Optimal Discrete-time Integral Sliding Mode Control for Piecewise Affine Systems

  • Olfa JeddaEmail author
  • Ali Douik
Regular Papers Control Theory and Applications


This paper presents an optimal discrete-time integral sliding mode control for constrained piecewise affine systems. The proposed scheme is developed on the basis of linear quadratic regulator approach and differential evolution algorithm in order to ensure the stability of the closed-loop system in discrete-time sliding mode and the optimization of response characteristics in presence of control input constraints. Moreover, the controller is designed such that chattering phenomenon is avoided and finite-time convergence to the sliding surface is guaranteed. The follow-up of a reference model is also ensured. The efficiency of the proposed method is illustrated with an inverted pendulum system.


Differential evolution algorithm discrete-time integral sliding mode control inverted pendulum system piecewise affine systems 


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  1. [1]
    R. L. Grossman, A. Nerode, A. P. Ravn, and H. Rischel, Hybrid Systems, Springer-Verlag, Berlin Heidelberg, 1993.CrossRefzbMATHGoogle Scholar
  2. [2]
    W. P. M. H. Heemels, J. M. Schumacher, and S. Weiland, “Linear complementarity systems,” SIAM Journal on Applied Mathematics, vol. 60, no. 4, pp. 1234–1269, March 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. J. van der Schaft and J. M. Schumacher, “Complementarity modeling of hybrid systems.” IEEE Trans. on Automatic Control, vol. 43, no. 4, pp. 483–490, April 1998.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A. Bemporad, and M. Morari, “Control of systems integrating logic, dynamics, and constraints,” Automatica, vol. 35, no. 3, pp. 407–427, March 1999.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    E. Sontag, “Nonlinear regulation: the piecewise linear approach,” IEEE Trans. on Automatic Control, vol. 26, no. 2, pp. 346–358, April 1981.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    B. De Schutter and T. Van den Boom, “On model predictive control for max-min-plus-scaling discrete event systems,” Control Systems Engineering, October 2000.Google Scholar
  7. [7]
    E. D. Sontag, “Interconnected automata and linear systems: a theoretical framework in discrete-time,” Hybrid Systems III: Verification and Control, Springer-Verlag, Berlin Heidelberg, pp. 436–448, 1996.CrossRefGoogle Scholar
  8. [8]
    A. Bemporad, G. Ferrari-Trecate, and M. Morari, “Observability and controllability of piecewise affine and hybrid systems,” IEEE Trans, on Automatic Control, vol. 45, no. 10, pp. 1864–1876, October 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    W. P. M. h. Heemels, B. De Schutter, and A. Bemporad, “Equivalence of hybrid dynamical models,” Automatica, vol. 37, no. 7, pp. 1085–1091, July 2001.CrossRefzbMATHGoogle Scholar
  10. [10]
    P. Biswas, P. Grieder, J. Lofberg, and M. Morari, “A survey on stability analysis of discrete-time piecewise afflne systems,” IFAC Proceedings Volumes, vol. 38, no. 1, pp. 283–294, 2005.CrossRefGoogle Scholar
  11. [11]
    D. Mignone, G. Ferrari-Trecate, and M. Morari, “Stability and stabilization of piecewise afflne and hybrid systems: an LMI approach,” Proc. of the 39th Conf. Decision and Control, vol. 1, pp. 504–509, 2000.CrossRefGoogle Scholar
  12. [12]
    G. Ferrari-Trecate, M. Muselli, D. Liberati, and M. Morari, “Identification of piecewise afflne and hybrid systems,” Proc. of the 2001 American Control Conference, vol. 5, pp. 3521–3526, 2001.Google Scholar
  13. [13]
    G. Ferrari-Trecate, M. Muselli, D. Liberati, and M. Morari, “A clustering technique for the identification of piecewise afflne systems,” Automatica, vol. 39, no. 2, pp. 205–217, February 2003.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    F. Borrelli, Constrained Optimal Control of Linear and Hybrid Systems, Springer, 2003.zbMATHGoogle Scholar
  15. [15]
    D. Q. Mayne, and S. Raković, “Model predictive control of constrained piecewise afflne discrete-time systems,” International Journal of Robust and Nonlinear Control, vol. 13, no. 3–4, pp. 261–279, February 2003.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    C. Milosavljevic, “General conditions for the existence of a quasi-sliding mode on the switching hyperplane in discrete variable structure systems,” Automation and Remote control, vol. 46, no. 3, pp. 307–314, March 1985.zbMATHGoogle Scholar
  17. [17]
    S. V. Drakunov, and V. I. Utkin, “On discrete-time sliding modes,” Nonlinear Control Systems Design, pp. 273–278, June 1989.Google Scholar
  18. [18]
    K. Furuta, “Sliding mode control of a discrete system,” Systems & Control Letters, vol. 14, no. 2, pp. 145–152, February 1990.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    W. Gao, Y. Wang, and A. Homaifa, “Discrete-time variable structure control systems,” IEEE Trans. on Industrial Electronics, vol. 42, no. 2, pp. 117–122, April 1995.CrossRefGoogle Scholar
  20. [20]
    G. Golo, and Č. Milosavljević, “Robust discrete-time chattering free sliding mode control,” Systems & Control Letters, vol. 41, no. 1, pp. 19–28, September 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    S. Z. Sarpturk, Y. Istefanopulos, and O. Kaynak, “On the stability of discrete-time sliding mode control systems,” IEEE Trans. on Automatic Control, vol. 32, no. 10, pp. 930–932, October 1987.CrossRefzbMATHGoogle Scholar
  22. [22]
    V. Utkin, “Variable structure systems with sliding modes,” IEEE Trans. on Automatic control, vol. 22, no. 2, pp. 212–222, April 1977.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer’s guide to sliding mode control,” Proc. 1996 IEEE International Workshop on Variable Structure Systems (VSS’96), pp. 1–14, 1996.Google Scholar
  24. [24]
    B. Bandyopadhyay, F. Deepak, and K. S. Kim, Sliding Mode Control Using Novel Sliding Surfaces, Springer-Verlag, Berlin Heidelberg, 2009.CrossRefGoogle Scholar
  25. [25]
    O. Jedda, J. Ghabi, and A. Douik, “Sliding mode control of an inverted pendulum,” Applications of Sliding Mode Control, Springer, Singapore, pp. 105–118, 2017.CrossRefGoogle Scholar
  26. [26]
    G. Bartolini, A. Ferrara, and V. I. Utkin, “Adaptive sliding mode control in discrete-time systems,” Automatica, vol. 31, no. 5, pp. 769–773, May 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    K. Abidi, J. X. Xu, and Y. Xinghuo, “On the discrete-time integral sliding-mode control,” IEEE Trans. on Automatic Control, vol. 52, no. 4, pp. 709–715, April 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    V. Utkin, and J. Shi, “Integral sliding mode in systems operating under uncertainty conditions,” Proc. of the 35th Conf. Decision and Control, vol. 4, pp. 4591–4596, 1996.CrossRefGoogle Scholar
  29. [29]
    M. C. Pai, “Discretetime variable structure control for robust tracking and model following,” Journal of the Chinese Institute of Engineers, vol. 31, no. 1, pp. 167–172, 2008.CrossRefGoogle Scholar
  30. [30]
    M. C. Pai, “Robust tracking and model following of uncertain dynamic systems via discrete-time integral sliding mode control,” International Journal of Control, Automation and Systems, vol. 7, no. 3, pp. 381–387, May 2009.CrossRefGoogle Scholar
  31. [31]
    M. C. Pai, “Robust discrete-time sliding mode control for multi-input uncertain time-delay systems,” Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 226, no. 7, pp. 927–935, June 2012.Google Scholar
  32. [32]
    M. C. Pai, “Discrete-time sliding mode control for robust tracking and model following of systems with state and input delays,” Nonlinear Dynamics, vol. 76, no. 3, pp. 1769–1779, January 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    R. Storn, and K. Price, “Differential evolution a simple and efficient heuristic for global optimization over continuous spaces,” Journal of Global Optimization, vol. 11, no. 4, pp. 341–359, December 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” IEEE Trans. on Evolutionary Computation, vol. 13, no. 2, pp. 398–417, April 2009.CrossRefGoogle Scholar
  35. [35]
    A. W. Mohamed, H. Z. Sabry, and M. Khorshid, “An alternative differential evolution algorithm for global optimization,” Journal of Advanced Research, vol. 3, no. 2, pp. 149–165, April 2012.CrossRefGoogle Scholar
  36. [36]
    J. Tvrdk, “Competitive differential evolution and genetic algorithm in GA-DS toolbox,” Technical Computing Prague, Praha, Humusoft, pp. 99–106, 2006.Google Scholar
  37. [37]
    A. D. Lilla, M. A. Khan, and P. Barendse, “Comparison of differential evolution and genetic algorithm in the design of permanent magnet generators,” Proc. of IEEE International Conference on Industrial Technology (ICIT), pp. 266–271, 2013.Google Scholar
  38. [38]
    F. D. Torrisi, and A. Bemporad, “HYSDEL-a tool for generating computational hybrid models for analysis and synthesis problems,” IEEE Trans, on Control Systems Technology, vol. 12, no. 2, pp. 235–249, March 2004.CrossRefGoogle Scholar
  39. [39]
    F. J. Christophersen, Optimal Control of Constrained Piecewise Affine Systems, Springer, 2007.zbMATHGoogle Scholar
  40. [40]
    E. C. Kerrigan, and D. Q. Mayne, “Optimal control of constrained, piecewise affine systems with bounded disturbances,” Proc. of the 41st Conf. Decision and Control, vol. 2, pp. 1552–1557, 2002.CrossRefGoogle Scholar
  41. [41]
    J. Qiu, Y. Wei, and L. Wu, “A novel approach to reliable control of piecewise affine systems with actuator faults,” IEEE Trans. on Circuits and Systems II: Express Briefs, vol. 64, no. 8, pp. 957–961, August 2017.CrossRefGoogle Scholar
  42. [42]
    T. H. Hopp, and W. E. Schmitendorf, “Design of a linear controller for robust tracking and model following,” Journal of Dynamic Systems, Measurement, and Control, vol. 112, no. 5, pp. 552–558, December 1990.CrossRefzbMATHGoogle Scholar
  43. [43]
    K. K. Shyu, and Y. C. Chen, “Robust tracking and model following for uncertain time-delay systems,” International Journal of Control, vol. 62, no. 3, pp. 589–600, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    S. Das, A. Abraham, and A. Konar, “Particle swarm optimization and differential evolution algorithms: technical analysis, applications and hybridization perspectives,” Advances of Computational Intelligence in Industrial Systems, Springer-Verlag, Berlin Heidelberg, pp. 1–38, 2008.Google Scholar
  45. [45]
    C. E. Garcia, D. M. Prett, and M. Morari, “Model predictive control: theory and practicea survey,” Automatica, vol. 25, no. 3, pp. 335–348, May 1989.CrossRefzbMATHGoogle Scholar
  46. [46]
    A. Bemporad, Hybrid Toolbox-User’s Guide, 2003.Google Scholar
  47. [47]
    S. V. Rakovic, P. Grieder, M. Kvasnica, D. Q. Mayne, and M. Morari, “Computation of invariant sets for piece-wise affine discrete time systems subject to bounded disturbances,” In 43rd Conf. on Decision and Control (CDC), vol. 2, pp. 1418–1423, 2004.Google Scholar
  48. [48]
    J. Qiu, H. Tian, Q. Lu, and H. Gao, “Nonsynchronized robust filtering design for continuous-time TS fuzzy affine dynamic systems based on piecewise Lyapunov functions,” IEEE Trans. on Cybernetics, vol. 43, no. 6, pp. 1755–1766, December 2013.CrossRefGoogle Scholar
  49. [49]
    J. Qiu, H. Gao, and S. X. Ding, “Recent advances on fuzzy-model-based nonlinear networked control systems: A survey,” IEEE Trans. on Industrial Electronics, vol. 63, no. 2, pp. 1207–1217, February 2016.CrossRefGoogle Scholar

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© ICROS, KIEE and Springer 2019

Authors and Affiliations

  1. 1.Electrical Engineering Department, National Engineering School of MonastirUniversity of MonastirMonastirTunisia
  2. 2.Computer Engineering Department, National Engineering School of SousseUniversity of SousseSousseTunisia

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