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Sliding Modes after the First Decade of the 21st Century pp 93–112Cite as

Frequency-Domain Methods in Conventional and Higher-Order Sliding Mode Control

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Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 412))

Abstract

Sliding mode control has become a mature theory and found a number of useful applications. The theory of sliding mode control is based on mostly state space models and Lyapunov approach to analysis of the convergence of the system states to the sliding surface. This approach often limits the analysis to the second-order systems. Frequency-domain methods could potentially overcome the above-mentioned limitation of the state-space approach. Yet they find limited applications in sliding mode control theory. The present article is aimed at giving an overview of some available and emerging frequency domain methods of analysis of systems having conventional and second-order sliding modes. The method of analysis of transient oscillations is given in detail. A frequency-domain criterion of finite-time convergence is presented.

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References

  1. Aguilar, L., Boiko, I., Fridman, L., Iriarte, R.: Generating self-excited oscillations via two-relay controller. IEEE Transaction on Automatic Control 54(2), 416–420 (2009)

    Article  MathSciNet  Google Scholar 

  2. Atherton, D.: Nonlinear Control Engineering - Describing Function Analysis and Design. Van Nostrand Company Limited, Workingham (1975)

    Google Scholar 

  3. Bartolini, G., Ferrara, A., Levant, A., Usai, E.: On Second Order Sliding Mode Controllers. In: Young, K., Ozguner, U. (eds.) Variable Structure Systems, Sliding Mode and Nonlinear. LNCIS, vol. 247, Springer, London (1999)

    Google Scholar 

  4. Bartolini, G., Ferrara, A., Usai, E.: Chattering Avoidance by Second-Order Sliding Mode Control. IEEE Transactions on Automatic Control 43(2), 241–246 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boiko, I.: Discontinuous Control Systems: Frequency-Domain Analysis and Design. Birkhauser, Boston (2009)

    MATH  Google Scholar 

  6. Boiko, I., Fridman, L.: Analysis of chattering in continuous sliding-mode controllers. IEEE Transaction on Automatic Control 50(9), 1442–1446 (2005)

    Article  MathSciNet  Google Scholar 

  7. Boiko, I., Fridman, L., Castellanos, M.: Analysis of second order sliding mode algorithms in the frequency domain. IEEE Transaction on Automatic Control 49(6), 946–950 (2004)

    Article  MathSciNet  Google Scholar 

  8. Fridman, L.: Singularly perturbed analysis of chattering in relay control systems. IEEE Transactions on Automatic Control 47(12), 2079–2084 (2002)

    Article  MathSciNet  Google Scholar 

  9. Gelb, A., Velde, W.V.: Multiple-Input Describing Functions and Nonlinear System Design. McGraw-Hill, New York (1968)

    MATH  Google Scholar 

  10. Hamel, B.: Contribution a l’etude mathematique des systemes de reglage par tout-ou-rien. Service Technique Aeronautique 17 (1949)

    Google Scholar 

  11. Krylov, N., Bogoliubov, N.: Introduction to Nonlinear Mechanics. Princeton University Press, Princeton (1947)

    Google Scholar 

  12. Levant, A.: Sliding order and sliding accuracy in sliding mode control. International Journal of Control 58(6), 1247–1263 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Levant, A.: Universal SISO sliding-mode controllers with finite-time convergence. IEEE Transactions on Automatic Control 46(9), 1447–1451 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Loeb, J.M.: Advances in nonlinear servo theory. In: Frequency Response, pp. 260–268 (1956)

    Google Scholar 

  15. Man, Z., Poplinsky, A., Wu, H.: A robust MIMO terminal sliding mode control for rigid robotic manipulators. IEEE Transaction on Automatic Control 39(12), 2464–2468 (1994)

    Article  MATH  Google Scholar 

  16. Shtessel, Y., Shkolnikov, I., Brown, M.: An asymptotic second-order smooth sliding mode control. Asian Journal of Control 5(4), 498–504 (2003)

    Article  Google Scholar 

  17. Starikova, M.: Analysis of Control Systems Having Logical Control Devices. Mashinostroenie, Moscow (1978)

    Google Scholar 

  18. Tsypkin, Y.: Relay Control Systems. Cambridge University Press, Cambridge (1984)

    MATH  Google Scholar 

  19. Utkin, V.: Sliding Modes in Control and Optimization. Springer, Heidelberg (1992)

    MATH  Google Scholar 

  20. Yu, X., Wu, Y., Man, Z.: On global stabilization of nonlinear dynamical systems. In: Young, K., Ozguner, U. (eds.) Variable Structure Systems, Sliding Mode and Nonlinear Control. LNCIS, vol. 247, pp. 109–122. Springer, London (1999)

    Chapter  Google Scholar 

  21. Zhiltsov, K.: Approximate Methods for Variable Structure Systems Analysis. Energia, Moscow (1974)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Honeywell and University of Calgary, 2500 University Dr. NW, Calgary, Alberta, T2N 1N4, Canada

    Igor M. Boiko

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  1. Igor M. Boiko
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Editors and Affiliations

  1. Departamento de Ingenieria de Control y Robotica, Division de Ingenieria Electrica, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria, Instituto de Ingenieria, 04510, Mexico, D.F., Mexico

    Leonid Fridman

  2. Instituto de Ingenieria, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria, 04510, Mexico, D.F., Mexico

    Jaime Moreno

  3. Instituto de Ingenieria, Departamento de Ingenieria de Control y Robotica, Division de Ingenieria Electrica, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria,, 04510, Mexico, D.F., Mexico

    Rafael Iriarte

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Boiko, I.M. (2011). Frequency-Domain Methods in Conventional and Higher-Order Sliding Mode Control. In: Fridman, L., Moreno, J., Iriarte, R. (eds) Sliding Modes after the First Decade of the 21st Century. Lecture Notes in Control and Information Sciences, vol 412. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22164-4_3

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  • DOI: https://doi.org/10.1007/978-3-642-22164-4_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22163-7

  • Online ISBN: 978-3-642-22164-4

  • eBook Packages: EngineeringEngineering (R0)

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