Sliding Mode Control for Systems with Fast Actuators: Singularly Perturbed Approach

  • Leonid M. Fridman
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 274)


Singularly perturbed relay control systems with second order sliding modes are considered for the modeling of sliding mode control systems with fast actuators. For sliding mode control systems with fast actuators, sufficient conditions for the exponential decreasing of the amplitude of chattering and unlimited growth of frequency are found. The connection between the stability of actuators and the stability of the plant on the one hand and the stability of the sliding mode system as the whole on the other hand is investigated. The algorithm for correction of sliding mode equations is suggested for taking into account the presence of fast actuators. Algorithms are proposed to solve the problem of eigenvalues assignment or optimal stabilization for sliding motions using the additional dynamics of fast actuators.


Slide Mode Control Slow Motion Sliding Mode Relay System Integral Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anosov D.V. (1959) On stability of equilibrium points of relay systems. Automatica i telemechanica ( Automation and Remote Control) 10, 135–149 (Russian)MathSciNetGoogle Scholar
  2. 2.
    Bartolini G. (1990) Chattering phenomena in discontinuous control systems. Int. Journ. of System Science, 20, 2471–2481CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bondarev A.G., Bondarev S.A. et al. (1985) On stability of equilibrium points of relay systems. Automatica i telemechanika ( Automation and Remote Control) 46, 679–684zbMATHGoogle Scholar
  4. 4.
    Filippov A.F. (1988) Differential Equations with Discontinuous Right Hand Side. Kluwer Publishers, DodrechtGoogle Scholar
  5. 5.
    Fridman L., Levant A. (1996) Higher order sliding modes as the natural phenomenon in control theory. In: F. Garafalo and L. Glielmo. (Eds.) Robust Control via Variable Structure and Lyapunov Techniques, Lecture notes in computer and information science, 217, Springer-Verlag, Berlin, 107–133CrossRefGoogle Scholar
  6. 6.
    Fridman L., Levant A. (2001) Higher order sliding modes. in: J.P Barbot, W. Perruguetti (Eds.) Sliding mode control in engineering, Marcel Dekker: New York, 51–100.Google Scholar
  7. 7.
    Fridman L.M. (1999) The problem of chattering: an averaging approach. In: Young K.K.D., Ozguner U. (Eds.) Variable Structure Systems, Sliding Mode and Nonlinear Control, Lecture Notes in Control and Information Sciences, 247, Springer-Verlag, Berlin, 363–386CrossRefGoogle Scholar
  8. 8.
    Fridman L.M. (2001) An averaging approach to chattering. IEEE Transaction of Automatic Control, 46(8), 1260–1265zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Heck B.S. (1991) Sliding mode control for singularly perturbed system. Int. Journ. of Control, 53, 985–1001zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Krupp D., Shkolnikov I.A., Y.B. Shtessel (2000) 2-sliding mode control for nonlinear plants with parametric and dynamic. Proc. of AIAA Guidance Navigation and Control Conference, Denver, CO, August, 2000, AIAA paper 2000–3965Google Scholar
  11. 11.
    Kokotovic P.V., Khalil H.K. et al. (1986) Singular Perturbation Methods in Control: Analysis and Design. Academic Press, LondonzbMATHGoogle Scholar
  12. 12.
    Levant A. (1998) Robust exact differentiation via sliding mode technique. Automatica, 34, 379–384zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Sira-Ramires H. (1988) Sliding regimes on slow manifolds of systems with fast actuators. Int. J. of System Science, 37, 875–887Google Scholar
  14. 14.
    Sira-Ramires H. (1989) Sliding regimes in general non-linear systems: a relative degree approach. Int. J. of Control. 50, 1487–1506CrossRefGoogle Scholar
  15. 15.
    Sobolev V.A. (1992) Singular perturbation in linear-quadratic problem of optimal control. Automation and Remote Control. 53, 53–63MathSciNetGoogle Scholar
  16. 16.
    Su W.-C. (1999) Sliding mode control for singularly perturbed systems. Int. J. of Control. 72, 990–995zbMATHCrossRefGoogle Scholar
  17. 17.
    Utkin V.I. (1992) Sliding Modes in Control Optimization. Springer Verlag, BerlinzbMATHGoogle Scholar
  18. 18.
    Utkin V., Guldner J. et al. (1999) Sliding Modes in Electromechanical Systems. Taylor and Francis, LondonGoogle Scholar
  19. 19.
    Vasilieva A.B., Butusov V.F. et al. (1995) The Boundary Layer Method for Singular Perturbation Problems. SIAM, PhiladelphiiaGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Leonid M. Fridman
    • 1
  1. 1.Chihuahua Institute of TechnologyChihuahua, Chih.Mexico

Personalised recommendations