Robust fractional PID controller synthesis approach for the permanent magnetic synchronous motor

  • Rochdi Bachir Bouiadjra
  • Moussa Sedraoui
  • Abdelaziz Younsi
Original Article


In this work, we propose the synthesis method of different robust fractional order PID controllers, which are synthesized from solving the constrained optimization algorithm for the permanent magnet synchronous motor (PMSM) speed control problem. This problem is formulated through the time and frequency-domain specifications where the obtained fractional controller should simultaneously satisfy some conflicted goals, such as good tracking dynamic behavior of the imposed set-point reference, good attenuation of the plant uncertainties, good suppression of the sensor noise effect and ensure an enough trade-off between the nominal performances and the robust stability of the feedback control system. The previous properties are satisfied not only for the nominal plant-model used in synthesis controller step, but also for a set of the neighboring plant uncertainties. Moreover, the integral time absolute error criterion should be minimized to satisfy the imposed time-domain requirements. However, the weighted-mixed sensitivity problem based upon the different proposed adjustable weights is minimized to ensure the imposed frequency-domain specifications where the load disturbances, sensor noises and the neglected nonlinear and fast PMSM dynamics are considered. The PMSM speed drive is controlled where its dynamic behavior is modeled by the unstructured-multiplicative uncertainty. The obtained simulation results are compared in time and frequency domains by those given with the conventional robust \(\mathcal {H}_\infty\) controller to demonstrate the effectiveness of the proposed fractional controllers.


Robust stability (RS) Nominal performances (NP) Robust fractional controller (RFC) Permanent magnetic synchronous motor (PMSM) 



The authors would like to thank the referees and the editor for detailed comments that have helped significantly improve the quality of presentation.


  1. 1.
    Freescale Semiconductor, Inc. (2016) Sensorless PMSM field oriented control. Design Reference Manual, Document Number: DRM148, Rev. 1 02/2016.
  2. 2.
    Liu H, Li S (2012) Speed control for PMSM servo system using predictive functional control and extended state observer. IEEE Trans Ind Electron 59(2):1171–1183CrossRefGoogle Scholar
  3. 3.
    Kim H, Son J, Lee J (2011) A high-speed sliding-mode observer for the sensorless speed control of a PMSM. IEEE Trans Ind Electron 58(9):4069–4077CrossRefGoogle Scholar
  4. 4.
    Li S, Li Z (2009) Adaptive speed control for permanent-magnet synchronous motor system with variations of load inertia. IEEE Trans Ind Electron 56(8):3050–3059CrossRefGoogle Scholar
  5. 5.
    Yadav AK, Gaur P, Saxena P (2016) Robust stability analysis of PMSM with parametric uncertainty using Kharitonov theorem. J Electr Syst 12(2):258–277Google Scholar
  6. 6.
    Zhou K (1995) Robust and optimal control. Prentice Hall, Upper Saddle River,Google Scholar
  7. 7.
    Zhou K (1999) Essentials of Robust control. Prentice Hall, Upper Saddle RiverGoogle Scholar
  8. 8.
    Doyle J, Francis B, Tannenbaum A (1990) Feedback control theory. Macmillan Publishing Co, New YorkGoogle Scholar
  9. 9.
    Francis BA (1987) A course in \(H_\infty\) control theory. In: Lecture notes in control and information sciences, vol 88. Springer, BerlinGoogle Scholar
  10. 10.
    Zhu M, Liu J, Lin Z, Meng H (2016) Mixed \({H}_2/{H}_\infty\) pitch control of wind turbine generator system based on global exact linearization and regional pole placement. Int J Mach Learn Cybern 7(5):921–930CrossRefGoogle Scholar
  11. 11.
    Ball JA, Helton JW (1989) \({H}_\infty\) optimal control for nonlinear plants: connection with differential games. In: Proceedings of the 28th IEEE conference on decision and control, Tampa, FL, USA, 13–15 December 1989, pp 956–962. doi: 10.1109/CDC.1989.70268
  12. 12.
    Ma Y, Jing Y (2015) Robust \({H}_\infty\) synchronization of chaotic systems with unmatched disturbance and time-delay. Int J Mach Learn Cybern. doi: 10.1007/s13042-015-0468-9 Google Scholar
  13. 13.
    Hua M, Tan H, Fei J, Ni J (2015) Robust stability and \({H}_\infty\) filter design for neutral stochastic neural networks with parameter uncertainties and time-varying delay. Int J Mach Learn Cybern 8(2):511–524CrossRefGoogle Scholar
  14. 14.
    Safonov MG, Chiang RY (1989) A Schur method for balanced model reduction. IEEE Trans Autom Control 2:729–733MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Safonov MG, Chiang RY, Limebeer DJN (1990) A Schur method for balanced model reduction. IEEE Trans Autom Control 35:496–502CrossRefMATHGoogle Scholar
  16. 16.
    Maruta I, Kim T-H, Sugie T (2009) Fixed-structure \({H}_\infty\) controller synthesis: a meta-heuristic approach using simple constrained particle swarm optimization. Automatica 4:553–559MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gumussoy S, Overton ML (2008) Fixed-order \({H}_\infty\) controller design via HIFOO, a specialized nonsmooth optimization package. 2008 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA, June 11–13Google Scholar
  18. 18.
    Rajasekhar A, Abraham A, Pant M (2014) A hybrid differential artificial bee colony algorithm based tuning of fractional order controller for permanent magnet synchronous motor drive. Int J Mach Learn Cybern 5(3):327–337CrossRefGoogle Scholar
  19. 19.
    Zang H, Qin Z, Dai Y (2014) Robust H-infinity space vector model of permanent magnet synchronous motor based on genetic algorithm. J Comput Inf Syst 10(14):5897–5905Google Scholar
  20. 20.
    Saptarshi D, Indranil P, Shantanu D et al (2012) Improved model reduction and tuning of fractional-order \(PI^\lambda D^\mu\) controllers for analytical rule extraction with genetic programming. ISA Trans 51:237–261CrossRefGoogle Scholar
  21. 21.
    Podlubny I (1999) Fractional order systems and \({PI^\lambda D^\mu }\) controllers. IEEE Trans Autom Control 44:208–214MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Oustaloup A, Levron F, Mathieu B (2000) Frequency-band complex non-integer differentiator: characterization and synthesis. IEEE Trans Circ Syst Fundam Theory Appl 47:25–39CrossRefGoogle Scholar
  23. 23.
    Aidoud M, Sedraoui M, Lachouri A, Boualleg A (2016) A robustication of the two degree-of-freedom controller based upon multivariable generalized predictive control law and robust \({H}_\infty\) control for a doubly-fed induction generator. Trans Inst Meas Control. doi: 10.1177/0142331216673425 Google Scholar
  24. 24.
    Sedraoui M, Amieur T, Bachir Bouiadjra R, Sahnoun M (2015). A Robustified fractional-order controller based on adjustable fractional weights for a doubly fed induction generator. Trans Inst Meas Control (published online before print November 14, 2016, doi:  10.1177/0142331215617236)
  25. 25.
    Zhang BT, Pi Y (2012) Robust fractional order proportion-plus-differential controller based on fuzzy inference for permanent magnet synchronous motor. IET Control Theory Appl 6(6):829–837MathSciNetCrossRefGoogle Scholar
  26. 26.
    Hwang C-L, Fang W-L, Shih C-L (2017) Globally neural-adaptive simultaneous position and torque variable structure tracking control for permanent magnet synchronous motors. J Vibr Control 14(1):147–163MathSciNetCrossRefGoogle Scholar
  27. 27.
    Chang W, Tong S (2016) Adaptive fuzzy tracking control design for permanent magnet synchronous motors with output constraint. J Nonlinear Dyn. doi: 10.1007/s11071-016-3043-3
  28. 28.
    Lundstrm P, Skogestad S, Wang ZQ (1991) Performance weight selection for \({H}\)-infinity and \(\mu\)-control methods. Trans Inst Meas Control 13:241–252CrossRefGoogle Scholar
  29. 29.
    Oloomi H, Shafai B (2003) Weight selection in mixed sensitivity robust control for improving the sinusoidal tracking performance. In: 42nd IEEE conference on decision and control, vol 1, pp 300–305Google Scholar
  30. 30.
    Ortega MG, Rubio FR (2004) Systematic design of weighting matrices for the \(\cal{H}_\infty\) mixed sensitivity problem. J Process Control 14:89–98CrossRefGoogle Scholar
  31. 31.
    Sarath SN (2011) Automatic weight selection algorithm for designing H-infinity controller for active magnetic bearing. Int J Eng Sci Technol 3:122–138CrossRefGoogle Scholar
  32. 32.
    Mazandarani M, Kamyad AV (2013) Modified fractional Euler method for solving Fuzzy fractional initial value problem. Commun Nonlinear Sci Numer Simul 18:12–21MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Mazandarani M, Najariyan M (2014) Type-2 fuzzy fractional derivatives. Commun Nonlinear Sci Numer Simul 19:2354–2372MathSciNetCrossRefGoogle Scholar
  34. 34.
    Li X, Wang Y, Li N, Han M, Tang Y, Liu F (2016) Optimal fractional order PID controller design for automatic voltage regulator system based on reference model using particle swarm optimization. Int J Mach Learn Cybern. doi: 10.1007/s13042-016-0530-2 Google Scholar
  35. 35.
    Singiresu SR (2009) Engineering optimization: theory and practice, 4th edn. Wiley, HobokenGoogle Scholar
  36. 36.
    Xu WJ (2012) Permanent magnet synchronous motor with linear quadratic speed controller. Energy Procedia 14:364–369CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Computer Science Department, Faculty of Exact SciencesUniversity Mustapha StambouliMascaraAlgeria
  2. 2.Laboratory of Research in Industrial Computing and NetworksUniversity Ahmed Benbella Oran 1OranAlgeria
  3. 3.Laboratoire des TélécommunicationsUniversity of 8 Mai 1945GuelmaAlgeria
  4. 4.Electronic and Telecommunication DepartementUniversity of 8 May 1945GuelmaAlgeria

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