Bifurcation surfaces and multi-stability analysis of state feedback control of PMSM

  • Wahid SouhailEmail author
  • Hedi Khammari
  • Mohamed Faouzi Mimouni


The paper presents a thorough bifurcation analysis of detailed permanent magnet machine with feedback drive, showing the effect of different control and system parameters on the bifurcation and the associated system stability. This study addresses the nonlinear behavior of the PMSM under parametric variation. The analysis of the phase space and parametric singularities are developed in order to identify some specific properties of the PMSM either uncontrolled or under state feedback control. A bifurcation structure of the uncontrolled PMSM in both motor and generator operating modes is identified in the speed-torque chart. The analysis of the complex dynamics of state feedback control of the PMSM permitted to define analytically the bifurcation surfaces of saddle-node (Limit point) and Hopf bifurcations. Seeking broader areas of parametric singularities led to establish the analytical existence conditions of bifurcation manifolds and particularly the bifurcation surfaces. The appearance of intermittent chaotic areas interspersed with cascading bifurcations by doubling the period of limit cycle has been observed in the controlled PMSM. The analysis of the attraction basins in the phase plane, and the synchronization regions in the parametric plane with constant initialization, pave the way to further characterization of the PMSM dynamics with different types of control methods.


PMSM Bifurcation surface Hopf Saddle-node Neutral-saddle Bogdanov-Takens 


  1. 1.
    Harb Ahmad M, Zaher Ashraf A (2004) Nonlinear control of permanent magnet stepper motors. Commun Nonlinear Sci Numer Simul 9:443–458MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chu Y-D, Zhang J-G, Li X-F, Chang Y-X, Luo G-W (2008) Chaos and chaos synchronization for a non-autonomous rational machine systems. Nonlinear Anal Real World Appl 9:1378–1393MathSciNetCrossRefGoogle Scholar
  3. 3.
    Diyi C, Peng S, Xiaoyi M (2012) Control and synchronization of chaos in an induction motor system. Int J Innov Comput Inf Control 8(10(B)):7237–7248Google Scholar
  4. 4.
    Zhong L, Jin Bae P, Young Hoon J, Bo Z, Guanrong C (2002) Bifurcations and chaos in permanent-magnet synchronous motor. In: IEEE transactions on circuits and systems-1: fundamental Theory and Application, Vol 49, No 3,Google Scholar
  5. 5.
    Mohamed Z, Ahmed O, Nejib S (2009) Controlling chaos in permanent magnet synchronous motor. Chaos Solitions Fractals 41:1266–1276CrossRefGoogle Scholar
  6. 6.
    Du Qu W, Xiao Shu L, Bing Hong W, Jin Qing F (2007) Robust adaptive dynamic surface control of chaos in permanent magnet synchronous motor. Phys Lett A 363:71–77 Science DirectGoogle Scholar
  7. 7.
    Li C-L, Yu S-M, Luo X-S (2012) Fractional- order permanent magnet magnet synchronous motor and its adaptive chaotic control. Chin Phys B 21(10):10056Google Scholar
  8. 8.
    Ali HN, Ahmed MH, Char-Ming C, (1996) Bifurcations in a power system model. Int J Bifurc Chaos 6:497–512Google Scholar
  9. 9.
    Charles KT, Raymond GG (1975) Stepping motor failure model. IEEE Trans Ind Electron 3:37585Google Scholar
  10. 10.
    Moosavi SS, Djerdir A, Amirat YA, Khaburi DA (2015) Demagnetization fault diagnosis in permanent magnet synchronous motors: a review of the state-of-the-art. J Magn Magn Mater 391:203212Google Scholar
  11. 11.
    Coria LN, Starkov KE (2009) Bounding a domain containing all compact invariant sets of the permanent magnet motor system. Commun Nonlinear Sci Numer Simul 14(11):3879–3888MathSciNetCrossRefGoogle Scholar
  12. 12.
    QI D-l, SONG Y-z (2006) Passive control of Permanent Magnet Synchronous Motor chaotic system based on state observer. J Zhejiang Univ SCIENCE A 7(12):1979–1983CrossRefGoogle Scholar
  13. 13.
    Viktor A, Michael S (2006) On multi-parametric bifurcations in a scalar piecewise-linear map. Nonlinearity 19(3):531552MathSciNetGoogle Scholar
  14. 14.
    Michael S (2000) Critical bifurcation surfaces of 3D discrete dynamics. Discret Dyn Nat Soc 4((C)):333–343Google Scholar
  15. 15.
    Henk B, Carles S, Renato V (2008) Hopf saddle-node bifurcation for fixed points of 3D- diffeomorphisms: anazlysis of resonance bubble. Physica D 273:1773–1799Google Scholar
  16. 16.
    Dirk S, Thilo G, Ralf S, Ulrike F (2007) Computation and Visualization of Bifurcation Surfaces. Int J Bifurc Chaos 18(8):2191–2206MathSciNetGoogle Scholar
  17. 17.
    Vaithianathan V, Heinz S, John Z (1995) Local bifurcation and feasibility regions in differential-algebraic systems. IEEE Trans Autom Ccntrol 40(12):1992–2013MathSciNetCrossRefGoogle Scholar
  18. 18.
    Thilo G, Ulrike F (2004) Analytical search for bifurcation surfaces in parameter space. Physica D Nonlinear Phenom 195(3–4):292302MathSciNetGoogle Scholar
  19. 19.
    Lanchares V, Inarrea M, Salas JP, Sierra JD (1995) Surfaces of bifurcation in triparametric quadratic Hamiltonian. Phys Rev E 52(5):5540MathSciNetCrossRefGoogle Scholar
  20. 20.
    Fuchs EF, Masoum MAS (2011) Block diagrams of electromechanical systems. Power Conversion of Renewable Energy Systems, Springer Science. CrossRefGoogle Scholar
  21. 21.
    Azeddine K (2000) Etude dune commande non-linaire adaptative dune machine synchrone aimants permanents. Thse, NovembreGoogle Scholar
  22. 22.
    Zhang K, Li J, Ouyang M, Gu J, Ma Y (2011) Electric braking performance analysis of pmsm for electric vehicle applications. In: International conference on electronic and mechanical engineering and information technology (EMEIT), vol 5, pp 2596 2599Google Scholar
  23. 23.
    Guangzhao L, Zhe C, Yantao D, Manfeng D, W eiguo L (2009) Research on braking of battery-supplied interior permanent magnet motor driving system. Vehicle power and propulsion conference, VPPC 09. IEEE, pp 270–274,Google Scholar
  24. 24.
    Qing-Chang Z (2010) Four-quadrant operation of AC machines powered by inverters that mimic synchronous generators. In: 5th international conference on power electronics, machines and drivesGoogle Scholar
  25. 25.
    Bezruchko BP, Prokhorov MD, Seleznev YP (2003) Oscillation types, multistability, and basins of attractors in symmetrically coupled period-doubling systems. Chaos Solitons Fractals 15:695–711Google Scholar
  26. 26.
    Harb AM, Widyan MS (2002) Controlling chaos and bifurcation of subsynchronous resonance in power system. Nonlinear Anal Model Control 7(2):15–36Google Scholar
  27. 27.
    Subhadeep C, Eric K, Asok R, Jeffrey M (2013) Detection and estimation of demagnetization faults in permanent magnet synchronous motors. Electr Power Syst Res 96:225236Google Scholar
  28. 28.
    Peter ME (2015) Reflecting saturation in the equivalent circuit of a three-phase induction motor: a numerical and comparative detailing. Int J Eng Technol 9,Google Scholar
  29. 29.
    Govaerts W (2000) Numerical bifurcation analysis for ODEs. J Comput Appl Math 125(1–2):57–68MathSciNetCrossRefGoogle Scholar
  30. 30.
    Gian-Italo B, Michael K (2003) Multistability and path dependence in a dynamic brand competition model. Chaos Solitons Fractals 18:561–576Google Scholar
  31. 31.
    Aria A, Rasool S (2006) Chaotic motions and fractal basin boundaries in spring-pendulum system. Nonlinear Anal Real World Appl 7:81–95MathSciNetCrossRefGoogle Scholar
  32. 32.
    Botella-Soler V, Oteo JA, Ros J (2012) Coexistence of periods in a bifurcation. Chaos Solitons Fractals 45:681–686MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sharma PR, Shrimali MD, Prasad A, Kuznetsov NV, Leonov GA (2015) Control of multistability in hidden attractors. Eur Phys J Spec Top 224:1485–1491CrossRefGoogle Scholar
  34. 34.
    Sura HCM, Sergio RL, Ricardo LV (2003) Boundary crises, fractal basin boundaries, and electric power collapses. Chaos Solitons Fractals 15:417–424CrossRefGoogle Scholar
  35. 35.
    Sprott JC, Xiong A (2015) Classifying and quantifying basins of attraction. Chaos 25:1054–1500MathSciNetCrossRefGoogle Scholar
  36. 36.
    Fassoni AC, Takahashi LT, dos Santos LJ (2014) Basins of attraction of the classic model of competition between two populations. Ecol Complex 18:39–48CrossRefGoogle Scholar
  37. 37.
    Chunbiao L, Julien Clinton S (2014) Multistability in the Lorenz system: a broken butterfly. Int J Bifurc Chaos 24:1450131MathSciNetCrossRefGoogle Scholar
  38. 38.
    Belardinelli P, Lenci S (2016) An efficient parallel implementation of Cell mapping methods for MDOF systems. Nonlinear Dyn 86:22792290MathSciNetCrossRefGoogle Scholar
  39. 39.
    Belardinelli P, Lenci S (2016) A first parallel programming approach in basins of attraction computation. Int J Non-linear Mech 80:76–81Google Scholar
  40. 40.
    Qiang M, Tao Z, Jing-feng H, Jing-yan S, Jun-wei H (2010) Dynamic modeling of a 6-degree-of-freedom Stewart platform driven by a permanent magnet synchronous motor. Front Inf Technol Electron Eng 11(10):751–761Google Scholar
  41. 41.
    Qi-huai C, Qing-feng W, Tao W (2015) Optimization design of an interior permanent magnet synchronous machine for a hybrid hydraulic excavator. Front Inf Technol Electron Eng 16(11):957–968Google Scholar
  42. 42.
    Gunpyo M, Han HC (2013) Adaptive sliding mode control of a chaotic nonsmooth-air-gap permanent magnet synchronous motor with uncertainties. Nonlinear Dyn 74:571–580MathSciNetCrossRefGoogle Scholar
  43. 43.
    Zhujun J, Yu C, Yu C, Guanrong C (2004) Complex dynamics in permanent-magnet synchronous motor model. Chaos Solitons Fractals 22:831–848MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Wahid Souhail
    • 1
    Email author
  • Hedi Khammari
    • 2
  • Mohamed Faouzi Mimouni
    • 1
  1. 1.Laboratory of Automation, Electrical System and Environment (LAESE)National Engineering School of MonastirMonastirTunisia
  2. 2.College of Computers and ITTaif UniversityTa’ifSaudi Arabia

Personalised recommendations