Skip to main content
SpringerLink
Log in

Characterization of hysteresis processes

  • Original Article
  • Published:
Mathematics of Control, Signals, and Systems Aims and scope Submit manuscript

Abstract

Hysteresis is a nonlinear phenomenon exhibited by systems stemming from various science and engineering areas. To detect experimentally the presence of hysteresis in a system, its graph output versus input is plotted for different frequencies of the input. For hysteresis systems, these graphs converge to a limit set when frequency goes to zero. Moreover, this limit approaches asymptotically a periodic orbit. The relevance of hysteresis in applications makes it important to characterize it mathematically, which is the purpose of this paper. The systems that are considered are operators that map an input signal and initial condition to an output signal, all belonging to specified sets. The main result of this paper is a criterion for the mathematical characterization of hysteresis. The tools introduced in this paper are illustrated by means of a case study.

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

Similar content being viewed by others

Notes

  1. \(L^1_{loc}(\mathbb R _+,\mathbb R ^n)\) is the space of locally integrable functions \(\mathbb R _+ \rightarrow \mathbb R ^n\).

  2. That is such that there exist \(t_1,t_2 \in \mathbb R _+\) such that \(u(t_1) \ne u(t_2)\).

  3. A null set is a set with measure zero.

  4. That is \(\Xi =\emptyset \).

  5. If \({\lim \nolimits _{w \downarrow 0}\bar{g}_+(w)=a_+\ne 0}\) and \({\lim \nolimits _{w \uparrow 0}\bar{g}_-(w)=-a_- \ne 0}\), the constants \(a_+\) and \(a_-\) are incorporated into the matrices \(A_+\) and \(A_-\) respectively.

  6. A matrix is Hurwitz if and only if all its eigenvalues have negative real parts.

  7. Note that the initial condition \(\zeta _\infty (0)\) may be different from \(x_0\).

References

  1. Adams RA, Fournier JJF (2003) Sobolev spaces. Elsevier, Amsterdam

    MATH  Google Scholar 

  2. Brokate M, Sprekels J (1996) Hysteresis and phase transitions. Springer, New York

    Book  MATH  Google Scholar 

  3. Bliman, P-A, Sorine, M (1995) Easy-to-use realistic dry friction models for automatic control. In: Proceedings of the 3rd European control conference, Roma, Italy, pp 3788–3794

  4. Dong R, Tan Y, Chen H, Xie Y (2008) A neural networks based model for rate-dependent hysteresis for piezoceramic actuators. Sens Actuators A 143:370–376

    Article  Google Scholar 

  5. Edgar GA (1990) Measure, topology and fractal geometry. Springer, New York

    Book  MATH  Google Scholar 

  6. Enachescu C et al (2006) Rate-dependent light-induced thermal hysteresis of [Fe(PM-BiA)2(NCS)2] spin transition complex. J Appl Phys 99:08J504

    Google Scholar 

  7. Filippov AF (1988) Differential equations with discontinuous righthand sides. Kluwer Academic Publishers, Dordrecht

  8. Fuzi J, Ivanyi A (2001) Features of two rate-dependent hysteresis models. Physica B 306:137–142

    Article  Google Scholar 

  9. Ghost I (2007) Ask the experts. IEEE Control Syst Mag 27(5):16–17

    Article  Google Scholar 

  10. Ikhouane F, Rodellar J (2005) On the hysteretic Bouc–Wen model. Part I: forced limit cycle characterization. Nonlinear Dyn 42(1):63–78

    Article  MathSciNet  MATH  Google Scholar 

  11. Ikhouane F, Rodellar J (2007) Systems with hysteresis analysis, identification and control using the Bouc–Wen model. Wiley, Chichester

    Book  MATH  Google Scholar 

  12. Krasnosel’skii MA, Pokrovskii AV (1989) Systems with hysteresis. Springer, Berlin

    Book  MATH  Google Scholar 

  13. Logemann H, Ryan EP, Shvartsmann I (2008) A class of differential-delay systems with hysteresis: asymptotic behaviour of solutions. Nonlinear Anal 69:363–391

    Article  MathSciNet  MATH  Google Scholar 

  14. Mayergoyz I (2003) Mathematical models of hysteresis. Elsevier Series in Electromagnetism, New York

  15. Macki JW, Nistri P, Zecca P (1993) Mathematical models for hysteresis. SIAM Rev 35(1):94–123

    Article  MathSciNet  MATH  Google Scholar 

  16. Oh J, Bernstein DS (2005) Semilinear Duhem model for rate-independent and rate-dependent hysteresis. IEEE Trans Autom Control 50(5):631–645

    Article  MathSciNet  Google Scholar 

  17. Rudin W (1987) Real and complex analysis, 3rd edn. McGraw-Hill Series in Higher Mathematics, USA

  18. Varberg DE (1965) On absolutely continuous functions. Am Math Mon 72(8):831–841

    Article  MathSciNet  MATH  Google Scholar 

  19. Visintin A (1994) Differential models of hysteresis. Springer, Berlin

    Book  MATH  Google Scholar 

  20. Wiggins S (1990) Introduction to applied nonlinear dynamical systems and chaos. Springer, New York

    Book  MATH  Google Scholar 

Download references

Acknowledgments

Supported by Grant DPI2011-25822 of the Spanish Ministry of Economy and Competitiveness.

Author information

Authors and Affiliations

  1. Departament de Matemàtica Aplicada III, Escola Universitària d’Enginyeria Tècnica Industrial de Barcelona, Universitat Politècnica de Catalunya, Comte d’Urgell, 187, 08036 , Barcelona, Spain

    Fayçal Ikhouane

Authors
  1. Fayçal Ikhouane
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fayçal Ikhouane.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ikhouane, F. Characterization of hysteresis processes. Math. Control Signals Syst. 25, 291–310 (2013). https://doi.org/10.1007/s00498-012-0099-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00498-012-0099-6

Keywords

Search

Navigation