Parametric Identification of Nonlinear Systems Using Chaotic Excitation

[+] Author and Article Information
M. D. Narayanan, S. Narayanan, Chandramouli Padmanabhan

Machine Design Section, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India

J. Comput. Nonlinear Dynam 2(3), 225-231 (Feb 13, 2007) (7 pages) doi:10.1115/1.2727489 History: Received May 18, 2006; Revised February 13, 2007

The use of a time series, which is the chaotic response of a nonlinear system, as an excitation for the parametric identification of single-degree-of-freedom nonlinear systems is explored in this paper. It is assumed that the system response consists of several unstable periodic orbits, similar to the input, and hence a Fourier series based technique is used to extract these nearly periodic orbits. Criteria to extract these orbits are developed and a least-squares problem for the identification of system parameters is formulated and solved. The effectiveness of this method is illustrated on a system with quadratic damping and a system with Duffing nonlinearity.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

Schematic of the identification procedure using chaotic excitation

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Figure 2

Generator system response with (a) phase plane (cycles 51–200) and (b) Poincaré section (cycles 51–2000)

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Figure 3

The response of system S with quadratic damping to a chaotic excitation. (a) Phase plane plot(cycles 51–200 cycles) and (b) Poincaré section (cycles 51–2000).

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Figure 4

Distribution of periods of extracted orbits from data in Fig. 3

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Figure 5

An extracted orbit having nearly seven periods; square represents discontinuity

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Figure 6

Parametric error as a function of noise to signal ratio for the quadratic nonlinearity example

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Figure 7

The periods of the extracted orbits for different sc values. (a) Complete set, (b) after eliminating nearly identical period orbits. The numbers 1–5 in (a) denote sc values from 0.1 to 400.1 with an increment of 100.

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Figure 8

The variation of Ep with orbit number (period) for different values of sc, which varies from 0.1 to 400.1 in steps of 100. Numbers 1–5 denote the sc values in ascending order.

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Figure 9

System with Duffing-type nonlinearity. (a) Extracted orbit having nearly one period; (b) Corresponding orbit of Gn; square locates the discontinuity in orbits.

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Figure 10

The variation of Ep with orbit number for different values of sc, which varies from 0.1 to 8.1 in steps of 2



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