This paper presents a novel fault diagnosis method for analog circuits using ensemble empirical mode decomposition (EEMD), relative entropy, and extreme learning machine (ELM). First, nominal and faulty response waveforms of a circuit are measured, respectively, and then are decomposed into intrinsic mode functions (IMFs) with the EEMD method. Second, through comparing the nominal IMFs with the faulty IMFs, kurtosis and relative entropy are calculated for each IMF. Next, a feature vector is obtained for each faulty circuit. Finally, an ELM classifier is trained with these feature vectors for fault diagnosis. Via validating with two benchmark circuits, results show that the proposed method is applicable for analog fault diagnosis with acceptable levels of accuracy and time cost.
Numerous researches have indicated that analog circuit fault diagnosis is a significant fundamental for design validation and performance evaluation in the integrated circuit manufacturing fields [
The procedure of fault diagnosis for analog circuits can be generally classified into four stages: data acquisition, feature extraction, fault detection, and fault identification and isolation. As one of the foremost stages in fault diagnosis, feature extraction methods are closely related to the efficiency of fault diagnosis. Many feature extraction methods have been proposed such as correlation function technique [
However, there are some problems which should be considered and solved in feature extraction. Firstly, how to select features to train classifiers should be considered because different features with different classifiers for analog fault diagnosis have different results. Secondly, we find that most of the aforementioned methods were validated with some discrete simulations data. That is, they only considered a CUT to be faulty when a component value is higher or lower than its nominal value by 50%. It means this method has low fault coverage. Thirdly, some methods should take the influence of tolerance and the continuity of faulty parameters into account.
In our work, therefore, we use the techniques of EEMD, kurtosis, and relative entropy to construct new feature vectors to train an ELM classifier to improve the diagnosability and reduce time cost. As an adaptive time frequency data analysis method ensemble empirical mode decomposition (EEMD) is suitable for linear, nonlinear, and no-stationary signals [
As a consequence, in this paper, we decomposed impulse responses of a CUT into IMFs using EEMD method and then adopting kurtosis and relative entropy techniques to obtain feature vectors. These features vectors can be used for diagnosis of faulty components among various variation possibilities. For this purpose, a classifier is needed. We selected extreme learning machine (ELM) classifier because it is proven to have excellent generalization performance and low computational cost [
This paper is organized as follows: Section
In the work, we combined EEMD, relative entropy, and ELM to perform fault diagnosis of analog circuits. Fundamentals of EEMD, relative entropy, and ELM are introduced firstly as follows.
Ensemble empirical mode decomposition, based on empirical mode decomposition (EMD), is to solve the aliasing in time frequency distribution with Gaussian white noise [ It has the same number of extrema and zero crossing or has the difference no more than one between them. The mean value of the envelopes defined by the local maxima and minima is zero.
From (
Given a signal
Let
Repeat the above procedure
Subtract
The residue, which contains useful information, is considered as main signal and Steps
When the residue
From the procedure, we can see that IMFs represent the degree of oscillation of signal in amplitude and frequency. It means that these IMFs contain much time frequency information of the signal. Thus, the authors in [
Let
In order to accurately and quickly diagnose faults, in our work, extreme learning machine (ELM) is adopted. ELM is one kind of fast algorithm of single hidden-layer feedforward networks (SLFN) as shown in Figure
SLFN.
Suppose
The diagnostic procedure based on EEMD, relative entropy, and ELM is shown in Figure
Block diagram of fault diagnosis.
The procedure of feature extraction of the proposed method is described as follows.
Every fault (including fault-free status) of CUT is simulated in PSPICE. And the relevant output waveforms are obtained.
Decompose each waveform with EEMD into
Calculate kurtosis and relative entropy of each IMF.
Calculate total energy of each IMF by where Calculate probability distribution. According to [ where According to the relative entropy theory, the definition of relative entropy of each IMFs is where
Segments of IMF.
A feature vector for each fault can be given as
To verify the capacity of fault diagnosis with the proposed method, the first example circuit is a second-order Sallen-Key bandpass filter circuit, which is a benchmark circuit and is used as a CUT in [
Fault configuration for Sallen-Key filter circuit.
Fault ID | Faults | Nominal values | Faulty parameter ranges |
---|---|---|---|
F0 | No-fault | ||
F1 |
|
5.18k | [0.5, 4.92k] |
F2 |
|
5.18k | [5.44k, 51meg] |
F3 |
|
1k | [0.1, 0.95k] |
F4 |
|
1k | [1.05k, 10meg] |
F5 |
|
2k | [0.2, 1.9k] |
F6 |
|
2k | [2.1k, 20meg] |
F7 |
|
4k | [0.4, 3.8k] |
F8 |
|
4k | [4.2k, 40meg] |
F9 |
|
4k | [0.4, 3.8k] |
F10 |
|
4k | [4.2k, 40meg] |
F11 |
|
5n | [0.5p, 4.75n] |
F12 |
|
5n | [5.25n, 50u] |
F13 |
|
5n | [0.5p, 4.75n] |
F14 |
|
5n | [5.25n, 50u] |
Second-order Sallen-Key bandpass filter circuit.
According to the fault classes in Table
In order to close to the actual circuit characteristic, every fault class will be simulated 150 times in faulty parameter ranges using Monte Carlo analysis method in time domain and a total of 2250 corresponding impulse response waveforms are obtained. Some related waveforms are shown in Figure
Examples of waveforms of fault-free status and fault status for the Sallen-Key bandpass filter circuit. (a) It is fault-free waveform, and (b), (c), (d), (e), and (f) are fault waveforms for
The simulation data in PSpice are recorded and imported into Matlab, and then their feature vectors are constructed with kurtosis and relative entropy to train an ELM classifier. The detail is described as follows.
First, to construct the feature vectors, we decompose these stored responses data into IMF components with EEMD method based on the discussion in Section
EEMD decomposition results of two response signals. In the decomposition, noise for standard deviation 0.2 is added, and the ensemble number is 800. (a) The nominal response signal decomposition results; (b) EEMD results of the response signal of
Next, kurtosis of each IMF for a certain fault circuit is calculated. Meanwhile each IMF waveform (300
PDF of the nominal IMFs of fault-free circuit.
Fault name | IMFs | ||||||
---|---|---|---|---|---|---|---|
|
|
|
|
|
| ||
Fault-free |
|
0.9710 | 0.0061 | 0.0047 | 0.0058 | 0.0063 | 0.0060 |
|
0.9683 | 0.0316 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |
|
0.4853 | 0.4481 | 0.0657 | 0.0009 | 0.0000 | 0.0000 | |
|
0.1316 | 0.0892 | 0.3222 | 0.3515 | 0.0982 | 0.0073 | |
|
0.5588 | 0.3277 | 0.0419 | 0.0011 | 0.0228 | 0.0476 | |
|
0.1302 | 0.2072 | 0.0276 | 0.2257 | 0.3032 | 0.1062 |
Feature vector of
Fault name | IMFs | Relative entropy |
Normalized kurtosis | Feature vector |
---|---|---|---|---|
|
|
0.0099 | 0.6162 |
|
|
0.0109 | 0.8729 | ||
|
0.2660 | 0.9813 | ||
|
0.7400 | 0.8713 | ||
|
0.1266 | 0.9203 | ||
|
0.4903 | 0.7989 |
Finally, for every fault class of the Sallen-Key circuit, 150 samples are split into two parts. The first 100 fault feature vectors are adopted to train an ELM classifier and the remaining 50 fault feature vectors are used to test the ELM. Because the testing accuracy is sensitive to the selection of activation functions, the RBF function is proper for the diagnostic and the number of neurons is set as 250.
In order to show the performance of the proposed diagnostic method, we compare our method with other existing feature extraction methods which are presented in [
Results of fault classification of the Sallen-Key filter for single faults.
Fault number | Fault name | Test accuracy | |||
---|---|---|---|---|---|
Proposed | Wavelet [ |
Lifting wavelet [ |
[ | ||
F0 | No-fault | 1.0000 | 0.8600 | 1.0000 | 1.0000 |
F1 |
|
0.9700 | 0.9800 | 1.0000 | 0.9800 |
F2 |
|
1.0000 | 0.8700 | 1.0000 | 1.0000 |
F3 |
|
1.0000 | 1.000 | 1.0000 | 0.9300 |
F4 |
|
1.0000 | 0.9800 | 1.0000 | 1.0000 |
F5 |
|
1.0000 | 0.8700 | 1.0000 | 1.0000 |
F6 |
|
0.9700 | 0.7800 | 0.9500 | 0.8200 |
F7 |
|
0.9900 | 0.9400 | 1.0000 | 1.0000 |
F8 |
|
1.0000 | 0.8400 | 1.0000 | 1.0000 |
F9 |
|
1.0000 | 0.9600 | 1.0000 | 1.0000 |
F10 |
|
1.0000 | 0.8200 | 1.0000 | 1.0000 |
F11 |
|
1.0000 | 0.8000 | 1.0000 | 1.0000 |
F12 |
|
0.9700 | 0.6400 | 0.9400 | 0.9600 |
F13 |
|
1.0000 | 1.0000 | 1.0000 | 1.0000 |
F14 |
|
1.0000 | 0.9800 | 1.0000 | 1.0000 |
Average accuracy | 0.9940 | 0.8880 | 0.9930 | 0.9790 |
For reducing time cost, we adopt an ELM algorithm as a classifier because it is one of the best classification algorithms and it also can provide higher performance in time cost. Table
Comparison of time cost and test accuracy between ELM-based method and SVM-based method for the Sallen-Key bandpass filter.
Feature extraction method | Time (s)/accuracy (%) | |
---|---|---|
SVM classifier | ELM classifier | |
Wavelet coefficients [ |
11.2/93.1 | 0.0289/88.8 |
Kurtosis and entropy [ |
7.1/98.6 | 0.0213/97.9 |
Lifting wavelet [ |
14.6/99.2 | 0.0350/99.3 |
Proposed method | 9.3/99.8 | 0.0275/99.4 |
The second example circuit is a leapfrog filter, which is used as a CUT in [
Fault classes of the leapfrog filter for the multifaults.
Fault ID | Nominal value | Fault value |
---|---|---|
F1 |
|
|
F2 |
|
|
F3 |
|
|
F4 |
|
|
F5 |
|
|
F6 |
|
|
F7 |
|
|
F8 |
|
|
F9 |
|
|
F10 |
|
|
Diagnostic results of the leapfrog filter for the multiparametric faults.
Diagnostic method | Fault ID | Average accuracy (%) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | ||
Reference [ |
100 | 100 | 51 | 46 | 86 | 100 | 100 | 100 | 98 | 100 | 88.1 |
Reference [ |
100 | 70 | 100 | 98 | 70 | 74 | 100 | 100 | 95 | 98 | 90.5 |
Reference [ |
100 | 100 | 52 | 34 | 82 | 100 | 100 | 100 | 100 | 100 | 86.8 |
Our method | 100 | 98 | 100 | 100 | 94 | 96 | 100 | 100 | 100 | 100 | 98.8 |
Schematic of a leapfrog filter.
Through the two experiments, the results of the proposed method can be summarized as follows: The proposed method in the paper has better accuracy than other methods such as the first wavelet coefficients technique and the lifting wavelet method. For multifaults diagnosis, the method adopting EEMD, kurtosis, and relative entropy to construct feature vectors has better classification accuracy than the traditional method used in [ ELM classifiers with the techniques of EEMD, kurtosis, and relative entropy sometimes get the same better classification results as SVM classifiers with the same original feature vectors. Meanwhile, ELM-based method has much lower classification time than SVM-based method.
To sum up, the proposed method in the paper is acceptable from two aspects: test accuracy and time cost. It has higher test accuracy and fast classification capacity.
In this paper, a combinational diagnostic method for analog circuit with EEMD, relative entropy, and ELM is proposed. The proposed method makes good use of the EEMD, kurtosis, and relative entropy technique to construct fault feature vectors, and then faults classification on CUTs are performed using the ELM classifier. The effectiveness of the proposed method has been validated with the classical two benchmark circuits for single and multifault diagnosis. The results of experiments show that the method can distinguish effectively different faults of circuit with the higher testing accuracy (99.4%) and the lower testing time.
The authors declare that there are no competing interests regarding the publication of this paper.
This work was supported in part by the Fundamental Research Funds for the Central Universities of China (Grant no. ZYGX2015J074), Science and Technology Support Project of Sichuan Province, China (2014FZ0037, 2015FZ0111), and Support Project of CDTU (KY1311018B).