Algorithm for calculating galerkin integrals for the "dead zone" nonlinearity in the synthesis problem of automatic control systems

Аuthors

Grechkin N. L.

Saint Petersburg State University of Aerospace Instrumentation, 67, Bolshaya Morskaya str., Saint Petersburg, 190000, Russia

e-mail: space.suai@bk.ru

Abstract

The article is devoted to the study and development of methods for synthesizing nonlinear automatic control systems (ACS) with characteristics of the "dead zone" type. The relevance of the problem of synthesizing such systems is considered, caused by the growing demand for complex control systems operating under conditions of strong nonlinearities and external disturbances. The article emphasizes that linear models are not always able to adequately reflect reality, so special attention is paid to the creation and study of nonlinear ACS. It is proposed to use the generalized Galerkin method as a mathematical apparatus. The advantage of this approach is that it allows synthesizing control laws for a wide class of systems, regardless of their complexity and the order of differential equations describing their dynamics. The mechanism of occurrence of dead zones in the characteristics of various elements of systems, caused by design features, frictional forces and external loads, is considered in detail. It has been established that the presence of such zones has a serious impact on the dynamics of the system: in some cases, causing instability and self-oscillations, in others, contributing to improved stability and suppression of unwanted oscillations.
An algorithm for calculating Galerkin integrals for a piecewise linear approximation of the "dead zone" nonlinearity is proposed. The algorithm implements a step-by-step procedure for calculating integrals based on determining the moments of nonlinearity switching and summing up intermediate results. The key conclusion is that the use of the proposed method allows increasing the accuracy of the synthesis of nonlinear ACS, reducing the risk of adverse consequences associated with the appearance of dead zones. It is emphasized that further development of this approach will provide the possibility of its wider integration into industrial and scientific areas, contributing to the growth of the quality and reliability of control systems.

Keywords:

dead zone, nonlinear systems, polynomial approximation, generalized Galerkin’s method, algorithm

References

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, 2018, vol. 79, no. 2, pp. 258-264. DOI: 10.1134/S0005117918020054
2. Mikheenko, A. M. Analysis of stability of a pulse-width system with negative feedback / A. M. Mikheenko, E. S. Abramova, S. S. Abramov // Electrosvyaz. - 2013. - No. 8. - p. 20-22.
3. Lovchikov, A. N. Analysis and synthesis of pulse-width systems / A. N. Lovchikov, E. E. Noskova // Bulletin of the Siberian State Aerospace University named after Academician M. F. Reshetnev. - 2010. - No. 4 (30). - p. 57-61
4. Smolyakov, V. N. Methodology for analyzing the absolute stability of nonlinear pulse systems using a computer based on the inner approach / V. N. Smolyakov // Engineering Bulletin of the Don. - 2017. - No. 1 (44). - p. 48-51.
5. Pshikhopov, V. Kh. Block synthesis of robust automatic systems with constraints on controls and state coordinates / V. Kh. Pshikhopov, M. Yu. Medvedev // Mechatronics, automation, control. - 2011. - No. 1. - p. 2-8
6. Panteleev A.V. Application of global optimization methods for parametric synthesis of a generalized proportional-integral-differential controller in a flight control problem / A.V. Panteleev, T.A. Letova, E.A. Pomazueva // Proceedings of MAI. 2015. No. 79.
7. Kantulev A.N. Study of stability of autonomous nonlinear dynamic systems / A.N. Kantulev, A.Yu. Kunetsov // Proceedings of MAI. 2010. No. 40.
8. Pyrkin A., Bobtsov A., Kolyubin S., Faronov M. Output Controller for Uncertain Nonlinear Systems with Structural, Parametric, and Signal Disturbances // IEEE Multi-Conference on Systems and Control, 2012. DOI: 10.1109/CCA.2012.6402352
9. Bernal Miguel, Sala Antonio, Lendek Zsofia, Guerra Thierry-Marie. Analysis and Synthesis of Nonlinear Control Systems: A Convex Optimisation Approach. 2022. DOI: 10.1007/978-3-030-90773-0
10. Lendek Zs., Guerra T., Lauber J. Controller design for TS models using delayed nonquadratic Lyapunov functions // IEEE Transactions on Cybernetics, 2015, vol. 45 (3), pp. 453–464. DOI: 10.1109/TCYB.2014.2327657
11. Vladimir Sinyakov, Antoine Girard. Controller Synthesis for Nonlinear Systemswith Reachability Specifications Using Monotonicity // IEEE Conferenceon Decisionand Control, 2019, Nice, France. DOI: 10.1109/CDC40024.2019.9029740
12. Shashihin V.N. Synthesis of Control for Nonlinear Systems // Synthesis of Control for Nonlinear Systems, 2019, vol. 53, pp. 97–106. DOI: 10.3103/s0146411619020068
13. Yang Z., Zhang L., Zeng X., Tang X., Peng C., Zeng Z. Hybrid Controller Synthesis for Nonlinear Systems Subject to Reach-Avoid Constraints. In: Enea, C., Lal, A. (eds) Computer Aided Verification. CAV 2023. Lecture Notes in Computer Science, vol. 13964. Springer, Cham. DOI: 10.1007/978-3-031-37706-8_16
14. Wang L., Ortega R., Bobtsov A. Observability is sufficient for the design of globally exponentially stable state observers for state-affine nonlinear systems // Automatica, 2023, vol. 149, pp. 11083.
15. Grechkin, N. L. Algorithm for determining the switching points of a nonlinear element "dead zone" / N. L. Grechkin, E. Yu. Vataeva, V. F. Shishlakov // Metrological support of innovative technologies: Collection of articles of the VII International Forum, St. Petersburg, March 04, 2025. - St. Petersburg: St. Petersburg State University of Aerospace Instrumentation, 2025. - P. 143-144. 14. 
16. Grechkin, N. L. Algorithm for implementing the "pulse-amplitude modulator" module / N. L. Grechkin, E. Yu. Vataeva, S. N. Trubeneva // Mathematical methods and models in high-tech production: Collection of abstracts of reports of the IV International Forum. In 2 parts, St. Petersburg, November 06, 2024. – Saint Petersburg: Saint Petersburg State University of Aerospace Instrumentation, 2024. – P. 242-245.
17. Grechkin, N. L. Algorithm for calculating the switching points of the nonlinearity "two-position relay with hysteresis" / N. L. Grechkin, E. Yu. Vataeva, V. F. Shishlakov // Radio Engineering. – 2024. – Vol. 88, No. 8. – P. 24-34. – DOI 10.18127/j00338486-202408-03.
18. Gnilitsky, V. Exploring sensitivity of mathematical model for the system of voltage symmetrization / V. Gnilitsky, A. Polishchuk, R. Petrosyan // Eastern-European Journal of Enterprise Technologies. – 2016. – Vol. 5, No. 8(83). – P. 4-8. – DOI 10.15587/1729-4061.2016.80066.
19. Shakhrai, E. A. Synthesis of a modified PID controller under conditions of multi-mode operation of a noisy object / E. A. Shakhrai, V. F. Lubentsov, E. V. Lubentsova // Electronic online polythematic journal "Scientific Works of KubSTU". - 2021. - No. 1. - p. 99-108.
20. Salama, B. Automatic control system of the oil pressure regulator for hydraulic press assembly of tension joints / B. Salama // Intelligent systems in production. - 2018. - Vol. 16, No. 1. - pp. 67-71. - DOI 10.22213/2410-9304-2018-1-67-71.
21. Dedesh, V. T. Stability and self-oscillations of nonlinear single-loop automatic control systems / V. T. Dedesh // Scientific notes of TsAGI. - 2010. - Vol. 41, No. 3. - pp. 82-92.
22. Nikitin A.V., Shishlakov V.F. Parametricheskii sintez nelineinykh sistem avtomaticheskogo upravleniya: monografiya (Parametric synthesis of nonlinear automatic control systems), Saint Petersburg, GUAP, 2003, 358 p.
23. Vataeva E.Yu., Parametric synthesis of a low-power potentiometric tracking system. Trudy MAI, 2024, no. 134. URL: https://trudymai.ru/eng/published.php?ID=178477
24. Vataeva E.Yu. Parametric synthesis of ACS control operators with polynomial approximation of the characteristics of nonlinear elements. Trudy MAI, 2023, no. 128. DOI: 10.34759/trd-2023-128-16

Download