New Implementation of Discrete-Time Fractional-Order PI Controller by Use of Model Order Reduction Methods

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1196)


The paper presents new results in implementation of a discrete-time fractional-order PI controller by use of computationally simple and accurate Model Order Reduction-based approximation of a fractional-order integrator. The main advantage of the introduced method is elimination of the steady-state control error, the feature outperforming other finite-length implementations of discrete-time fractional-order integrators. Simulation experiments confirm the effectiveness of the presented methodology, both in terms of high accuracy and computational effectiveness of the introduced approximation.


Fractional-order PID controller Model order reduction Discrete time system 


  1. 1.
    Ahmed, M.F., Dorrah, H.T.: Design of gain schedule fractional PID control for nonlinear thrust vector control missile with uncertainty. Automatika 59(3–4), 357–372 (2018). Scholar
  2. 2.
    Baranowski, J., Bauer, W., Zagorowska, M.: Stability properties of discrete time-domain Oustaloup approximation. In: Theoretical Developments and Applications of Non-integer Order Systems. Lecture Notes in Electrical Engineering. Springer (2016)Google Scholar
  3. 3.
    Barbosa, R., Tenreiro, J.A., Ferreira, I.M.: Tuning of PID controllers based on Bode’s ideal transfer function. Nonlinear Dyn. 38(1–4), 305–321 (2004)CrossRefGoogle Scholar
  4. 4.
    Cervera, A., Baños, A., Monje, C.A.: Tuning of fractional PID controllers by using QFT. In: Proceedings of the 32nd Annual Conference of the IEEE Industrial Electronics Society (IECON 2006), Paris, France (2006)Google Scholar
  5. 5.
    da Costa, J.S.: An introduction to fractional control. In: Control, Robotics and Sensors, Institution of Engineering and Technology (2012).
  6. 6.
    Hosseinnia, S.H., Tejado, I., Milanés, V., Villagrá, J., Vinagre, B.M.: Experimental application of hybrid fractional-order adaptive cruise control at low speed. IEEE Trans. Control Syst. Technol. 22(6), 2329–2336 (2014)CrossRefGoogle Scholar
  7. 7.
    HosseinNia, S.H., Tejado, I., Vinagre, B.M.: A method for the design of robust controllers ensuring the quadratic stability for switching systems. J. Vibr. Control 20(7), 1085–1098 (2014). Scholar
  8. 8.
    Keyser, R.D., Muresan, C.I., Ionescu, C.M.: A novel auto-tuning method for fractional order PI/PD controllers. ISA Trans. 62, 268–275 (2016). sI: Control of Renewable Energy Systems.
  9. 9.
    Krajewski, W., Viaro, U.: A method for the integer-order approximation of fractional-order systems. J. Franklin Inst. 351(1), 555–564 (2014)CrossRefGoogle Scholar
  10. 10.
    Liu, L., Zhang, S.: Robust fractional-order PID controller tuning based on Bode’s optimal loop shaping. Complexity 2018(6570560) (2018)Google Scholar
  11. 11.
    Lopes, A.M., Machado, J.T.: Discrete-time generalized mean fractional order controllers. IFAC-PapersOnLine 51(4), 43–47 (2018). 3rd IFAC Conference on Advances in Proportional-Integral-Derivative Control PID 2018.
  12. 12.
    Maâmar, B., Rachid, M.: IMC-PID-fractional-order-filter controllers design for integer order systems. ISA Trans. 53(5), 1620–1628 (2014). iCCA 2013.
  13. 13.
    Monje, C., Chen, Y., Vinagre, B., Xue, D., Feliu, V.: Fractional-order Systems and Controls: Fundamentals and Applications. Series on Advances in Industrial Control. Springer, London (2010)CrossRefGoogle Scholar
  14. 14.
    Monje, C.A., Vinagre, B.M., Feliu, V., Chen, Y.: Tuning and auto-tuning of fractional order controllers for industry applications. Control Eng. Pract. 16(7), 798–812 (2008).
  15. 15.
    Mozyrska, D., Ostalczyk, P.: Generalized fractional-order discrete-timeintegrator. Complexity 2017(3452409) (2017)Google Scholar
  16. 16.
    Muresan, C.I., Dutta, A., Dulf, E.H., Pinar, Z., Maxim, A., Ionescu, C.M.: Tuning algorithms for fractional order internal model controllers for time delay processes. Int. J. Control 89(3), 579–593 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Oprzędkiewcz, K., Mitkowski, W., Gawin, E.: The plc implementation of fractional-order operator using cfe approximation. In: Advances in Intelligent Systems and Computing, vol. 550. Springer (2017)Google Scholar
  18. 18.
    Oprzedkiewicz, K., Stanisławski, R., Gawin, E., Mitkowski, W.: A new algorithm for a CFE-approximated solution of a discrete-time non integer-order state equation. Bull. Pol. Acad. Sci. Tech. Sci. 65(4), 429–437 (2017)Google Scholar
  19. 19.
    Padula, F., Visioli, A.: Tuning rules for optimal PID and fractional-order PID controllers. J. Process Control 21(1), 69–81 (2011). Scholar
  20. 20.
    Rydel, M., Stanisławski, R.: A new frequency weighted Fourier-based method for model order reduction. Automatica 88, 107–112 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Stanisławski, R., Latawiec, K.J.: Normalized finite fractional differences - the computational and accuracy breakthroughs. Int. J. Appl. Math. Comput. Sci. 22(4), 907–919 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Stanisławski, R., Latawiec, K.J., Łukaniszyn, M.: A comparative analysis of Laguerre-based approximators to the Grünwald-Letnikov fractional-order difference. Math. Problems Eng. 2015, 1–10 (2015). Article ID: 512104CrossRefGoogle Scholar
  23. 23.
    Stanisławski, R., Rydel, M., Latawiec, K.J.: Modeling of discrete-time fractional-order state space systems using the balanced truncation method. J. Franklin Inst. 354(7), 3008–3020 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Tejado, I., HosseinNia, S.H., Vinagre, B.M., Chen, Y.: Efficient control of a smartwheel via internet with compensation of variable delays. Mechatronics 23(7), 821–827 (2013). 1. Fractional Order Modeling and Control in Mechatronics 2. Design, control, and software implementation for distributed MEMS (dMEMS).
  25. 25.
    Tejado, I., Vinagre, B.M., Traver, J.E., Prieto-Arranz, J., Nuevo-Gallardo, C.: Back to basics: meaning of the parameters of fractional order PID controllers. Mathematics 7(6) (2019).
  26. 26.
    Vilanova, R., Visioli, A.: PID Control in the Third Millennium. Lessons Learned and New Approaches. Advances in Industrial Control. Springer, London (2012)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Electrical, Control and Computer EngineeringOpole University of TechnologyOpolePoland

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