This study addresses the identification of nonlinear systems. It is assumed that the function form in the nonlinear system is known, leaving some unknown parameters to be estimated. Since Haar wavelets can form a complete orthogonal basis for the appropriate function space, they are used to expand all signals. In doing so, the state equation can be transformed into a set of algebraic equations in unknown parameters. The technique of Kronecker product is utilized to simplify the expressions of the associated algebraic equations. Together with the least square method, the unknown system parameters are estimated. The proposed method is applied to the identification of an experimental two-well chaotic system known as the Moon beam. The identified model is validated by comparing the chaotic characteristics, such as the largest Lyapunov exponent and the correlation dimension, of the experimental data with that of the numerical results. The simple least square approach is also performed for comparison. The results indicate that the proposed method can reliably identify the characteristics of the nonlinear chaotic system.

References

References
1.
Juditsky
,
A.
,
Hjalmarsson
,
H.
,
Benveniste
,
A.
,
Delyon
,
B.
,
Ljung
,
L.
, Sjö
berg
,
J.
, and
Zhang
,
Q.
, 1995, “
Nonlinear Black Box Models in System Identification Mathematical Foundations
,”
Automatica
,
31
(
12
), pp.
1725
1750
.
2.
Nelles
,
O.
, 2001,
Nonlinear System Identification
,
Springer-Verlag
,
Berlin-Heidelberg
.
3.
Sjöberg
,
J.
,
Zhang
,
Q.
,
Ljung
,
L.
,
Benveniste
,
A.
,
Deylon
,
B.
,
Glorennec
,
P.-Y.
,
Hjalmarsson
,
H.
, and
Juditsky
,
A.
, 1995, “
Nonlinear Black Box Models in System Identification: A Unified Overview
,”
Automatica
,
31
(
12
), pp.
1691
1724
.
4.
Chen
,
S.
, and
Lu
,
J.
, 2002, “
Parameters Identification and Synchronization of Chaotic Systems Based Upon Adaptive Control
,”
Phys. Lett. A
,
299
, pp.
353
358
.
5.
Dedieu
,
Herv´e and Ogorzałek
, and
Maciej
,
J.
, 1997, “
Identifiability and Identification of Chaotic Systems based on Adaptive Synchronization
,”
IEEE Trans. Circuits Syst.
,
44
(
10
), pp.
948
962
.
6.
Feldman
,
M.
, 1997, “
Nonlinear Free Vibration Identification via the Hilbert Transformation
,”
J. Sound. Vib.
,
208
, pp.
475
489
.
7.
Gottlieb
,
O.
, and
M.
Feldman
, 1997, “
Application of a Hilbert-Transform Based Algorithm for Parameter Estimation of a Nonlinear Ocean System Roll Model
,”
J. Offshore Mech. Arct. Eng.
,
119
, pp.
239
243
.
8.
Grosu
,
J.
, 2004, “
Estimation of Model Parameters from Time Series by OPCL Autosynchronization
,”
Int. J. Bifurcation Chaos
,
14
(
6
), pp.
2133
2141
.
9.
Stry
,
G.
, and
Mook
,
D.
, 1992, “
An Experimental Study of Nonlinear Dynamic System Identification
,”
Nonlinear Dyn.
,
3
, pp.
1
11
.
10.
Voss
,
H. U.
,
Timmer
,
J.
, and
Kurths
,
J.
, 2004, “
Nonlinear Dynamical System Identification from Uncertain and Indirect Measurements
,”
Int. J. Bifurcation Chaos
,
14
(
6
), pp.
1905
1933
.
11.
Feeny
,
B. F.
,
Yuan
,
C. M.
, and
Cusumano
,
J. P.
, 2001, “
Parametric Identification of an Experimental Magneto-Elastic Oscillator
,”
J. Sound Vib.
,
247
(
5
), pp.
785
806
.
12.
Nayfeh
,
A.
, 1985, “
Parametric Identification of Nonlinear Dynamic Systems
,”
Comput. Struct.
,
20
, pp.
487
493
.
13.
Yasuda
,
K.
,
Kawamura
,
S.
, and
Watanabe
,
K.
, 1988, “
Identification of Nonlinear Multi-Degree-of-Freedom Systems (Presentation of an Identification Technique)
,”
JSME Int. J., Ser. III
,
31
, pp.
8
14
.
14.
Yasuda
,
K.
,
Kawamura
,
S.
, and
Watanabe
,
K.
, 1988, “
Identification of Nonlinear Multi-Degree-of-Freedom Systems (Identification under Noisy Measurements)
,”
JSME Int. J., Ser. III
,
31
, pp.
302
309
.
15.
Palanthandalam-Madapusi
,
H. J.
,
Hoagg
,
J. B.
, and
Bernstein
,
D. S.
, 2004, “
Basis-Function Optimization for Subspace-Based Nonlinear Identification of Systems With Measured-Input Nonlinearities
,”
Proceedings of the American Control Conference,
Boston
,
Massachusetts
, June 30–July 2, pp.
4788
4793
.
16.
Chen
,
C. F.
, and
Hsiao
,
C. H.
, 1997, “
Haar Wavelet Method for Solving Lumped and Distributed Parameter System
,”
IEE Proc.-D: Control Theory Appl.
.,
144
, pp.
87
94
.
17.
Chen
,
S.-L.
,
Lai
,
H.-C.
, and
Ho
,
K.-C.
, 2006, “
Identification of Linear Time Varying Systems by Haar Wavelets
,”
Int. J. Syst. Sci.
,
37
(
9
), pp.
619
628
.
18.
Ghanem
,
R.
, and
Romeo
,
F.
, 2000, “
A Wavelet-Based Approach for the Identification of Linear Time-Varying Dynamical Systems
,”
J. Sound Vib.
,
234
(
4
), pp.
555
576
.
19.
Ghanem
,
R.
, and
Romeo
,
F.
, 2001, “
A Wavelet-based Approach for Model and Parameter Identification of Nonlinear Systems
,”
Int. J. Nonlinear Mech.
,
36
, pp.
835
859
.
20.
Hsiao
,
C. H.
, and
Wang
,
W. J.
, 2000, “
State Analysis and Parameter Estimation of Bilinear System via Haar Wavelets
,”
IEEE Trans. Circuits Syst., I: Fundam. Theory Appl.
,
47
(
2
), pp.
246
250
.
21.
Chui
,
C. K.
, 1992,
An Introduction to Wavelets
,
Academic Press
,
Boston
.
22.
Brewer
,
J. W.
, 1978, “
Kronecker Products and Matrix Calculus in System Theory
,”
IEEE Trans. Circuits Syst.
,
25
(
9
), pp.
772
781
.
23.
Moon
,
F. C.
, 1980, “
Experiments on Chaotic Motions of a Forced Nonlinear Oscillator: Strange Attractor
,”
ASME Trans. J. Appl. Mech.
,
47
, pp.
638
644
.
24.
Moon
,
F. C.
, and
Holmes
,
P. J.
, 1979, “
A Magnetoelastic Strange Attractor
,”
J. Sound Vib.
,
65
(
2
), pp.
275
296
.
25.
Moon
,
F. C.
, 1987,
Chaotic Vibrations: An Introduction for Applied Scientist and Engineers
,
John Wiley and Sons
,
New York
.
26.
Kantz
,
H.
, and
Schreiber
,
T.
, 1997,
Nonlinear Time Series Analysis
,
Cambridge University Press
,
Cambridge, MA
.
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