Mathematical Problems in Engineering

Volume 2016, Article ID 7305702, 7 pages

http://dx.doi.org/10.1155/2016/7305702

## Multifractal Analysis for Soft Fault Feature Extraction of Nonlinear Analog Circuits

The Higher Educational Key Laboratory for Measuring & Control Technology and Instrumentations of Heilongjiang Province, Harbin University of Science and Technology, Harbin 150080, China

Received 1 March 2016; Accepted 20 April 2016

Academic Editor: Anna Vila

Copyright © 2016 Xinmiao Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Aiming at the nonstationarity and nonlinearity of soft fault signals of nonlinear analog circuits, the use of multifractal detrended fluctuation analysis can effectively reveal the dynamic behavior hidden in multiscale nonstationary signals. This paper adopts a new method that uses multifractal detrended fluctuation analysis to calculate the multifractal singularity spectrum of soft fault signals of nonlinear analog circuits. Moreover, this method endows the parameters of the spectrum with definite physical meanings including width, maximum singular index, minimum singular index, and corresponding singularity index of the extreme point. Therefore, this method can be applied to characterize the internal dynamic mechanism of the soft fault signals of nonlinear analog circuits, making it suitable for the feature extraction of fault circuits. All multifractal feature parameters can be organized into a feature set, which will be then input to a support vector machine, and fault detection for the nonlinear analog circuit can be conducted via the support vector machine.

#### 1. Introduction

The testing and fault diagnosis of nonlinear analog circuits have always been a hot research topic in the field of circuit testing. Although many important results have been achieved, the study of nonlinear systems is still in the stage of approximation process and numerical calculation. System modeling analysis, diagnosis theory, and its practicality all await further study. The nature of fault diagnosis of nonlinear analog circuits is pattern recognition, and the key to diagnosis is to construct fault features that reflect the nature of the tested circuit. Different approaches can be used to describe nonlinear analog circuits, including the Volterra series method, Wiener series method, and wavelet analysis-based feature extraction method of recent years.

The Volterra series can more accurately establish common fault models of nonlinear analog circuits. However, owing to the problem of dimensionality involved in using the Volterra series, in practice, only the low-order finite term can be taken. Furthermore, for some nonanalytic nonlinear systems, Volterra series expansion cannot be used. In addition, the terms in the Volterra series are not orthogonal to each other, making it difficult to extract the fault features of the circuit to be diagnosed, which reduces the effect of the fault diagnosis [1–3].

Lin et al. [4] solved this problem using the Wiener series. Gaussian white noise excitation is applied to a nonlinear system with unknown characteristic parameters, and an orthogonal functional series expansion is calculated for its response in order to obtain a set of functions that characterize system features. A time-correlation function is used to estimate the correlation function of each order to solve the Wiener kernels of the system, and a BP neural network is used to realize fault diagnosis. However, this method must be improved to obtain the Wiener kernels of the tested circuit more accurately and more efficiently [5].

The feature extraction theory based on wavelet analysis is the most utilized theory in the fault diagnosis of analog circuits. For the fault diagnosis of analog circuits, the biggest advantage of wavelet analysis and artificial neural networks is that the time-domain localized features are excellent, high frequencies and low frequencies are separated, the system is insensitive to circuit noise, and the noise removal ability is satisfactory, especially when there are many nonlinear components in the tested analog circuit [6–8]. When the ambient noise is particularly high, the frequency separation feature of wavelet analysis can function effectively, greatly reducing the difficulty of extracting circuit fault features. However, this method also requires a great amount of experimental data in order to form an accurate classification of the fault phenomena [9–11].

Chaotic dynamics and fractal theory developed in recent years provide a good direction for the study of nonlinear systems. Fractal theory is not only at the leading edge and an important branch of nonlinear science, but also an emerging transdisciplinary science. The use of the fractal method can accurately extract characteristics of the nonlinear signal. The majority of researchers use a single fractal to represent the overall irregular performance of signals. Zhang et al. [12] used the wavelet fractal method to perform a fault diagnosis of a three-phase bridge rectifier circuit. Mao et al. [13] also achieved satisfactory results by using analog circuit fault diagnosis based on a support vector machine (SVM) and fractal characteristics.

Peng et al. [14] proposed a detrended fluctuation analysis (DFA) that analyzes power-law functions; this analysis can effectively characterize the long-range correlation of the nonlinear time series of complex dynamic systems. However, the DFA method uses only a second-order wave function to analyze the scaling behavior of a sequence of events, obtaining only a single scaling exponent. Therefore, this method is only suitable for nonstationary time series with single scaling behavior (also known as a single fractal), which lacks a description of local singular characteristics. “Multifractal” not only describes the irregularity of signals as a whole but also more finely reflects the description of the local characteristics of the signal. Multifractal depicts the local scaling characteristics of the fractal subsets of different scales and scaling exponents distributed over the subsets.

Kantelhardt et al. [15] not only extended a one-dimensional fractal DFA but also proposed an MF-DFA method by combining the one-dimensional fractal DFA with the standard partition function-based multifractal formalism. This method can effectively eliminate the interference trends and fluctuation scale of time series measurement, thus effectively overcoming the defects in DFA methods when analyzing multiscale multifractal nonstationary time series. In addition, this method can more accurately estimate the multifractal spectrum.

This paper introduces the MF-DFA method into nonlinear analog circuits and applies the MF-DFA method to analyze the collected signals for fault diagnosis based on the characteristic parameters of the multifractal spectrum of the fault signals.

#### 2. Estimating Multifractal Spectrum Using the MF-DFA Method

Assume that the time series to be detected has a signal length . The DFA method is conducted as follows.

*Step 1. *Calculate the average value of the original time series. Sum the difference of the terms between the given time series and the average value, producing a new contour sequence :where and is the average value of the time sequence .

*Step 2. *Divide the new sequence obtained from the transform into nonoverlapping subintervals of equal length so that , which is a rounding process for . Because may not be a multiple of , to avoid the loss of information from the original time series, the new sequence is also divided in the same way, so that all figures in series are in the calculation. Finally, the 2 subintervals of equal set length can be obtained.

*Step 3. *Remove the fluctuation trend of each small section using the least squares method. Assume that is the th order of the least squares fitting polynomial of the th small interval. The function is the th-order polynomial fitting of the th subinterval for observations, and the fitting equation obtained is as follows:In the formula, is the fitting polynomial, and is the order of the fitting polynomial.

*Step 4. *Eliminate the trend within each subinterval, and calculate the mean square error

When , there isWhen , there is

*Step 5. *For the detrended intervals, take the average value of to obtain the th wave function: When , MF-DFA degenerates into DFA; different values describe the impact on from different degrees of fluctuation. The mean square error can eliminate the trend of each small section using the polynomial fitting of each subinterval, which is more beneficial for the singularity recognition of local fluctuation.

*Step 6. *By analyzing the relationship in the double logarithmic plot ~, the scaling exponent of the wave function can be determined, so a power-law relationship exists:Each corresponds to a wave function . The wave function value is an increasing function of . Draw the relationship plot of with respect to , and the slope obtained from the linear fitting of the least squares method is , also known as the generalized Hurst exponent. When is constant, the time series appears as a single fractal. When , the time series appears either without a correlation or with a short correlation. When has a nonlinear relationship with , the time series appears to have multifractal characteristics.

*Step 7. *Between the generalized Hurst exponent from the discrete time series and the mass index from classic multifractal theory, there existsIn this way, the relationship between the generalized Hurst exponent and the scaling index from the partition function-based method can be obtained. With the Legendre transform from statistical physics, the relationship between the multifractal spectral index , , and the generalized Hurst exponent is obtained:

#### 3. Feature Extraction of Multifractal Spectrum

The multifractal spectrum obtained from the MF-DFA method is a set of parameters that can finely portray the dynamic behavior of a multifractal time series. This paper will adopt six types of commonly used multifractal spectrum parameters as the fault characteristic values of soft fault signals of a nonlinear analog circuit. The parameters of multifractal singularity spectrum ~ are as follows:

(1) The width of the multifractal singularity spectrum : corresponds to the minimum probability subset, corresponds to the maximum probability subset, and represents the multifractal strength of a time series, a greater value of which would mean a more uneven distribution of probability measure of the time series and a more intense change in the signals. can describe the degree of uneven distribution of circuit detection signals.

(2) The dimensional difference : represents the fractal dimension of the maximum fluctuation point set, and represents the fractal dimension of the minimum fluctuation point set. reflects the frequency ratio of the maximum fluctuation to the minimum fluctuation. The value of reflects the proportion of the number of most stable subsets and most volatile subsets in the fluctuation in the circuit monitor signals.

#### 4. Experimental Simulation and Analysis

The MF-DFA method can be used to evaluate the fault feature extraction based on six parameters of the multifractal spectrum. To assess the performance of MF-DFA for soft fault signal analysis of nonlinear analog circuits, an experimental fault diagnosis simulation of chaotic circuits previously described in the literature [16] is conducted. The structure and parameters of the circuit are shown in Figure 1.