Nonlinear Dynamics

, Volume 73, Issue 3, pp 1557–1563 | Cite as

A fractional model for robust fractional order Smith predictor

  • Djalil Boudjehem
  • Moussa Sedraoui
  • Badreddine Boudjehem
Original Paper


In this paper, we propose to use a fractional order model to predict the process output in Smith predictor. The parameters of the model are determined by minimizing the error between its output and one of the processes using a genetic algorithm. After determining the model’s parameters, a fractional PID controller is proposed to improve the controlled system performances. The parameters of the controller are also determined in an optimal way by minimizing the position error taking into account the sensitivity and the complementary sensitivity conditions. Applications on a dead time and multiple lags processes have been performed, where the simulation results show that the proposed Smith predictor enhance the closed loop control system.


Smith predictor Fractional calculus Fractional controller Fractional model Sensitivity function Complementary sensitivity function Optimization 


  1. 1.
    Ho, W.K., Hang, C.C., Cao, L.S.: Tuning of PID controllers based on gain and phase margin specifications. Automatica 31(3), 497–502 (1995) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Smith, O.J.M.: Closer control of loops with dead time. Chem. Eng. Prog. 53, 217–219 (1957) Google Scholar
  3. 3.
    Åstrom, K.J., Hang, C.C., Lim, B.C.: A new Smith predictor for controlling a process with an integrator and long dead time. IEEE Trans. Autom. Control 39, 343–345 (1994) MATHCrossRefGoogle Scholar
  4. 4.
    Zhang, W., Xu, X.: Analytical design and analysis of mismatched Smith predictor. ISA Trans. 40(2), 133–138 (2001) CrossRefGoogle Scholar
  5. 5.
    Feliu-Batlle, V., Rivas Pérez, R., Castillo, F.J., Sanchez Rodriguez, L.: Smith predictor based robust fractional order control: application to water distribution in a main irrigation canal pool. J. Process Control 19, 506–519 (2009) CrossRefGoogle Scholar
  6. 6.
    Jesus Isabel, S., Tenreiro Machado, J.A.: Fractional control of heat diffusion systems. Nonlinear Dyn. 54(3), 263–282 (2008) MATHCrossRefGoogle Scholar
  7. 7.
    Rapaić, M.R., Jeličić, Z.D.: Optimal control of a class of fractional heat diffusion systems. Nonlinear Dyn. 62(1–2), 39–51 (2010) MATHCrossRefGoogle Scholar
  8. 8.
    Castillo, F.J., Feliua, V., Rivas, R., Sánchez, L.: Design of a class of fractional controllers from frequency specifications with guaranteed time domain behavior. Comput. Math. Appl. 59, 1656–1666 (2010) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Boudjehem, D., Boudjehem, B.: A fractional model predictive control for fractional order systems. In: Baleanu, D. (ed.) Fractional Dynamics and Control 1st, pp. 49–57. Springer, Berlin (2012) CrossRefGoogle Scholar
  10. 10.
    Monje Concepción, A., Calderon Antonio, J., Vinagre Blas, M., Chen, Y., Feliu, V.: On fractional PIλ controllers: some tuning rules for robustness to plant uncertainties. Nonlinear Dyn. 38(1–4), 369–381 (2004) MATHCrossRefGoogle Scholar
  11. 11.
    Boudjehem, B., Boudjehem, D.: Parameter tuning of a fractional-order PI controller using the ITAE criteria. In: Baleanu, D. (ed.) Fractional Dynamics and Control 1st, pp. 49–57. Springer, Berlin (2012) CrossRefGoogle Scholar
  12. 12.
    Podlubny, I.: Fractional-order systems and PI Λ D μ controllers. IEEE Trans. Autom. Control 44, 208–214 (1999) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Wang, Q.G., Bi, Q., Zhang, Y.: Re-design of Smith predictor systems for performance enhancement. ISA Trans. 39(1), 79–92 (2000) CrossRefGoogle Scholar
  14. 14.
    Tenreiro Machado, J.A.: Optimal tuning of fractional controllers using genetic algorithms. Nonlinear Dyn. 62(1–2), 447–452 (2010) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Tenreiro Machado, J.A.: Fractional order modelling of fractional-order holds. Nonlinear Dyn. 70(1), 789–796 (2012) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Valério, D., da Costa, J.S.: Tuning of fractional PID controllers with Ziegler–Nichols-type rules. Signal Process. 86, 2771–2784 (2006) MATHCrossRefGoogle Scholar
  17. 17.
    Shantanu, D.: Functional Fractional Calculus for System Identification and Controls. Springer, Berlin (2008) MATHGoogle Scholar
  18. 18.
    Rao, A.S., Rao, V.S.R., Chidambaram, M.: Set point weighted modified Smith predictor for integrating and double integrating processes with time delay. ISA Trans. 46(1), 59–71 (2007) CrossRefGoogle Scholar
  19. 19.
    Baiyu, O., Lei, S., Chunlei, C.: Tuning of fractional PID controllers by using radial basis function neural network. In: Poceed. of IEEE International Conference on Control and Automation, pp. 1239–1244 (2010) Google Scholar
  20. 20.
    Jesus Isabel, S., Tenreiro Machado, J.A.: Fractional control with a Smith predictor. J. Comput. Nonlinear Dyn. 6(3), 031014 (2011) CrossRefGoogle Scholar
  21. 21.
    Clarke, T., Achar, B.N.N., Hanneken, J.W.: Mittag–Leffler functions and transmission lines. J. Mol. Liq. 114, 159–163 (2004) CrossRefGoogle Scholar
  22. 22.
    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999) MATHGoogle Scholar
  23. 23.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) MATHGoogle Scholar
  24. 24.
    Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, San Diego (1974) MATHGoogle Scholar
  25. 25.
    Oustaloup, A.: La Commande CRONE: Commande Robuste D’ordre Non Entier. Hermès, Paris (1991) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Djalil Boudjehem
    • 1
  • Moussa Sedraoui
    • 1
  • Badreddine Boudjehem
    • 1
  1. 1.Advanced Control Laboratory (LabCAV)University of 8 Mai 1945 of GuelmaGuelmaAlgeria

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