PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł artykułu

Automated Linearization of a System of Nonlinear Ordinary Differential Equations

Treść / Zawartość
Identyfikatory
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This paper investigates the possibility of automatically linearizing nonlinear models. Constructing a linearised model for a nonlinear system is quite labor-intensive and practically unrealistic when the dimension is greater than 3. Therefore, it is important to automate the process of linearisation of the original nonlinear model. Based on the application of computer algebra, a constructive algorithm for the linearisation of a system of non-linear ordinary differential equations was developed. A software was developed on MatLab. The effectiveness of the proposed algorithm has been demonstrated on applied problems: an unmanned aerial vehicle dynamics model and a twolink robot model. The obtained linearized models were then used to test the stability of the original models. In order to account for possible inaccuracies in the measurements of the technical parameters of the model, an interval linearized model is adopted. For such a model, the procedure for constructing the corresponding interval characteristic polynomial and the corresponding Hurwitz matrix is automated. On the basis of the analysis of the properties of the main minors of the Hurwitz matrix, the stability of the studied system was analyzed.
Twórcy
  • Al-Farabi Kazakh NationalUniversity, Kazakhstan
  • Al-Farabi Kazakh NationalUniversity, Kazakhstan
  • Lublin Technical University, Poland
  • Institute of Information and Computational Technologies CS MES RK, Al-Farabi Kazakh National University, Kazakhstan
  • Institute of Information and Computational Technologies CS MES RK, Al-Farabi Kazakh National University, Kazakhstan
Bibliografia
  • [1] O.I. Borisov, “Methods of control of robotics applications” St. Petersburg: ITMO University, 2016.
  • [2] A.G. Bulgakov, V.A. Vorobyov, “Industrial robots. Kinematics, dynamics, control and management”, Moscow: Solon-Press, 2011.
  • [3] V.A. Kolemaev, “Mathematical Economics”, Moscow: Unity-Dana, 2002.
  • [4] V.V. Khristianovsky, V.P. Shcherbina, “Economic and mathematical methods and models: theory and practice”, Donetsk: DonNU, 2010.
  • [5] B.L. Teush, “Ordinary differential equations: A course for future engineers”, Moscow: Lenand, 2022.
  • [6] [6] V.G. Borisov Applied Problems of the Theory of Ordinary Differential Equations. Mechanical motion. - Kemerovo: Kemerovo State University, 2015. - 130 р.
  • [7] F. Boarotto, M. Sigalott, “Dwell-time control sets and applications to the stability analysis of linear switched systems,” Journal of Differential Equations, vol. 268, no. 4, pp. 1345-1378, 2020. https://doi.org/10.48550/arXiv.1902.03757
  • [8] V. Ayala, A. Da Silva, P. Jouan, G. Zsigmond, “Control sets of linear systems on semi-simple Lie groups,” Journal of Differential Equations, vol. 269. no. 1, pp. 449-466. 2020. https://doi.org/10.1016/j.jde.2019.12.010
  • [9] W. Zhang, Q-L. Han, Y. Tang, Y. Liu, “Sampled-data control for a class of linear time-varying systems,” Automatica, vol. 103, pp. 126-134. 2019. https://doi.org/10.1016/j.automatica.2019.01.027
  • [10] T. Mazakov, W. Wójcik, Sh. Jomartova, N. Karymsakova, G. Ziyatbekova, A. Tursynbai, “The Stability Interval of the Set of Linear System,” International Journal of Electronics and Telecommunications, vol. 67, no. 2, pр. 155-161, 2021. https://doi.org/10.24425/ijet.2021.135957
  • [11] A. Mazakova, S. Jomartova, T. Mazakov, T. Shormanov, B. Amirkhanov, “Controllability of an unmanned aerial vehicle,” IEEE 7th International Energy Conference (ENERGYCON), pp. 1-5. 2022. https://doi.org/10.1109/ENERGYCON53164.2022.9830244
  • [12] A.A. Shcheglova, “On controllability of nonlinear differential-algebraic equations,” Proc. of International Conference "Dynamic Systems: Stability, Control, Optimization", Minsk, pр.166-168, 2008.
  • [13] A.M. Molchanov, “On the stability of nonlinear systems,” Pushchino: Institute of Mathematical Problems of Biology RAS, 2013.
  • [14] V.Z. Aladyev, V.K. Boyko, E.A. Rovba, “Programming and Development of Applications in Maple”, Grodno: GrSU., Tallinn: International Noosphere Academy, 2007.
  • [15] Y. Lazarev, “Modeling Processes and Systems in Matlab,” St. Petersburg: Peter: VNV Publishing Group, 2005.
  • [16] V.P. Dyakonov, “Systems of computer algebra Derive,” Moscow: Solon-Press, 2002.
  • [17] M.V. Grosheva, G.B. Efimov, V.A. Samsonov, “History of using analytical calculations in mechanics problems,” Moscow: The Keldysh Institute for Applied Mechanics of the Russian Academy of Sciences, 2005.
  • [18] V.P. Dyakonov, “Encyclopedia of computer algebra”, Moscow: DMK Press, 2010.
  • [19] R.N. Kvyetnyy, O.Y. Sofina, A.V. Lozun, A. Smolarz, O. Zhirnova, “Modification of fractal coding algorithm by a combination of modern technologies and parallel computations,” Proc. SPIE, vol. 9816, art. no. 98161R, 2015. https://doi.org/10.1117/12.2229009
  • [20] A.N. Kvitko, “Methods for solving boundary value problems for controllable systems of ordinary differential equations and their application in solving problems of controlling the motion of the center of mass of an aircraft,” PhD. thesis, 2000.
  • [21] S.F. Burdakov, “Mathematical models and identification of robots with elastic elements,” Leningrad: LSTU. 1990.
  • [22] L.B. Freidovich, “Stability and control of manipulating robots” PhD. thesis in Physics and Mathematics, St. Petersburg, 1999.
  • [23] Sh.A. Jomartova, G.N. Karymsakova, A.S. Tursynbai, “Application of analytical calculations system for the derivation of equations of dynamics of robotic systems,” Bulletin of KazATK, no.2, pp. 207-213, Almaty 2020.
  • [24] N.T. Karymsakova, “Development of controllability criteria for dynamic systems with limited control,” PhD thesis, Almaty 2022.
Uwagi
1. Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).
2. The work was performed at the Research Institute of Mathematics and Mechanics of Kazakh National University named after Al-Farabi at the expense of the program-targeted funding of scientific research for 2023-2025 under the project IRN AR19676966 «Development of hardware-software system for psychophysiological selection and rehabilitation of snipers».
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-f8221f7b-efa0-4c0e-8490-bca45f0cd4b3
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.