THE SYNTHESIS OF MATHEMATICAL MODELS OF NONLINEAR DYNAMIC SYSTEMS USING VOLTERRA INTEGRAL EQUATION

Borys Mokin

borys.mokin@gmail.com
Vinnytsia National Technical University, Faculty of Intelligent Information Technologies and Automation (Ukraine)
http://orcid.org/0000-0002-5906-6122

Vitalii Mokin


Vinnytsia National Technical University, Faculty of Intelligent Information Technologies and Automation (Ukraine)
http://orcid.org/0000-0003-1946-0202

Oleksandr Mokin


Vinnytsia National Technical University, Faculty of Intelligent Information Technologies and Automation (Ukraine)
http://orcid.org/0000-0002-9277-3312

Orken Mamyrbaev


Al Farabi Kazakh National University, Institute of Information and Computer Technologies (Kazakhstan)
http://orcid.org/0000-0001-8318-3794

Saule Smailova


D. Serikbayev East Kazakhstan Technical University (Kazakhstan)
http://orcid.org/0000-0002-8411-3584

Abstract

The problem of creating mathematical models of nonlinear dynamical systems does not have an unambiguous solution and requires the creation of a separate synthesis method for each such object. To develop a method for synthesizing mathematical models of an extensive class of nonlinear dynamical systems with polynomial nonlinearities. The work uses a method based on the solution of the Volterra integral equation in the ideology set forth in Van Trees H.L., according to which the structure of a nonlinear dynamical object present47s a series connection of the linear part, characterizing the inertial properties of the system, and the nonlinear element, given by static characteristic. The difference of the suggested version of the method from the classical one, proposed in the works of Van Trees H.L., is an expansion of their input and output signals into Fourier series and a representation of the inertial part of these systems by their Bode plots, connected into one structure with input and output signals and non-linearity by Volterra integral equation. The algorithm of the proposed method is disclosed by the example of solving the problem of identifying a nonlinear dynamical system which impulse response of the inertial part satisfies the separability requirement, the order of the polynomial nonlinearity is three, and the model of the input signal has the form of a sinusoid "raised" over the time axis on a priori given constant level. A computational experiment was carried out on the example of nonlinear dynamical systems with the third order of the nonlinear characteristic and the first and second orders of the model of the inertial part of these systems with the specified algorithms of their parametric identification. The suggested method allows to synthesis the mathematical model of a nonlinear dynamical system with the polynomial static characteristic to the case when the input signal has an arbitrary number of harmonics, and the model of the inertial part and the nonlinear polynomial function have an arbitrary order.


Keywords:

nonlinear dynamical system, mathematical model, polynomial nonlinearity function, Bode plot, Fourier series, Volterra integral equation

Chua L. O., Ng C-Y.: Frequency domain analysis of nonlinear systems: general theory. Electronic Circuits and Systems 3(2), 1979, 165–185.
DOI: https://doi.org/10.1049/ij-ecs.1979.0030   Google Scholar

Chua L. O., Ng C-Y.: Frequency domain analysis of nonlinear systems: formulation of transfer functions. Electronic Circuits and Systems 3(4), 1979, 257–269.
DOI: https://doi.org/10.1049/ij-ecs.1979.0045   Google Scholar

Halas M., Huba M., Kotta Ü.: An overview of transfer function formalism for nonlinear systems. Journal of Cybernetics and Informatics 8(3), 2009, 28–35.
  Google Scholar

Halas M.: An algebraic framework generalizing the concept of transfer functions to nonlinear systems. Automatica 44(2), 2008, 1181–1190 [http://doi.org/10.1016/j.automatica.2007.09.008].
DOI: https://doi.org/10.1016/j.automatica.2007.09.008   Google Scholar

Halas М., Kotta Ü.: A transfer function approach to the realization problem of nonlinear systems. International Journal of Control 85(1), 2012, 320–331 [http://doi.org/10.1080/00207179.2011.651748].
DOI: https://doi.org/10.1080/00207179.2011.651748   Google Scholar

Kerschen G., Worden K., Vakakis A. F. et al.: Past, present and future of nonlinear system identification in structural dynamics. Mechanical Systems and Signal Processing 20(3), 2006, 505–592 [http://doi.org/10.1016/j.ymssp.2005.04.008].
DOI: https://doi.org/10.1016/j.ymssp.2005.04.008   Google Scholar

Mokin A. B., Mokin V. B., Mokin B. I. et al.: Determining the Conditions and Designing the Methods for Description of Processes in Complex Dynamic Objects by Equivalent Models not Higher than the Third-Order. Journal of Automation and Information Sciences 48(3), 2016, 83–97 [http://doi.org/10.1615/JAutomatInfScien.v48.i3.90].
DOI: https://doi.org/10.1615/JAutomatInfScien.v48.i3.90   Google Scholar

Mokin B. I., Mokin O. B.: The Fourier Integral Method in the Problems of Identification and Input Signal Renewal of Nonlinear Dynamical Systems. Visnyk of Vinnytsia Polytechnical Institute 3, 2000, 107–112.
  Google Scholar

Mokin B. I.: Vossstanovleniye vkhodnykh signalov snelineynymi kharakteristikami preobrazovaniya. Metody teorii identifikatsii v zadachakh izmeritel'noy tekhniki i metrologii: III Vsesoyuznyy simpozium. 1982, 207–209.
  Google Scholar

Mokin O. B., Mokin B. I., Khomiuk Ya. V.: Conditions of Equivalentiation of Nonlinear Dynamic Systems with Power Nonlinearities in the Frequency Domain. Visnyk of Vinnytsia Polytechnical Institute 5, 2016, 40–44.
  Google Scholar

Mokin O. B., Mokin B. I.: Modeling and optimization of movement of multi-mass electric vehicles with difficult terrain surfaces. Vinnytsia National Technical University, Vinnytsia 2013.
  Google Scholar

Mokin O. B., Mokin B. I.: Renewal of input signals of nonlinear Measuring converters by Fourier-integral method. International Measurement Confederation (IMEKO): XVII World Congress of the (Metrology in the 3rd Millennium), Dubrovnik 2003, 468–471.
  Google Scholar

Nassirharand А., Mousavi Firdeh S.R.: Design of nonlinear controllers using describing functions with application to servomechanism. Asian Journal of Control 11(3), 2009, 446–450.
DOI: https://doi.org/10.1002/asjc.124   Google Scholar

Nassirharand А., Mousavi Firdeh S.R.: Design of nonlinear lead and/or lag compensators. International Journal of Control, Automation and Systems 6(3), 2008, 394–400.
  Google Scholar

Nassirharand А., Teh S.H.: Describing function-based identification of nonlinear transfer functions for nonlinear systems from experimental/simulation data. Int. J. Modelling, Identification and Control 25(2), 2016, 93–101 [http://doi.org/10.1504/IJMIC.2016.075270].
DOI: https://doi.org/10.1504/IJMIC.2016.075270   Google Scholar

Pavlenko V. Speranskyy V.: Polyharmonic test signals application for identification of nonlinear dynamical systems based on volterra model. International Conference on Information and Telecommunication Technologies and Radio Electronics (UkrMiCo), 2017, 1–5 [http://doi.org/10.1109/UkrMiCo.2017.8095372].
DOI: https://doi.org/10.1109/UkrMiCo.2017.8095372   Google Scholar

Rijlaarsdam D., Nuij P., Schoukens J., Steinbuch M.: A comparative overview of frequency domain methods for nonlinear systems. Mechatronics 42, 2017, 11–24 [http://doi.org/10.1016/j.mechatronics.2016.12.008].
DOI: https://doi.org/10.1016/j.mechatronics.2016.12.008   Google Scholar

Van Trees H. L.: Synthesis of Optimum Non-Linear Control Systems. Massachusetts Inst. of Technology, Cambridge 1962.
  Google Scholar

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Published
2022-06-30

Cited by

Mokin, B. ., Mokin, V., Mokin, O., Mamyrbaev, O. ., & Smailova, S. . (2022). THE SYNTHESIS OF MATHEMATICAL MODELS OF NONLINEAR DYNAMIC SYSTEMS USING VOLTERRA INTEGRAL EQUATION. Informatyka, Automatyka, Pomiary W Gospodarce I Ochronie Środowiska, 12(2), 15–19. https://doi.org/10.35784/iapgos.2947

Authors

Borys Mokin 
borys.mokin@gmail.com
Vinnytsia National Technical University, Faculty of Intelligent Information Technologies and Automation Ukraine
http://orcid.org/0000-0002-5906-6122

Authors

Vitalii Mokin 

Vinnytsia National Technical University, Faculty of Intelligent Information Technologies and Automation Ukraine
http://orcid.org/0000-0003-1946-0202

Authors

Oleksandr Mokin 

Vinnytsia National Technical University, Faculty of Intelligent Information Technologies and Automation Ukraine
http://orcid.org/0000-0002-9277-3312

Authors

Orken Mamyrbaev 

Al Farabi Kazakh National University, Institute of Information and Computer Technologies Kazakhstan
http://orcid.org/0000-0001-8318-3794

Authors

Saule Smailova 

D. Serikbayev East Kazakhstan Technical University Kazakhstan
http://orcid.org/0000-0002-8411-3584

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