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Second-Order Sliding Mode Controllers and Differentiators

  • Yuri Shtessel
  • Christopher Edwards
  • Leonid Fridman
  • Arie Levant
Part of the Control Engineering book series (CONTRENGIN)

Abstract

As we have seen, classic sliding modes provide robust and high-accuracy solutions for a wide range of control problems under uncertainty conditions. However, two main restrictions remain. First, the constraint to be held at zero in conventional sliding modes has to be of relative degree 1, which means that the control needs to explicitly appear in the first time derivative of the constraint.

Keywords

Finite Time Relative Degree Slide Mode Control Virtual Control Noise Magnitude 
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.

References

  1. 12.
    Baccioti, A., Rosier, L.: Lyapunov Functions and Stability in Control Theory, 2nd edn. Springer, New York (2005)CrossRefGoogle Scholar
  2. 15.
    Bartolini, G., Pisano, A., Usai, E.: First and second derivative estimation by sliding mode technique. J. Signal Proc. 4(2), 167–176 (2000)Google Scholar
  3. 16.
    Bartolini, G., Pydynowski, P.: An improved, chattering free, V.S.C. scheme for uncertain dynamical systems. IEEE Trans. Automat. Contr. 41(8), 1221–1226 (1996)Google Scholar
  4. 17.
    Bartolini, G., Ferrara, A., Usai, E.: Output tracking control of uncertain nonlinear second-order systems. Automatica 33(12), 2203–2212 (1997)MATHMathSciNetCrossRefGoogle Scholar
  5. 18.
    Bartolini, G., Ferrara, A., Usai, E.: Chattering avoidance by second-order sliding mode control. IEEE Trans. Automat. Contr. 43(2), 241–246 (1998)MATHMathSciNetCrossRefGoogle Scholar
  6. 19.
    Bartolini, G., Ferrara A., Levant, A., Usai, E.: On second order sliding mode controllers. In: Young, K.D., Ozguner, U. (eds.) Variable Structure Systems, Sliding Mode and Nonlinear Control. Lecture Notes in Control and Information Sciences, vol. 247, pp. 329–350. Springer, Berlin (1999)CrossRefGoogle Scholar
  7. 20.
    Bartolini, G., Pisano, A., Punta, E., Usai, E.: A survey of applications of second-order sliding mode control to mechanical systems. Int. J. Control 76(9/10), 875–892 (2003)MATHMathSciNetCrossRefGoogle Scholar
  8. 29.
    Boiko, I.: Discontinuous Control Systems: Frequency-Domain Analysis and Design. Birkhauser, Boston (2009)Google Scholar
  9. 31.
    Boiko, I., Fridman, L.: Analysis of chattering in continuous sliding-mode controllers. IEEE Trans. Automat. Contr. 50(9), 1442–1446 (2005)MathSciNetCrossRefGoogle Scholar
  10. 33.
    Boiko,I., Iriarte, R., Pisano, A., Usai, E.: Parameter tuning of second-order sliding mode controllers for linear plants with dynamic actuators. Automatica 42(5), 833–839 (2006)MATHMathSciNetCrossRefGoogle Scholar
  11. 35.
    Boiko, I., Fridman, L., Pisano, A., Usai, E.: Performance analysis of second-order sliding-mode control systems with fast actuators. IEEE Trans. Automat. Contr. 52(6), 1053–1059 (2007)MathSciNetCrossRefGoogle Scholar
  12. 51.
    Cruz-Zavala, E., Moreno, J., Fridman, L.: Uniform robust exact differentiator. IEEE Trans. Automat. Contr. 56(11), 2727–2733 (2011)MathSciNetCrossRefGoogle Scholar
  13. 75.
    Emelyanov, S.V., Korovin, S.K. Levantovsky, L.V.: Higher order sliding modes in the binary control systems. Sov. Phys. Dokl. 31(4), 291–293 (1986)Google Scholar
  14. 87.
    Fridman, L.: The problem of chattering: an averaging approach. In: Young, K., Ozguner, U. (eds.) Variable Structure, Sliding Mode and Nonlinear Control. Lecture Notes in Control and Information Science, vol. 247, pp. 363–386. Springer, London (1999)CrossRefGoogle Scholar
  15. 90.
    Fridman, L.: Chattering analysis in sliding mode systems with inertial sensors. Int. J. Control 76(9/10), 906–912 (2003)MATHMathSciNetCrossRefGoogle Scholar
  16. 98.
    Furuta, K., Pan, Y.: Variable structure control with sliding sector. Automatica 36(2), 211–228 (2000)MATHMathSciNetCrossRefGoogle Scholar
  17. 101.
    Gonzalez, T., Moreno, J., Fridman, L.: Variable Gain Super-Twisting Sliding Mode Control. IEEE Trans. Automat. Contr. 57(8), 2100–2105 (2012)MathSciNetCrossRefGoogle Scholar
  18. 122.
    Krupp, D.R., Shkolnikov, I.A., Shtessel, Y.B.: 2-sliding mode control for nonlinear plants with parametric and dynamic uncertainties. In: Proceedings of Conference on Guidance Navigation and Control, AIAA paper, pp. 2000–3965, Denver, CO (2000)Google Scholar
  19. 123.
    Levant, A.: Robust exact differentiation via sliding mode technique. Automatica 34(3), 379–384 (1998)MATHMathSciNetCrossRefGoogle Scholar
  20. 125.
    Levant, A.: Higher-order sliding modes, differentiation and output-feedback control. Int. J. Control 76(9/10), 924–941 (2003).MATHMathSciNetCrossRefGoogle Scholar
  21. 126.
    Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica 41(5), 823–830 (2005)MATHMathSciNetCrossRefGoogle Scholar
  22. 127.
    Levant, A.: Quasi-continuous high-order sliding-mode controllers. IEEE Trans. Automat. Contr. 50(11) 1812–1816 (2006)MathSciNetCrossRefGoogle Scholar
  23. 128.
    Levant, A.: Construction principles of 2-sliding mode design. Automatica 43(4), 576–586 (2007)MATHMathSciNetCrossRefGoogle Scholar
  24. 132.
    Levant, A., Levantovsky, L.V.: Sliding order and sliding accuracy in sliding mode control. Int. J. Control 586, 1247–1263 (1993)CrossRefGoogle Scholar
  25. 138.
    Man Z., Paplinski, A.P., Wu, H.R.: A robust MIMO terminal sliding mode control for rigid robotic manipulators. IEEE Trans. Automat. Contr. 39(12), 2464–2468 (1994)MATHMathSciNetCrossRefGoogle Scholar
  26. 142.
    Moreno, J.A.: Lyapunov approach to analysis and design of second order sliding mode algorithms. In: Fridman, L., Moerno, J., Iriarte, R. (eds.) Sliding Modes after the first Decade of the 21st Century. Lecture Notes in Control and Information Science, vol. 412, pp. 115–149. Springer, Berlin (2011)Google Scholar
  27. 143.
    Moreno, J., Osorio, M.: Strict lyapunov functions for the super-twisting algorithm. IEEE Trans. Automat. Contr. 57(4), 1035–1040 (2012)MathSciNetCrossRefGoogle Scholar
  28. 152.
    Polyakov, A.: Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans. Automat. Contr. 57(8), 2106–2110 (2012)MathSciNetCrossRefGoogle Scholar
  29. 153.
    Polyakov, A., Poznyak, A.: Reaching time estimation for super-twisting second order sliding mode controller via Lyapunov function designing. IEEE Trans. Automat. Contr. 54(8), 1951–1955 (2009)MathSciNetCrossRefGoogle Scholar
  30. 171.
    Slotine, J.-J., Li, W.: Applied Nonlinear Control. Prentice Hall, New Jersey (1991)MATHGoogle Scholar
  31. 196.
    Zubov, V.I.: Methods of A. M. Lyapunov and Their Applications. Noordhoff International, Groningen (1964)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Yuri Shtessel
    • 1
  • Christopher Edwards
    • 2
  • Leonid Fridman
    • 3
  • Arie Levant
    • 4
  1. 1.Department of Electrical and Computer EngineeringUniversity of Alabama in HuntsvilleHuntsvilleUSA
  2. 2.College of Engineering, Mathematics and Physical ScienceUniversity of ExeterExeterUK
  3. 3.Department of Control Division of Electrical EngineeringFaculty of Engineering National Autonomous University of MexicoFederal DistrictMexico
  4. 4.Department of Applied Mathematics School of Mathematical SciencesTel-Aviv UniversityTel-AvivIsrael

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