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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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Fractional PI Stabilization of Delay Systems: Application to a Thermal System

Journal of Applied Nonlinear Dynamics 8(3) (2019) 509--518 | DOI:10.5890/JAND.2019.09.012

Aymen Rhouma, Sami Hafsi, Kaouther Laabidi

Shaqra University, Kingdom of Saudi Arabia

Université deTunis El Manar, Ecole Nationale d’Ingénieurs deTunis, Tunisie

LR11ES20 Laboratoire Analyse, Conception et Commande des Syst`emes, Tunis 1002, Tunisie

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In this paper, an application of fractional-order PIλ controller is given as an alternative to solve some control problems that can arise. It aims to apply the analytical tuning procedure to control the heat flow systems. This system, modeled by first-order system involving time delay, is one with open loop characteristic equations are fractional order quasi-polynomials. Using the proposed method, the entire stability region of PI λ controllers is obtained and visualized in the plane (Kp,Ki, λ ). The simulation was carried out on thermal systems and the results demonstrate the effectiveness of the proposed type of controllers and the tuning rule.


  1. [1]  Machado, J.A.T. (1997), Analysis and design of fractional-order digital control systems, SAMS, 27, 107-122.
  2. [2]  Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., and Feliu, V. (2010), Fractional-order Systems and Controls: Fundamentals and Applications (Advances in Industrial Control). Springer-Verlag London Limited 2010.
  3. [3]  Tan, N., Ozguven, O.F., and Ozyetkin, M.M. (2009), Robust stability analysis of fractional order interval polynomials, ISA Transactions, 48, 166-172.
  4. [4]  Azar, A.T., Vaidyanathan, S., and Ouannas, A. (2017), Fractional Order Control and Synchronization of Chaotic Systems, Studies in Computational Intelligence, 688, Springer-Verlag, Germany. ISBN 978-3-319- 50248-9.
  5. [5]  Azar, A.T. and Serrano, F.E. (2016), Stabilization of mechanical systems with backlash by PI loop shaping, International Journal of System Dynamics Applications (IJSDA), 5, 20-47.
  6. [6]  Ouannas, A. (2016), On inverse generalized synchronization of continuous chaotic dynamical systems, International Journal of Applied and Computational Mathematics, 2(1), 1-11.
  7. [7]  Ouannas, A. and Al-sawalha, M.M. (2016), Synchronization between different dimensional chaotic systems using two scaling matrices, Optik Int. J. for Light and Electron Optics, 127, 959-963.
  8. [8]  Azar, A.T. and Vaidyanathan, S. (2016), Advances in Chaos Theory and Intelligent Control, Studies in Fuzziness and Soft Computing, 337, Springer-Verlag, Berlin, Germany, ISBN 978-3-319-30338-3.
  9. [9]  Azar, A.T., Vaidyanathan, S., and Ouannas, A. (2017), Fractional Order Control and Synchronization of Chaotic Systems, Studies in Computational Intelligence, 688, Springer-Verlag, Berlin, Germany, ISBN 978- 3-319-50248-9.
  10. [10]  Dingyu, X. and Chen, Y.Q. (2002), A comparative introduction of four fractional order controllers, in Proceedings of the 4th World Congress on Intelligent Control and Automation, 4, 3228-3235, Shanghai, China.
  11. [11]  Oustaloup, A. (1991), La commande crone, Herm.
  12. [12]  Podlubny, I. (1999), Fractional-order systems and PIλDμ controllers, IEEE Transactions on Automatic Control, 44, 208-214.
  13. [13]  Suman, S., Saptarshi, D., Ratna, G., Bhaswati, G., Balasubramanian, R., Chandra, A.K., Shantanu, D., and Amitava, G. (2010), Fractional order phase shaper design with Bode’s integral for iso-damped control system, ISA Transactions, 49, 196-206.
  14. [14]  Podlubny, I., Dorcak, L., and Kostial, I. (1997), On fractional derivatives, fractional-order dynamic systems and PIλDμ controllers, Proceeedings of the 36th IEEE Conference on Decision and Control, pp. 4985-4990.
  15. [15]  Petra, I. and Vinagre, B.M. (2002), Practical application of digital fractional order controller to temperature control, Acta Montanistica Slovaca, 7(2), 131-137.
  16. [16]  Bhambhani, V. and Chen, Y.Q. (2008), Experimental study of fractional order proportional integral (FOPI) controller for water lever control. In: Proceedings of the 47th IEEE Conference on Decision and Control, 9-11 December, Cancun, Mexcio.
  17. [17]  Caponetto, R., Dongola, G., Fortuna, L., and Petras, I. (2010), Fractional Order Systems: Modeling and Control Applications, World Scientific Series on Nonlinear Science, Series A, 72.
  18. [18]  Caponetto, R., Dongola, G., Fortuna, L., and Gallo, A. (2010), New results on the synthesis of FO-PID controllers, Communs Nonlinear Sci. Numer. Simulation, 15, 997-1007.
  19. [19]  Hafsi, S., Laabidi, K., and Farkh, R. (2013), Synthesis of fractional PI controller for a first-order time delay system, Transactions of the Institute of Measurement and Control, 35(8), 997-1007.
  20. [20]  Silva, G.J., Datta, A., and Bhattacharyya, S.P. (2002), New synthesis of PID controller, IEEE transactions on automatic control, 47(2).
  21. [21]  Silva, G.J., Datta, A., and Bhattacharyya, S.P. (2001), PI stabilization of first-order systems with time-delay, Automatica, 37(12), 2025-2031.
  22. [22]  Hafsi, S., Laabidi, K., and Farkh, R. (2015), A new tuning method for stabilization time delay systems using PIλDμ controllers, Asian Journal of Control, 17(3), 1-11.
  23. [23]  Malti, R., Victor, S., and Oustaloup, A. (2008), Advances in system identification using fractional models, ASME J. Comput. Nonlinear Dyn., 3(2), 21-40.
  24. [24]  Battaglia, J.L., Le Lay, L., Batsale, J.C., Oustaloup, A., and Cois, O. (2000), Heat flux estimation through inverted non integer identification models, Int. J. Thermal Science, 39(3), 374-389.
  25. [25]  Rhouma, A., Bouani, F., Bouzouita, B., and Ksouri, M. (2014), Model predictive control of fractional order systems, J. Comput. Nonlinear Dynam., 9(3).
  26. [26]  Rhouma, A. and Bouani, F. (2015), Robust model predictive control of uncertain fractional systems: a thermal application, IET Control Theory and Applications, 8(17).