. Bortakovskii A.S., Konovalova A.A. Teoriya i sistemy upravleniya, 2014, no. 5, pp. 38.
22. Agrachev A.A., Sachkov Yu.L. Geometricheskaya teoriya upravleniya (Geometric control theory), Moscow, Fizmatlit, 2005, 391 p.
23. Dmitruk A.V., Kaganovich A.M. The Hybrid Maximum Principle is a consequence of Pontryagin Maximum Principle, Systems & Control Letters, 2008, pp. 964–970. DOI:10.1016/j.sysconle.2008.05.006

Download