Improved frequency-domain design method for the fractional order proportional–integral–derivative controller optimal design: a case study of permanent magnet synchronous motor speed control

Improved frequency-domain design method for the fractional order proportional–integral–derivative controller optimal design: a case study of permanent magnet synchronous motor speed control

For access to this article, please select a purchase option:

Buy article PDF
(plus tax if applicable)
Buy Knowledge Pack
10 articles for £75.00
(plus taxes if applicable)

IET members benefit from discounts to all IET publications and free access to E&T Magazine. If you are an IET member, log in to your account and the discounts will automatically be applied.

Learn more about IET membership 

Recommend Title Publication to library

You must fill out fields marked with: *

Librarian details
Your details
Why are you recommending this title?
Select reason:
IET Control Theory & Applications — Recommend this title to your library

Thank you

Your recommendation has been sent to your librarian.

An improved frequency-domain method for the fractional order controller design is proposed in this study. A proportional relation between the integral gain and derivative gain of the controller is built, while the derivative order is set to be equal to the integral order. Applying the improved method, the controller parameters can be calculated analytically according to different design requirements. The proportional coefficient between the integral and derivative gains is studied and then the model of the optimal proportional coefficient for the commonly used permanent magnet synchronous motor speed control system is built. Based on the established model, the optimal controllers are obtained analytically applying the proposed method. Motor speed control simulations and experiments are performed, comparing the performance of the controllers obtained using the proposed method and those obtained using the current frequency-domain method and other design methods. Simulation and experimental results show that the control system obtained using the proposed method achieves the specified stability, robustness to gain variations and the optimal dynamic performance.


    1. 1)
      • 1. Podlubny, I.: ‘Fractional differential equations’ (Academic Press, San Diego, USA, 1999).
    2. 2)
      • 2. Monje, C.A., Chen, Y.Q., Vinagre, B.M., et al: ‘Fractional-order systems and controls: fundamentals and applications’ (Springer, London, UK, 2010).
    3. 3)
      • 3. Charef, A.: ‘Fractional-order PIλDμ and active disturbance rejection control of nonlinear two-mass drive system’, IEE Proc. Control Theory Appl., 2006, 153, (6), pp. 714720.
    4. 4)
      • 4. Vinagre, B.M., Petras, I., Podlubny, I.: ‘Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control’, Nonlinear Dynam., 2002, 29, (1–4), pp. 269279.
    5. 5)
      • 5. Hamamci, S.E.: ‘An algorithm for stabilization of fractional-order time delay systems using fractional-order PID controllers’, IEEE Trans. Autom. Control, 2007, 52, (10), pp. 19641969.
    6. 6)
      • 6. Charef, A., Assabaa, M., Ladaci, S., et al: ‘Fractional order adaptive controller for stabilised systems via high-gain feedback’, IET Control Theory A, 2013, 7, (6), pp. 822828.
    7. 7)
      • 7. Podlubny, I.: ‘Fractional-order systems and PIλDμ controllers’, IEEE. T. Autom. Control, 1999, 44, (1), pp. 208214.
    8. 8)
      • 8. Luo, Y., Chen, Y.Q.: ‘Stabilizing and robust FOPI controller synthesis for first order plus time delay systems’, Automatica, 2011, 48, (9), pp. 20402045.
    9. 9)
      • 9. Erenturk, K.: ‘Analogue realisation of fractional-order integrator, differentiator and fractional PIλDμ controller’, IEEE T. Ind. Electron., 2013, 60, (9), pp. 39063813.
    10. 10)
      • 10. Monje, C.A., Vinagre, B.M., Feliu, V., et al: ‘Tuning and auto-tuning of fractional order controllers for industry applications’, Control Eng. Pract., 2008, 16, (7), pp. 798812.
    11. 11)
      • 11. Luo, Y., Chen, Y.Q., Wang, C.Y., et al: ‘Tuning fractional order proportional integral controllers for fractional order systems’, J. Process. Control, 2010, 20, (7), pp. 823831.
    12. 12)
      • 12. Mercader, P., Banos, A., Vilanova, R.: ‘Robust proportional integral derivative design for processes with interval parametric uncertainty’, IET Control Theory A, 2017, 11, (7), pp. 10161023.
    13. 13)
      • 13. Luo, Y., Chen, Y.Q.: ‘Fractional order [proportional derivative] controller for a class of fractional order systems’, Automatica, 2009, 45, (10), pp. 24462450.
    14. 14)
      • 14. Zheng, W.J., Pi, Y.: ‘Study of the fractional order proportional integral controller for the permanent magnet synchronous motor based on the differential evolution algorithm’, ISA Transc., 2016, 63, pp. 287293.
    15. 15)
      • 15. Zamani, M., Karimi-Ghartemani, M., Sadati, N., et al: ‘Design of a fractional order PID controller for an AVR using particle swarm optimization’, Control Eng. Pract., 2009, 17, (12), pp. 13801387.
    16. 16)
      • 16. Lee, C.H., Chang, F.K.: ‘Fractional-order pid controller optimization via improved electromagnetism-like algorithm’, Exp. Syst. Appl., 2010, 37, (12), pp. 88718878.
    17. 17)
      • 17. Zheng, W.J., Luo, Y., Wang, X.H., et al: ‘Fractional order PIλDμ controller design for satisfying time and frequency domain specifications simultaneously’, ISA Transc, 2017, 68, pp. 212222.
    18. 18)
      • 18. Ruan, Y., Yang, Y., Chen, B.S.: ‘Control systems of electric drives – motion control systems (in Chinese)’ (China Machine Press, Beijing, P. R. China, 1981, 5th edn., 2016).
    19. 19)
      • 19. Saidi, B., Amairi, M., Najar, S., et al: ‘Bode shaping-based design methods of a fractional order PID controller for uncertain systems’, Nonlinear. Dyn., 2015, 80, pp. 18171838.
    20. 20)
      • 20. Oustaloup, A.: ‘La Dérivation non entière: théorie, synthèse et applications’ (Hermès, Paris, France, 1995).
    21. 21)
      • 21. Oustaloup, A., François, L., Benoît, M., Florence, M.: ‘Frequency-band complex noninteger differentiator: Characterization and synthesis’, IEEE Trans. Circuits I, 2000, 47, (1), pp. 2529.
    22. 22)
      • 22. Impulse response invariant discretization of fractional order integrators/differentiators’,, accessed 16 March 2018..
    23. 23)
      • 23. Martín, F., Monje, C., Moreno, L., et al: ‘DE-based tuning of PIλDμ controllers’, ISA Transc., 2015, 59, pp. 398407.

Related content

This is a required field
Please enter a valid email address