Analog-circuit fault diagnosis method based on continuous wavelet analysis and ELM neural network

Abstract

An analog-circuit fault diagnosis method based on continuous wavelet analysis and an ELM network comprises: data acquisition: performing data sampling on output responses of an analog circuit respectively through Multisim simulation to obtain an output response data set; feature extraction: performing continuous wavelet analysis by taking the output response data set of the circuit as training and testing data sets respectively to obtain a wavelet time-frequency coefficient matrix, dividing the coefficient matrix into eight sub-matrixes of the same size, and performing singular value decomposition on the sub-matrixes to calculate a Tsallis entropy for each sub-matrix to form feature vectors of corresponding faults; and fault classification: submitting the feature vector of each sample to the ELM network to implement accurate and quick fault classification. The method of the invention has a better effect on extracting the circuit fault features and can be used to implement circuit fault classification accurately and efficiently.

Claims

1. An analog-circuit fault diagnosis method based on a continuous wavelet analysis and an ELM neural network, comprising three steps of data acquisition, feature extraction and fault classification, wherein the step of data acquisition comprises: performing a data sampling on an output end of an analog circuit to obtain an output response data set, wherein the step of feature extraction comprises: performing a continuous wavelet transform by taking the output response data set as a training data set and a testing data set respectively to obtain a wavelet time-frequency coefficient matrix of fault signals; dividing the wavelet time-frequency coefficient matrix into eight sub-matrixes of the same size; performing a singular value decomposition on the sub-matrixes to obtain a plurality of singular values of each sub-matrix; and calculating Tsallis entropy values for the plurality of singular values of each sub-matrix, wherein the Tsallis entropy values form corresponding circuit response fault feature vectors, and wherein the step of fault classification comprises: inputting the circuit response fault feature vectors into the ELM neural network to implement a fault classification for the analog circuit; wherein the eight sub-matrixes obtained by dividing the wavelet time-frequency coefficient matrix are represented by the following formula: $\begin{array}{cc}{W}_x\left(\tau ,a\right)={\left[\begin{array}{cccc}{\left(B_1\right)}_{\frac{m}{2}\times \frac{n}{4}}& {\left(B_2\right)}_{\frac{m}{2}\times \frac{n}{4}}& {\left(B_3\right)}_{\frac{m}{2}\times \frac{n}{4}}& {\left(B_4\right)}_{\frac{m}{2}\times \frac{n}{4}}\\ {\left(B_5\right)}_{\frac{m}{2}\times \frac{n}{4}}& {\left(B_6\right)}_{\frac{m}{2}\times \frac{n}{4}}& {\left(B_7\right)}_{\frac{m}{2}\times \frac{n}{4}}& {\left(B_8\right)}_{\frac{m}{2}\times \frac{n}{4}}\end{array}\right]}_{m\times n},& \left(3\right)\end{array}$ wherein W.sub.x(τ,a) represents an m×n-dimension wavelet time-frequency coefficient matrix, and B.sub.1,B.sub.2,B.sub.3,B.sub.4,B.sub.5,B.sub.6,B.sub.7, B.sub.8 represent the eight sub-matrixes obtained through division; wherein the plurality of singular values obtained by performing the singular value decomposition on the sub-matrixes are represented by the following formula:
B.sub.c×d=U.sub.c×lA.sub.l×lV.sub.l×d  (4), wherein B.sub.c×d represents c×d-dimension sub-matrixes obtained after the division via the formula (3), U.sub.c×l represents a c×l-dimension left singular matrix, V.sub.l×d represents an l×d-dimension right singular matrix, and a plurality of principal diagonal elements λ.sub.i (i=1, 2, . . . ,l) of A.sub.l×l are the singular values of B.sub.c×d with λ.sub.1≥λ.sub.2≥ . . . ≥λ.sub.l≥0, wherein l is the number of non-zero singular values, the step of calculating the Tsallis entropy values for the singular values of each sub-matrix is represented by the following formula: $\begin{array}{cc}{W}_{\mathrm{TSE}}=\frac{c}{q-1}\left(1-\underset{i=1}{\overset{l}{.Math.}}\Delta P_i^q\right),q\in R,& \left(5\right)\end{array}$ wherein W.sub.TSE represents the Tsallis entropy values as calculated, $\Delta P_i=-\left(\frac{{\lambda }_i}{\underset{j=1}{\overset{l}{.Math.}}{\lambda }_j}\right)\log \left(\frac{{\lambda }_i}{\underset{j=1}{\overset{l}{.Math.}}{\lambda }_j}\right),$ q represents a non-extensive parameter, and R represents a real number, with c=1 and q=1.2, wherein the Tsallis entropy values of the singular values of respective sub-matrixes as calculated with the formula (5) are combined together to form the corresponding circuit response fault feature vectors.

2. The analog-circuit fault diagnosis method based on the continuous wavelet analysis and the ELM neural network according to claim 1, wherein the wavelet time-frequency coefficient matrix is obtained from the following formula:
W.sub.x(τ,a)=√{square root over (a)}∫.sub.−∞.sup.−∞x(t)φ(a(t−τ))dt=<x(t),φ.sub.T,a(t)>  (1) wherein W.sub.x(τ, a) represents a continuous wavelet transform time-frequency coefficient matrix of a signal x(t), τ and a represent a time parameter and a frequency parameter for the continuous wavelet transform respectively, with a>0, a(t−τ) represents the relation between the expansion and contraction of a wavelet mother function on the frequency axis and a translation on the time axis, φ(t) represents a wavelet generating function, and φ.sub.τ,a(t) represents a wavelet basis function which is a set of function series formed by dilation and translation of the wavelet generating function φ(t) and satisfies the following formula: $\begin{array}{cc}{\phi }_{\tau ,a}\left(t\right)=\frac{1}{\sqrt{a}}\phi \left(\frac{t-\tau }{a}\right).& \left(2\right)\end{array}$

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a flow chart of a fault diagnosis method;

(2) FIG. 2 is a circuit diagram of a four-operation-amplifier low-pass filter;

(3) FIG. 3 is a structural diagram of an ELM neural network;

(4) FIG. 4(a) is a diagram showing the Tsallis entropy fault features of sub-matrixes B.sub.1,B.sub.2,B.sub.3,B.sub.4 of a four-operation-amplifier low-pass filter;

(5) FIG. 4(b) is a diagram showing the Tsallis entropy fault features of sub-matrixes B.sub.5,B.sub.6,B.sub.7,B.sub.8 of the four-operation-amplifier low-pass filter; and

(6) FIG. 5 is a fault classification diagram of the four-operation-amplifier low-pass filter.

DETAILED DESCRIPTION OF THE INVENTION

(7) The invention will be further described in detail below in conjunction with the accompanying drawings and particular embodiments.

(8) 1. Fault Diagnosis Method

(9) As shown in FIG. 1, the specific steps of the analog-circuit fault diagnosis method based on a continuous wavelet transform and an ELM neural network are as follows:

(10) data acquisition: performing a data sampling on output responses of an analog circuit through Multisim simulation to obtain an output response data set;

(11) feature extraction: performing a continuous wavelet transform by taking the output response data set of the circuit obtained through simulation, as a training data set and a testing data set to obtain a wavelet time-frequency coefficient matrix of fault signals, dividing the coefficient matrix into eight sub-matrixes of the same size, performing a singular value decomposition on the respective sub-matrixes to obtain singular values, and calculating a Tsallis entropy for the singular values of each sub-matrix, the Tsallis entropy values form corresponding circuit response fault feature vectors; and

(12) fault classification: inputting the circuit response fault feature vectors into an ELM network to implement the accurate and quick fault classification.

(13) The core technologies, i.e., continuous wavelet analysis, singular value decomposition, Tsallis entropy and ELM neural network, in the fault diagnosis method of the invention will be further illustrated in detail below.

(14) 1.1 Continuous Wavelet Analysis

(15) The continuous wavelet analysis originates from wavelet analysis. Continuous wavelets are characterized by continuously changing scales, and capable of more finely describing the local form of a signal. Continuous wavelet transform coefficients of a circuit response x(t) can be represented with the formula below: (1),

(16) here, W.sub.x(τ, a) is a continuous wavelet transform time-frequency coefficient matrix; τ is a time parameter, and a is a frequency parameter, with a>0; a(t−τ) represents the relation between the expansion and contraction of the wavelet mother function on the frequency axis and the translation on the time axis, φ(t) is a wavelet generating function; φ.sub.τ,a (t) is a wavelet basis function which is a set of function series formed by dilation and translation of the wavelet generating function φ(t), that is,

(17) $\begin{array}{cc}{\phi }_{\tau ,a}\left(t\right)=\frac{1}{\sqrt{a}}\phi \left(\frac{t-\tau }{a}\right).& \left(2\right)\end{array}$

(18) The continuous wavelet transform maps the signals to a time-frequency plane by means of the continuously changing time and scale, and the coefficient matrix W.sub.x(τ, a) measures the level of similarity between the signals and wavelets, reflecting the feature information of the signals.

(19) 1.2 Singular Value Decomposition and Tsallis Entropy Calculation

(20) First, the time-frequency coefficient matrix W.sub.x(τ,a) obtained is equally divided into eight fractions, that is, the eight sub-matrixes B.sub.1,B.sub.2,B.sub.3,B.sub.4,B.sub.5,B.sub.6,B.sub.7,B.sub.8 are obtained according to

(21) $\begin{array}{cc}{W}_x\left(\tau ,a\right)={\left[\begin{array}{cccc}{\left(B_1\right)}_{\frac{m}{2}\times \frac{n}{4}}& {\left(B_2\right)}_{\frac{m}{2}\times \frac{n}{4}}& {\left(B_3\right)}_{\frac{m}{2}\times \frac{n}{4}}& {\left(B_4\right)}_{\frac{m}{2}\times \frac{n}{4}}\\ {\left(B_5\right)}_{\frac{m}{2}\times \frac{n}{4}}& {\left(B_6\right)}_{\frac{m}{2}\times \frac{n}{4}}& {\left(B_7\right)}_{\frac{m}{2}\times \frac{n}{4}}& {\left(B_8\right)}_{\frac{m}{2}\times \frac{n}{4}}\end{array}\right]}_{m\times n}.& \left(3\right)\end{array}$
According to the theory of singular value decomposition, the sub-matrixes are decomposed as follows:
B.sub.c×d=U.sub.c×lA.sub.l×lV.sub.l×d  (4),

(22) here, B.sub.c×d represents c×d-dimension sub-matrixes obtained after the division by the formula (3), U.sub.c×l represents the c×l-dimension left singular matrix, V.sub.l×d represents the l×d-dimension right singular matrix, and the principal diagonal elements λ.sub.i(i=1,2, . . . ,l) of a diagonal matrix A.sub.l×l are the singular values of B.sub.c×d with λ.sub.1≥λ.sub.2≥ . . . ≥λ.sub.l≥0, l is the number of non-zero singular values of the matrix B.sub.c×d.

(23) Said calculating a Tsallis entropy for the singular values of each sub-matrix is represented by the following formula:

(24) $\begin{array}{cc}{W}_{\mathrm{TSE}}=\frac{c}{q-1}\left(1-\underset{i=1}{\overset{l}{.Math.}}\Delta P_i^q\right),q\in R,\mathrm{here},\Delta P_i=-\left(\frac{{\lambda }_i}{\underset{j=1}{\overset{l}{.Math.}}{\lambda }_j}\right)\log \left(\frac{{\lambda }_i}{\underset{j=1}{\overset{l}{.Math.}}{\lambda }_j}\right),& \left(5\right)\end{array}$
q is a non-extensive parameter, and R represents a real number, with C=1 and q=1.2.

(25) The Tsallis entropy values of the singular values of respective sub-matrixes as calculated with the formula (5) are combined together to form corresponding circuit response fault feature vectors.

(26) 1.3 ELM Neural Network

(27) The extreme learning machine is a new neural network based on single-hidden layer feed forward networks, which have been widely applied in practice due to their high learning speed and simple network structure. Research has shown that for the single-hidden layer feed forward networks, there is no need to either adjust the randomly initialized w.sub.i and b.sub.i or deviate the output layer as long as an excitation function g(s) is infinitely derivable in any real number interval, the output weight value βi is calculated with a regularization principle to approach any continuous system, and there is almost no need to learn.

(28) The ELM neural network lacks the output layer deviation, moreover, the input weight w.sub.i and the hidden layer deviation b.sub.i are generated randomly and need no adjustment, only the output weight β.sub.i in the whole network needs to be determined.

(29) For each neuron in FIG. 3, the output of the ELM neural network can be uniformly represented in model as follows:

(30) $\begin{array}{cc}{f}_L\left(s\right)=\underset{i=1}{\overset{L}{.Math.}}{\beta }_{i}g\left(s\right)\left(w_i.Math.s_i+b_i\right),& \left(6\right)\end{array}$

(31) here, s.sub.i=[s.sub.i1,s.sub.i2, . . . ,s.sub.ip].sup.TϵR.sup.p,w.sub.iϵR.sup.p, β.sub.iϵR.sup.q, S is an input feature vector; p is the number of network input nodes, that is the dimension of the input feature vector; q is the number of network output node; L represents hidden layer nodes; and g(s) represents an excitation function. w.sub.1=[w.sub.i1, w.sub.i2, . . . , w.sub.ip].sup.T represents the input weights from the input layer to the i th hidden layer node; b.sub.i represents the deviation of the i th hidden node; and β.sub.i=[β.sub.i1, β.sub.i2, . . . , β.sub.iq].sup.T is the output weight connecting the i th hidden layer node.

(32) When the feed forward neural network having L hidden layer nodes can approach the sample with zero error, then w.sub.i, b.sub.i and β.sub.i exist, allowing:

(33) $\begin{array}{cc}{f}_L\left(s\right)=\underset{i=1}{\overset{L}{.Math.}}{\beta }_{i}g\left(s\right)\left(w_i.Math.s_i+b_i\right)=y_i,i=1,2,\mathrm{.Math.},L,& \left(7\right)\end{array}$
here, y.sub.i=[y.sub.i1, y.sub.i2, . . . , y.sub.ip].sup.TϵR.sup.q represents the output of the network.

(34) The formula (7) can be simplified into Hβ=Y, wherein,

(35) 0$H\left(w_1,\mathrm{.Math.},w_L;b_1,\mathrm{.Math.},b_L;s_1,\mathrm{.Math.},s_p\right)={.Math.\begin{array}{ccc}g\left(a_1.Math.s_1+b_1\right)& \mathrm{.Math.}& g\left(a_L.Math.s_1+b_L\right)\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ g\left(a_1.Math.s_p+b_1\right)& \mathrm{.Math.}& g\left(a_L.Math.s_p+b_L\right)\end{array}.Math.}_{p\times L}=Y={.Math.\begin{array}{c}{y}_1^T\\ \mathrm{.Math.}\\ y_L^T\end{array}.Math.}_{p\times q}$

(36) here, p is the number of network input nodes, that is, the dimension of the input feature vector; q is the number of network output node;

(37) H represents the hidden layer output matrix of the network, and the output weight matrix can be obtained from the formula below:
β=H.sup.+Y  (8),

(38) here, H.sup.+ is a Moore-Penrose generalized inverse matrix of H.

(39) 2. Exemplary Circuit and Method Application:

(40) FIG. 2 shows a four-operation-amplifier biquad high-pass filter, with the nominal values of respective elements marked in the figure. By taking this circuit as an example, the whole process flow of the fault diagnosis method provided by the invention is demonstrated, where pulse waves with the duration of 10 us and the amplitude of 10V are used as an excitation source, and fault time-domain response signals are obtained at the output end of the circuit. The tolerance range of the circuit elements is set as 5%.

(41) R1, R2, R3, R4, C1 and C2 are selected as test objects, and Table 1 gives the fault code, fault type, nominal value and fault value of each circuit element under test, where ↑ and ↓ represent being above and below the nominal value respectively, and NF represents no fault. 60 data are sampled for each of the fault types respectively and divided into two parts, the former 30 data are used to establish the ELM neural network fault diagnosis model based on continuous wavelet transform, and the latter 30 data are used to test the performance of this fault diagnosis model.

(42) TABLE-US-00001 TABLE 1 Fault code, fault type, nominal value and fault value Fault Code Fault Type Nominal Value Fault Value F0 NF F1 R1↓ 6200Ω 3000Ω F2 R1↑ 6200Ω 15000Ω  F3 R2↓ 6200Ω 2000Ω F4 R2↑ 6200Ω 18000Ω  F5 R3↓ 6200Ω 2700Ω F6 R3↑ 6200Ω 12000Ω  F7 R4↓ 1600Ω  500Ω F8 R4↑ 1600Ω 2500Ω F9 C1↓ 5 nF 2.5 nF F10 C1↑ 5 nF  10 nF F11 C2↓ 5 nF 1.5 nF F12 C2↑ 5 nF  15 nF

(43) Data Acquisition:

(44) In the four-operation-amplifier biquad high-pass filter, the applied excitation response is a pulse sequence with the amplitude of 10V and the duration of 10 us. The output response of the circuit under different fault modes is subjected to Multisim simulation.

(45) Feature Extraction:

(46) The continuous wavelet transform is used below to analyze the output responses of the circuit, where the complex Morlet wavelet is selected as the wavelet basis for wavelet analysis. The output response coefficient matrix obtained is divided into eight sub-matrixes, which are then subjected to singular value decomposition according to the formulae (4) and (5) for calculating Tsallis entropy features.

(47) As is known, the greater the feature value difference among different faults or between the faults and the normal status, the more significant the signal response difference among different faults or between the faults and the normal status, and the more beneficial this fault feature is to the fault diagnosis. As can be known from FIGS. 4(a) and 4(b), the numeric difference between the response features of the circuit in fault and the response features of the circuit in normal status as obtained with the method of the invention, as well as the numeric differences of the features of the circuit under different fault modes are significant, which fully demonstrate the effectiveness of the fault feature extraction of the invention.

(48) Fault Classification:

(49) The Tsallis entropy feature set obtained is divided into two parts, i.e., a training set and a testing set. The training set is input into the ELM neural network to train the ELM classifier model, and after the completion of the training, the testing set is input into the ELM classifier model, with the fault diagnosis results as shown in FIG. 5. The ELM classifier model successfully identifies all the faults, with the overall success rate up to 100% for the fault diagnosis.