Skip to main content
SpringerLink
Log in

Study of the effects of structural uncertainties on a fractional system of the first kind – application in vibration isolation with the CRONE suspension

  • Original Paper
  • Published:
Signal, Image and Video Processing Aims and scope Submit manuscript

Abstract

In this paper, we deal with the effects of the uncertainties on a fractional system of the first kind, mainly on the frequency-domain and the time-domain responses. For the structural uncertainties, two main aspects are studied: the nonlinearities of the physical components used to realize the fractional system and the consideration of the previously neglected dynamics of the system. Both uncertainties are introduced for the hydropneumatic CRONE suspension, previously synthesized and realized without taking into consideration these uncertainties. So, the novel approach treated in this work is to find whether the uncertainties, which were previously neglected in the synthesis and the realization phases, alter the behaviour of the system or not. The results show that the fractional order system behaviour is not affected.

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

Similar content being viewed by others

References

  1. Dugowson, S.: Les différentielles métaphysiques : histoire et philosophie de la généralisation de l’ordre de dérivation - Ph.D. Thesis, Université Paris Nord (1994)

  2. Malti R., Moreau X., Khemane F., Oustaloup A.: Stability and resonance conditions of elementary fractional transfer functions Automatica. J. IFAC Int. Federation Automat. Control 47(11), 2462–2467 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Oldham K.B., Spanier J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  4. Miller K.S., Ross B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, New York (1993)

    MATH  Google Scholar 

  5. Dauphin-Tanguy G.: Les Bond-Graphs. Edition Hermès, Paris (2000)

    Google Scholar 

  6. Krishna B.T.: Studies on fractional order differentiators and integrators: a survey. Signal Process. 91, 386–426 (2011)

    Article  MATH  Google Scholar 

  7. Le Méhauté A., Nigmatullin R., et Nivanen L.: Flêche du temps et géométrie Fractale. Edition Hermès, Paris (1998)

    Google Scholar 

  8. Hartley T.T., Lorenzo C.F.: Dynamics and control of initialized fractional-order systems. Nonlinear Dyn. 29, 201–233 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lorenzo, C.F., Hartley, T.T.: Initialization of fractional differential equations: background and theory. In: Proceedings of the ASME 2007, DETC2007-34810, Las Vegas, Nevada (2007)

  10. Lorenzo, C.F., Hartley, T.T.: Initialization of fractional differential equations: theory and application. In: Proceedings of the ASME 2007, DETC2007-34814, Las Vegas, Nevada (2007)

  11. Oustaloup A.: La dérivation non entière : théorie, synthèse et Applications. Edition Hermès, Paris (1995)

    MATH  Google Scholar 

  12. Charef, A., Fergani, N.: PIλDμ Controller Tuning For Desired Closed-Loop Response Using Impulse Response. In: Workshop on Fractional Deviation and Applications, Badajoz, Spain, October (2010)

  13. Trigeassou, J.C., Poinot, T., Lin, J., Oustaloup, A., Levron, F.: Modeling and identification of a non integer order system. In: Proceedings of ECC’99, European Control Conference, Karlsruhe, Germany (1999)

  14. Serrier, P., Moreau, X., Oustaloup, A.: Synthesis of a limited-bandwidth fractional differentiator made in hydropneumatic technology. In: Proceedings of IDETC/CIE 2005: ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Long Beach, California, USA, September 24–28 (2005)

  15. Abi Zeid Daou R., Francis C., Moreau X.: Synthesis and implementation of non-integer integrators using RLC devices. Int. J. Electron. 96(12), 1207–1223 (2009)

    Article  Google Scholar 

  16. Serrier, P., Moreau, X., Oustaloup, A.: Volterra series based analysis of components nonlinearities in a limited-bandwidth fractional differentiator achieved in hydropneumatic technology. In: Proceedings of 2nd IFAC Workshop on Fractional Differentiation and its Applications, FDA 06, Porto, Portugal (2006)

  17. Abi Zeid Daou, R., Francis, C., Moreau, X.: Study of the CRONE suspension RLC components’ nonlinearities based on Volterra series. Part 1: background and theory. In: The 4th IFAC Workshop on Fractional Differentiation and its Applications, FDA10, Badajoz, Spain, October 18–20 (2010)

  18. Abi Zeid Daou, R., Moreau, X., Francis, C.: Study of the CRONE suspension RLC components’ nonlinearities based on Volterra series. Part 2: simulation results. In: The 4th IFAC Workshop on Fractional Differentiation and its Applications, FDA10, Badajoz, Spain, October 18–20 (2010)

  19. Nise N.: Control Systems Engineering. Wiley, New York (2010)

    Google Scholar 

  20. Abi Zeid Daou, R.: Etude de l’influence des incertitudes sur le comportement d’un système dynamique non entier de première espèce - Ph.D. Thesis, Université Bordeaux 1 (2011)

  21. Serrier, P.: Analyse de l’influence des non-linéarités dans l’approche CRONE: Application en isolation vibratoire. Ph.D. Thesis, Université Bordeaux 1, 30 Septembre (2008)

  22. Volterra V.: Theory of functionals and of integrals and integro-differential equations. Dover, New York (1959)

    Google Scholar 

  23. Wiener N.: Nonlinear Problems in Random Theory. Wiley, New York (1958)

    MATH  Google Scholar 

  24. Schetzen M.: The Volterra and Wiener Theories of Non-Linear Systems. Wiley-Intersciences, New York (1980)

    Google Scholar 

  25. Mirri D., Iuculano G., Filicori F., Pasini G., Vannini G., Gualtieri G.P.: A modified volterra series approach for nonlinear dynamic systems modeling. IEEE Trans. Circuits Syst. I Fundament. Theory Appl. 49(8), 1118–1128 (2002)

    Article  Google Scholar 

  26. Boyd S., Tang Y.S., Chua L.O.: Measuring Volterra Kernels. IEEE Trans. Circuits Syst. 30, 571–577 (1983)

    Article  MATH  Google Scholar 

  27. Evans, C., Rees, D., Jones, L., Weiss, M.: Probing signals for measuring nonlinear Volterra Kernels. In: Proceedings of instrumentation and measurement technology conference (IMTC’ 95), Boston, MA, 10–15, June (1995)

  28. Worden K., Manson G.: Random vibrations of a duffing oscillator using the Volterra series. J. Sound Vib. 217(4), 781–789 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rugh W.J.: Nonlinear System Theory the Volterra Wiener Approach Baltimore, MD. Johns Hopkins University Press, Baltimore (1981)

    Google Scholar 

  30. George, D.A.: Continuous nonlinear systems. Technical Report 355, Research Laboratory of Electronics, M. I. T. (1959)

Download references

Author information

Authors and Affiliations

  1. Laboratoire IMS, Department LAPS, University of Bordeaux I, 33405, Talence Cedex, Bordeaux, France

    Roy Abi Zeid Daou & Xavier Moreau

  2. Faculty of Public Health, Biomedical Technologies Department, Lebanese German University, Sahel Alma, P.O. Box: 206, Jounieh, Lebanon

    Roy Abi Zeid Daou

  3. Faculty of Engineering I, Lebanese University, Tripoli, Lebanon

    Clovis Francis

Authors
  1. Roy Abi Zeid Daou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xavier Moreau
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Clovis Francis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roy Abi Zeid Daou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abi Zeid Daou, R., Moreau, X. & Francis, C. Study of the effects of structural uncertainties on a fractional system of the first kind – application in vibration isolation with the CRONE suspension. SIViP 6, 463–478 (2012). https://doi.org/10.1007/s11760-012-0333-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11760-012-0333-1

Keywords

Search

Navigation