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On second order sliding mode controllers

  • Giorgio Bartolini
  • Antonella Ferrara
  • Arie Levant
  • Elio Usai
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 247)

Abstract

In this paper a collection of control algorithms which are able to give rise to 2-sliding modes have been presented, and for each of them the sufficient convergence conditions are given. Furthermore, the real sliding behaviour is briefly considered, and, in some cases, the upper bound of the convergence time is given.

2-sliding mode control seems to be an effective tool for the control of uncertain nonlinear systems since it overcomes the main drawbacks of the classical sliding mode control, i.e., the chattering phenomenon and the large control effort. Its real implementation implies very simple control laws and assures an improvement of the sliding accuracy with respect to real 1-sliding mode control.

The main difficulty in using 2-sliding mode controllers is the tuning of the parameters which characterize the various algorithms. Their values depend on the bounds of the uncertain dynamics and on the chosen sliding manifold, and only sufficient conditions for the convergence to the sliding behaviour are known. These conditions are very conservative, and, in practice, the parameters are heuristically tuned. It depends on the engineer’s experience to define which of the presented algorithms is more suitable for the specific control problem, even if, in the authors’ opinion, the super-twisting and the sub-optimal ones seem to be able to cover a large class of control problems with a remarkable implementation easiness.

Keywords

Finite Time Slide Mode Control Uncertain System Uncertain Nonlinear System Finite Time Stabilization 
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.

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References

  1. 1.
    G. Bartolini, P. Pydynowski, “An improved chattering free VSC scheme for uncertain dynamical systems,” IEEE Trans. Automatic Control, vol. 41, pp. 1220–1226, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    G. Bartolini, A. Ferrara, E. Usai, “Second order VSC for non linear systems subjected to a wide class of uncertainty conditions”, Proc. of the 1996 IEEE Int. Workshop on Variable Structure Systems — VSS'96, Seiken Symposium No. 19, pp. 49–54, Tokyo, Japan, December 1996.Google Scholar
  3. 3.
    G. Bartolini, A. Ferrara, E. Usai, “Applications of a sub-optimal discontinuous control algorithm for uncertain second order systems”, Int. J. of Robust and Nonlinear Control, vol. 7, no. 4, pp. 299–319, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    G. Bartolini, A. Ferrara, E. Usai, “Output Tracking Control of Uncertain Nonlinear Second-Order Systems”, Automatica, vol. 33, no. 12, pp. 2203–2212, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    G. Bartolini, A. Ferrara, A. Pisano, E. Usai, “Adaptive reduction of the control effort in chattering free sliding mode control of uncertain nonlinear systems”, Applied Mathematics and Computer Science, vol. 8, no. 1, pp. 51–71, 1998.zbMATHMathSciNetGoogle Scholar
  6. 6.
    G. Bartolini, A. Ferrara, E. Usai, “Chattering Avoidance by Second Order Sliding Mode Control”, IEEE Trans. Automatic Control, vol. 43, no. 2, pp. 241–246, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    G. Bartolini, A. Ferrara, E. Usai, “Real-time output derivatives estimation by means of higher order sliding modes”, Proc. of CESA’ 98 IMACS Multiconference, pp. 782–787, Nabeul-Hammamet, Tunisia, April 1998.Google Scholar
  8. 8.
    G. Bartolini, A. Ferrara, E. Usai, V. I. Utkin, “Second order chattering-free sliding mode control for some classes of multi-input uncertain nonlinear systems”, Proc. of the 6th IEEE Mediterranean Conf. on Control and Systems, Alghero, Italy, June 1998, to be published.Google Scholar
  9. 9.
    G. Bartolini, A. Pisano, E. Usai, “Digital Second Order Sliding Mode Control of SISO Uncertain Nonlinear Systems', Proc. of the 1998 American Control Conference ACC'98, vol. 1, pp. 119–124, Philadelfia, Pensilvania, June 1998.Google Scholar
  10. 10.
    ML.W. Chang, “A MIMO sliding control with second order sliding condition,” ASME W.A.M., paper no. 90-WA/DSC-5, Dallas, Texas, 1990.Google Scholar
  11. 11.
    M. Corless, and G. Leitmann, “Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems,’ IEEE Trans. Automatic Control, vol. 26, pp. 1139–1141, 1981.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    M. Corless, G. Leitmann, “Exponential Convergence for Uncertain Systems with Component-Wise Bounded Controllers”, Robust Control via variable structure and Lyapunov techniques. F. Garofalo and L. Glielmo eds., Lecture Notes in Control and Information Science no. 217, pp. 175–196, Springer-Verlag, London, 1996.CrossRefGoogle Scholar
  13. 13.
    S.V. Emel'yanov, S.K. Korovin, L.V. Levantovsky, “Higher-Order Sliding Modes in the Binary Control Systems”, Soviet Physics, Dokady, vol. 31, no. 4, pp. 291–293, 1986.zbMATHGoogle Scholar
  14. 14.
    S.V. Emel'yanov, S.K. Korovin, L.V. Levantovsky, “Drift Algorithm in Control of Uncertain Processes”, Problems of Control and Information Theory, vol. 15, no. 6, pp. 425–438, 1986.MathSciNetGoogle Scholar
  15. 15.
    A.F. Filippov, Differential Equations with Discontinuous Righthand Side, Kluwer Academic Publishers, Dordrecht, 1988.Google Scholar
  16. 16.
    A. Isidori, Nonlinear control systems, Springer-Verlag, New York, 1989.zbMATHGoogle Scholar
  17. 17.
    D.E. Kirk, Optimal control theory, Prentice Hall, Englewood Cliffs, NJ, 1970.Google Scholar
  18. 18.
    G. Leitmann, The calculus of variations and optimal control, Plenum Press, New York, 1981.zbMATHGoogle Scholar
  19. 19.
    L. Fridman, A. Levant, “Higher Order Sliding Modes as a Natural Phenomenon in Control Theory”, in Robust Control via variable structure and Lyapunov techniques. F. Garofalo and L. Glielmo eds., Lecture Notes in Control and Information Science no. 217, pp. 107–133, Springer-Verlag, London, 1996.CrossRefGoogle Scholar
  20. 20.
    A. Levant (Levantovsky L.V.), “Sliding order and sliding accuracy in sliding mode control”, Int. J. of Control, vol. 58, no. 6, pp. 1247–1263, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    A. Levant, “Robust exact differentiation via sliding mode technique”, Automatica, vol.34, no. 3, pp. 379–384, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    A. Levant, “Arbitrary-order sliding modes with finite time convergence”, Proc. of the 6th IEEE Mediterranean Conf. on Control and Systems, Alghero, Italy, June 1998, to be published.Google Scholar
  23. 23.
    C. Milosavljevic, “General conditions for the existence of a quasisliding mode on the switching hyperplane in discrete variable systems”, Automation Remote Control, vol. 43, no. 1, pp. 307–314, 1985.Google Scholar
  24. 24.
    K.S. Narendra, A.M. Annaswamy, Stable adaptive systems, Prentice-Hall, Englewood Cliffs, NJ, 1989.Google Scholar
  25. 25.
    H. Sira-Ramirez, “On the sliding mode control of nonlinear systems,” System and Control Letters, vol. 19, pp. 303–312, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    V.I. Utkin, Sliding modes in control and optimization, Springer Verlag, Berlin, 1992.zbMATHGoogle Scholar
  27. 27.
    M. Zhihong, A.P. Paplinski, H.R. Wu, “A robust MIMO terminal sliding mode control for rigid robotic manipulators”, IEEE Trans. Automatic Control, vol. 39, no. 12, pp. 2464–2468, 1994.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • Giorgio Bartolini
    • 1
  • Antonella Ferrara
    • 2
  • Arie Levant
    • 3
  • Elio Usai
    • 1
  1. 1.Department of Electrical and Electronic EngineeringUniversity of CagliariCagliariItaly
  2. 2.Department of Communication, Computer and System SciencesUniversity of GenovagenovaItaly
  3. 3.Institute for Industrial MathematicsBeer-ShevaIsrael

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