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Thermal system identification using fractional models for high temperature levels around different operating points

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Abstract

This work aims at the preparation of an experiment for the thermal modeling of an ARMCO iron sample (iron of the American Rolling Mill COmpany) for small temperature variations around different operating points. Fractional models have proven their efficacy for modeling thermal diffusion around the ambient temperature and for small variations. Due to their compactness, as compared to rational models and to finite element models, they are suitable for modeling such diffusive phenomena. However, for large temperature variations, thermal characteristics such as thermal conductivity and specific heat vary along with the temperature. In this context, the thermal diffusion obeys a nonlinear partial differential equation and cannot be modeled by a single linear model. In this paper, thermal diffusion of the iron sample is modeled around different operating points for temperatures ranging from 400 to 1070 K, which is above the Curie point (In physics and materials science, the Curie temperature (T C), or Curie point, is the temperature at which a ferromagnetic or a ferrimagnetic material becomes paramagnetic.) showing that for a large range of temperature variations, a nonlinear model is required. Identification and validation data are generated by finite element methods using COMSOL Software.

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References

  1. Adolfsson, K.: Nonlinear fractional order viscoelasticity at large strains. Nonlinear Dyn. 38, 233–246 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agrawal, O.: Application of fractional derivatives in thermal analysis of disk brakes. Nonlinear Dyn. 38, 191–206 (2004)

    Article  MATH  Google Scholar 

  3. Aoun, M., Malti, R., Levron, F., Oustaloup, A.: Synthesis of fractional Laguerre basis for system approximation. Automatica 43(9), 1640–1648 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Battaglia, J.L.: Heat Transfer in Materials Forming Processes. Wiley, Chippenham (2008)

    Google Scholar 

  5. Battaglia, J.L., Kusiak, A.: Estimation of heat fluxes during high-speed driling. Int. J. Adv. Manuf. Technol. 39, 750–758 (2005)

    Article  Google Scholar 

  6. Battaglia, J.L., Le Lay, L., Batsale, J.C., Oustaloup, A., Cois, O.: Heat flux estimation through inverted non-integer identification models. Int. J. Therm. Sci. 39(3), 374–389 (2000)

    Article  Google Scholar 

  7. Benchellal, A., Bachir, S., Poinot, T., Trigeassou, J.C.: Identification of a non-integer model of induction machines. In: 1st IFAC Workshop on Fractional Differentiation and its Applications (FDA), Bordeaux, France (2004)

    Google Scholar 

  8. Cois, O., Oustaloup, A., Battaglia, E., Battaglia, J.L.: Non-integer model from modal decomposition for time domain system identification. In: 12th IFAC Symposium on System Identification, SYSID, Santa Barbara, CA (2000)

    Google Scholar 

  9. Cois, O., Oustaloup, A., Poinot, T., Battaglia, J.L.: Fractional state variable filter for system identification by fractional model. In: IEEE 6th European Control Conference (ECC’2001), Porto, Portugal (2001)

    Google Scholar 

  10. Cugnet, M., Sabatier, J., Laruelle, S., Grugeon, S., Sahut, B., Oustaloup, A., Tarascon, J.M.: On lead-acid battery resistance and cranking capability estimation. IEEE Trans. Ind. Electron. 57(3), 9.9–917 (2010)

    Article  Google Scholar 

  11. Dugowson, S.: Les différentielles métaphysiques: histoire et philosophie de la généralisation de l’ordre de dérivation. PhD thesis, Université Paris XIII (1994)

  12. Gabano, J., Poinot, T.: Estimation of thermal parameters using fractional modelling. Signal Process. 91(4), 938–948 (2011)

    Article  MATH  Google Scholar 

  13. Gabano, J.D., Poinot, T.: Fractional modelling and identification of thermal systems. Signal Process. 91(3), 531–541 (2011)

    Article  MATH  Google Scholar 

  14. Grünwald, A.: Ueber begrenzte Derivationen und deren Anwendung. Z. Angew. Math. Phys. 12, 441–480 (1867)

    Google Scholar 

  15. Heymans, N.: Fractional calculus description of non-linear viscoelastic behaviour of polymers. Nonlinear Dyn. 38, 221–231 (2004)

    Article  MATH  Google Scholar 

  16. Liouville, J.: Mémoire sur quelques questions de géométrie et de mécanique et sur un nouveau genre de calcul pour résoudre ces équations. J. Éc. Polytech. 13, 71–162 (1832)

    Google Scholar 

  17. Maachou, A., Malti, R., Melchior, P., Battaglia, J.L., Oustaloup, A., Hay, B.: Thermal identification using fractional linear models in high temperatures. In: The 4th IFAC Workshop on Fractional Differentiation and its Applications (FDA), Badajoz, Spain (2010)

    Google Scholar 

  18. Malti, R., Sabatier, J., Akçay, H.: Thermal modeling and identification of an aluminum rod using fractional calculus. In: The 15th IFAC Symposium on System Identification (SYSID), Saint Malo, France (2009)

    Google Scholar 

  19. Malti, R., Victor, S., Oustaloup, A., Garnier, H.: An optimal instrumental variable method for continuous-time fractional model identification. In: The 17th IFAC World Congress, Seoul, South Korea (2008)

    Google Scholar 

  20. Malti, R., Victor, S., Oustaloup, A.: Advances in system identification using fractional models. J. Comput. Nonlinear Dyn. 3(2), 021401 (2008)

    Article  Google Scholar 

  21. Malti, R., Moreau, X., Khemane, F., Oustaloup, A.: Stability and resonance conditions of elementary fractional transfer functions. Automatica 47(11), 2462–2467 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mathieu, B., Oustaloup, A., Levron, F.: Transfer function parameter estimation by interpolation in the frequency domain. In: EEC’95, Rome, Italy (1995)

    Google Scholar 

  23. Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley–Interscience, New York (1993)

    MATH  Google Scholar 

  24. Oldham, K., Spanier, J.: The remplacement of Fick’s laws by a formulation involving semi-differentiation. J. Electroanal. Chem. Interfacial Electrochem. 26, 331–341 (1970)

    Article  Google Scholar 

  25. Oldham, K., Spanier, J.: A general solution of the diffusive equation for semi-infinite geometries. J. Math. Anal. Appl. 39, 655–669 (1972)

    Article  MathSciNet  Google Scholar 

  26. Oldham, K., Spanier, J.: Diffusive transport to planar, cylindrical and spherical electrodes. J. Electroanal. Chem. Interfacial Electrochem. 41, 351–358 (1973)

    Article  Google Scholar 

  27. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1995)

    Google Scholar 

  28. Poinot, T., Trigeassou, J.C.: Identification of fractional systems using an output-error technique. Nonlinear Dyn. 38, 133–154 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Riemann, B.: Versuch einer allgemeinen Auffassung der Integration und Differentiation. In: Gesammelte Mathematische Werke und Wissenschaftlicher, pp. 331–344 (1876)

    Google Scholar 

  30. Rodrigues, S., Munichandraiah, N., Shukla, A.K.: A review of state of charge indication of batteries by means of A.C. impedance measurements. J. Power Sources 87, 12–20 (2000)

    Article  Google Scholar 

  31. Sabatier, J., Aoun, M., Oustaloup, A., Grégoire, G., Ragot, F., Roy, P.: Fractional system identification for lead acid battery sate charge estimation. Signal Process. 86(10), 2645–2657 (2006)

    Article  MATH  Google Scholar 

  32. Sommacal, L., Melchior, P., Cabelguen, J.M., Oustaloup, A., Ijspeert, J.: Fractional multi-models of the gastrocnemius frog muscle. J. Vib. Control 14(9–10), 1415–1430 (2008)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. IPB/ENSEIRB-MATMECA, IMS UMR 5218 CNRS, Université Bordeaux 1, 351 cours de la Libération, Bât A4, 33405, Talence Cedex, France

    Asma Maachou, Rachid Malti & Pierre Melchior

  2. ENSAM, TREFLE UMR 8508 CNRS, Esplanade des Arts et Metiers, Université de Bordeaux 1, 33405, Talence Cedex, France

    Jean-Luc Battaglia

  3. LNE, Centre Metrologie et Instrumentation, 29 avenue Roger Hennequin, 78197, Trappes Cedex, France

    Bruno Hay

Authors
  1. Asma Maachou
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  2. Rachid Malti
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  3. Pierre Melchior
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  4. Jean-Luc Battaglia
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  5. Bruno Hay
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Correspondence to Asma Maachou.

Additional information

A preliminary version of this paper was presented at the 4th IFAC Workshop on Fractional Differentiation and its Applications, Badajoz, Spain, 2010 [17]. Here is a revisited and enhanced version of this preliminary paper.

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Maachou, A., Malti, R., Melchior, P. et al. Thermal system identification using fractional models for high temperature levels around different operating points. Nonlinear Dyn 70, 941–950 (2012). https://doi.org/10.1007/s11071-012-0507-y

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  • DOI: https://doi.org/10.1007/s11071-012-0507-y

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