Frontiers in Energy

, Volume 9, Issue 4, pp 413–425 | Cite as

Classical state feedback controller for nonlinear systems using mean value theorem: closed loop-FOC of PMSM motor application

  • Abrar AllagEmail author
  • Abdelhamid Benakcha
  • Meriem Allag
  • Ismail Zein
  • Mohamed Yacine Ayad
Research Article


The problem of state feedback controllers for a class of Takagi-Sugeno (T-S) Lipschitz nonlinear systems is investigated. A simple systematic and useful synthesis method is proposed based on the use of the differential mean value theorem (DMVT) and convex theory. The proposed design approach is based on the mean value theorem (MVT) to express the nonlinear error dynamics as a convex combination of known matrices with time varying coefficients as linear parameter varying (LPV) systems. Using the Lyapunov theory, stability conditions are obtained and expressed in terms of linear matrix inequalities (LMIs). The controller gains are then obtained by solving linear matrix inequalities. The effectiveness of the proposed approach for closed loop-field oriented control (CL-FOC) of permanent magnet synchronous machine (PMSM) drives is demonstrated through an illustrative simulation for the proof of these approaches. Furthermore, an extension for controller design with parameter uncertainties and perturbation performance is discussed.


Takagi-Sugeno (T-S) fuzzy systems sector nonlinearity nonlinear controller linear matrix inequality (LMI) approach differential mean value theorem (DMVT) field oriented control (FOC) linear parameter varying (LPV) 


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  1. 1.
    Ichalal D, Arioui H, Mammar S. Observer design for two-wheeled vehicle: a Takagi-Sugeno approach with unmeasurable premise variables. In: Proceedings of the 19th Mediterranean Conference on Control and Automation. Corfu, Greece, 2011, 1–6Google Scholar
  2. 2.
    Pertew A M, Marquez H J, Zhao Q. H1 observer design for Lipschitz nonlinear systems. IEEE Transactions on Automatic Control, 2006, 51(7): 1211–1216CrossRefMathSciNetGoogle Scholar
  3. 3.
    Luenberger D G. An introduction to observers. IEEE Transactions on Automatic Control, 1971, 16(6): 596–602CrossRefMathSciNetGoogle Scholar
  4. 4.
    Rajamani R. Observers for Lipschitz nonlinear systems. IEEE Transactions on Automatic Control, 1998, 43(3): 397–401zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bara G I, Daafouz J, Kratz F, Ragot J. Parameter dependent state observer design for affine LPV systems. International Journal of Control, 2001, 74(16): 1601–1611zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Zemouche A, Boutayeb M, Bara G I. Observer design for nonlinear systems: an approach based on the differential mean value theorem. In: Proceedings of the 44th IEEE Conference on Decision and Control. Seville, Spain, 2005, 6353–6358Google Scholar
  7. 7.
    Phanomchoeng G, Rajamani R, Piyabongkarn D. Nonlinear observer for bounded Jacobian systems, with applications to automotive slip angle estimation. IEEE Transactions on Automatic Control, 2011, 56(5): 1163–1170CrossRefMathSciNetGoogle Scholar
  8. 8.
    Tanaka K, Wang H O. Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. New York: Wiley-Interscience, 2001CrossRefGoogle Scholar
  9. 9.
    Baranyi P. TP model transformation as a way to LMI-based controller design. IEEE Transactions on Industrial Electronics, 2004, 51(2): 387–400CrossRefGoogle Scholar
  10. 10.
    Merzoug M, Naceri F. Comparison of field-oriented control and direct torque control for permanent magnet synchronous motor (PMSM). International Journal of Electrical, Computer, Energetic, Electronics and Communication Engineering, 2008, 2(9): 107–112Google Scholar
  11. 11.
    Molavi R, Khaburi D A. Optimal control strategies for speed control of permanent-magnet synchronous motor drives. Proceedings of World Academy of Science Engineering and Technology, 2008, 44: 428Google Scholar
  12. 12.
    Štulrajter M, Hrabovcov¨¢ V, Franko M. Permanent magnets synchronous motor control theory. Journal of Electrical Engineering, 2007, 58(2): 79–84Google Scholar
  13. 13.
    Novotny D, Lipo T. Vector Control and Dynamics of AC Drives. New York: Oxford University Press, 1996Google Scholar
  14. 14.
    Solsona J, Valla M I, Muravchik C. Nonlinear control of a permanent magnet synchronous motor with disturbance torque estimation. IEEE Transactions on Energy Conversion, 2000, 15(2): 163–168CrossRefGoogle Scholar
  15. 15.
    Shi J L, Liu T H, Chang Y C. Optimal controller design of a sensorless PMSM control system. Industrial Electronics Society, 2005. IECON 2005. 31st Annual Conference of IEEE, 2005.Google Scholar
  16. 16.
    Zemouche A, Boutayeb M, Bara G I. Observers for a class of Lipschitz systems with extension to H1 performance analysis. Systems & Control Letters, 2008, 57(1): 18–27zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Sahoo P K, Riedel T. Mean Value Theorems and Functional Equations. New Jersey:World Scientific Publishing Company, 1999Google Scholar
  18. 18.
    Bergsten P, Palm R, Driankov D. Observers for Takagi-Sugeno fuzzy systems. IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics, 2002, 32(1): 114–121CrossRefGoogle Scholar
  19. 19.
    Raghavan S, Hedrick J K. Observer design for a class of nonlinear systems. International Journal of Control, 1994, 59(2): 515–528zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Rahman M A, Vilathgamuwa D M, Uddin M N, Tseng K J. Nonlinear control of interior permanent-magnet synchronous motor. IEEE Transactions on Industry Applications, 2003, 39(2): 408–416CrossRefGoogle Scholar
  21. 21.
    Ren H, Liu D. Nonlinear feedback control of chaos in permanent magnet synchronous motor. IEEE Transactions on Circuits and Systems II: Express Briefs, 2006, 53(1): 45–50CrossRefGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Abrar Allag
    • 1
    Email author
  • Abdelhamid Benakcha
    • 1
  • Meriem Allag
    • 1
  • Ismail Zein
    • 2
  • Mohamed Yacine Ayad
    • 2
  1. 1.LGEB Laboratory, Department of Electrical EngineeringUniversity of BiskraBiskraAlgeria
  2. 2.R&D, Industrial Hybrid Vehicle ApplicationsBelfortFrance

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