A bad initialization of output-error (OE) technique can lead to an inappropriate identification results. In this paper, we introduce a solution to this problem; the basic idea is to estimate the parameters and the fractional order of the noninteger system by a new approach of least-squares (LS) method based on repeated fractional integration to initialize OE technique. It will be shown that LS method offers a good initialization to OE algorithm and leads to acceptable identification results. The performance of the proposed method is shown through numerical simulation examples.

References

1.
Liouville
,
J.
,
1932
, “
Some Questions About Geometry and Mechanics and a New Genre of Computing to Solve These Equations (Mémoire sur Quelques Questions de Géométrie et de Mécanique et sur un Nouveau Genre de Calcul Pour Résoudre ces Equations)
,”
J. Ec. Polytech.
,
13
, pp.
71
162
.
2.
Riemann
,
B.
,
1876
, “
Testing of a General Conception of Integration and Derivation (Essai d'une Conception Générale de L'intégration et de la Dérivation)
,”
Works Math. Res.
, pp.
331
344
.
3.
Oldham
,
K. B.
, and
Spanier
,
J.
,
1973
, “
Diffusive Transport to Planar, Cylindrical and Spherical Electrodes
,”
Electroanal. Chem. Interfacial Electrochem
,
41
(
3
), pp.
351
358
.
4.
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
.
5.
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
.
6.
Benchellal
,
A.
,
Bachir
,
S.
,
Poinot
,
T.
, and
Trigeassou
,
J. C.
,
2004
, “
Identification of a Non-Integer Model of Induction Machines
,”
1st IFAC Workshop on Fractional Differentiation and its Applications
,
Bordeaux, France
, pp.
400
407
.
7.
Le Lay
,
L.
,
1998
, “
Identification Fréquentielle et Temporelle par Modèle Non-Entier
,” Ph.D. thesis, Université de Bordeaux 1, Bordeaux, France.
8.
Lin
,
J.
,
2001
, “
Modélisatoin et Identification de Systemes d'ordre non Entier
,” Ph.D. thesis, Université de Poitiers, Poitiers, France.
9.
Cois
,
O. A.
,
Oustaloup
,
A.
,
Poinot
,
T.
, and
Battaglia
,
J. L.
,
2001
, “
Fractional State Variable Filter for System Identification by Fractional Model
,”
ECC Conference
,
Porto, Portugal
.
10.
Maiti
,
D.
,
Acharya
,
A.
,
Janarthanan
,
R.
, and
Konar
,
A.
,
2008
, “
Complete Identification of a Dynamic Fractional Order System Under Non-Ideal Conditions Using Fractional Differ Integral Definitions
,”
16th International Conference on Advanced Computing and Communications (ADCOM)
,
Chennai, India
.
11.
Victor
,
S.
,
Malti
,
R.
, and
Oustaloup
,
A.
,
2009
, “
Instrumental Variable Method With Optimal Fractional Differentiation Order for Continuous-Time System Identification
,”
15th IFAC Symposium on System Identification
,
Saint-Malo, France
.
12.
Benchellal
,
A.
,
Poinot
,
T.
, and
Trigeassou
,
J. C.
,
2005
, “
Approximation and Identification of Fractional Systems
,”
ASME
Paper No. DETC2005-84784.
13.
Trigeassou
,
J. C.
,
Poinot
,
T.
,
Lin
,
J.
,
Oustaloup
,
A.
, and
Levron
,
F.
,
1999
, “
Modeling and Identification of a Non-Integer Order System
,”
ECC’99, European Control Conference
,
Karlsruhe, Germany
.
14.
Djamah
,
T.
,
Mansouri
,
R.
,
Djennoune
,
S.
,
Guermah
,
S.
, and
Bettayeb
,
M.
,
2007
, “
Identification of Fractional System With Optimal Reduced Integer Order Model
,”
2nd International Conference on Modeling, Simulation and Applied Optimization ICMSAO’07
,
Abu Dhabi, UAE
.
15.
Poinot
,
T.
, and
Trigeassoui
,
J. C.
,
2004
, “
Identification of Fractional Systems Using an Output-Error Technique
,”
Nonlinear Dyn.
,
38
(
1–4
), pp.
133
154
.
16.
Ljung
,
L.
,
1987
,
System Identification—Theory for the User
,
Prentice Hall
,
Englewood Cliffs, NJ
.
17.
Pearson
,
A. E.
,
1988
, “
Least Squares Parameter Identification of Nonlinear Differential I/O Models
,” 27th
IEEE
Conference on Decision and Control
, Austin, TX, Dec. 7–9, pp.
1931
1935
.
18.
Van den Hof
,
P. M. J.
,
1989
, “
Criterion Based Equivalence for Equation Error Models
,”
IEEE Trans. Autom. Control
,
34
(
2
), pp.
191
193
.
19.
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.
20.
Khadrahoui
,
A.
,
Jelassi
,
K.
, and
Trigeassou
,
J. C.
,
2013
, “
Identification of a Fractional Order Model by a Least Squares Technique: Hn1,n2 Model
,”
Sciences and Techniques of Automatic Control and Computer Engineering
(
STA
), Sousse, Dec. 20–22, pp.
461
467
.
21.
Oldham
,
K. B.
, and
Spanier
,
J.
,
1974
,
The Fractional Calculus
,
Academic Press
,
New York
.
22.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
San Diego, CA
.
23.
Montseny
,
G.
,
1998
, “
Diffusive Representation of Pseudo Differential Time Operators
,”
ESSAIM
, Vol.
5
, pp.
159
175
.
24.
Sabatier
,
J.
,
Merveillaut
,
M.
,
Malti
,
R.
, and
Oustaloup
,
A.
,
2008
, “
On a Representation of Fractional Order Systems: Interests for the Initial Condition Problem
,”
3rd IFAC Workshop, FDA’08
,
Anhara, Turkey
.
25.
Heleschewitz
,
D.
, and
Matignon
,
D.
,
1998
, “
Diffusive Realizations of Fractional Integro-Differential Operators: Structural Analysis Under Approximation
,”
Conference IFAC, System, Structure and Control
, Vol.
2
,
Nantes, France
, pp.
243
248
.
26.
Helechewitz
,
D.
,
2000
, “
Analyse et Simulation de Systèmes Différentiels Fractionnaires et Pseudo-Différentiels Sous Representation Diffusive
,” Ph.D. thesis, ENST Paris, Paris, France.
27.
Trigeassou
,
J. C.
,
Poinot
,
P.
,
Lin
,
J.
,
Oustaloup
,
A.
, and
Levron
,
F.
,
1999
, “
Modelling and Identification of a Non-Integer Order System
,”
ECC’99 European Control Conference
,
Karlsruhe, Germany
.
28.
Jelloul
,
A.
,
Jelassi
,
K.
,
Trigeassou
,
J. C.
, and
Melchior
,
P.
,
2011
, “
Comparison of Fractional Identification Techniques for Rotor Skin Effect in Induction Machines
,”
IJCSI
,
8
(
3
).
29.
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
.
30.
Trigeassou
,
J. C.
,
Maamri
,
N.
,
Sabatier
,
J.
, and
Oustaloupa
,
A.
,
2013
, “
The Infinite State Approach: Origin and Necessity
,”
Comput. Math. Appl.
,
66
(
5
), pp.
892
907
.
31.
Richalet
,
J.
,
Rault
,
A.
, and
Pouliquen
,
R.
,
1971
,
Identification of Process by the Output Error Method (Identification des Processus par la Méthode du Modéle)
,
Gordon and Breach
,
Paris, France
.
32.
Marquardt
,
D. W.
,
1963
, “
An Algorithm for Least-Squares Estimation of Non-Linear Parameters
,”
J. Soc. Ind. Appl. Math.
,
11
(
2
), pp.
431
441
.
33.
Jelloul
,
A.
,
Jelassi
,
K.
, and
Trigeassou
,
J. C.
,
2012
, “
Fractional Modeling and Identification of Rotor Skin Effect in Induction Machines (Modelisation et Identification des Effets de Frequence dans la Machine Asynchrone par Approche d'ordre non entier)
,” Ph.D. thesis, ENIT, Tunis, Tunisia.
34.
Trigeassou
,
J. C.
,
1988
,
Computer-Assisted of Experimental Models Search (Recherche des Modeles Experimentaux Assistée par Ordinateur)
,
Lavoisier-Tec et Doc
,
Paris, France
.
35.
Khadrahoui
,
A.
,
Jelassi
,
K.
, and
Trigeassou
,
J. C.
,
2014
, “
Least Squares and Instrumental Variable Techniques for Global Identification of Fractional Differential Equation
,”
Electrical Sciences and Technologies in Maghreb (CISTEM) International Conference
,
Tunis
, Tunisia.
You do not currently have access to this content.