Nonlinear System Modeling with Invariance to Fourier Transform for Fault Diagnosis: Application to Power Transformers

Authored by: Gerasimos Rigatos , Pierluigi Siano

Measurement, Instrumentation, and Sensors Handbook

Print publication date:  February  2014
Online publication date:  February  2014

Print ISBN: 9781439848913
eBook ISBN: 9781439848937
Adobe ISBN:


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A neural network with basis functions that remain invariant under the Fourier transform is proposed for fault diagnosis of nonlinear systems. The considered neural network is of the feedforward type and uses Gauss–Hermite polynomial basis functions. This neural model follows the concept of wavelet networks [13] and employs basis functions that are localized both in space and frequency, thus allowing better approximation of the multifrequency characteristics of monitored nonlinear system [48]. Gauss–Hermite basis functions have some interesting properties [9,10]: (1) They remain almost unchanged by the Fourier transform and satisfy an orthogonality property, which means that the weights of the associated neural network demonstrate the energy that is distributed to the various eigenmodes of the nonlinear system’s dynamics, and (2) unlike wavelet basis functions, the Gauss–Hermite basis functions have a clear physical meaning since they represent the solutions of differential equations describing stochastic oscillators (see [11]) and each neuron can be regarded as the frequency filter of the respective eigenfrequency.

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