Fractional Modelling and Identification of a Thermal Process

Abstract

Heat transfer problems are subject to diffusion phenomenon, and can be modelled with the help of fractional systems. The simulation of these particular systems is based on a fractional integrator, where the non-integer behaviour occurs only within a limited spectral band. A first model, based on the use of one fractional integrator, has been already defined and tested on a thermal pilot. Starting from frequency considerations derived from the analysis of a diffusion problem, a more general approximation of the fractional system is proposed here. The new model is based on the use of two fractional integrator operators. This makes it possible to define a state-space model for simulation of transients, and to use an output-error technique in order to estimate the parameters of the model. Experimental results obtained for a thermal process illustrate the improvements obtained using the proposed model.

Keywords

Fractional systems non-integer systems fractional operator modelling estimation output-error identification heat transfer diffusive interfaces

Get full access to this article

View all access options for this article.

References

Battaglia, J.L. , 2002, "Méthodes d'identification de modèles à dérivées d'ordres non entiers et de réduction modale. Application à la résolution de problèmes thermiques inverses dans des systèmes industriel," HDR Thesis (in French), Université de Bordeaux I, Talence , France.

Benchellal, A. , Bachir, S. , Poinot, T. , and Trigeassou, J.C. , 2004, "Identification of a non-integer model of induction machines," in Proceedings of the 1st IFAC Workshop on Fractional Differentiation and its Applications, Bordeaux, France, July 19-21, pp. 400-407.

Canat, S. and Faucher, J. , 2004, "Modeling and simulation of induction machine with fractional derivative," in Proceedings of the 1st IFAC Workshop on Fractional Differentiation and its Applications, Bordeaux , France, July 19-21, pp. 393-399.

Cois, O. , 2002, "Systèmes linéaires non entiers et identification par modèle non entier: Application en thermique," Ph.D. Thesis (in French), Université de Bordeaux I, Talence, France.

Khaorapapong, T. , 2001, "Modélisation d'ordre non entier des effets de fréquence dans les barres rotoriques d'une machine asynchrone," Ph.D. Thesis (in French), INP de Toulouse, Toulouse, France.

Lin, J. , 2001, "Modélisation et identification de systèmes d'ordre non entier," Ph.D. Thesis (in French), Université de Poitiers , Poitiers, France.

Lin, J. , Poinot, T. , Trigeassou, J.C. , and Ouvrard, R. , 2000, "Parameter estimation of fractional systems: Application to the modeling of a lead-acid battery," in Proceedings of the IFAC Symposium on System Identification, Santa Barbara, CA, June 21-23.

Ljung, L. , 1987, System Identification - Theory for the User, Prentice Hall, Englewood Cliffs, NJ.

Marquardt, D.W. , 1963, "An algorithm for least-squares estimation of non-linear parameters," Journal of the Society for Industrial and Applied Mathematics 11(2), 431-441.

10.

Montseny, G. , 1998, "Diffusive representation of pseudo-differential time-operators ," ESAIM: Proceedings 5, 159- 175.

11.

Oustaloup, A. , 1995, La Dérivation non Entière: Théorie, Synthèse et Applications , (in French) Hermès, Paris.

12.

Richalet, J. , Rault, A. , and Pouliquen, R. , 1971, Identification des Processus par la Méthode du Modèle, (in French) Gordon and Breach, London, UK.

13.

Riu, D. and Retière, N. , 2004, "Implicit half-order systems utilisation for diffusion phenomenon modelling," in Proceedings of the 1st IFAC Workshop on Fractional Differentiation and its Applications, Bordeaux, France, July 19-21, pp. 387-392.

14.

Trigeassou, J.C. , Poinot, T. , Lin, J. , Oustaloup, A. , and Levron, F. , 1999, "Modeling and identification of a non integer order system ," in Proceedings of the European Control Conference , Karlsruhe, Germany, August 31-September 3.