Skip to main content

A fractional model for robust fractional order Smith predictor

Abstract

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

References

  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)

    MathSciNet  MATH  Article  Google Scholar 

  2. Smith, O.J.M.: Closer control of loops with dead time. Chem. Eng. Prog. 53, 217–219 (1957)

    Google Scholar 

  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)

    MATH  Article  Google Scholar 

  4. Zhang, W., Xu, X.: Analytical design and analysis of mismatched Smith predictor. ISA Trans. 40(2), 133–138 (2001)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  6. Jesus Isabel, S., Tenreiro Machado, J.A.: Fractional control of heat diffusion systems. Nonlinear Dyn. 54(3), 263–282 (2008)

    MATH  Article  Google Scholar 

  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)

    MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    MATH  Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  12. Podlubny, I.: Fractional-order systems and PI Λ D μ controllers. IEEE Trans. Autom. Control 44, 208–214 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  13. Wang, Q.G., Bi, Q., Zhang, Y.: Re-design of Smith predictor systems for performance enhancement. ISA Trans. 39(1), 79–92 (2000)

    Article  Google Scholar 

  14. Tenreiro Machado, J.A.: Optimal tuning of fractional controllers using genetic algorithms. Nonlinear Dyn. 62(1–2), 447–452 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  15. Tenreiro Machado, J.A.: Fractional order modelling of fractional-order holds. Nonlinear Dyn. 70(1), 789–796 (2012)

    MathSciNet  Article  Google Scholar 

  16. Valério, D., da Costa, J.S.: Tuning of fractional PID controllers with Ziegler–Nichols-type rules. Signal Process. 86, 2771–2784 (2006)

    MATH  Article  Google Scholar 

  17. Shantanu, D.: Functional Fractional Calculus for System Identification and Controls. Springer, Berlin (2008)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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. Jesus Isabel, S., Tenreiro Machado, J.A.: Fractional control with a Smith predictor. J. Comput. Nonlinear Dyn. 6(3), 031014 (2011)

    Article  Google Scholar 

  21. Clarke, T., Achar, B.N.N., Hanneken, J.W.: Mittag–Leffler functions and transmission lines. J. Mol. Liq. 114, 159–163 (2004)

    Article  Google Scholar 

  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)

    MATH  Google Scholar 

  23. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  24. Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, San Diego (1974)

    MATH  Google Scholar 

  25. Oustaloup, A.: La Commande CRONE: Commande Robuste D’ordre Non Entier. Hermès, Paris (1991)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Djalil Boudjehem.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Boudjehem, D., Sedraoui, M. & Boudjehem, B. A fractional model for robust fractional order Smith predictor. Nonlinear Dyn 73, 1557–1563 (2013). https://doi.org/10.1007/s11071-013-0885-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-013-0885-9

Keywords

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