Synthesis algorithm for a phase shift control system using generalized Galerkin method

A.V. Statkevich

---

Abstract. This paper examines a pressing engineering problem: synthesizing the parameters of a system with a variable structure. The primary subject of the study is a control system incorporating a phase-shifting filter. Phase-shifting filters are widely used in modern digital signal processing. Their key advantages – flexible configuration and high accuracy in achieving the required characteristics – make them indispensable in areas such as telecommunications; radar; audio equipment; and automatic control systems. The objective is to synthesize system parameters that ensure the following conditions: operational stability; required dynamic characteristics; minimization of tracking errors; and adaptability to changing external conditions.
Aim. The study is to develop an effective approach to tuning control systems containing phase-shifting filters.
Results. A mathematical model of a non-stationary system with a variable structure was constructed; a generalized Galerkin method was proposed for synthesizing an automatic control system with a variable structure; an algorithm for synthesizing an automatic control system with a variable structure was developed using the generalized Galerkin method as an example of a phase-shifting filter.
Conclusions. The proposed approach opens new possibilities for creating efficient, flexible, and reliable control systems with phase-shifting filters.

Keywords: phase-shifting filter, sliding motion, variable-structure system, generalized Galerkin method, variable-structure system, non-stationary system, synthesis of automatic control systems, mathematical description, differential equation of a control system, automatic control system

For citation. Statkevich A.V. Synthesis algorithm for a phase shift control system using generalized Galerkin method. News of the Kabardino-Balkarian Scientific Center of RAS. 2026. Vol. 28. No. 2. Pp. 25–33. DOI: 10.35330/1991-6639-2026-28-2-25-33

©   Statkevich A.V., 2026

References

1. Kazakov I.E. Statisticheskaya dinamika sistem s peremennoy strukturoy [Statistical dynamics of systems with variable structure]. Moscow: Nauka, glavnaya redaktsiya fiziko-matematicheskoy literatury izdatel’stva «Nauka», 1977. 416 p. (In Russian)
2. Steinberger M. (ed.), Horn M. (ed.), Fridman L. (ed.). Variable-structure systems and sliding-mode control: from theory to practice. Springer International Publishing AG, 2020. 462 p. (Studies in Systems, Decision and Control). DOI: 10.1007/978-3-030-36621-6
3. Barth A., Weise C., Reger J. Application of higher-order sliding-modes to a ball and plate system in Proc. of 15-th International Workshop on Variable Structure Systems (VSS), Graz, Austria, July 2018. Pp. 192–197.
4. Yemelyanov S.V. Sistemy avtomaticheskogo upravleniya s peremennoy strukturoy [Automatic control systems with variable structure]. Moscow: Nauka, 1967, 366 p. (In Russian)
5. Utkin V.I., Orlov Yu.V. Teoriya beskonechnomernykh sistem upravleniya na skol’zyashchikh rezhimakh [Theory of infinite-dimensional sliding mode control systems]. Moscow: Nauka, 133 p. (In Russian)
6. Statkevich A.V., Shishlakov V.F. Application of phase-shifting filters for solving the synthesis problem. Matematicheskiye metody i modeli v vysokotekhnologichnom proizvodstve [Mathematical Methods and Models in High-Tech Production]: collection of abstracts, IV International Forum in 2 parts. St. Petersburg, November 6, 2024. Pp. 281–283. (In Russian)
7. Statkevich A.V., Shishlakov V.F. Systems with variable structure: an example of a sliding mode of operation. XXVII Int. Sci. Conf. “Volnovaya elektronika i infokommunikatsionnyye sistemy” [Wave Electronics and Infocommunication Systems]: St. Petersburg, June 3–7, 2024, collection of articles in 3 parts. Part 3, 329 p. (In Russian)
8. Utkin V.I. Skol’zyashchiy rezhim i ikh primeneniya v sistemakh s peremennoy strukturoy [Sliding mode and its applications in systems with variable structure]. Moscow: Nauka, glavnaya redaktsiya fiziko-matematicheskoy literatury, 1974. 272 p. (In Russian)
9. Reshetnikova N.V. Application of the electronic switching method in the synthesis of non-stationary automatic control systems. Inzhenernyy vestnik Dona [Engineering Bulletin of the Don]. 2025. No. 7(127). Pp. 445–454. EDN: DZYATZ. (In Russian)
10. Vataeva E.Yu. Parametric synthesis for a low-power potentiometric tracking system. Trudy MAI [Proceedings of the MAI]. 2024. No. 134. EDN: FUSCJQ. (In Russian)
11. Shishlakov D.V., Shishlakov V.F., Grechkin N.L., Vataeva E.Yu. Synthesis of multivariable ACS with polynomial approximation of nonlinear characteristics. IV-th International Forum Matematicheskiye metody i modeli v vysokotekhnologichnom proizvodstve [Mathematical Methods and Models in High-Tech Production]: collection of abstracts in 2 parts. St. Petersburg, November 6, 2024. Pp. 422–424. (In Russian)
12. Shishlakov V.F., Shishlakov A.V., Timofeev S.S. Sintez SAU pri razlichnykh vidakh approksimatsii nelineynykh kharakteristik: teoriya i praktika [Synthesis of ACS with different types of approximation of nonlinear characteristics: theory and practice]: monograph. St. Petersburg: GUAP, 2017. (In Russian)
13. Shishlakov V.F. Sintez impul’snykh sistem avtomaticheskogo upravleniya vo vremennoy oblasti [Synthesis of pulse automatic control systems in the time domain]. Diss. … Doctor of Technical Sciences: 05.13.01. St. Petersburg, 2001. 325 p. (In Russian)
14. Grechkin N.L., Vataeva E.Yu., Shishlakov D.V. Solution of the problem of synthesis of linear automatic control systems using the generalized Galerkin method. Zavalishinskiye chteniya 23: Sbornik dokladov XVIII Mezhdunarodnoy konferentsii po elektromekhanike i robototekhnike [Zavalishinsky Readings 23: collection of papers of the XVIII International Conference on Electromechanics and Robotics] St. Petersburg, April 18–19, 2023. Pp. 96–98. (In Russian)
15. Krivolapchuk I.G., Vataeva E.Yu., Reshetnikova N.V., Shishlakov V.F. Regulator based on imprecise sets. Sbornik statey v 2-h chastyah XXII Mezhdunarodnoy nauchnoy konferentsii: “Volnovaya elektronika i infokommunikatsionnyye sistemy” [Collection of papers in 2 parts of the XXII International Scientific Conference: “Wave Electronics and Infocommunication Systems”] St. Petersburg, June 3–7, 2019. Part 2. Pp. 257–262. (In Russian)

Information about the author

Anastasia V. Statkevich, Senior Lecturer, Department of Management in Technical Systems, SaintPetersburg State University of Aerospace Instrumentation;
67, Let. A, Bolshaya Morskaia street, Saint-Petersburg, 190000, Russia;
astat19@yandex.ru, ORCID: https://orcid.org/0009-0006-5413-4938, SPIN-code: 7825-7540

Funding

This work was supported by the Ministry of Science and Higher Education of the Russian Federation, agreement No. FSRF-2023-0003, “Fundamental principles for constructing interference-resistant systems of space and satellite communications, relative navigation, machine vision, and aerospace monitoring”.