Practical Relative Degree Approach in Sliding-Mode Control

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


The high-order sliding-mode approach offers a robust way to solve numerous output-regulation problems when the system relative degree is known. Still the difficult cases remain when the relative degree does not exist, is very high, or the mathematical model is not reliable. The notion of practical relative degree is proposed, which generalizes the standard relative-degree notion for the cases of uncertain systems lacking certain mathematical model. Practical output regulation is ensured. Computer simulation and practical results confirm the theoretical approach.


Relative Degree Slide Mode Control Homogeneity Degree Transmission Unit Finite Time Stability 
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  1. 1.
    Bacciotti, A., Rosier, L.: Liapunov functions and stability in control theory. Springer, London (2005)zbMATHGoogle Scholar
  2. 2.
    Bartolini, G., Ferrara, A., Usai, E.: Chattering avoidance by second-order sliding mode control. IEEE Trans. Automat. Control 43(2), 241–246 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bartolini, G., Pisano, A., Punta, E., 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
  4. 4.
    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
  5. 5.
    Bejarano, F.J., 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
  6. 6.
    Bhat, S.P., Bernstein, D.S.: Finite time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Boiko, I., Fridman, L.: Analysis of chattering in continuous sliding-mode controllers. IEEE Trans. Automatic Control 50(9), 1442–1446 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dinuzzo, F., Ferrara, A.: Higher order sliding mode controllers with optimal reaching. IEEE Trans. Automatic Control 54(9), 2126–2136 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Defoort, M., Floquet, T., Kokosy, A., Perruquetti, W.: A novel higher order sliding mode control scheme. Systems & Control Letters 58, 102–108 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dorel, L.: Glucose level regulation via integral high-order sliding modes. Mathematical Biosciences and Engineering 8(2), 549–560 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Evangelista, C., Puleston, P., Valenciaga, F.: Wind turbine efficiency optimization. Comparative study of controllers based on second order sliding modes. International Journal of Hydrogen Energy (2010), available online January 8, 2010Google Scholar
  12. 12.
    Edwards, C., Spurgeon, S.K.: Sliding mode control: theory and applications. Taylor & Francis (1998)Google Scholar
  13. 13.
    Filippov, A.F.: Differential equations with discontinuous right-hand side. Kluwer, Dordrecht (1988)Google Scholar
  14. 14.
    Floquet, T., Barbot, J.-P., Perruquetti, W.: Higher-order sliding mode stabilization for a class of nonholonomic perturbed systems. Automatica 39, 1077–1083 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Fridman, L.: Chattering analysis in sliding mode systems with inertial sensors. International Journal of Control 76(9/10), 906–912 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Furuta, K., Pan, Y.: Variable structure control with sliding sector. Automatica 36, 211–228 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Gallardo-Hernandez, A.G., Fridman, L., Levant, A., Shtessel, Y., Leder, R., Islas-Andrade, S., Revilla-Monsalve, C.: High-order sliding-mode control of blood glucose concentration via practical relative degree identification. In: Proc. IEEE CDC 2011, Orlando, FL, USA, December 12-15 (2011)Google Scholar
  18. 18.
    Isidori, A.: Nonlinear control systems, 2nd edn. Springer, New York (1989)zbMATHGoogle Scholar
  19. 19.
    Kaveh, P., Shtessel, Y.B.: Blood glucose regulation using higher-order sliding mode control. International Journal of Robust and Nonlinear Control 18(4-5), 557–569 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kokotovic, P.V., Khalil, H.K., O’Reilly, J.: Singular perturbation methods in control: analysis and design. SIAM (1999)Google Scholar
  21. 21.
    Kolmogoroff, A.N.: On inequalities between upper bounds of consecutive deriva-tives of an arbitrary function defined on an infinite interval. Amer. Math. Soc. Transl. 2, 233–242 (1962)Google Scholar
  22. 22.
    Levant, A., Levantovsky, L.V.: Sliding order and sliding accuracy in sliding mode control. International Journal of Control 58(6), 1247–1263 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Levant, A.: Robust exact differentiation via sliding mode technique. Automatica 34(3), 379–384 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Levant, A.: Higher-order sliding modes, differentiation and output-feedback control. International Journal of Control 76(9/10), 924–941 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica 41(5), 823–830 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Levant, A.: Quasi-continuous high-order sliding-mode controllers. IEEE Trans. Automat. Control 50(11), 1812–1816 (2006)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Levant, A.: Robustness of homogeneous sliding modes to relative degree fluctuations. In: Proc. of 6th IFAC Symposium on Robust Control Design, Haifa, Israel, June 16-18 (2009)Google Scholar
  28. 28.
    Levant, A.: Chattering analysis. IEEE Transactions on Automatic Control 55(6), 1380–1389 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Levant, A.: Digital sliding-mode-based differentiation. In: Proc. 11th Scientific Workshop VSS 2012, Mumbay, India, January 12-14 (2012)Google Scholar
  30. 30.
    Levant, A., Alelishvili, L.: Integral high-order sliding modes. IEEE Trans. Automat. Control 52(7), 1278–1282 (2007)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Levant, A., Fridman, L.: Accuracy of homogeneous sliding modes in the presence of fast actuators. IEEE Transactions on Automatic Control 55(3), 810–814 (2010)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Levant, A., Michael, A.: Adjustment of high-order sliding-mode controllers. International Journal of Robust and Nonlinear Control 19(15), 1657–1672 (2009)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Levant, A., Pavlov, Y.: Generalized homogeneous quasi-continuous controllers. International Journal of Robust and Nonlinear Control 18(4-5), 385–398 (2008)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Levant, A., Pridor, A., Gitizadeh, R., Yaesh, I., Ben-Asher, J.Z.: Aircraft pitch control via second order sliding technique. J. of Guidance, Control and Dynamics 23(4), 586–594 (2000)CrossRefGoogle Scholar
  35. 35.
    Kaveh, P., Shtessel, Y.B.: Blood glucose regulation using higher-order sliding mode control. International Journal of Robust and Nonlinear Control 18(4-5), 557–569 (2008)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Man, Z., Paplinski, A.P., Wu, H.R.: A robust MIMO terminal sliding mode control for rigid robotic manipulators. IEEE Trans. Automat. Control 39(12), 2464–2468 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Massey, T., Shtessel, Y.: Continuous traditional and high order sliding modes for satellite formation control. AIAA J. Guidance, Control, and Dynamics 28(4), 826–831 (2005)CrossRefGoogle Scholar
  38. 38.
    Orlov, Y.: Finite time stability and robust control synthesis of uncertain switched systems. SIAM J. Cont. Optim. 43(4), 1253–1271 (2005)zbMATHCrossRefGoogle Scholar
  39. 39.
    Pisano, A., Davila, J., Fridman, L., Usai, E.: Cascade control of PM DC drives via second-order sliding-mode technique. IEEE Transactions on Industrial Electronics 55(11), 3846–3854 (2008)CrossRefGoogle Scholar
  40. 40.
    Sira-Ramirez, H.: On the dynamical sliding mode control of nonlinear systems. International Journal of Control 57(5), 1039–1061 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    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
  42. 42.
    Utkin, V.I.: Sliding modes in optimization and control problems. Springer, New York (1992)CrossRefGoogle Scholar
  43. 43.
    Yu, X., Xu, J.X.: An adaptive signal derivative estimator. Electronic Letters 32(16) (1996)Google Scholar
  44. 44.
    Yu, S., Yu, X., Shirinzadeh, B., Man, Z.: Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica 41(11), 1957–1964 (2005)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Tel-Aviv UniversityTel-AvivIsrael

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