Stability of the program manifold of automatic indirect control systems taking into account the external load

Yıl 2023, Cilt: 7 Sayı: 2, 405 - 412, 23.07.2023

Sailaubay Zhumatov Sandugash Mynbayeva

https://doi.org/10.31197/atnaa.1200890

Öz

The problems of stability system that arises in the construction of different automatic systems of indirect control are considered. It is known that a given program is not always exactly performed, since there are always initial, constantly acting perturbations.
Therefore, it is also reasonable to require the stability of the program manifold itself with respect to some function. In the first part, the stability being investigated of automatic indirect control systems with rigid and tachometric feedback. Necessary and sufficient conditions for the absolute stability of a program manifold are established separately. In the second part, the automatic systems of indirect control taking into account the external load are considered. The equations of the hydraulic actuator, taking into account the action of an external load, are presented in a convenient form for research. Then it reduces to studying the stability of the system of equations with respect to a given program manifold. By constructing LyapunovВ’s functions for the system in canonical form, sufficient conditions are obtained for the absolute stability of the program manifold. The results obtained can be used in the construction of stable automatic indirect control systems.

Anahtar Kelimeler

program manifold, indirect control system, external load, rigid and tachometric feedback, Lyapunov function

Destekleyen Kurum

Institute of Mathematics and Mathematical modelling, Almaty, Kazakhstan

Proje Numarası

AP 09258966

Teşekkür

This research is funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan

Kaynakça

• 1. Erugin N.P. Construction all the set of systems of differential equations, possessing by given integral curve, Prikladnaya Matematika i Mecanika, 1952, 6, 659-670. (In Russian)
• 2. Galiullin A.S., Mukhametzyanov I.A., Mukharlyamov R.G. A survey of investigating on analytic construction of program motion's systems, Vestnik RUDN. 1994, No. 1., 5-21. (In Russian)
• 3. Galiullin A.S. Methods of solving of dynamics inverse problems. Nauka, Moskow (1986).
• 4. Galiullin A.S., Mukhametzyanov I.A., Mukharlyamov R.G. Review of researches on the analytical construction of the systems programmatic motions. Vestnik RUDN. , No.1 (1994) 5-21.
• 5. Mukametzyanov I.A. On stability of a program manifold. I. , Differential Equations. 1973. Vol. 9. No 5. P. 846-856. (In Russian)
• 6. Mukametzyanov I.A. On stability of a program manifold. II., Differential Equations. 1973. Vol. 9. No 6. P. 1057-1048. (In Russian)
• 7. Tleubergenov M.T. On the inverse stochastic reconstruction problem // Differential Equations. 2014. Vol. 50. No 2. P. 274-278. https://doi.org/10.1134/s 0012266 11402 0165
• 8. Mukharlyamov R.G. Simulation of Control Processes, Stability and Stabilization of Systems with Program Contraints. Journal of Computer and Systems Sciences International. 2015. 54, No.1., 13-26.
• 9. Tleubergenov M.T., Ibraeva G.T. On the restoration problem with degenerated diffusion // in TWMS Journal of pure and applied mathematics: 2015. Vol. 6, Issue: 1. P. 93-99.
• 10. Vasilina G.K., Tleubergenov M.T. Solution of the problem of stochastic stability of an integral manifold by the second Lyapunov method // Ukrainian Mathematical Journal. 2016. Vol.68. No 1. P.14-28. https://doi.org/10.1007/s 11253-016-1205-6
• 11. Llibre J., Ramirez R. Inverse Problems in Ordinary Differential Equations and Applications. Springer International Publishing Switzerland. 2016.
• 12. Maygarin B.G. Stability and quality of process of nonlinear automatic control system, Alma-Ata. Nauka. 1981. (In Russian)
• 13. Zhumatov S.S., Krementulo B.B., Maygarin B.G. Lyapunov's second method in the problems of stability and control by motion. Almaty. Gylym. 1999. (In Russian)
• 14. Zhumatov S.S. Stability of a program manifold of control systems with locally quadratic relationsw, Ukrainian Mathematical Journal. 2009. Vol.61. No 3. P.500-509. https://doi.\-org/10.1007/s 11253-008-0224-y
• 15. Zhumatov S.S. Exponential stability of a program manifold of indirect control systems // Ukrainian Mathematical Journal. 2010. Vol.62. No 6. P.907-915. https://doi.org/10.\-10\-07/s 11253-010-0399-2
• 16. Zhumatov S.S. On an instability of the indirect control systems in the neighborhood of program manifold // Mathematical Journal.- Almaty, 2017. Vol.17. No 1. P.91-97.
• 17. Zhumatov S.S. On a program manifrold's stability of one contour automatic control systems// Open Engineering, 2017. Vol. 7, Issue 1, Pages 479-484, ISSN (Online) 2391-5439, DOI: https://doi.org/10.1515/eng-2017-0051.
• 18. Samoilenko A.M., Stanzhytsskj O.M. The reduction principle in stability theory of invariant sets for stochastic Ito type systems, Differentialnye uravnenya. - 2001. - 53(2). - P. 282-285.
• 19. Zhumatov S.S. Absolute stability of a program manifold of non-autonomous basic control systems News of the NAS RK. Physico-mathematical series. - 2018. -V.6. - No 6. - P.37-43.
• 20. Zhumatov S.S. Stability of a program manifold of indirect control systems with variable coefficients, Mathematical Journal.- Almaty, 2019. Vol.19. No 2. P.121-130.
• 21. Zhumatov S.S. On the stability of a program manifold of control systems with variable coefficients, Ukrainian Mathematical Journal. - 2020. -Vol. 71, No 8. - pp. 1202-1213 . DOI: https://doi.org/10.1007/s11253-019-01707-7
• 22. Zhumatov S.S. Stability of the program manifold of different automatic indirect control systems, News Of the Khoja Akhmet Yassawi Kazakh-Turkish International University. Mathematics, physics, computer science series. ,16, No. 1 (2021), 69-82.
• 23. Khokhlov I.A. Electrohydraulic servo drive. V., Nauka. 1966. (in Russian)
• 24. Letov A.M. Stability of nonlinear control systems, M.: Nauka. - 1962. 484 p.(in Russian)