Discrete-Time Sliding-Mode-Based Differentiation

  • Arie Levant
  • Miki Livne
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 440)


Homogeneous sliding-mode-based differentiators provide for the high accuracy robust finite-time-exact estimation of derivatives. It is shown that their discrete-time implementation misses the homogeneity, and respectively features worse accuracy with respect to the sampling time interval. Detailed analysis of the asymptotic accuracy is provided in both cases of constant and variable sampling intervals.


Sampling Interval Slide Mode Control Differential Inclusion Slide Mode Observer Variable Sampling Interval 
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  1. 1.
    Atassi, A., Khalil, H.: Separation results for the stabilization of nonlinear sys-tems using different high-gain observer designs. Systems & Control Letters 39(3), 183–191 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bacciotti, A., Rosier, L.: Liapunov Functions and Stability in Control Theory. Springer, London (2005)zbMATHGoogle Scholar
  3. 3.
    Bartolini, G.: Chattering phenomena in discontinuous control systems. Internat. J. Systems Sci. 20, 2471–2481 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bartolini, G., Pisano, A.E.P., Usai, E.: A survey of applications of second-order sliding mode control to mechanical systems. International Journal of Control 76(9/10), 875–892 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bartolini, G., Pisano, A., Usai, E.: First and second derivative estimation by sliding mode technique. Journal of Signal Processing 4(2), 167–176 (2000)Google Scholar
  6. 6.
    Bejarano, F., Fridman, L.: High order sliding mode observer for linear systems with unbounded unknown inputs. International Journal of Control 83(9), 1920–1929 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cruz-Zavala, E., Moreno, J.: L., F.: Uniform second-order sliding mode observer for mechanical systems. In: Proc. of 11th International Workshop VSS, June 26-28, pp. 14–19 (2010)Google Scholar
  8. 8.
    Edwards, C., Spurgeon, S.: Sliding Mode Control: Theory and Applications. systems and control book series. Taylor & Francis. Taylor & Francis (1998)Google Scholar
  9. 9.
    Filippov, A.: Differential Equations with Discontinuous Right-Hand Sides. Mathematics and Its Applications. Kluwer Academic Publishers (1988)Google Scholar
  10. 10.
    Fridman, L.: Chattering analysis in sliding mode systems with inertial sensors. International Journal of Control 76(9/10), 906–912 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kobayashi, S., Suzuki, S., Furuta, K.: Frequency charachteristics of levant’s differentiator and adaptive sliding mode differentiator. International Journal of Systems Science 38(10), 825–832 (2007)zbMATHCrossRefGoogle Scholar
  12. 12.
    Kolmogoroff, A.N.: On inequalities between upper bounds of consecutive derivatives of an arbitrary function defined on an infinite interval. Amer. Math. Soc. Transl. 2, 233–242 (1962)Google Scholar
  13. 13.
    Levant, A.: Sliding order and sliding accuracy in sliding mode control. International J. Control 58, 1247–1263 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Levant, A.: Robust exact differentiation via sliding mode technique. Automatica 34(3), 379–384 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Levant, A.: Higher order sliding modes, differentiation and output-feedback control. International J. Control 76(9/10), 924–941 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica 41(5), 823–830 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Levant, A.: Quasi-continuous high-order sliding-mode controllers. IEEE Trans. Aut. Control 50(11), 1812–1816 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Levant, A.: Chattering analysis. IEEE Trans. Aut. Control 55(6), 1380–1389 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Levant, A.: Discretization issues of high-order sliding modes. In: Proc. of 18th IFAC Congress, Milano, Italy (2011)Google Scholar
  20. 20.
    Levant, A., Pridor, A., Gitizadeh, R., Yaesh, I., Ben-Asher, J.Z.: Aircraft pitch control via second-order sliding technique. AIAA Journal of Guidance, Control 23(4), 586–594 (2000)CrossRefGoogle Scholar
  21. 21.
    Plestan, F., Glumineau, A., Laghrouche, S.: A new algorithm for high-order sliding mode control. International Journal of Robust and Nonlinear Control 18(4/5), 441–453 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Shtessel, Y.B., Shkolnikov, I.A.: Aeronautical and space vehicle control in dynamic sliding manifolds. International Journal of Control 76(9/10), 1000–1017 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Utkin, V.I.: Sliding modes in control and optimization. Springer Verlag, Berlin, Germany (1992)Google Scholar
  24. 24.
    Yu, X., Xu, J.X.: Nonlinear derivative estimator. Electronic Letters 32(16) (1996)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Ramat-AvivTel-Aviv UniversityTel-AvivIsrael

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