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Automation and Remote Control

, Volume 80, Issue 9, pp 1671–1680 | Cite as

Control Systems with Vector Relays

  • V. I. UtkinEmail author
  • Yu. V. OrlovEmail author
Topical Issue

Abstract

The paper presents the evolution of discontinuous control systems starting from a relay with only two output constant values. The relay systems were widely used at the early stage of the feedback control system history. The analysis and design methods for them were developed by Ya. Tsypkin and discussed in his monograph “Theory of relay control systems,” published in 1956. It is shown how a relay function is modified in the so-called variable structure systems, when the relay output cab be equal to one of two continuous state functions. The next step is made in the framework of variable structure systems with vector control. The design procedure for systems with vector relay control relies on selection of a discontinuity surface for each control component. High efficiency of such designed systems results from enforcing sliding modes on the surfaces. Finally, the vector relay unit control is offered. The method is free of the component-wise design and proved to be applicable for infinite-dimensional systems.

Keywords

sliding mode discontinuity surface regularization convex set 

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References

  1. 1.
    Kulebakin, V., To the Theory of Automatic Vibrational Regulators for Electric Motors, Teor. Eksp. Elektron., 1932, no. 4, pp. 3–21.Google Scholar
  2. 2.
    Flugge-Lotz, I., Discontinuous Automatic Control, New Jersey: Princeton Univ. Press, 1953.CrossRefzbMATHGoogle Scholar
  3. 3.
    Tsypkin, Ya., Teoriya releinykh sistem avtomaticheskogo regulirovaniya (Theory of Relay Control Systems), Moscow: Gostekhizdat, 1955.Google Scholar
  4. 4.
    Nikolskii, G.N., On Automatic Stability of a Ship on the Given Course, Tr. Tsentr. Lab. Provodnoi Svyazi, 1934, no. 1, pp. 34–75.Google Scholar
  5. 5.
    Proc. 2nd All Union Control Conf., Discussion, 1955, vol. v, pp. 460–462.Google Scholar
  6. 6.
    Neymark, Yu.I., Note on A. Filippov’s Paper, Proc. 1st IFAC Congr., London: Butterworth, 1960.Google Scholar
  7. 7.
    Teoriya sistem s peremennoi strukturoi (Theory of Variable Structure Systems), Emelyanov, S., Ed., Moscow: Nauka, 1970.Google Scholar
  8. 8.
    Utkin, V., Skol’zyashchie rezhimy i ikh primenenie v sistemakh s peremennoi strukturoi (Sliding Mode and Their Application in Variable Structure Systems, Moscow: Nauka, 1974.Google Scholar
  9. 9.
    Luk’yanov, A.G. and Utkin, V.I., Methods of Reducing Equations of Dynamic Systems to the Regular Form, Autom. Remote Control, 1981, vol. 42, no. 3, pp. 413–420.zbMATHGoogle Scholar
  10. 10.
    Gutman, S., Uncertain Dynamic Systems—A Lyapunov Min-Max Approach, IEEE Trans. Autom. Control, 1979, vol. AC-24, pp. 437–449.CrossRefzbMATHGoogle Scholar
  11. 11.
    Gutman, S. and Leitmann, G., Stabilizing Feedback Control for Dynamic Systems with Bounded Uncertainties, Proc. Conf. Decision Control, 1976, pp. 94–99.Google Scholar
  12. 12.
    Drazenovic, B., The Invariance Conditions in Variable Structure Systems, Automatica, 1969, vol. 5, no. 3, pp. 287–295.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Barbashin, E.A., Vvedenie v teoriyu ustoichivosti (Introduction to Stability Theory), Moscow: Nauka, 1967.Google Scholar
  14. 14.
    Orlov, Y.V. and Utkin, V.I., Use of Sliding Modes in Distributed System Control Problems, Autom. Remote Control, 1982, vol. 43, no. 9, pp. 1127–1135.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Orlov, Y.V., Application of Lyapunov Method in Distributed Systems, Autom. Remote Control, 1983, vol. 44, no. 4, pp. 426–430.MathSciNetzbMATHGoogle Scholar
  16. 16.
    Orlov, Y.V., Discontinuous Systems: Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions, Communications and Control Engineering Series, Berlin: Springer, 2009.zbMATHGoogle Scholar
  17. 17.
    Perrollaz, V. and Rosier, L., Finite-time Stabilization of 2x2 Hyperbolic Systems on Tree-shaped Networks, SIAM J. Control Optim., 2014, vol. 52, no. 1, pp. 143–163.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Curtain, R. and Zwart, H., An Introduction to Infinite-Dimensional Linear Systems Theory. Texts in Applied Mathematics, New York: Springer, 1995.CrossRefzbMATHGoogle Scholar
  19. 19.
    Orlov, Y. and Utkin, V., Unit Sliding Mode Control in Infinite-Dimensional Systems, J. Appl. Math. Comput. Sci., 1998, vol. 8, pp. 7–20.MathSciNetzbMATHGoogle Scholar
  20. 20.
    Rudin, W., Functional Analysis, New York: McGraw-Hill, 1991, 2nd ed.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.The Ohio State UniversityColumbusUSA
  2. 2.CICESEEnsenadaMexico
  3. 3.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia

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