Skip to main content

Sliding Mode Enforcement after 1990: Main Results and Some Open Problems

  • Chapter
Sliding Modes after the First Decade of the 21st Century

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

Abstract

The objective of this chapter is to try to analyze the main stages in the development of sliding mode enforcing control algorithms, starting from the first Variable Structure Systems workshop (VSS90). I would like to underline that this is my personal opinion, I am just trying to understand the steps we have made as a community during the last twenty years after VSS90 as well as which problems still remain open. Of course, generally I will concentrate the chapter on results in open problems I have discovered working with my group and coauthors.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ahmed-Ali, T., Lamnabhi-Lagarrigue, F.: Sliding observer-controller design for uncertain triangular nonlinear systems. IEEE Transaction on Automatic Control 44(6), 1244–1249 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Akian, M., Bliman, P., Sorine, M.: Control of delay systems with relay. IMA Journal on Mathematical Control and Information 19, 133–155 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Levant, A.: Chattering analysis. IEEE Trans. Autom. Control 55(6), 1380–1389 (2010)

    Article  MathSciNet  Google Scholar 

  4. Angulo, M., Levant, A.: On robust output based finite-time control of LTI systems using HOSMs. International Journal of Systems Science (2011), doi: http://dx.doi.org/10.1080/00207721.2011.564676

  5. Angulo, M.T., Fridman, L.: Output-based finite time control of LTI systems with matched perturbations using HOSM. In: Proceedings of the 48th IEEE Conference on Decision and Control 2009, pp. 6095–6100 (2009)

    Google Scholar 

  6. Angulo, M.T., Fridman, L., Moog, C., Moreno, J.: Output feedback design for exact state stability of flat nonlinear systems. In: Proceedings of 11th Workshop on Varibale Structure Systems (VSS 2010), pp. 32–38 (2010)

    Google Scholar 

  7. Atherton, D.P.: Nonlinear Control Engineering - Describing Function Analysis and Design. Van Nostrand Company Limited, Workingham (1975)

    Google Scholar 

  8. Barbot, J., Saadaoui, H., Djema, M., Manamanni, N.: Nonlinear observer for autonomous switching systems with jumps. Nonlinear Analysis: Hybrid Systems 1(4), 537–547 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Barbot, J.P., Boutat, D., Floquet, T.: An observation algorithm for nonlinear systems with unknown inputs. Automatica 45, 1970–1974 (2009)

    Article  MATH  Google Scholar 

  10. Bartolini, G., Ferrara, A., Pisano, A., Usai, E.: Adaptive reduction of the control effort in chattering free sliding mode control of uncertain nonlinear systems. Journal of Applied Mathematics and Computer Science 8(1), 51–71 (1998)

    MathSciNet  MATH  Google Scholar 

  11. Bartolini, G., Ferrara, A., Punta, E.: Multi-input second-order sliding-mode hybrid control of constrained manipulators. Dyn. Control 10(3), 277–296 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bartolini, G., Ferrara, A., Usai, E.: Output tracking control of uncertain nonlinear second-order systems. Automatica 33(12), 2203–2212 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bartolini, G., Ferrara, A., Usai, E., Utkin, V.: On multi-input chattering-free second order sliding mode control. IEEE Transactions on Automatic Control 45(9), 1711–1717 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bartolini, G., Levant, A., Pisano, A., Usai, E.: 2-sliding mode with adaptation. In: Proceedings of the Seventh IEEE Mediterranean Conference on Control and Systems (1999)

    Google Scholar 

  15. Bartolini, G., Pillosu, S., Pisano, A., Usai, E.: Time-optimal stabilization for a third-order integrator: a robust state-feedback implementation. In: Colunius, F., Grne, L. (eds.) Bifurcation and Control. LNCIS, vol. 273, pp. 83–108. Springer, Heidelberg (2002)

    Google Scholar 

  16. 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 45(9/10), 875–892 (2003)

    Article  MathSciNet  Google Scholar 

  17. 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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bejarano, F., Fridman, L.: State exact reconstruction for switched linear systems via a super-twisting algorithm. International Journal of Systems Science 42(5), 717–724 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bejarano, F., Fridman, L., Poznyak, A.: Exact state estimation for linear systems with unknown inputs based on hierarchical super-twisting algorithm. International Journal of Robust and Nonlinear Control 17(18), 1734–1753 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bejarano, F., Fridman, L., Poznyak, A.: Hierarchical second-order sliding-mode observer for linear time invariant systems with unknown inputs. International Journal of Systems Science 38(10), 793–802 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bejarano, F., Fridman, L., Poznyak, A.: Unknown input and state estimation for unobservable systems. SIAM Journal on Control and Optimization 48(2), 1155–1178 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bejarano, F., Pisano, A., Usai, E.: Finite-time converging jump observer for linear switched systems with unknown inputs. Nonlinear Analysis: Hybrid Systems 5(2), 174–188 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bhat, S., Bernstein, D.: Finite-time stability of continuous autonomous systems. SIAM Journal on Control and Optimization 38(3), 751–766 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Boiko, I.: Discontinuous Control Systems. Birkhäuser, Boston (2009)

    MATH  Google Scholar 

  25. Boiko, I.: Oscillations and transfer properties of relay feedback systems with time-delay linear plants. Automatica 45(12), 1127–1135 (2009)

    Article  MathSciNet  Google Scholar 

  26. Boiko, I., Fridman, L.: Analysis of chattering in continuous sliding-mode controllers. IEEE Transactions on Automatic Control 50(9), 1442–1446 (2005)

    Article  MathSciNet  Google Scholar 

  27. Boiko, I., Fridman, L., Castellanos, M.I.: Analysis of second order sliding mode algorithms in the frequency domain. IEEE Transactions on Automatic Control 49(6), 946–950 (2004)

    Article  MathSciNet  Google Scholar 

  28. Boiko, I., Fridman, L., Pisano, A., Usai, E.: Analysis of chattering in systems with second order sliding modes. IEEE Trans. Autom. Control 52(11), 2085–2102 (2007)

    Article  MathSciNet  Google Scholar 

  29. Boiko, I., Fridman, L., Pisano, A., Usai, E.: Performance analysis of second-order sliding-mode control systems with fast actuators. IEEE Trans. Autom. Control 52(6), 1053–1059 (2007)

    Article  MathSciNet  Google Scholar 

  30. Boiko, I., Fridman, L., Pisano, A., Usai, E.: On the transfer properties of the generalized sub-optimal second-order sliding mode control algorithm. IEEE Transactions on Automatic Control 54(2), 399–403 (2009)

    Article  MathSciNet  Google Scholar 

  31. Bondarev, A.G., Bondarev, S.A., Kostylyeva, N.Y., Utkin, V.I.: Sliding modes in systems with asymptotic state observers. Automatica i Telemechanica (Automation and Remote Control) 46(5), 679–684 (1985)

    MATH  Google Scholar 

  32. Bregeault, V., Plestan, F., Shtessel, Y., Poznyak, A.: Adaptive sliding mode control for an electropneumatic actuator. In: Proceedings of VSS, pp. 260–265 (2010)

    Google Scholar 

  33. Cannas, B., Cincotti, S., Usai, E.: An algebraic observability approach to chaos synchronisation by sliding differentiators. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 49(7), 1000–1006 (2002)

    Article  Google Scholar 

  34. Cao, W.J., Xu, J.X.: Nonlinear integral-type sliding surface for both matched and unmatched uncertain systems. IEEE Trans. Automatic Control 49(8), 1355–1360 (2004)

    Article  MathSciNet  Google Scholar 

  35. Castaños, F., Fridman, L.: Analysis and design of integral sliding manifolds for systems with unmatched perturbations. IEEE Trans. Automatic Control 51(5), 853–858 (2006)

    Article  Google Scholar 

  36. Choi, S.B., Hedrick, J.K.: Robust throttle control of automotive engines. ASME Journal of Dynamic Systems, Measurement, and Control 18, 92–98 (1996)

    Article  Google Scholar 

  37. Cruz, E., Moreno, J., Fridman, L.: Uniform robust exact differentiator. In: Proceedings of 2010 49th IEEE Conference on Decision and Control, pp. 102–107 (2010)

    Google Scholar 

  38. Cruz, E., Moreno, J., Fridman, L.: Uniform second-order sliding mode observer for mechanical systems. In: Proceedings of 11th IEEE Workshop in Variable Structure Sytems, pp. 14–19 (2010)

    Google Scholar 

  39. Davila, A., Moreno, J., Fridman, L.: Optimal Lyapunov function selection for reaching time estimation of super twisting algorithm. In: Proc. 48th Conference on Decision and Control, pp. 8405–8410 (2009)

    Google Scholar 

  40. Davila, A., Moreno, J., Fridman, L.: Optimal Lyapunov function selection for reaching time estimation of super twisting algorithm. In: Proc. American Control Conference (ACC2010), pp. 968–973 (2010)

    Google Scholar 

  41. Davila, J., Fridman, L., Levant, A.: Second order sliding mode observer for mechanical systems. IEEE Transactions on Automatic Control 50(11), 1785–1789 (2005)

    Article  MathSciNet  Google Scholar 

  42. Davila, J., Fridman, L., Pisano, A., Usai, E.: Finite-time state observation for non-linear uncertain systems via higher-order sliding modes. International Journal of Control 82(8), 1564–1574 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Davila, J., Fridman, L., Poznyak, A.: Observation and identification of mechanical systems via second order sliding modes. International Journal of Control 79(10), 1251–1262 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. Defoort, M., Floquet, T., Kokosy, A., Perruquetti, W.: A novel higher order sliding mode control scheme. Syst. Control Lett. 58(2), 102–108 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Defoort, M., Perruquetti, T.F.W.: Decentralized tracking for a class of interconnected nonlinear systems using variable structure control. Informatics in Control, Automation and Robotics 13, 277–288 (2006)

    Google Scholar 

  46. Drakunov, S.V., Izosimov, D.B., Loukyanov, A.G., Utkin, V.A., Utkin, V.I.: Block control principle. Automation and Remote Control 51(5), 601–609 (1990)

    MathSciNet  MATH  Google Scholar 

  47. Drakunov, S.V., Utkin, V.I.: Sliding mode control in dynamic systems. International Journal of Control 55(7), 1029–1037 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  48. Drazenovic, B.: The invariance conditions in variable structure systems. Automatica 5(3), 287–295 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  49. Edwards, C., Floquet, T., Spurgeon, S.: Circumventing the relative degree condition in sliding mode design. In: Modern Sliding Mode Control Theory New Perspectives and Applications. LNCIS, vol. 375, pp. 137–158 (2008)

    Google Scholar 

  50. Edwards, C., Spurgeon, S., Hebden, R.: On development and applications of sliding mode observers. In: Xu, J., Xu, Y. (eds.) Variable Structure Systems:Towards XXIst Century. LNCIS, pp. 253–282. Springer, Berlin (2002)

    Chapter  Google Scholar 

  51. Emel’yanov, S.V., Korovin, S.K., Levantovsky, L.V.: Second order sliding modes in controlling uncertain processes. Soviet Journal of Computer and System Science 24(4), 63–68 (1986)

    MATH  Google Scholar 

  52. Estrada, A., Fridman, L.: Exact compensation of unmatched perturbations via HOSM. In: Proceedings of the 47th IEEE Conference on Decision and Control, pp. 278–282 (2008)

    Google Scholar 

  53. Fei, J., Batur, C.: Adaptive sliding mode control with sliding mode observer for a micro-electro-mechanical vibratory gyroscope. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 222(8), 839–847 (2008)

    Article  Google Scholar 

  54. Feng, Y., Yu, X., Man, Z.: Control of nonlinear systems using terminal sliding modes. In: Proc. Conference on Decision and Control (CDC 2001), Orlando, pp. 4021–4026 (2001)

    Google Scholar 

  55. Feng, Y., Yu, X., Man, Z.: Non-singular terminal sliding mode control of rigid manipulators. Automatica 38, 2159–2167 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  56. Ferreira, A., Bejarano, F.J., Fridman, L.: Robust control with exact uncertainties compensation: With or without chattering? IEEE Transactions of Control Systems Technology (2011), doi: http://dx.doi.org/10.1109/TCST.2010.2064168

  57. Filippov, A.: Differential equations with discontinuous right-hand side. Kluwer, Dordrecht (1988)

    Google Scholar 

  58. Floquet, T., Barbot, J.: A canonical form for the design of unknown input sliding mode observers. In: Edwards, C., Fossas, E., Fridman, L. (eds.) Advances in Variable Structure and Sliding Mode Control. LNCIS, vol. 334, pp. 271–292. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  59. Floquet, T., Edwards, C., Spurgeon, S.: On sliding mode observers for systems with unknown inputs. Int. J. Adapt. Control Signal Process. 21(8-9), 638–656 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  60. Fridman, L.: An averaging approach to chattering. IEEE Trans. Autom. Control 46(8), 1260–1265 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  61. Fridman, L.: Singularly perturbed analysis of chattering in relay control systems. IEEE Transactions on Automatic Control 47(12), 2079–2084 (2002)

    Article  MathSciNet  Google Scholar 

  62. Fridman, L., Acosta, P., Polyakov, A.: Robust eigenvalue assignment for uncertain delay control systems. In: Proceedings of 3rd IFAC Workshop on Time Delay Systems, pp. 239–244 (2001)

    Google Scholar 

  63. Fridman, L., Davila, J., Levant, A.: High-order sliding-mode observation for linear systems with unknown inputs. Nonlinear Analysis: Hybrid Systems (2011), doi: http://dx.doi.org/10.1016/j.nahs.2010.09.003

  64. Fridman, L., Fridman, E., Shustin, E.: Steady modes and sliding modes in relay control systems with delay. In: Barbot, J., Perruquetti, W. (eds.) Sliding Mode Control In Engineering, pp. 263–294. Marcel Dekker, New York (2002)

    Google Scholar 

  65. Fridman, L., Levant, A.: Higher order sliding modes as the natural phenomena of control theory. In: Proceedings of the Workshop on Variable Structure and Lyapunov Technique, pp. 302–309 (1994)

    Google Scholar 

  66. Fridman, L., Levant, A.: Higher order sliding modes as the natural phenomena of control theory. In: Garofalo, F., Glielmo, G. (eds.) Robust Control Variable Structure and Lyapunov Techniques. LNCIS, vol. 217, pp. 107–133. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  67. Fridman, L., Levant, A., Davila, J.: Observation and identification via high-order sliding modes. In: Bartolini, G., Fridman, L., Pisano, A., Usai, E. (eds.) Modern Sliding Mode Control Theory New Perspectives and Applications. LNCIS, vol. 375, pp. 293–320

    Google Scholar 

  68. Fridman, L., Levant, A., Davila, J.: Observation of linear systems with unknown inputs via high-order sliding-mode. International Journal of Systems Science 38(10), 773–791 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  69. Fridman, L., Shtessel, Y., Edwards, C., Yan, X.G.: Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems. International Journal of Robust and Nonlinear Control 18(4-5), 399–412 (2008)

    Article  MathSciNet  Google Scholar 

  70. Fridman, L., Strygin, V., Polyakov, A.: Stabilization of amplitude of oscillations via relay delay control. Int. J. Control 76(8), 770–780 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  71. Fridman, L., Strygin, V., Polyakov, A.: Stabilization via delayed relay control rejecting uncertainty in a time delay. Int. J. Robust Nonlinear Control 14(1), 15–37 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  72. Gutman, S.: Uncertain dynamic systems - a Lyapunov min-max approach. IEEE Trans. Autom. Control 24(3), 437–449 (1979)

    Article  MATH  Google Scholar 

  73. Gutman, S., Leitmann, G.: Stabilizing feedback control for dynamic systems with bounded uncertainties. In: Proceedings of the 15th Conference on Decision and Control, pp. 94–99 (1976)

    Google Scholar 

  74. Hall, C., Shtessel, Y.: Sliding mode disturbance observer-based control for a reusable launch vehicle. AIAA J. Guidance, Control and Dynamics 29(6), 1315–1328 (2006)

    Article  Google Scholar 

  75. Hashimoto, H., Utkin, V., Xu, J., Suzuki, H., Harashima, F.: VSS observer for linear time varying system. In: Procedings of IECON 1990, pp. 34–39. Pacific Grove, CA (1990)

    Google Scholar 

  76. Hautus, M.: Strong detectability and observers. Linear Algebra and its Applications 50, 353–468 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  77. Hsu, L., Lizarralde, F., Araújo, A.D.: New results on output-feedback variable structure model-reference adaptive control: design and stability analysis. IEEE Trans. Autom. Control 42(3), 386–393 (1997)

    Article  MATH  Google Scholar 

  78. Hsu, L., Peixoto, A.J., Cunha, J.P.S., Costa, R.R., Lizarralde, F.: Output feedback sliding mode control for a class of uncertain multivariable systems with unmatched nonlinear disturbances. In: Variable Structure Systems, Sliding Mode and Nonlinear Control, Advances in Variable Structure and Sliding Mode Control, pp. 195–225. Springer, Berlin (2006)

    Google Scholar 

  79. Huang, Y.J., Kuo, T.C., Chang, S.H.: Adaptive sliding-mode control for nonlinear systems with uncertain parameters. IEEE Trans. Syst. Man, and Cybernetics-Part B: Cybernetics 38(2), 534–539 (2008)

    Article  Google Scholar 

  80. Boiko, I., Castellanos, M., Fridman, L.: Describing function analysis of second-order sliding mode observers. Intern. Journ. of Systems Science 38(10), 817–824 (2007)

    Article  MATH  Google Scholar 

  81. Boiko, I., Castellanos, M., Fridman, L.: Analysis of response of second-order sliding mode controllers to external inputs in frequency domain. International Journal on Robust and Nonlinear Control 38(4-5), 502–514 (2008)

    Article  MathSciNet  Google Scholar 

  82. Khoo, S., Trinh, H., Man, Z., Shen, W.: Fast finite-time consensus of a class of high-order uncertain nonlinear systems. In: Proc. 5th IEEE Conference on Industrial Electronics and Applications (ICIEA 2010), pp. 2076–2081 (2010)

    Google Scholar 

  83. Khoo, S., Xie, L., Man, Z.: Robust finite-time consensus tracking algorithm for multirobot systems. IEEE/ASME Transactions on Mechatronics 14(2), 219–228 (2009)

    Article  Google Scholar 

  84. Kolmogorov, A.N.: On inequalities between upper bounds of consecutive derivatives of an arbitrary function defined on an infinite interval. Amer. Math. Soc. Transl. 2(9), 233–242 (1962)

    Google Scholar 

  85. Laghrouche, S., Plestan, F., Glumineau, A.: Higher order sliding mode control based on integral sliding mode. Automatica 43(3), 531–537 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  86. Lee, H., Utkin, V.: Chattering suppression methods in sliding mode control systems. Annual Reviews in Control 31, 179–188 (2007)

    Article  Google Scholar 

  87. Levaggi, L.: Infinite dimensional systems sliding motions. European Journal of Control 8, 508–518 (2002)

    Article  Google Scholar 

  88. Levaggi, L.: Infinite dimensional systems sliding motions. Differential Integral Equations 15, 167–189 (2002)

    MathSciNet  MATH  Google Scholar 

  89. Levant, A.: Sliding order and sliding accuracy in sliding mode control. International Journal of Control 58(6), 1247–1263 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  90. Levant, A.: Robust exact differentiation via sliding mode technique. Automatica 34(3), 379–384 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  91. Levant, A.: Universal SISO sliding-mode controllers with finite-time convergence. IEEE Transactions on Automatic Control 46(9), 1447–1451 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  92. Levant, A.: High-order sliding modes: differentiation and output feedback control. International Journal of Control 76(9-10), 924–941 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  93. Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica 41(5), 823–830 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  94. Levant, A.: Quasi-continuous high-order sliding-mode controllers. IEEE Trans. Autom. Control 50(11), 1812–1816 (2005)

    Article  MathSciNet  Google Scholar 

  95. Levant, A.: Homogeneous quasi-continuous sliding-mode control. In: Advances in Variable Structure and Sliding Mode Control. LNCIS, vol. 334, pp. 143–168 (2006)

    Google Scholar 

  96. Levant, A.: Principles of 2-sliding mode design. Automatica 43(4), 576–586 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  97. 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)

    Article  MathSciNet  Google Scholar 

  98. Levant, A., Michael, A.: Adjustment of high-order sliding-mode controllers. International Journal of Robust and Nonlinear Control 19(15), 1657–1672 (2009)

    Article  MathSciNet  Google Scholar 

  99. Li, X., Yurkovitch, S.: Sliding mode control of systems with delayed states and controls. In: Young, K.D., Ozguner, U. (eds.) Variable Structure Systems, Sliding Mode and Nonlinear Control. LNCIS, pp. 93–108. Springer, London (1999)

    Chapter  Google Scholar 

  100. Lin, W.S., Chen, C.S.: Robust adaptive sliding mode control using fuzzy modeling for a class of uncertain MIMO nonlinear systems. IEE Proceedings of Control Theory Applications 149, 193–201 (2002)

    Article  Google Scholar 

  101. Loukianov, A., Fridman, L., Canedo, J., Sanchez, E., Soto-Cota, A.: Higher order sm block-control of nonlinear systems with unmodeled actuators: Application to electric power systems and electrohydraulic servo-drives. In: Bartolini, G., Fridman, L., Pisano, A., Usai, E. (eds.) Modern Sliding Mode Control Theory New Perspectives and Applications. LNCIS, pp. 401–426. Springer, London (2008)

    Chapter  Google Scholar 

  102. Loukianov, A.G., Espinosa-Guerra, O., Castillo-Toledo, B., Utkin, V.A.: Integral sliding mode control for systems with time delay. In: Proceedings of 9th IEEE Workshop on Variable Structure Systems, pp. 256–261 (2006)

    Google Scholar 

  103. Luenberguer, D.G.: Canonical forms for linear multivariable systems. IEEE Transactions on Automatic Control 12(3), 290–293 (1967)

    Article  Google Scholar 

  104. Malek-Zaverei, M., Jamshidi, M.: Time-Delay Systems Analysis Optimization and Applications. Elsevier Science Ltd, New York (1987)

    Google Scholar 

  105. Man, Z., Paplinski, A.P., Wu, H.R.: A robust MIMO terminal sliding mode control scheme for rigid robotic manipulators. IEEE Trans. Automatic Control 39, 2464–2469 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  106. Man, Z., Yu, X.: Adaptive terminal sliding mode tracking control for rigid robotic manipulators with uncertain dynamics. JSME Int. J. of Mechanical Systems, Machine Elements and Manufacturing 40(3), 493–502 (1997)

    MATH  Google Scholar 

  107. Man, Z., Yu, X., O’Day, M.: A robust adaptive terminal sliding mode control for rigid robotic manipulators. J. of Intelligent and Robotic Systems 24(3), 23–41 (1999)

    MATH  Google Scholar 

  108. Manitius, A., Olbrot, A.: Finite spectrum assignment problem for systems with delay. IEEE Trans. on Automatic Control 24(4), 541–553 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  109. Matthews, G.P., DeCarlo, R.A.: Decentralized tracking for a class of interconnected nonlinear systems using variable structure control. Automatica 24, 187–193 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  110. Mirkin, B., Gutman, P.O., Shtessel, Y.: Adaptive continuous control with sliding mode for plants under nonlinear perturbations, external disturbances and actuator failures. In: Proceedings of VSS, pp. 250–255 (2010)

    Google Scholar 

  111. Moreno, J.: A linear framework for the robust stability analysis of a generalized supertwisting algorithm. In: Proc. 6th Int. Conf. Elect. Eng., Comp. Sci. and Aut. Cont. (CCE 2009), Mexico, pp. 12–17 (2009)

    Google Scholar 

  112. Newman, W.: Robust sub-optimal control. IEEE Transactions on Automatic Control 35(7), 841–844 (1990)

    Article  MATH  Google Scholar 

  113. Nunes, E.V.L., Hsu, L., Lizarralde, F.: Global exact tracking for uncertain systems using output-feedback sliding mode control. IEEE Trans. Autom. Control 54(5), 1141–1147 (2009)

    Article  MathSciNet  Google Scholar 

  114. Oliveira, T., Peixoto, A., Nunes, E., Hsu, L.: Control of uncertain nonlinear systems with arbitrary relative degree and unknown control direction using sliding modes. Int. J. Adaptive Control Signal Process. 21(8/9), 692–707 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  115. Oliveira, T.R., Hsu, L.: Dwell-time and monitoring schemes for peaking avoidance in high-gain observer based output-feedback control. In: Proc. of 48th IEEE Conf. on Decision and Control (CDC 2009), Shanghai, pp. 7557–7562 (2009)

    Google Scholar 

  116. Oliveira, T.R., Peixoto, A.J., Costa, R.R., Hsu, L.: New results on output-feedback variable structure model-reference adaptive control: design and stability analysis. Dynamics of Continuous, Discrete and Impulsive Systems. Series B: Applications and Algorithms 17(6), 839–874 (2010)

    MathSciNet  MATH  Google Scholar 

  117. Oliveira, T.R., Peixoto, A.J., Hsu, L.: Sliding mode control of uncertain multivariable nonlinear systems with unknown control direction via switching and monitoring function. IEEE Trans. Autom. Control 55(4), 1028–1034 (2010)

    Article  MathSciNet  Google Scholar 

  118. Orlov, Y.: Discontinuous unit feedback control of uncertain infinite-dimensional systems. IEEE Transactions on Automatic Control 45(5), 834–843 (2000)

    Article  MATH  Google Scholar 

  119. Orlov, Y.: Discontinuous Systems: Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions. Springer, London (2008)

    Google Scholar 

  120. Orlov, Y., Dochain, D.: Discontinuous feedback stabilization of minimum-phase semilinear infinite-dimensional systems with application to chemical tubular reactor. IEEE Trans. Autom. Control 47(8), 1293–1304 (2002)

    Article  MathSciNet  Google Scholar 

  121. Orlov, Y., Lou, Y., Christofides, P.: Sliding mode control synthesis of uncertain time-delay systems. Int. J. Contr. 77, 1115–1136 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  122. Orlov, Y., Pisano, A., Usai, E.: Continuous state-feedback tracking of an uncertain heat diffusion process. Systems and Control Letters 59(12), 754–759 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  123. Orlov, Y., Pisano, A., Usai, E.: Exponential stabilization of the uncertain wave equation via distributed dynamic input extension. IEEE Transactions on Automatic Control 56(1), 212–217 (2011)

    Article  MathSciNet  Google Scholar 

  124. Orlov, Y., Pisano, A., Usai, E.: Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques. SIAM Journal on Control and Optimization 49(2), 363–382 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  125. Orlov, Y., Utkin, V.: Sliding mode control in infinite-dimensional systems. Automatica 23, 1127–1135 (1987)

    Article  MathSciNet  Google Scholar 

  126. Orlov, Y., Utkin, V.: Unit sliding mode control in infinite-dimensional systems. Applied Math. and Computer Science 8, 7–20 (1987)

    MathSciNet  Google Scholar 

  127. Paulo, J., Cunha, V.S., Hsu, L., Costa, R., Lizarralde, F.: Output-feedback model-reference sliding mode control of uncertain multivariable system. IEEE Transactions on Automatic Control 48(12), 2245–2250 (2003)

    Article  Google Scholar 

  128. Peixoto, A.J., Oliveira, T.R., Hsu, L.: Global tracking output-feedback sliding mode control design via norm estimators and variable high gain observer. In: Proc. of 48th IEEE Conf. on Decision and Control (CDC 2009), Shanghai, pp. 6083–6088 (2009)

    Google Scholar 

  129. Polyakov, A.: Robust eigenvalue assignment for uncertain delay control systems. In: Proc. 47th Conference on Decision and Control, pp. 5306–5311 (2008)

    Google Scholar 

  130. Polyakov, A., Poznyak, A.: Lyapunov function design for finite-time convergence analysis: Twisting controller for second order sliding mode realization. Automatica 45, 444–448 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  131. Polyakov, A., Poznyak, A.: Reaching time estimation for super-twisting second order sliding mode controller via Lyapunov function designing. IEEE Trans. Automatic Control 45(8), 1951–1955 (2009)

    Article  MathSciNet  Google Scholar 

  132. Poznyak, A.: Deterministic output noise effects in sliding mode observation. In: Sabanovic, A., Fridman, L., Spurgeon, S. (eds.) Variable Structure Systems: from Principles to Implementation. IEE Control Engineering Series, pp. 45–80. IEE, London (2004)

    Google Scholar 

  133. Roh, Y.H., Oh, J.H.: Robust stabilization of uncertain input delay systems by sliding mode control with delay compensation. Automatica 35(12), 1861–1865 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  134. Rosales, J., Boiko, I., Fridman, L.: Design of compensators for second order sliding modes. In: Proc. of 11th International Workshop on Variable Structure Systems (VSS 2010), Mexico, pp. 20–25 (2010)

    Google Scholar 

  135. Rubagotti, M., Estrada, A., Castaños, F., Ferrara, A., Fridman, L.: Optimal disturbance rejection by integral sliding mode control for systems in regular form. In: Proc. of the Variable Structure Systems Workshop, Mexico City, Mexico, pp. 78–82 (2010)

    Google Scholar 

  136. Saadaoui, H., Manamanni, N., Djema, M., Barbot, J., Floquet, T.: Exact differentiation and sliding mode observers for switched Lagrangian systems. Nonlinear Analysis: Hybrid Systems and Applications 65, 1050–1069 (2006)

    MATH  Google Scholar 

  137. Saif, M., Xiong, Y.: Sliding mode observers and their application in fault diagnosis. In: Caccavale, F., Villani, L. (eds.) Fault Diagnosis and Fault Tolerance for Mechatronic Systems: Recent Advances. Springer Tracts in Advanced Robotics, pp. 1–52. Springer, Berlin (2003)

    Chapter  Google Scholar 

  138. Shtessel, Y., Edwards, C., Spurgeon, S., Kochalummoottil, J.: Adaptive continuous finite reaching time control and second order sliding modes. In: Proceedings of VSS, pp. 266–271 (2010)

    Google Scholar 

  139. Shtessel, Y., Moreno, J., Plestan, F., Fridman, L., Poznyak, A.: Super-twisting adaptive sliding mode control: a Lyapunov design. In: 49th IEEE Conference on Decision and Control, pp. 5109–5113 (2010)

    Google Scholar 

  140. Shtessel, Y., Zinober, A., Shkolnikov, I.: Sliding mode control for nonlinear systems with output delay via method of stable system centre. ASME Journal of Dynamic Systems, Measurement and Control 25(2), 253–257 (2003)

    Article  Google Scholar 

  141. Shustin, E., Fridman, L., Fridman, E., Castanos, F.: Robust semiglobal stabilization of the second order system by relay feedback with an uncertain variable time delay. SIAM J. Control Optim. 47(1), 196–217 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  142. Sing, K.N.: Comments on robust stabilization of uncertain input delay systems by sliding mode control with delay compensation. Automatica 37, 73–84 (2001)

    Google Scholar 

  143. Song, G., Cai, L., Wang, Y., Longman, R.: A sliding mode based smooth adaptive robust controller for friction compensation. Int. J. Robust Nonlinear Control 8, 725–739 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  144. Spurgeon, S.: Sliding mode observers - a survey. International Journal of Systems Science 39(8), 751–764 (2008)

    Article  MathSciNet  Google Scholar 

  145. Tan, C., Edwards, C.: Sliding mode observers for robust detection and reconstruction of actuator and sensor faults. International Journal of Robust and Nonlinear Control 13, 443–463 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  146. Tan, C., Yu, X., Man, Z.: Terminal sliding mode observers for a class of nonlinear systems. Automatica 46(8), 1401–1404 (2010)

    Article  MATH  Google Scholar 

  147. Tsypkin, Y.: Relay Control Systems. Cambridge University Press, Cambridge (1984)

    MATH  Google Scholar 

  148. Utkin, V.: First stage of VSS: People and events. In: Yu, X., Xu, J.-X. (eds.) Variable Structure Systems: Towards the 21st Century. LNCIS, vol. 274, pp. 1–32. Springer, London (2002)

    Chapter  Google Scholar 

  149. Utkin, V., Guldner, J., Shi, J.: Relay Control Systems. Taylor and Francis, Taylor (1999)

    Google Scholar 

  150. Utkin, V., Shi, J.: Integral sliding mode in systems operating under uncertainty conditions. In: Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, pp. 4591–4596 (1996)

    Google Scholar 

  151. Utkin, V.I. (ed.): Sliding Modes in Control Optimization. Springer, Heidelberg (1992)

    MATH  Google Scholar 

  152. Venkataraman, S.T., Gulati, S.: Control of nonlinear systems using terminal sliding modes. In: Proc. American Control Conference, pp. 891–893 (1992)

    Google Scholar 

  153. Wu, Y., Yu, X., Man, Z.: Terminal sliding mode control design for uncertain dynamic systems. Systems and Control Letters 34, 281–287 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  154. Xiong, Y., Saif, M.: Sliding mode observer for nonlinear uncertain systems. IEEE Transaction on Automatic Control 46(12), 2012–2017 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  155. Yu, L., Barbot, J.P., Boutat, D., Benmerzouk, D.: Observability normal forms for a class of switched systems with Zeno phenomena. In: ACC 2009: Proceedings of the 2009 American Control Conference, pp. 1766–1771. IEEE Press, Piscataway (2009)

    Chapter  Google Scholar 

  156. Yu, S., Yu, X., Man, Z.: A fuzzy neural network approximator with fast terminal sliding mode and its applications. Fuzzy Sets and Systems 148(3), 469–486 (2005)

    Article  MathSciNet  Google Scholar 

  157. Yu, X., Kaynak, O.: Sliding mode control with soft computing: A survey. IEEE Transactions on Industrial Electronics 56(9), 3275–3285 (2010)

    Google Scholar 

  158. Yu, X., Man, Z.: Model reference adaptive control systems with terminal sliding modes. Int. J. Control 64, 1165–1176 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  159. Yu, X., Man, Z.: Model reference adaptive control systems with terminal sliding modes. IEEE Trans. Circuits and Systems-I 44(11), 1065–1070 (1997)

    Article  MathSciNet  Google Scholar 

  160. Yu, X., Man, Z.: Non-singular terminal sliding mode control of rigid manipulators. IEEE Transactions on Circuits and Systems, Part 1 49(2), 261–264 (2002)

    Article  MathSciNet  Google Scholar 

  161. Yu, X., Wang, B., Batsukh, B., Wang, L., Man, Z.: An improved training algorithm for feedforward neural network learning based on terminal attractors. Journal of Global Optimization (2011), doi: 10.1007/s10898–010–9597–6

    Google Scholar 

  162. Zak, S.: Terminal atractors for addressable memory in neural network. Physics Letters A 133(1,2), 18–22 (1988)

    Article  Google Scholar 

  163. Zak, S.: Terminal attractors in neural networks. Neural Networks 133(2), 259–274 (1989)

    Article  Google Scholar 

  164. Zhihong, M., O’day, M., Yu, X.: A robust adaptive terminal sliding mode control for rigid robotic manipulators. J. Intell. Robotics Syst. 24(1), 23–41 (1999)

    Article  MATH  Google Scholar 

  165. Zolezzi, T.: Variable structure control of semilinear evolution equations. In: Partial differential equations and the calculus of variations: Essays in Honor of Ennio De Giorgi, pp. 997–1018. Birkhäuser, Basel (1989)

    Google Scholar 

  166. Zubov, V.: Methods of A.M. Lyapunov and their applications. Noordhoff, Groningen (1964)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Fridman, L. (2011). Sliding Mode Enforcement after 1990: Main Results and Some Open Problems. In: Fridman, L., Moreno, J., Iriarte, R. (eds) Sliding Modes after the First Decade of the 21st Century. Lecture Notes in Control and Information Sciences, vol 412. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22164-4_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-22164-4_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22163-7

  • Online ISBN: 978-3-642-22164-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics