V.F. Shishlakov1, N.V. Reshetnikova2, E.Yu. Vataeva3, D.V. Shishlakov4

1–4 Saint-Petersburg State University of Aerospace Instrumentation (St. Petersburg, Russia)

The development and widespread introduction of automatic control systems (ACS) in various branches of technology, increasing the requirements for their accuracy and reliability, necessitates the improvement of design and research methods.The theory of automatic control systems (ACS) with nonlinear properties is currently not fully developed than the theory of linear automatic systems. Special attention is paid to the problem of synthesis and modeling of nonlinear ACS. Applying the classical control theory for the synthesis of systems, the statement is used that the mathematical model of an object is known and adequately describes its behavior. However, modern approaches to the formulation and solution of problems of synthesis of control systems are characterized by the presence of a provision on the uncertainty of specifying the model of an object, in particular, on the inaccuracy of knowing its parameters.

Modern methods of analysis and synthesis of nonlinear automatic control systems are based mainly on solving the control problem with a significant simplification of the physical and mathematical dependencies that characterize the ACS processes, which is not the perfection of the research apparatus used. When solving the problem of synthesizing a nonlinear ACS of any degree of complexity, it is important to draw up an adequate mathematical model. It is known that when constructing a mathematical model, all the main and most essential features and properties of the synthesized ACS should be preserved. Therefore, the issues related to the approximation of nonlinear links included in the ACS are extremely relevant. The complexity of this issue lies in the fact that there are no standard rules and algorithms for the idealization of nonlinear elements. Thus, to solve the problem of approximating nonlinear characteristics, it is necessary to choose an approximating function and determine the corresponding coefficients. The paper proposes to use a polynomial approximation of nonlinear links included in the ACS. To further solve the problem of synthesizing the control laws of the controller, it is proposed to use the inversion of the direct variational analysis method – the generalized Galerkin method – to solve the problem. This makes it possible to completely algebraize the solution of the problem posed for nonlinear control systems, the dynamics of which is described by nonlinear differential equations of arbitrary order.

Shishlakov V.F., Reshetnikova N.V., Vataeva E.Yu., Shishlakov D.V. Polynomial approximation of nonlinear elements in solving the problem of ACS synthesis. Science Intensive Technologies. 2021. V. 22. № 8. P. 69−74. DOI: https://doi.org/10.18127/j19998465-20210810 (in Russian)

1. Wang J., Aranovskiy S.V., Bobtsov A.A., Pyrkin A.A., Kolyubin S.A. A Method to Provide Conditions for Sustained Excitation. Automation and Remote Control, IET. 2018. V. 79. № 2. P. 258–264.
2. Bazylev D., Vukosavic S., Bobtsov A., Pyrkin A., Stankovic A., Ortega R. Sensorless Control of PM Synchronous Motors with a Robust Nonlinear Observer. Proceedings 2018 IEEE Industrial Cyber-Physical Systems (ICPS), IET. 2018. P. 304–309.
3. Pyrkin A., Bobtsov A., Kolyubin S., Faronov M. Output Controller for UncertainNonlinear Systems with Structural, Parametric, and Signal Disturbances. IEEE Multi-Conference on Systems and Control. Scopus. 2012.
4. Borgul A., Bobtsov A., Kolyubin S., Zimenko K., Rabyish E.,Pyrkin A. Mechatronic and Robotic Setups for Modern Control Theory Workshops. Preprints of ACE2012: 9th IFAC Symposium on Advances in Control Education. Scopus. 2012. P. 348–353.
5. Ismatovich F.S. Synthesis of an automatic control system with pulse-width modulation according to the speed criterion. 2020 International Conference on Information Science and Communications Technologies, ICISCT 2020, Tashkent, 04–06 November 2020. Tashkent, 2020. P. 9351393. DOI 10.1109/ICISCT50599.2020.9351393.
6. Shishlakov V.F., Shishlakov A.V., Timofeev S.S. Sintez SAU pri razlichnyh vidah approksimacii nelinejnyh harakteristik: teoriya i praktika: Monografiya. Pod red. V.F. Shishlakova. S.-Peterb. gos. un-t aerokosm. priborostroeniya. SPb.: Izd-vo GUAP. 2017. 151 s. (in Russian).
7. Nikitin A.V., Shishlakov V.F. Parametricheskij sintez nelinejnyh sistem avtomaticheskogo upravleniya: Monografiya. Pod red. V.F. Shishlakova. S.-Peterb. gos. un-t aerokosm. priborostroeniya. SPb.: Izd-vo GUAP. 2003. 353 s. (in Russian).
8. Shishlakov V.F., Reshetnikova N.V., Vataeva E.Yu., Shishlakov D.V. Obshchaya skhema resheniya zadachi sinteza nelinejnyh nestacionarnyh SAU vo vremennoj oblasti. Datchiki i sistemy. 2020. № 7(249). S. 12–16. DOI 10.25728/datsys.2020.7.3 (in Russian).
9. Shishlakov V.F., Anisimova E.V. Sintez parametrov SAU pri approksimacii nelinejnyh harakteristik irracional'noj funkciej obobshchennym metodom Galerkina. Zavalishinskie chteniya 14: Sbornik dokladov, Sankt-Peterburg, 07–11 aprelya 2014 goda. SPb.: SanktPeterburgskij gosudarstvennyj universitet aerokosmicheskogo priborostroeniya. 2014. S. 111–115 (in Russian).
10. Shishlakov V.F., Anisimova E.V., Shishlakov A.V., Shishlakov D.V. Sintez parametrov zakona upravleniya dlya nelinejnyh SAU pri razlichnyh vidah approksimacii harakteristik. Izv. vuzov. Ser. Priborostroenie. 2015. T. 58. № 9. S. 701–706. DOI 10.17586/0021-34542015-58-9-701-706 (in Russian).