Advertisement

Finite-Time Stability and High Relative Degrees in Sliding-Mode Control

  • Arie Levant
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 412)

Abstract

Establishing and exactly keeping constraints of high relative degrees is a central problem of the modern sliding-mode control. Its solution in finite-time is based on so-called high-order sliding modes, and is reduced to finite-time stabilization of an auxiliary uncertain system. Such stabilization is mostly based on the homogeneity approach. Robust exact differentiators are also developed in this way and are used to produce robust output-feedback controllers. The resulting controllers feature high accuracy in the presence of sampling noises and delays, ultimate robustness to the presence of unaccounted-for fast stable dynamics of actuators and sensors, and to small model uncertainties affecting the relative degrees. The dangerous types of the chattering effect are removed artificially increasing the relative degree. Parameters of the controllers and differentiators can be adjusted to provide for the needed convergence rate, and can be also adapted in real time. Simulation results and applications are presented in the fields of control, signal and image processing.

Keywords

IEEE Transaction Relative Degree Differential Inclusion Homogeneity Degree Homogeneity Approach 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aguilar-Lopez, R., Martinez-Guerra, R., Puebla, H., Hernandez-Suerez, R.: High order sliding-mode dynamic control for chaotic intracellular calcium oscillations. Nonlinear Analysis: Real World Applications 11(1), 217–231 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Atassi, A.N., Khalil, H.K.: Separation results for the stabilization of nonlinear systems using different high-gain observer designs. Systems & Control Letters 39 (2000)Google Scholar
  3. 3.
    Bacciotti, A., Rosier, L.: Liapunov functions and stability in control theory. Springer, London (2005)zbMATHGoogle Scholar
  4. 4.
    Bartolini, G., Ferrara, A., Usai, E.: Chattering avoidance by second-order sliding mode control. IEEE Transaction on Automatic Control 43(2), 241–246 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bartolini, G., Ferrara, A., Usai, E., Utkin, V.: On multi-input chattering-free second-order sliding mode control. IEEE Transaction on Automatic Control 45(9), 1711–1717 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    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
  8. 8.
    Bartolini, G., Pisano, A., Usai, E.: Global stabilization for nonlinear uncertain systems with unmodeled actuator dynamics. IEEE Transaction on Automatic Control 46(11), 1826–1832 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bartolini, G., Punta, E., Zolezzi, T.: Approximability properties for second-order sliding mode control systems. IEEE Transaction on Automatic Control 52(10), 1813–1825 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bejarano, F., Fridman, L.: Unbounded unknown inputs estimation based on high-order sliding mode differentiator. In: Proc. of 48th IEEE CDC, Shanghai, China (2009)Google Scholar
  11. 11.
    Beltran, B., Ahmed-Ali, T., Benbouzid, M.: High-order sliding-mode control of variable-speed wind turbines. IEEE Transaction on Automatic Control 56(11), 3314–3321 (2009)Google Scholar
  12. 12.
    Benallegue, A., Mokhtari, A., Fridman, L.: High-order sliding-mode observer for a quadrotor UAV. International Journal of Robust Nonlinear Control 18(4), 427–440 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bhat, S., Bernstein, D.: Finite time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Boiko, I., Fridman, L.: Analysis of chattering in continuous sliding-mode controllers. IEEE Transaction on Automatic Control 50(9), 1442–1446 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Daniele, B., Capisani, M., Ferrara, A., Pisu, P.: Fault Detection for Robot Manipulators via Second-Order Sliding Modes. IEEE Transactions on Industrial Electronics 55(11), 3954–3963 (2008)CrossRefGoogle Scholar
  16. 16.
    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
  17. 17.
    Dinuzzo, F., Ferrara, A.: Higher order sliding mode controllers with optimal reaching. IEEE Transaction on Automatic Control 54(9), 2126–2136 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Edwards, C., Spurgeon, S.K.: Sliding Mode Control: Theory and Applications. Taylor & Francis, Abington (1998)Google Scholar
  19. 19.
    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 35(11), 5934–5939 (2010)CrossRefGoogle Scholar
  20. 20.
    Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Side. Kluwer, Dordrecht (1988)Google Scholar
  21. 21.
    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
  22. 22.
    Fridman, L.: An averaging approach to chattering. IEEE Transactions on Automatic Control 46, 1260–1265 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Fridman, L.: Chattering analysis in sliding mode systems with inertial sensors. International Journal of Control 76(9/10), 906–912 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Furuta, K., Pan, Y.: Variable structure control with sliding sector. Automatica 36, 211–228 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Isidori, A.: Nonlinear Control Systems, 2nd edn. Springer, New York (1989)zbMATHGoogle Scholar
  26. 26.
    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
  27. 27.
    Kobayashi, S., Suzuki, S., Furuta, K.: Adaptive vs differentiator, advances in variable structure systems. In: Proc. of the 7th VSS Workshop. Sarajevo (2002)Google Scholar
  28. 28.
    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
  29. 29.
    Krupp, D., Shkolnikov, I.A., Shtessel, Y.B.: 2-sliding mode control for nonlinear plants with parametric and dynamic uncertainties. In: Proceedings of AIAA Guidance, Navigation, and Control Conference, Denver, CO (2000)Google Scholar
  30. 30.
    Levant, A.: Sliding order and sliding accuracy in sliding mode control. International Journal of Control 58(6), 1247–1263 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Levant, A.: Robust exact differentiation via sliding mode technique. Automatica 34(3), 379–384 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Levant, A.: Higher-order sliding modes, differentiation and output-feedback control. International Journal of Control 76(9/10), 924–941 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica 41(5), 823–830 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Levant, A.: Exact differentiation of signals with unbounded higher derivatives. In: Procedings of the 45th IEEE Conference on Decision and Control CDC 2006, San-Diego, CA, USA (2006)Google Scholar
  35. 35.
    Levant, A.: Quasi-continuous high-order sliding-mode controllers. IEEE Transaction on Automatic Control 50(11), 1812–1816 (2006)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Levant, A.: Chattering analysis. IEEE Transactions on Automatic Control 55(6), 1380–1389 (2010)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Levant, A.: Construction principles of 2-sliding mode design. Automatica 43(4), 576–586 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Levant, A.: Finite differences in homogeneous discontinuous control. IEEE Transaction on Automatic Control 52(7), 1208–1217 (2007)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Levant, A.: Robustness of homogeneous sliding modes to relative degree fluctuations. In: Proceeding of the 6th IFAC Symposium on Robust Control Design, Haifa, Israel (2009)Google Scholar
  40. 40.
    Levant, A., Alelishvili, L.: Integral high-order sliding modes. IEEE Transaction on Automatic Control 52(7), 1278–1282 (2007)MathSciNetCrossRefGoogle Scholar
  41. 41.
    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
  42. 42.
    Levant, A., Michael, A.: Adjustment of high-order sliding-mode controllers. International Journal of Robust Nonlinear Control 19(15), 1657–1672 (2009)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Levant, A., Pavlov, Y.: Generalized homogeneous quasi-continuous controllers. International Journal of Robust Nonlinear Control 18(4-5), 385–398 (2008)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Levant, A., Pridor, A., Gitizadeh, R., Yaesh, I., Ben-Asher, J.: Aircraft pitch control via second order sliding technique. J. of Guidance, Control and Dynamics 23(4), 586–594 (2000)CrossRefGoogle Scholar
  45. 45.
    Man, Z., Paplinski, A., Wu, H.: A robust MIMO terminal sliding mode control for rigid robotic manipulators. IEEE Transaction on Automatic Control 39(12), 2464–2468 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    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
  47. 47.
    Orlov, Y.: Finite time stability and robust control synthesis of uncertain switched systems. SIAM J. Control Optim. 43(4), 1253–1271 (2005)zbMATHCrossRefGoogle Scholar
  48. 48.
    Orlov, Y., Aguilar, L., Cadiou, J.C.: Switched chattering control vs. back-lash/friction phenomena in electrical servo-motors. International Journal of Control 76(9/10), 959–967 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    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
  50. 50.
    Saks, S.: Theory of the Integral. Dover Publ. Inc., New York (1964)zbMATHGoogle Scholar
  51. 51.
    Shtessel, Y., Shkolnikov, I.: Aeronautical and space vehicle control in dynamic sliding manifolds. International Journal of Control 76(9/10), 1000–1017 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Sira-Ramirez, H.: On the dynamical sliding mode control of nonlinear systems. International Journal of Control 57(5), 1039–1061 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Sira-Ramirez, H.: Dynamic Second-Order Sliding Mode Control of the Hovercraft Vessel. IEEE Transactions On Control Systems Technology 10(6), 860–865 (2002)CrossRefGoogle Scholar
  54. 54.
    Slotine, J.J., Li, W.: Applied Nonlinear Control. Prentice-Hall Inc., London (1991)zbMATHGoogle Scholar
  55. 55.
    Spurgeon, S., Goh, K., Jones, N.: An application of higher order sliding modes to the control of a diesel generator set (genset). In: Yu, X., Xu, J.X. (eds.) Proc. of the 7th VSS Workshop on Advances in Variable Structure Systems ( 2002)Google Scholar
  56. 56.
    Utkin, V.: Sliding Modes in Optimization and Control Problems. Springer, New York (1992)Google Scholar
  57. 57.
    Yu, X., Xu, J.: An adaptive signal derivative estimator. Electronic Letters 32(16) (1996)Google Scholar
  58. 58.
    Zinober, A.: Variable Structure and Lyapunov Control. Springer, Berlin (1994)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Arie Levant
    • 1
  1. 1.Applied Mathematics DepartmentTel-Aviv UniversityTel-AvivIsrael

Personalised recommendations