N.L. Grechkin1, E.Yu. Vataeva2, V.F. Shishlakov3

1,2,3 Saint-Petersburg State University of Aerospace Instrumentation (St. Petersburg, Russia)

1 space.suai@bk.ru, 2 elivat@guap.ru, 3 svfmail@yandex.ru

The problem. Modern dynamic objects in industry, aviation, power engineering, and robotics are described by high-order nonlinear differrential equations. The absence of a universal analytical method for solving them leads to significant difficulties in the synthesis of automatic control systems (ACS). The generalized Galerkin method is one of the most effective approaches for the parametric synthesis of both continuous and impulsive nonlinear ACS. However, it remains highly labor-intensive because it requires approximating nonlinear characteristics, specifying the desired program motion, solving systems of equations with constraints, and repeatedly calculating the Galerkin integrals. Without automation, these operations are time-consuming and increase the probability of errors in ACS synthesis tasks.

Target. The development of a module algorithm that automatically determines the Galerkin integrals Bqi when the desired program motion is specified with respect to the error coordinate and with respect to the output coordinate. These algorithms form part of the software package being created for the complete automation of parametric synthesis of control operators in nonlinear ACS.

Results. In this paper, two block-modular algorithms for determining the Galerkin integrals have been developed. The input data include the parameters of the desired program motion, the argument of the coordinate function pq, and the coefficients of the polynomial P approximating the nonlinear characteristic. Next, recurrent relations for the integrals Bqi are computed for the two variants of specifying the desired motion – relative to the error coordinate and relative to the output coordinate of the system. The obtained integral values are element-wise multiplied by the coefficients of the approximating polynomial, summed, and transferred to the main part of the software package for further calculation of the objective function. The proposed algorithms fully automate the process of computing the Galerkin integrals when solving the problem of parametric synthesis of nonlinear ACS.

Practical significance. The developed module significantly simplifies and accelerates the design process of nonlinear ACS, enabling the consideration of the complex structure of modern technological processes. Due to its support for various types of approximation of nonlinear elements, the tool is applicable in mechanical engineering, electric power engineering, aviation, and transport construction. In the future, the module will be extended for adaptation to specific industries, which will increase the efficiency of automated ACS synthesis and reduce the time required for controller development.

Grechkin N.L., Vataeva E.Yu., Shishlakov V.F. Software module for computing Galerkin integrals // Achievements of Modern Radioelectronics. 2026. V. 80. № 9. P. 76−84. DOI: https://doi.org/10.18127/j20700784-202609-08

1. Lovchakov V.I., Sibyakin O.A. Multidimensional Linearization of Control Objects with Polynomial Nonlinearities. 2020 International Multi-Conference on Industrial Engineering and Modern Technologies (FarEastCon). Vladivostok, Russia. 2020. P. 1–4. doi: 10.1109/FarEastCon50210.2020.9271438
2. Conger L., Vernon S., Mazumdar E. Designing System Level Synthesis Controllers for Nonlinear Systems with Stability Guarantees. Proceedings of Machine Learning Research (PMLR). 2023. V. 211. P. 1–12.
3. Ma T. Filtered adaptive constrained sampled-data control for uncertain multivariable nonlinear systems. International Journal of Adaptive Control and Signal Processing. 2020. V. 34. № 10. P. 1433–1448. DOI: 10.1002/acs.3140
4. Santra S., Joby M., Sathishkumar M., Anthoni S.M. LMI approach-based sampled-data control for uncertain systems with actuator saturation: application to multi-machine power system. Nonlinear Dynamics. 2022. V. 107. № 2. P. 1731–1746. DOI: 10.1007/s11071-021-06995-y
5. Huang Q., Sun J. Control of impulsive systems via high-order fully actuated system approach. Nonlinear Dynamics. 2023. V. 111. № 14. P. 13279–13294. DOI: 10.1007/s11071-023-08749-4
6. Yu Y., Yuan Y., Yang H., Liu H. Nonlinear sampled-data ESO-based active disturbance rejection control for networked control systems with actuator saturation. Nonlinear Dynamics. 2019. V. 96. № 2. P. 1405–1422. DOI: 10.1007/s11071-018-4636-9
7. Liu W., Lu J., Xu S., Li Y., Zhang Z. Sampled-data controller design and stability analysis for nonlinear systems with input saturation and disturbances. Applied Mathematics and Computation. 2019. V. 363. P. 124609. DOI: 10.1016/j.amc.2019.05.016
8. Ignat`ev V.V. Avtomaticheskij sintez intellektual`ny`x regulyatorov na osnove algoritma samoorganizacii robastny`x baz znanij. Programmny`e produkty` i sistemy`. 2025. № 4. URL: https://cyberleninka.ru/article/n/avtomaticheskiy-sintez-intellektualnyh-regulyatorov-na-osnove-algoritma-samoorganizatsii-robastnyh-baz-znaniy (data obrashheniya: 23.03.2026). (in Russian).
9. Nikitin A.V., Shishlakov V.F. Parametricheskij sintez nelinejny`x sistem avtomaticheskogo upravleniya: monografiya. SPb.: SPbGUAP. 2003. 358 s. (in Russian).
10. Grechkin N.L., Vataeva E.Yu., Shishlakov V.F. Algoritm vy`chisleniya tochek pereklyucheniya nelinejnosti «dvuxpoziczonnoe rele s gisterezisom». Radiotexnika. 2024. T. 88. № 8. S. 24–34. DOI 10.18127/j00338486-202408-03 (in Russian).
11. Grechkin N.L., Vataeva E.Yu. Algoritmy` modulej dlya realizacii amplitudno-impul`snoj modulyacii. Vestnik Dagestanskogo gosudarstvennogo texnicheskogo universiteta. Texnicheskie nauki. 2025. T. 52. № 3. S. 29–37. DOI 10.21822/2073-6185-2025-52-3-29-37 (in Russian).