Parametric synthesis of continuous control systems with hysteresis nonlinearities

N.L. Grechkin, E.Yu. Vataeva

---

Abstract. In modern automatic control theory, nonlinear elements with ambiguous (hysteresis) characteristics are widely encountered. Such nonlinearities are typical of magnetic amplifiers, position sensors, electromechanical actuators, and a number of other devices. Classical methods of parametric synthesis developed for linear or single-valued nonlinear automatic control systems (ACS) are either not directly applicable to systems with ambiguous characteristics, require complex computations, or fail to guarantee the achievement of specified quality indicators of transient processes.
Aim. The aim of this study is to demonstrate the effectiveness of the proposed algorithm for modeling smooth hysteresis characteristics in solving the parametric synthesis problem, illustrated by the example of a specific nonlinear ACS containing a hysteresis element.
Methods. To solve the parametric synthesis problem for a continuous ACS, a generalized Galerkin method is applied; in addition, the algorithm for modeling smooth hysteresis characteristics developed by the authors is used.
Results. Specific values of the controller parameters (k1, k2, T) have been obtained that ensure the system’s transient response meets the prescribed quality criteria. Simulation modeling has shown a high degree of agreement between the actual system behavior with the synthesized parameters and the desired reference motion, confirming both the correctness and the practical applicability of the proposed approach.
Conclusions. The practical significance of the method lies in its potential application in the design of controllers for radio engineering, electromechanical, and other systems that include hysteresis elements. The proposed methodology significantly reduces the time required for parameter calculation, enables achievement of the required dynamic characteristics without resorting to complex adaptive or nonlinear correction devices, and simplifies the subsequent technical implementation of automatic control systems.

Keywords: smooth hysteresis characteristics, hysteresis modeling, slope coefficient, saturation level, ambiguous nonlinearities, automatic control systems, backlash with saturation, parametric synthesis

For citation. Grechkin N.L., Vataeva E.Yu. Parametric synthesis of continuous control systems with hysteresis nonlinearities. News of the Kabardino-Balkarian Scientific Center of RAS. 2026. Vol. 28. No. 3. Pp. 23–33. DOI: 10.35330/1991-6639-2026-28-3-23-33

©   Grechkin N.L., Vataeva E.Yu., 2026

References

1. Gao J., Xie X. A weighted switching sequence optimization algorithm for static output feedback control synthesis of nonlinear systems. Applied Mathematics and Computation. Vol. 489. 2025. P. 129152. DOI: 10.1016/j.amc.2024.129152
2. Mushtaq T., Seiler P., Maziar S. Hemati. On the convexity of static output feedback control synthesis for systems with lossless nonlinearities. Automatica. Vol. 159. 2024. P. 111380. DOI: 10.1016/j.automa ica.2023.111380
3. Huang W.-Ch., Sun H.-F., Zeng J.-P. Robust control synthesis of polynomial nonlinear systems using sum of squares technique. Acta Automatica Sinica. Vol. 39. No. 6. 2013. Pp. 799–805. DOI: 10.1016/S1874-1029(13)60055-5
4. Ding P., Chen K., Ou L. et al. Controller synthesis for a class of nonlinear systems with time delay based on stability region of PID controllers. ISA Transactions. Vol. 162. 2025. Pp. 85–94. DOI: 10.1016/j.isatra.2025.04.015
5. Do H.-T., Blanchini F., Prodan I. Nonlinear constrained control systems. In: Zhengtao Ding, ed. Encyclopedia of Systems and Control Engineering. 1st ed. Elsevier. 2026. Pp. 112–128. DOI: 10.1016/B978-0-443-14081-5.00165-3
6. Rysin A.V. The use of mathematical models of overhead power transmission line elements for equipment. Journal of Instrument Engineering. 2025. Vol. 68. No. 6. Pp. 557–561. DOI: 10.17586/0021-3454-2025-68-6-557-561. (In Russian)
7. Pepe P. Nonlinear systems with delays. In: Zhengtao Ding, ed. Encyclopedia of Systems and Control Engineering, Elsevier. 2026. Pp. 44–60. ISBN: 978-0-443-14080-8. DOI: 10.1016/B978-0-443-14081-5.00154-9
8. Ferrara A., Incremona G.P. Nonlinear sliding mode control. In: Zhengtao Ding, ed. Encyclopedia of Systems and Control Engineering. Elsevier. 2026. Pp. 100–111. DOI: 10.1016/B978-0-443-14081-5.00146-X
9. Cissé A., Boutayeb M. On state feedback control of a class of nonlinear PDE systems in finite dimension. Communications in Nonlinear Science and Numerical Simulation. Vol. 130. P. 107751. DOI: 10.1016/j.cnsns.2023.107751
10. Grechkin N.L., Vataeva E.Yu. About modeling hysteresis characteristics in problems of nonlinear control system synthesis. Engineering Journal of Don. 2026. No. 1. URL: ivdon.ru/ru/magazine/archive/n1y 2026/10677 (open access). (In Russian)
11. Nikitin A.V., Shishlakov V.F. Parametricheskiy sintez nelineynykh sistem avtomaticheskogo upravleniya [Parametric synthesis of nonlinear automatic control systems]. St. Petersburg: SPbGUAP, 2003. 358 p. (In Russian)
12. Shishlakov V.F., Vataeva E.Yu., Grechkin N.L., Shishlakov D.V. Application of polynomial approximation in solving problems of synthesis of nonlinear ACS. Sensors & Systems. 2023.
    No. 4-2(270). Pp. 10–12. DOI: 10.25728/datsys.2023.4.2.2. (In Russian)
13. Shishlakov V.F., Vataeva E.Yu., Grechkin N.L., Shishlakov D.V. Parametric synthesis of a potentiometric low-power tracking system. Sensors & Systems. 2023. No. 4-1(269). Pp. 5–9. DOI: 10.25728/datsys.2023.4.1.1. (In Russian)
14. Grechkin N.L., Vataeva E.Yu., Shishlakov V.F. Algorithm for calculating nonlinearity the switching points “two-position relay with hysteresis”. Radioengineering. 2024. Vol. 88. No. 8. Pp. 24–34. DOI: 10.18127/j00338486-202408-03. (In Russian)
15. Grechkin N.L. Parametric synthesis of automatic control systems with ambiguous characteristics. Engineering Journal of Don. 2026. No. 1. URL: ivdon.ru/ru/magazine/archive/
    n1y2026/10678 (open access). (In Russian)

Information about the authors

Nikita L. Grechkin, Senior Lecturer, Department of Control in Technical Systems, Saint Petersburg State University of Aerospace Instrumentation;
67, Lit. A, Bolshaya Morskaya street, Saint Petersburg, 190000, Russia;
space.suai@bk.ru, ORCID: https://orcid.org/0009-0001-8477-0752, SPIN-code: 7170-8166
Elizaveta Yu. Vataeva, Candidate of Technical Sciences, Associate Professor, Department of Control in Technical Systems, Saint Petersburg State University of Aerospace Instrumentation;
67, Lit. A, Bolshaya Morskaia street, Saint Petersburg, 190000, Russia;
elizavetavataeva@mail.ru, ORCID: https://orcid.org/0000-0002-7007-4967, SPIN-code: 2234-6792

Funding

The paper was prepared with the financial support of the Ministry of Science and Higher Education of the Russian Federation under grant agreement No. FSRF-2023-0003, “Fundamental principles of the construction of noise-immune systems for space and satellite communications, relative navigation, technical vision, and aerospace monitoring”.