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.

References

References
1.
Hsia
,
T. C.
,
1977
,
System Identification: Least-Squares Methods
,
Lexington Books
,
Lexington, MA
.
2.
Strejc
,
V.
,
1980
, “
Least Squares Parameter Estimation
,”
Automatica
,
16
(
5
), pp.
535
550
.10.1016/0005-1098(80)90077-1
3.
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
.10.1080/00207179208934373
4.
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
.10.1080/00207178008961080
5.
Landau
,
I.
,
1976
, “
Unbiased Recursive Identification Using Model Reference Adaptive Techniques
,”
IEEE Trans. Autom. Control
,
21
(
2
), pp.
194
202
.10.1109/TAC.1976.1101195
6.
Ljung
,
L.
,
1977
, “
Analysis of Recursive Stochastic Algorithms
,”
IEEE Trans. Autom. Control
,
22
(
4
), pp.
551
575
.10.1109/TAC.1977.1101561
7.
Goodwin
,
G. C.
, and
Payne
,
R. L.
,
1977
,
Dynamic System Identification: Experiment Design and Data Analysis
,
Academic Press
,
New York
.
8.
Stoica
,
P.
, and
Soderstrom
,
T.
,
1982
, “
Bias Correction in Least-Squares Identification
,”
Int. J. Control
,
35
(
3
), pp.
449
457
.10.1080/00207178208922631
9.
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
.
10.
Ljung
,
L.
,
1987
,
System Identification: Theory for the User
,
Prentice Hall
,
Englewood Cliffs, NJ
.
11.
Van den Hof
,
P. M. J.
,
1989
, “
Criterion Based Equivalence for Equation Error Models
,”
IEEE Trans. Autom. Control
,
34
(
2
), pp.
191
193
.10.1109/9.21096
12.
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
.10.1016/j.sigpro.2006.02.030
13.
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
.10.1016/S0022-0728(73)80413-9
14.
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
.10.1016/S0017-9310(00)00310-0
15.
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.
16.
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.
17.
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
.
18.
Aoun
,
M.
,
Malti
,
R.
,
Levron
,
F.
, and
Oustaloup
,
A.
,
2007
, “
Synthesis of Fractional Laguerre Basis for System Approximation
,”
Automatica
,
43
(
9
), pp.
1640
1648
.10.1016/j.automatica.2007.02.013
19.
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
.10.1016/S1290-0729(00)00220-9
20.
Lay
,
L.
,
1998
, “
Identification frequentielle et temporelle par modele non entier
,” Ph.D. thesis,
Universite de Bordeaux I, Bordeaux Cedex
,
France
.
21.
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
.
22.
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
.
23.
Malti
,
R.
,
Victor
,
S.
, and
Oustaloup
,
A.
,
2008
, “
Advances in System Identification Using Fractional Models
,”
ASME J. Comput. Nonlinear Dyn.
,
3
(
2
), p.
021401
.10.1115/1.2833910
24.
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.
25.
Oldham
,
K. B.
, and
Spanier
,
J.
,
1974
,
The Fractional Calculus
,
Academic Press
,
New York
.
26.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
San Diego, CA
.
27.
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
.
28.
Helechewitz
,
D.
,
2000
, “
Analyse et simulation de systèmes différentiels fractionnaires et pseudo-différentiels sous representation diffusive
,” Ph.D. thesis,
ENST
,
Paris, France
.
29.
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
.
30.
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
.
31.
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
.10.1016/j.camwa.2012.03.099
32.
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
.10.1016/j.camwa.2012.11.020
33.
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
.
34.
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
.
35.
Victor
,
S.
,
Malti
,
R.
,
Garnier
,
H.
, and
Oustaloup
,
A.
,
2013
, “
Parameter and Differentiation Order Estimation in Fractional Models
,”
Automatica
,
49
(
4
), pp.
926
935
.10.1016/j.automatica.2013.01.026
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