Fault Diagnosis of Nonlinear Analog Circuit Based on Generalized Frequency Response Function and LSSVM Classifier Fusion

For fault diagnosis of nonlinear analog circuit, a novel method based on generalized frequency response function (GFRF) and least square support vector machine (LSSVM) classiﬁer fusion is presented. The sinusoidal signal is used as the input of analog circuit, and then, the generalized frequency response functions are estimated directly by the time-domain formulations. The discrete Fourier transform of measurement data is avoided. After obtaining the generalized frequency response functions, the amplitudes of the GFRFs are chosen as the fault feature parameters. A classiﬁer fusion algorithm based on least square support vector machine (LSSVM) is used for fault identiﬁcation. Two LSSVM multifault classiﬁers with diﬀerent kernel functions are constructed as subclassiﬁers. Fault diagnosis experiments of resistor-capacitance (RC) circuit and Sallen Key ﬁlter are carried out, respectively. The results show that the estimated GFRFs of the circuit are accurate, and the fault diagnosis method can get high recognition rate.


Introduction
Analog circuits are widely used in various electronic systems and play a very important role. When the fault occurred, they will affect the performance of the relevant electronic systems and even make the systems completely invalid. erefore, it is of great significance to research the fault diagnosis of analog circuits. In order to study the problem of fault diagnosis of analog circuits, a variety of diagnosis methods have been proposed [1][2][3]. A wavelet neural network diagnosis model is presented to improve the diagnosis capability of soft fault for analog circuit, and the adaptive unscented Kalman filter is used to optimize the parameters of wavelet neural network [4]. e fuzzy classifier is constructed to diagnose the faults of analog electronic circuit, and the simulation before test approach is used to obtain signature parameters by analyzing the frequency response [5]. By using kurtosis and entropy as feature information, the diagnosis of analog circuit is studied based on artificial neural network [6]. A modular fault diagnosis system for analog electronic circuits is designed based on wavelet transform and neural network [7]. e features are obtained by wavelet fractal analysis and kernel principal component analysis, and novel linear ridgelet network approach is used for analog circuit fault diagnosis [8]. e relationship between the input and output of analog circuit is very complex, which is often nonlinear. ere are still many problems in fault diagnosis of nonlinear analog circuits, especially in feature information acquisition and fault identification.
Volterra series is an important mathematical tool for theoretical analysis and practical application of nonlinear systems. Volterra frequency-domain kernel, also known as generalized frequency response function, is an extension of frequency response function of linear system in nonlinear system. It provides a description method of frequencydomain characteristics of nonlinear system, which can reflect the inherent characteristics of the system [9][10][11][12][13][14]. e generalized random signal is used as the input, and the first two-order GFRFs of the nonlinear system are solved according to the input spectrum and the output spectrum [15]. e measurement of Volterra frequency-domain kernel for nonlinear dynamic 3×3 multi-input multi-output system is studied, and the characteristic of Volterra frequency-domain kernel is analyzed [16]. In [17], a new complex orthogonal least square algorithm is used to truncate the Volterra series and estimate the generalized frequency response function. A novel method for highorder frequency response function measurement of nonlinear system is presented based on cross-correlation technology, and the system input is sinusoidal signal [18]. For the prediction of nonlinear aeroelastic vibration response and flutter, a frequency component identification method based on correlation analysis is used to estimate the GFRFs [19]. In [20], a two-step strategy is used to estimate the generalized frequency response function of the nonlinear controlled plant in the closed-loop state. For the above estimation methods, the input and output data need to be Fourier-transformed in the process of solving GFRFs. An estimation method based on time-domain data is introduced to calculate GFRFs of single-input single-output nonlinear systems [21]. It does not need Fourier transform for time-domain data and reduces the computational complexity. e feature information in frequency domain is obtained by the generalized frequency response function, which can be used for fault diagnosis of nonlinear system. In [22], a new fault diagnosis method for nonlinear analog circuit based on simplified estimation of GFRFs and hybrid-kernel least squares support vector machine is proposed.
After obtaining the frequency-domain characteristic information of nonlinear system by using the generalized frequency response function, it is very important to identify the fault accurately according to the data. As an effective machine learning method, support vector machine can be used to study classification problems [23,24]. It can realize small sample learning and has good generalization ability. It has been used in fault diagnosis of many systems [25][26][27][28]. In [29], for the real-time fault diagnosis of high-voltage circuit breakers, fault detection is carried out by adaptive KPCA, and then fault recognition is completed by SVM classifier. e mechanical fault diagnosis of high voltage circuit breaker is studied by combining genetic algorithm with support vector machine [30]. e wavelet theory is used to obtain feature information, and then the support vector machine is used to detect bearing fault of asynchronous motor [31]. e state of asynchronous motor is monitored based on Hilbert-Park transform and support vector machine [32]. By combining support vector machine and multisensor data fusion method, the fault diagnosis of the motor is studied [33]. In [34], the statistical features are classified by support vector machine, and the fault diagnosis of the hydraulic brake system is carried out. In [35], a modified support vector machine classifier is used to analyze the fault of high voltage circuit breaker based on the waveform of coil current and travel waveforms. A twostage scheme based on support vector machine is presented for the fault detection and classification of electrical pitch drives in offshore wind farms [36]. e least squares support vector machine is proposed by changing the risk function of support vector machine [37]. e training of LSSVM only needs to solve a system of linear equations, which reduces the computational complexity.
In this paper, a new fault diagnosis method of nonlinear analog circuit is proposed based on generalized frequency response function and LSSVM classifier fusion algorithm. In order to simplify calculation, the sinusoidal signal is used as the input, and the time-domain measurements are used to directly estimate the GFRFs. After obtaining the generalized frequency response functions, their amplitudes of GFRFs are chosen as the feature parameters of analog circuits. Two multifault classifiers are constructed as subclassifier using least square support vector machine, where the kernel functions are polynomial function and Gaussian radial basis function, respectively. By using a classifier fusion algorithm based on the binary LSSVM classification function of the subclassifiers, the diagnosis result can be derived. Two experiments for the fault diagnosis of RC circuit and Sallen Key filter verified the effectiveness of the proposed method.

Simplified Estimation of GFRFs Using Time-Domain Measurements
e Volterra series is an extension of Taylor series and is widely used to describe and analyze nonlinear systems. A single-input single-output nonlinear system can be expressed as a Volterra series: where u(t) ∈ R is the input, y(t) ∈ R is the output, and h n (τ 1 , τ 2 , . . . , τ n ) is the nth order Volterra kernel. e multidimensional Fourier transform of the nth order Volterra kernel h n (τ 1 , τ 2 , . . . , τ n ) is given by where H n (jω 1 , jω 2 , . . . , jω n ) represents the nth-order Volterra frequency-domain kernel, also known as the nthorder generalized frequency response function, which is a mapping form of the nth-order Volterra kernel in the frequency domain. e frequency-domain output of a single-input singleoutput nonlinear system can be expressed as a generalized frequency response function: where Y(jω) is the frequency-domain output of the system, Y n (jω) is the nth order frequency-domain output of the system, and U(·) is the input spectrum. e discrete form of the single-input single-output nonlinear system described by equations (1) and (2) can be expressed as If a weakly nonlinear system can be described by the first-order Volterra series, the frequency-domain form of the discrete Volterra model of the system is given by where H 1 (ω) is the first-order generalized frequency response function, H 2 (p, q) is the second order generalized frequency response function, and U(·) is the input spectrum. Let the input be a sinusoidal signal u(t) � A cos ωt, and then, the output of the nonlinear system can be expressed as [21] where Re(·) is the real part of a complex number, and e(k) is the zero mean Gaussian white noise. Define where R 1 (ω) and I 1 (ω) represent the real part and imaginary part of H 1 (ω), respectively, R 2 (ω, ω) and I 2 (ω, ω) represent the real part and imaginary part of H 2 (ω, ω), respectively, and R 0 is a constant. From the equations (7)∼(10), the following can be obtained: For k � 1, 2, . . . , N, equation (11) can be written as the matrix form: Equation (12) can be solved by using standard least squares principle: where θ � [R 1 , I 1 , R 2 , I 2 , R 0 ] T . e generalized frequency response function can reflect the essential characteristics of nonlinear systems. For a nonlinear system, the GFRF is different in different fault modes. erefore, fault diagnosis of nonlinear analog circuits can be realized by using GFRF to obtain characteristic information. After estimating the generalized frequency response functions of the nonlinear analog circuit by equation (14), their amplitudes are chosen as the feature parameters for fault diagnosis in this paper.

Multifault Classifier Design Based on LSSVM Classifier Fusion
Support vector machine is a machine learning method developed on the basis of statistical learning theory. It uses the principle of structural risk minimization to improve the Mathematical Problems in Engineering generalization ability, which can solve the problems of small samples and local minima and can effectively classify data. However, the support vector machine needs to solve the quadratic programming problem so that the calculation is very complicated. Least squares support vector machine is a deformed support vector machine algorithm. e training of LSSVM only requires solving a system of linear equations, which is easy to solve and has high training efficiency. Let the training sample data be S 1 : { } is the class label, and M is the sample size. e binary classification problem of least squares support vector machine can be described as where ω is the weight vector of the classification hyperplane, C > 0 is the penalty factor, ξ i is the relaxation variable, φ(·) is the nonlinear mapping, and b is the classification threshold. Define the Lagrange function as where α i ≥ 0 is the Lagrange multiplier.
Letting the partial derivatives of the Lagrange function for ω, b, ξ i and α i be zero, the constraint optimization problem of LSSVM can be transformed into a system of linear equations: where Y � [y 1 , y 2 , · · · , y M ] T , where x is the sample vector to be classified. For the classification of nonlinear data, the SVM classifier maps the feature data to a high-dimensional feature space using a nonlinear function and then completes the classification. erefore, when using the least squares support vector machine to construct a classifier, the choice of kernel function is very important and will directly affect the classification result. Commonly used kernel functions include polynomial kernel function and Gaussian radial basis kernel function. When traditional LSSVM deals with the multiclassification problem, one type of function is selected as the kernel function, and one classifier is constructed for classification. Because different kernel functions have different characteristics, their classification performance for the same feature data is also different.
In order to improve the fault recognition rate, a LSSVM classifier fusion method is adopted to identify faults in nonlinear analog circuits. Polynomial kernel function and Gaussian radial basis kernel function are used to construct two LSSVM classifiers as subclassifiers.
en, the two subclassifiers are fused to obtain the diagnosis result. e polynomial kernel function is a global kernel function, and the Gaussian radial basis kernel function is a local kernel function. erefore, this LSSVM fusion algorithm can take into account the global characteristics and local characteristics of the feature data obtained through the generalized frequency response functions of the nonlinear analog circuits.
e structures of the designed LSSVM subclassifiers are "one-against-one," and the voting method is used for fault decisions of subclassifiers. e structure of nonlinear circuit fault diagnosis based on LSSVM fusion is shown in Figure 1, where N is the number of categories of data to be classified. e fusion strategy of support vector machine has a great impact on the accuracy of the classifier. erefore, choosing an appropriate fusion algorithm is very important for fault identification.
According to decision function (18) of the binary SVM, define e function y ′ (x) represents the distance from sample x to the optimal hyperplane, which can indirectly reflect the degree of support for decision results. In this paper, the weight coefficients are obtained on the basis of the function y ′ (x) of each binary SVM in the subclassifier, and then the weighted fusion is performed to obtain the fault identification result.
Assume that the fault modes of the nonlinear analog circuit are f 1 , f 2 , . . . , f N , N is the number of modes, the obtained GFRFs amplitude data of the nonlinear analog circuit is x, and the decision results of the two LSSVM subclassifiers are D 1 and D 2 . Define the weight of the LSSVM subclassifier as where j � 1, 2, M is the training sample capacity for each binary LSSVM classifier in subclassifier j, V j is the number of binary LSSVM classifiers with a decision result of D j , α jki and y jki are the Lagrangian multiplier and class label of the k th binary LSSVM classifier in subclassifier j, and K jk (x, x jki ) and b jk are the kernel function and the classification threshold of the K-th binary LSSVM classifier in the subclassifier j. By fusing the decision results of the two subclassifiers, the fault identification result is given by where Q⊆ f 1 , f 2 , . . . , f N is the set of fault modes formed by the decision results of the two LSSVM subclassifiers. e result of subclassifier is given by where D jm is the decision result of the m th binary LSSVM classifier of subclassifier j, and p � N((N − 1)/2). After obtaining the frequency-domain feature information of the nonlinear analog circuit based on the generalized frequency response functions, the fusion algorithm of least square support vector machine is used for fault diagnosis. e fault diagnosis steps are as follows:

Experiment and Analysis
Two experiments of nonlinear analog circuits are used to verify the effectiveness of the diagnosis method. Figure 2, where u s is the supply voltage, R 1 and R 2 are the resistances, and C is the capacitance. e relationship between voltage and current of the nonlinear resistance R 2 is i 2 � a 1 u + a 2 u 2 , where a 1 � 1 × 10 − 3 and a 2 � 5 × 10 − 4 .

Resistor-Capacitance (RC) Circuit. A resistor-capacitance circuit is shown in
According to Kirchhoff's voltage law and current law, we can obtain where i 3 (t) � C(du(t)/dt). Let the power supply voltage u in (t) be the input and let the voltage u(t) on R 2 be the output. e nonlinear differential equation can be obtained as  Mathematical Problems in Engineering e nominal values of components in the analog circuit system are C � 10 − 4 F and R 1 � 10 3 Ω. e simulation experiment has been performed using MATLAB R2014a. e CPU clock speed of the computer is 2.3 GHz, and the main memory is 8 GB.
Let the input signal be u in (t) � cos 2πt. e sampling frequency is 100 Hz, and sampling length is 10 s. e output spectrum of the nonlinear circuit is shown in Figure 3. ere are significant peaks at frequencies 1 Hz and 2 Hz, which are 0.48 and 0.03, respectively. e output of analog circuit has obvious second harmonics so that the circuit has obvious nonlinear characteristics. Because the harmonic amplitudes above the third harmonic are approximately equal to 0, the analog circuit can be described by the first two-order generalized frequency response functions.
For the nonlinear circuit system described by Equation (26), the actual value of GFRFs can be derived [38]: When the output noise is Gaussian white noise with signal-to-noise ratios (SNR) of 35 dB, 30 dB, and 25 dB, respectively, the simplified estimation method is used to obtain the generalized frequency response functions of the analog circuit. e output of the nonlinear circuit is shown in Figure 4. As the SNR gradually decreases, the deformation of the output becomes more and more obvious. e estimation results are shown in Table 1, where ω � 2π. Under different signal-to-noise ratios, the GFRFs of nonlinear analog circuit can be accurately estimated. As the signal-to-noise ratio decreases, the estimation error gradually increases.
Assume that, under normal conditions, the variation ranges of C and R 1 are within 5%. If the actual value of C or R 1 differs from the nominal value by 50%, the component will fail. e analog circuit contains five fault modes, and the fault description is shown in Table 2.
e Monte Carlo method is used for fault diagnosis simulation experiment of the analog circuit. e output noise is Gaussian white noise, and the SNR is 25 dB. 200 sets of input and output sample data of the analog circuit are collected for each fault. e amplitudes of the GFRFs H 1 (ω), H 2 (ω, ω) and H 2 (ω, − ω) are solved by the proposed simplified estimation method. e amplitude distributions of the five fault types are shown in Figure 5. e 100 sets of sample data are used as training samples, and the remaining 100 sets are used as test samples. e fault diagnosis of the analog circuit is carried out by the LSSVM fusion algorithm and the traditional single LSSVM multiclassification algorithm, respectively. e order of the polynomial function is 2, and the width of the Gaussian radial basis function is 4 × 10 − 3 . e fault diagnosis steps of the LSSVM fusion algorithm are shown in Section 3, and the results are obtained using Step 1 to Step 4. e fault recognition rate is shown in Table 3. e recognition rates of the LSSVM fusion algorithm are higher than those of the traditional single LSSVM classifiers. Figure 6, where R 1 � 1 kΩ, R 2 � 3 kΩ, R 3 � 2 kΩ, R 4 � 4 kΩm, R 5 � 4 kΩ, C 1 � 5 nF, C 2 � 5 nF, and the input voltage is u � cos 2000πtV.

Sallen Key Filter. A Sallen Key filter circuit is shown in
Assuming that the actual value of R 4 or R 5 differs from the nominal value by 50%, the component will fail. ere are five fault modes, normal (F0), R4 increased by 50% (F1), R4 decreased by 50% (F2), R5 increased by 50% (F3), and R5 decreased by 50% (F4). e Monte Carlo method is used for fault diagnosis experiment of the Sallen Key filter circuit, and the tolerances of the circuit components are 5%. e input and output data of the circuit are obtained using Multisim10. e diagnosis algorithm has been performed using MAT-LAB R2014a. e output noise is Gaussian white noise, and the SNR is 40 dB. e sampling frequency is 105 Hz, and sampling length is 10 ms. e output of the Sallen Key filter in normal state is shown in Figure 7.
200 sets of input and output sample data of the analog circuit are collected for each fault. e amplitudes of the GFRFs H 1 (ω), H 2 (ω, ω) and H 2 (ω, − ω) are solved by the proposed simplified estimation method, where ω � 2000π. Figure 2: A resistor-capacitance circuit.         Mathematical Problems in Engineering e fault recognition rate is shown in Table 4. e recognition rates of the LSSVM fusion algorithm are significantly higher than those of the single LSSVM and multikernel LSSVM classifiers. erefore, the proposed diagnosis method for nonlinear analog circuit based on the generalized frequency response function and LSSVM fusion algorithm has good diagnostic performance.

Conclusions
According to generalized frequency response function and LSSVM classifier fusion, a fault diagnosis method for nonlinear analog circuits is presented in this paper. e sinusoidal signal is used as input, and the first two-order generalized frequency response functions of the circuit system are directly estimated by the time-domain measurements.
e amplitudes of the generalized frequency response functions are used as the feature parameters. In order to improve the accuracy of diagnosis, a classifier fusion algorithm based on LSSVM is presented for fault identification. e polynomial kernel function and Gaussian radial basis function are used to construct two different LSSVM classifiers, and then the decision fusion is carried out by the binary LSSVMs of these two classifiers. Two fault diagnosis experiments for RC circuit and Sallen Key filter are used to verify the effectiveness of the proposed method. e results show that the method can achieve good fault diagnosis performance.
For fault diagnosis of complex circuits in real time applications, the diagnosis can be carried out by the proposed method only when the input signal is sinusoidal. Furthermore, the fault diagnosis of analog circuit under the excitation of multitone signal will be studied.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.