Conventional Sliding Modes

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


This chapter considers the development of conventional sliding mode methods. The chapter describes the early work to define the notion of the solution of differential equations with discontinuous right-hand sides and the concept of “equivalent control” as a means to describe the reduced-order dynamics while a sliding motion is taking place. The main focus of the chapter is on the development of sliding mode design techniques for uncertain linear systems—specifically systems which can be thought of as predominantly linear in a characteristic, or nonlinear systems which can be modeled well (at least locally) by a linear system. For such systems, sliding surfaces formed from linear combinations of the states are considered (i.e., hyperplanes in the state space).


Slide Mode Control Switching Function Symmetric Positive Definite Matrix Equivalent Control Regular Form 
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  1. 1.
    Abe, M.: Vehicle dynamics and control for improving handling and active safety: from four-wheel steering to direct yaw moment control. Proc. Inst. Mech. Eng. 213(2), 87–101 (1999)Google Scholar
  2. 2.
    Ackermann, J., Utkin V.I.: Sliding mode control design based on Ackermann’s formula. IEEE Trans. Automat. Contr. 43(2), 234–237 (1998)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Acary, V., Brogliato, B., Orlov, Y.: Chattering-free digital sliding-mode control with state observer and disturbance rejection. IEEE Trans. Automat. Contr. 57(5), 1087–1101 (2012)MathSciNetCrossRefGoogle Scholar
  4. 6.
    Alwi, H., Edwards, C.: Fault tolerant control using sliding modes with online control allocation. Automatica 44, 1859–1866 (2008)MATHMathSciNetCrossRefGoogle Scholar
  5. 7.
    Alwi, H., Edwards, C., Tan, C.P.: Fault Detection and Fault-tolerant Control Using Sliding Modes. Advances in Industrial Control Series. Spring, Berlin (2011)MATHCrossRefGoogle Scholar
  6. 13.
    Bag, S.K., Spurgeon, S.K., Edwards, C.: Output feedback sliding mode design for linear uncertain systems. Proc. IEE, Part D 144, 209–216 (1997)MATHGoogle Scholar
  7. 14.
    Bandyopadhyay, B., Janardhanan, S.: Discrete-time Sliding Mode Control: A Multi-rate Output Feedback Approach. Lecture Notes in Control and Information Sciences, vol. 323. Springer, Berlin (2006).Google Scholar
  8. 21.
    Basin, M., Rodriguez, J., Fridman, L.: Optimal and robust control for linear state-delay systems. J. Franklin Inst. 344(6), 830–845 (2007)MATHMathSciNetCrossRefGoogle Scholar
  9. 26.
    Bejarano, J., Fridman, L., Poznyak, A.: Output integral sliding mode control based on algebraic hierarchical observer. Int. J. Control 80(3), 443–453 (2007)MATHMathSciNetCrossRefGoogle Scholar
  10. 27.
    Bejarano, J., Fridman, L., Poznyak, A.: Output integral sliding mode for min-max optimization of multi-plant linear uncertain systems. IEEE Trans. Automat. Contr. 54(11), 2611–2620 (2009)MathSciNetCrossRefGoogle Scholar
  11. 40.
    Burton, J.A., Zinober, A.S.I.: Continuous approximation of variable structure control. Int. J. Syst. Sci. 17(6), 876–885 (1986)CrossRefGoogle Scholar
  12. 43.
    Castanos, F., Fridman, L.: Analysis and design of integral sliding manifolds for systems with unmatched perturbations. IEEE Trans. Automat. Contr. 55(5), 853–858 (2006)MathSciNetCrossRefGoogle Scholar
  13. 45.
    Castanos, F., Xu, J.X., Fridman, L.: Integral sliding modes for systems with matched and unmatched uncertainties. In: Edwards, C., Colet, E.F., Fridman, L. (eds.) Advances in Variable Structure and Sliding Mode Control. Lecture Notes in Control and Information Sciences, vol. 334, pp. 227–246. Springer, Berlin (2006)CrossRefGoogle Scholar
  14. 49.
    Chen, M.-S., Hwang, Y.-R., Tomizuka, M.: A State-Dependent Boundary Layer Design for Sliding Mode Control. IEEE Trans. Automat. Contr. 47(10), 1677–1681 (2002)MathSciNetCrossRefGoogle Scholar
  15. 50.
    Craig, J.: Introduction to Robotics: Mechanics and Control. Addison-Wesley Publishing, Boston (1989)MATHGoogle Scholar
  16. 58.
    DeCarlo, R.A., Zak, S.H., Drakunov, S.V.: Variable structure, sliding mode controller design. In: The Control Handbook, 2nd edn. Electrical Engineering Handbook Series, pp. 50.1–50.21. CRC Press, Boca Raton, USA (2010)Google Scholar
  17. 59.
    Defoort, M., Floquet, T., Kokosy, A., Perruquetti, W.: Integral sliding mode control for trajectory tracking of a unicycle type mobile robot. Integr. Comput. Aid. E. 13(3), 277–288 (2006)Google Scholar
  18. 60.
    DeJager, B.: Comparison of methods to eliminate chattering and avoid steady state errors in sliding mode digital control. In: Proceedings of the IEEE VSC and Lyapunov Workshop, pp. 37–42, Sheffield, UK (1992)Google Scholar
  19. 63.
    Drakunov, S., Su, W., Ozguner, U.: Constructing discontinuity surfaces for variable structure systems: a Lyapunov approach. Automatica 32(6), 925–928 (1996)MATHMathSciNetCrossRefGoogle Scholar
  20. 66.
    Edwards, C., Spurgeon, S.K.: Sliding mode stabilisation of uncertain systems using only output information. Int. J. Control 62(5), 1129–1144 (1995)MATHMathSciNetCrossRefGoogle Scholar
  21. 67.
    Edwards, C., Spurgeon, S.: Sliding Mode Control: Theory and Applications. Taylor and Francis, London (1998)Google Scholar
  22. 68.
    Edwards, C., Spurgeon, S.: On the limitations of some variable structure output feedback controller designs. Automatica 36, 743–748 (2000)MATHMathSciNetCrossRefGoogle Scholar
  23. 71.
    Edwards, C., Spurgeon, S.K., Akoachere, A.: Sliding mode output feedback controller design using linear matrix inequalities. IEEE Trans. Automat. Contr. 46(1), 115–119 (2001)MATHMathSciNetCrossRefGoogle Scholar
  24. 72.
    Edwards, C., Spurgeon, S.K., Hebden, R.G.: On the design of sliding mode output feedback controllers. Int. J. Control 76, 893–905 (2003)MATHMathSciNetCrossRefGoogle Scholar
  25. 73.
    El-Khazali R., DeCarlo, R.A.: Output feedback variable structure control design. Automatica 31(6), 805–816 (1995)MATHMathSciNetCrossRefGoogle Scholar
  26. 81.
    Filippov, A.: Differential Equations with Discontinuous Right-hand Sides. Kluwer Academic Publishers, Dordrecht (1988)MATHCrossRefGoogle Scholar
  27. 86.
    Franklin, G.F., Powell, J.D., Emami-Naeini, A.: Feedback Control of Dynamic Systems. Prentice Hall, New Jersey (2002)Google Scholar
  28. 93.
    Fridman, L., Poznyak, A., Bejarano, J.: Decomposition of the min-max multimodel problem via integral sliding mode. Int. J. Robust. Nonlin. 15(13), 559–574 (2005)MATHMathSciNetCrossRefGoogle Scholar
  29. 97.
    Furuta, K.: Sliding mode control of a discrete system. Syst. Control Lett. 14(2), 145–152 (1990)MATHMathSciNetCrossRefGoogle Scholar
  30. 103.
    Gutman, S.: Uncertain dynamic Systems – a Lyapunov min-max approach. IEEE Trans. Automat. Contr. 24(3), 437–449 (1979)MATHCrossRefGoogle Scholar
  31. 105.
    Hamayun, M.T., Edwards, C. Alwi, H.: Design and analysis of an integral sliding mode fault-tolerant control scheme. IEEE Trans. Automat. Contr. 57, 1783–1789 (2012)MathSciNetCrossRefGoogle Scholar
  32. 108.
    Hebden, R.G., Edwards, C., Spurgeon, S.K.: Automotive stability in a split-mu manoeuvre using an observer based sliding mode controller. Department of Engineering Report 02–4, Leicester University (2002)Google Scholar
  33. 109.
    Heck, B.S., Ferri, A.A.: Application of output feedback to variable structure systems. J. Guid. Control Dynam. 12, 932–935 (1989)MATHCrossRefGoogle Scholar
  34. 110.
    Heck, B.S., Yallapragada, S.V., Fan, M.K.H.: Numerical methods to design the reaching phase of output feedback variable structure control. Automatica 31(2), 275–279 (1995)MATHMathSciNetCrossRefGoogle Scholar
  35. 136.
    Lukyanov, A.G.: Reducing dynamic systems: regular form. Automat. Rem. Contr. 41(3), 5–13 (1981)Google Scholar
  36. 137.
    Lukyanov, A.G., Utkin, V.I.: Methods for reducing equations for dynamic system to a regular form. Automat. Rem. Contr. 4, 14–18 (1981)Google Scholar
  37. 140.
    Matthews, G.P., DeCarlo, R.A.: Decentralized tracking for a class of interconnected nonlinear systems using variable structure control. Automatica 24(2), 187–193 (1988)MATHMathSciNetCrossRefGoogle Scholar
  38. 141.
    Milosavljevic, C.: Discrete-time VSS. In: Sabanovich, E., Spurgeon, S., Fridman, L. (eds.) Variable Structure systems: From Principles to Implementation. IEE Control Series, vol. 66, pp. 99–128. IEE-publisher Stevenage, UK (2004)CrossRefGoogle Scholar
  39. 149.
    Petersen, I.R.: A stabilization algorithm for a class of uncertain linear systems. Syst. Control Lett. 8(4), 351–357 (1987)MATHCrossRefGoogle Scholar
  40. 151.
    Plestan, F., Grizzle, J.W., Westervelt, E.R., Abba, G.: Stable walking of a 7-DOF biped robot. IEEE Trans. Robotic. Autom. 19(4), 653–668 (2009)CrossRefGoogle Scholar
  41. 154.
    Poznyak, A., Fridman, L., Bejarano, F.: Mini-max integral sliding mode control for multimodel linear uncertain systems. IEEE Trans. Automat. Contr. 49(1), 97–102 (2004)MathSciNetCrossRefGoogle Scholar
  42. 156.
    Rubagotti, M., Estrada, A., Castanos, F., Ferrara, A., Fridman, L.: Integral sliding mode control for nonlinear systems with matched and unmatched perturbations. IEEE Trans. Automat. Contr. 56(11), 2699–2704 (2011)MathSciNetCrossRefGoogle Scholar
  43. 157.
    Ryan, E.P., Corless, M.: Ultimate boundedness and asymptotic stability of a class of uncertain dynamical systems via continuous and discontinuous control. IMA J. Math. Control Inform. 1(3), 223–242 (1984)MATHCrossRefGoogle Scholar
  44. 158.
    Saks, S.: Theory of the integral. Dover Publ. Inc., New York (1964)MATHGoogle Scholar
  45. 162.
    Shtessel, Y., Buffington, J., Banda, S.: Tailless aircraft flight control using multiple time scale re-configurable sliding modes. IEEE Trans. Contr. Syst. Tech. 10, 288–296 (2002)CrossRefGoogle Scholar
  46. 171.
    Slotine, J.-J., Li, W.: Applied Nonlinear Control. Prentice Hall, New Jersey (1991)MATHGoogle Scholar
  47. 175.
    Steinberg, A., Corless, M.J.: Output feedback stabilisation of uncertain dynamical systems. IEEE Trans. Automat. Contr. 30(10), 1025–1027 (1985)MATHMathSciNetCrossRefGoogle Scholar
  48. 177.
    Strang, G.: Linear Algebra and its Applications. Harcourt Brace Jovanovich, London (1988)Google Scholar
  49. 182.
    Utkin, V.I.: Sliding Modes in Optimization and Control Problems. Springer, New York (1992)CrossRefGoogle Scholar
  50. 185.
    Utkin, V.I., Shi, J.: Integral sliding mode in systems operating under uncertainty conditions. In: Proceedings of the 35th IEEE Conference on Decision and Control, pp. 4591–4596, Kobe, Japan (1996)Google Scholar
  51. 186.
    Utkin, V., Guldner, J., Shi, J.: Sliding Mode Control in Electromechanical Systems. Taylor and Francis, London (1999)Google Scholar
  52. 187.
    Utkin, V.I, Guldner, J., Shi, J.: Sliding Mode Control in Electro-Mechanical Systems, 2nd edn. CRC Press, Boca Raton (2009)CrossRefGoogle Scholar
  53. 191.
    Xu, J.X., Pan, Y.J., Lee, T.H., Fridman, L.: On nonlinear hinfty sliding mode control for a class of nonlinear cascade systems. Int. J. Syst. Sci. 36(15), 983–992 (2005)MATHMathSciNetCrossRefGoogle Scholar
  54. 192.
    Xu, J.X., Abidi, K. On the discrete-time integral sliding-mode control. IEEE Trans. Automat. Contr. 52(4), 709–715 (2007)MathSciNetCrossRefGoogle 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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