Skip to main content
SpringerLink
Log in

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

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  1. Harb Ahmad M, Zaher Ashraf A (2004) Nonlinear control of permanent magnet stepper motors. Commun Nonlinear Sci Numer Simul 9:443–458

    Article  MathSciNet  Google Scholar 

  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 9:1378–1393

    Article  MathSciNet  Google Scholar 

  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)):7237–7248

  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,

  5. Mohamed Z, Ahmed O, Nejib S (2009) Controlling chaos in permanent magnet synchronous motor. Chaos Solitions Fractals 41:1266–1276

    Article  Google Scholar 

  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 Direct

  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):10056

    Google Scholar 

  8. Ali HN, Ahmed MH, Char-Ming C, (1996) Bifurcations in a power system model. Int J Bifurc Chaos 6:497–512

  9. Charles KT, Raymond GG (1975) Stepping motor failure model. IEEE Trans Ind Electron 3:37585

  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 391:203212

  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):3879–3888

    Article  MathSciNet  Google Scholar 

  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):1979–1983

    Article  Google Scholar 

  13. Viktor A, Michael S (2006) On multi-parametric bifurcations in a scalar piecewise-linear map. Nonlinearity 19(3):531552

    MathSciNet  Google Scholar 

  14. Michael S (2000) Critical bifurcation surfaces of 3D discrete dynamics. Discret Dyn Nat Soc 4((C)):333–343

  15. Henk B, Carles S, Renato V (2008) Hopf saddle-node bifurcation for fixed points of 3D- diffeomorphisms: anazlysis of resonance bubble. Physica D 273:1773–1799

    Google Scholar 

  16. Dirk S, Thilo G, Ralf S, Ulrike F (2007) Computation and Visualization of Bifurcation Surfaces. Int J Bifurc Chaos 18(8):2191–2206

    MathSciNet  Google Scholar 

  17. Vaithianathan V, Heinz S, John Z (1995) Local bifurcation and feasibility regions in differential-algebraic systems. IEEE Trans Autom Ccntrol 40(12):1992–2013

    Article  MathSciNet  Google Scholar 

  18. Thilo G, Ulrike F (2004) Analytical search for bifurcation surfaces in parameter space. Physica D Nonlinear Phenom 195(3–4):292302

    MathSciNet  Google Scholar 

  19. Lanchares V, Inarrea M, Salas JP, Sierra JD (1995) Surfaces of bifurcation in triparametric quadratic Hamiltonian. Phys Rev E 52(5):5540

    Article  MathSciNet  Google Scholar 

  20. Fuchs EF, Masoum MAS (2011) Block diagrams of electromechanical systems. Power Conversion of Renewable Energy Systems, Springer Science. https://doi.org/10.1007/978-1-4419-7979-7_2

    Book  Google Scholar 

  21. Azeddine K (2000) Etude dune commande non-linaire adaptative dune machine synchrone aimants permanents. Thse, Novembre

    Google Scholar 

  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 2599

  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,

  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 drives

  25. Bezruchko BP, Prokhorov MD, Seleznev YP (2003) Oscillation types, multistability, and basins of attractors in symmetrically coupled period-doubling systems. Chaos Solitons Fractals 15:695–711

  26. Harb AM, Widyan MS (2002) Controlling chaos and bifurcation of subsynchronous resonance in power system. Nonlinear Anal Model Control 7(2):15–36

  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 96:225236

    Google Scholar 

  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,

  29. Govaerts W (2000) Numerical bifurcation analysis for ODEs. J Comput Appl Math 125(1–2):57–68

    Article  MathSciNet  Google Scholar 

  30. Gian-Italo B, Michael K (2003) Multistability and path dependence in a dynamic brand competition model. Chaos Solitons Fractals 18:561–576

  31. Aria A, Rasool S (2006) Chaotic motions and fractal basin boundaries in spring-pendulum system. Nonlinear Anal Real World Appl 7:81–95

    Article  MathSciNet  Google Scholar 

  32. Botella-Soler V, Oteo JA, Ros J (2012) Coexistence of periods in a bifurcation. Chaos Solitons Fractals 45:681–686

    Article  MathSciNet  Google Scholar 

  33. Sharma PR, Shrimali MD, Prasad A, Kuznetsov NV, Leonov GA (2015) Control of multistability in hidden attractors. Eur Phys J Spec Top 224:1485–1491

    Article  Google Scholar 

  34. Sura HCM, Sergio RL, Ricardo LV (2003) Boundary crises, fractal basin boundaries, and electric power collapses. Chaos Solitons Fractals 15:417–424

    Article  Google Scholar 

  35. Sprott JC, Xiong A (2015) Classifying and quantifying basins of attraction. Chaos 25:1054–1500

    Article  MathSciNet  Google Scholar 

  36. Fassoni AC, Takahashi LT, dos Santos LJ (2014) Basins of attraction of the classic model of competition between two populations. Ecol Complex 18:39–48

    Article  Google Scholar 

  37. Chunbiao L, Julien Clinton S (2014) Multistability in the Lorenz system: a broken butterfly. Int J Bifurc Chaos 24:1450131

    Article  MathSciNet  Google Scholar 

  38. Belardinelli P, Lenci S (2016) An efficient parallel implementation of Cell mapping methods for MDOF systems. Nonlinear Dyn 86:22792290

    Article  MathSciNet  Google Scholar 

  39. Belardinelli P, Lenci S (2016) A first parallel programming approach in basins of attraction computation. Int J Non-linear Mech 80:76–81

  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):751–761

    Google Scholar 

  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):957–968

  42. Gunpyo M, Han HC (2013) Adaptive sliding mode control of a chaotic nonsmooth-air-gap permanent magnet synchronous motor with uncertainties. Nonlinear Dyn 74:571–580

    Article  MathSciNet  Google Scholar 

  43. Zhujun J, Yu C, Yu C, Guanrong C (2004) Complex dynamics in permanent-magnet synchronous motor model. Chaos Solitons Fractals 22:831–848

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Automation, Electrical System and Environment (LAESE), National Engineering School of Monastir, 5019, Monastir, Tunisia

    Wahid Souhail & Mohamed Faouzi Mimouni

  2. College of Computers and IT, Taif University, Ta’if, Saudi Arabia

    Hedi Khammari

Authors
  1. Wahid Souhail
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hedi Khammari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohamed Faouzi Mimouni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wahid Souhail.

Additional information

This paper was not presented at any IFAC meeting. Corresponding author Souhail Wahid. Tel. +21696812290. Fax +XXXIX-VI-mmmxxv.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Souhail, W., Khammari, H. & Mimouni, M.F. Bifurcation surfaces and multi-stability analysis of state feedback control of PMSM. Int. J. Dynam. Control 7, 276–294 (2019). https://doi.org/10.1007/s40435-018-0443-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-018-0443-x

Keywords

Search

Navigation