Review Article

Identification of Fractional Model by Least-Squares Method and Instrumental Variable

[+] Author and Article Information
Abir Khadhraoui

Laboratoire des systèmes Electriques (LSE),
Ecole Nationale d’Ingenieurs de Tunis,
Le Belvedere 1002
Tunis, Tunisia
e-mail: abbir_k2007@yahoo.fr

Khaled Jelassi

Laboratoire des systèmes Electriques (LSE),
Ecole Nationale d’Ingenieurs de Tunis,
Le Belvedere 1002
Tunis, Tunisia
e-mail: jelassi_2000@yahoo.fr

Jean-Claude Trigeassou

Laboratoire Intégration du Matériau
au Système (IMS-APS),
Université Bordeaux 1,
Bordeaus, France
e-mail: jean.claude.trigeassou@ims-bordeaux.fr

Pierre Melchior

Laboratoire Intégration du Matériau
au Système (IMS-APS),
Université Bordeaux 1,
Bordeaux, France
e-mail: pierre.melchior@laps.ims-bordeaux.fr

1Corresponding author.

Manuscript received November 27, 2014; final manuscript received February 20, 2015; published online April 8, 2015. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 10(5), 050801 (Sep 01, 2015) (10 pages) Paper No: CND-14-1301; doi: 10.1115/1.4029904 History: Received November 27, 2014; Revised February 20, 2015; Online April 08, 2015

This paper deals with fractional model identification using least-squares (LS) method and instrumental variable (IV) in a noisy output context. A new identification method, which extends LS techniques to fractional system to identify not only the parameters but also the unknown order, is presented. In order to eliminate the bias of identification results, IV method is chosen which permits unbiased parameter estimation. Monte Carlo simulation analyses are used to demonstrate the validity and the performance of the proposed fractional order system identification method.

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Hsia, T. C., 1977, System Identification: Least-Squares Methods, Lexington Books, Lexington, MA.
Strejc, V., 1980, “Least Squares Parameter Estimation,” Automatica, 16(5), pp. 535–550. [CrossRef]
Zhao, Z. Y., Sagara, S., and Tomizuka, M., 1992, “A New Bias-Compensating LS Method for Continuous System Identification in the Presence of Coloured Noise,” Int. J. Control, 56(6), pp. 1441–1452. [CrossRef]
Young, P. C., and Jakeman, A. J., 1980, “Refined Instrumental Variable Methods of Recursive Time-Series Analysis Part III. Extensions,” Int. J. Control, 31(4), pp. 741–764. [CrossRef]
Landau, I., 1976, “Unbiased Recursive Identification Using Model Reference Adaptive Techniques,” IEEE Trans. Autom. Control, 21(2), pp. 194–202. [CrossRef]
Ljung, L., 1977, “Analysis of Recursive Stochastic Algorithms,” IEEE Trans. Autom. Control, 22(4), pp. 551–575. [CrossRef]
Goodwin, G. C., and Payne, R. L., 1977, Dynamic System Identification: Experiment Design and Data Analysis, Academic Press, New York.
Stoica, P., and Soderstrom, T., 1982, “Bias Correction in Least-Squares Identification,” Int. J. Control, 35(3), pp. 449–457. [CrossRef]
Pearson, A. E., 1988, “Least Squares Parameter Identification of Nonlinear Differential I/O Models,” Proceedings of the 27th IEEE Conference on Decision and Control, Austin, TX, Dec. 7–9, pp. 1931–1835.
Ljung, L., 1987, System Identification: Theory for the User, Prentice Hall, Englewood Cliffs, NJ.
Van den Hof, P. M. J., 1989, “Criterion Based Equivalence for Equation Error Models,” IEEE Trans. Autom. Control, 34(2), pp. 191–193. [CrossRef]
Sabatier, J., Aoun, M., Oustaloup, A., Gregoire, G., Ragot, F., and Roy, P., 2006, “Fractional System Identification for Lead Acid Battery Sate Charge Estimation,” Signal Process., 86(10), pp. 2645–2657. [CrossRef]
Oldham, K. B., and Spanier, J., 1973, “Diffusive Transport to Planar, Cylindrical and Spherical Electrodes,” J. Electroanal. Chem. Interfacial Electrochem., 41(3), pp. 351–358. [CrossRef]
Battaglia, J. L., Cois, O., Puigsegur, L., and Oustaloup, A., 2001, “Solving an Inverse Heat Conduction Problem Using a Non-Integer Identified Model,” Int. J. Heat Mass Transfer, 44(14), pp. 2671–2680. [CrossRef]
Oustaloup, A., Lay, L., and Mathieu, B., 1996, “Identification of Non-Integer Order System in the Time-Domain,” Proceedings of the CESA’96 IMACS-IEEE/SMC Multi-Conference, Computational Engineering in Systems Applications, Lille, France, July 9–12.
Trigeassou, J. C., Poinot, T., Lin, J., Oustaloup, A., and Levron, F., 1999, “Modeling and Identification of a Non-Integer Order System,” Proceedings of the European Control Conference, Karlsruhe, Germany, Aug. 31–Sept. 3.
Cois, O., Oustaloup, A., Poinot, T., and Battaglia, J. L., 2001, “Fractional State Variable Filter for System Identification by Fractional Model,” Proceedings of the European Control Conference, Porto, Portugal, Sept. 4–7, pp. 417–433.
Aoun, M., Malti, R., Levron, F., and Oustaloup, A., 2007, “Synthesis of Fractional Laguerre Basis for System Approximation,” Automatica, 43(9), pp. 1640–1648. [CrossRef]
Battaglia, J. L., Lay, L., Batsale, J. C., Oustaloup, A., and Cois, O., 2000, “Heat Flow Estimate Through Inverted Not Integer Identification Models,” Int. J. Therm. Sci., 39(3), pp. 374–389. [CrossRef]
Lay, L., 1998, “Identification frequentielle et temporelle par modele non entier,” Ph.D. thesis, Universite de Bordeaux I, Bordeaux Cedex, France.
Maiti, D., Acharya, A., Janarthanan, R., and Konar, A., 2008, “Complete Identification of a Dynamic Fractional Order System Under Non-Ideal Conditions Using Fractional Differintegral Definitions,” Proceedings of 16th International Conference on Advanced Computing and Communications (ADCOM), Chennai, India, Dec. 14–17, pp. 285–292.
Victor, S., Malti, R., and Oustaloup, A., 2009, “Instrumental Variable Method With Optimal Fractional Differentiation Order for Continuous-Time System Identification,” Proceedings of the 15th IFAC Symposium on System Identification, Saint-Malo, France, July 6–8, pp. 904–909.
Malti, R., Victor, S., and Oustaloup, A., 2008, “Advances in System Identification Using Fractional Models,” ASME J. Comput. Nonlinear Dyn., 3(2), p. 021401. [CrossRef]
Cois, O., Oustaloup, A., Battaglia, E., and Battaglia, J. L., 2000, “Non Integer Model From Modal Decomposition for Time Domain System Identification,” Proceedings of the 12th IFAC SYSID’2000, Santa-Barbara, CA, June 21–23.
Oldham, K. B., and Spanier, J., 1974, The Fractional Calculus, Academic Press, New York.
Podlubny, I., 1999, Fractional Differential Equations, Academic Press, San Diego, CA.
Heleschewitz, D., and Matignon, D., 1998, “Diffusive Realisations of Fractional Integrodifferential Operators: Structural Analysis Under Approximation,” Proceedings of the IFAC Conference System, Structure and Control, Nantes, France, July 8–10, Vol. 2, pp. 243–248.
Helechewitz, D., 2000, “Analyse et simulation de systèmes différentiels fractionnaires et pseudo-différentiels sous representation diffusive,” Ph.D. thesis, ENST, Paris, France.
Trigeassou, J. C., and Oustaloup, A., 2011, “Fractional Integration: A Comparative Analysis of Fractional Integrators,” Proceedings of the 2011 8th International Multi-Conference on Systems, Signals and Devices (SSD), IEEE SSD’11, Sousse, Tunisia, Mar. 22–25, pp. 1–6.
Jelloul, A., Jelassi, K., Trigeassou, J. C., and Melchior, P., 2011, “Comparison of Fractional Identification Techniques for Rotor Skin Effect in Induction Machines,” Int. J. Comput. Sci. Issues, 8(3), pp. 57–68.
Trigeassou, J. C., Maamri, N., Sabatier, J., and Oustaloup, A., 2012, “State Variables and Transients of Fractional Order Differential Systems,” Comput. Math. Appl., 64(10), pp. 3117–3140. [CrossRef]
Trigeassou, J. C., Maamri, N., Sabatier, J., and Oustaloup, A., 2013, “The Infinite State Approach: Origin and Necessity,” Comput. Math. Appl., 66(5), pp. 892–907. [CrossRef]
Khadrahoui, A., Jelassi, K., and Trigeassou, J. C., 2013, “Identification of a Fractional Order Model by a Least Squares Technique: Hn Model,” Prog. Comput. Appl., 2(2), pp. 91–101.
Khadrhoui, A., Jelassi, K., and Trigeassou, J. C., 2013, “Identification of a Fractional Order Model by a Least Squares Technique: Hn1,n2 Model,” Proceedings of the 2013 14th International Conference Sciences and Techniques of Automatic Control and Computer Engineering (STA), Sousse, Tunisia, Dec. 20–22, pp. 461–467.
Victor, S., Malti, R., Garnier, H., and Oustaloup, A., 2013, “Parameter and Differentiation Order Estimation in Fractional Models,” Automatica, 49(4), pp. 926–935. [CrossRef]


Grahic Jump Location
Fig. 1

Frequency discretization of μ(w)

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Fig. 2

The modal representation (infinite state representation) of fractional integrator

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Fig. 3

Simulation of a one derivate FDE

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Fig. 4

Simulation of the two derivate DFE

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Fig. 5

Estimated and true response of Hn model

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Fig. 6

Quadratic criterion variation: Hn model

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Fig. 7

Error variation: Hn model

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Fig. 8

Output data of Hn model, and estimated response: SNR = 1

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Fig. 9

Output data of Hn1,n2 model, and estimated response: SNR = 15



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