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Research Papers

Parametric Identification of Nonlinear Systems by Haar Wavelets: Theory and Experimental Validation

[+] Author and Article Information
Shy-Leh Chen1

Advanced Institute of Manufacturing with High-Tech Innovations,Department of Mechanical Engineering,  National Chung Cheng University, 168 University Road, Minhsiung Township, Chiayi County, 62102, Taiwan, R.O.C. e-mail: imeslc@ccu.edu.tw

Jin-Wei Liang

Department of Mechanical Engineering,  Ming-Chi University of Technology, Taishan, New Taipei City, 24301,Taiwan, R.O.C.

Keng-Chu Ho

Advanced Institute of Manufacturing with High-Tech Innovations,Department of Mechanical Engineering,  National Chung Cheng University, 168 University Road, Minhsiung Township, Chiayi County, 62102, Taiwan, ROC

1

Corresponding author.

J. Vib. Acoust 134(3), 031005 (Apr 24, 2012) (12 pages) doi:10.1115/1.4006229 History: Received August 27, 2010; Revised January 30, 2012; Published April 23, 2012; Online April 24, 2012

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.

Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Illustration of some Haar basis functions

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Figure 3

Numerical identification results for the unforced responses: λ

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Figure 4

Numerical identification results for the unforced responses: β and δ

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Figure 5

The chaotic response of the Moon beam under forced excitation

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Figure 6

Numerical identification results for the chaotic responses: λ

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Figure 7

Numerical identification results for the chaotic responses: β and δ

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Figure 8

The spectrum of the forced response in the numerical example

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Figure 9

The experimental setup

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Figure 10

The spectrum of the free response in the experimental example

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Figure 11

Experimental identification results for the free responses: λ

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Figure 12

Experimental identification results for the free responses: β and δ

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Figure 13

Comparison of the free response between the experimental system and the identified models

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Figure 14

Poincaré map of the experimental system: φ=0 deg

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Figure 15

Poincaré map of the identified model: φ=0 deg

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Figure 16

Poincaré map of the experimental system: φ=90 deg

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Figure 17

Poincaré map of the identified model: φ=90 deg

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Figure 18

Correlation dimensions of the identified model and experimental system

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