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

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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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