Fault diagnosis using improved pattern spectrum and fruit fly optimization algorithm–support vector machine

Abstract

A fault diagnosis method using improved pattern spectrum and fruit fly optimization algorithm–support vector machine is proposed. Improved pattern spectrum is introduced for feature extraction by employing morphological erosion operator. Simulation analysis is processed, and the improved pattern spectrum curves present a steady distinction feature and smaller calculating amount than pattern spectrum method. Support vector machine with fruit fly optimization algorithm which can help seeking optimal parameters is employed for pattern recognition. Experiments were conducted, and the proposed method is verified by roller bearing vibration data including different fault types. The classification accuracy of the proposed approach reaches 87.5% (21/24) in training and 91.7% (44/48) in testing, showing an acceptable diagnosis effect.

Introduction

A roller bearing is a sort of key mechanical component, and it is meaningful to propose an efficient fault diagnosis technique for its reliability. The basic process of fault diagnosis involves data collection, feature extraction, and diagnostic model. ¹ Vibration signal processing is widely used because it is sensitive to structural dynamic condition variation. ² Therefore, it is feasible to extract features from vibration signals with efficient techniques.

As to the step of feature extraction, the non-linear feature is distinct due to many non-linear factors when roller bearing is working. ³ For this reason, the linear-based processing methods are not able to process vibration signals accurately. Some non-linear analytical techniques offer a feasible way in signal processing of roller bearing considering the non-linear character, such as information entropy analysis ⁴ and fractal dimensions. ⁵ Among the non-linear analytical techniques, the mathematical morphological particle analysis is able to describe the shape changing information on different scales, and it is apt to analyze the signal essence on different levels. ⁶ This method is widely used in image feature description, image segmentation, and image restoration,^7,8 in addition to this field, there are still some studies in fault feature extraction.^9,10 Pattern spectrum (PS) using opening operator is usually adopted in mechanical signal based on mathematical morphological particle analysis. Morphological erosion operator is introduced, and a new definition of improved pattern spectrum (IPS) is proposed for feature extraction by contrast with traditional PS.

As to the step of diagnostic method, typical artificial neural network (ANN) needs sufficient samples which are not practical. ¹¹ Support vector machine (SVM) is proposed based on statistical learning theory, and it is able to obtain an optimal solution. ¹² In order to get high accuracy with small sample number, SVM is brought into machine fault diagnosis.^13–15 In view of the influence from the parameters for the effect of SVM, ¹⁶ typical algorithms have been introduced in this field, such as particle swarm optimization (PSO) and genetic algorithm (GA) algorithm.^17–21 In order to get an optimal combination, fruit fly optimization algorithm (FOA) is introduced in parameters optimizing of SVM in this article.

In conclusion, a diagnostic method is developed based on IPS and fruit fly optimization algorithm–support vector machine (FOA-SVM). First, vibrating signal is processed based on IPS method and forming an efficient feature. And then, five statistic parameters are calculated as the feature sets. Finally, SVM is optimized with FOA, and the faulty samples can be identified with this model. The proposed method is verified by roller bearing vibration data including different fault types.

This article is organized as follows. In section “Theory of PS,” theory of PS is briefly introduced. In section “Proposition of feature extraction based on IPS,” feature extraction based on IPS is proposed. In section “Fault diagnosis flow using IPS and SVM,” the fault diagnosis flow using IPS and SVM is given, and in section “Application and discussion,” the proposed method is verified and the results are discussed. Finally, the conclusion of this article is given in section “Conclusion.”

Theory of PS

PS is used based on mathematical morphological particle analysis, which is an effective way in dealing with image granularity and shape features. The main idea is that analyzing images using structuring elements with different sizes and shapes to get the internal characteristics finally.

Mathematical morphological particle analysis is defined as follows. It is a set of image transformations meeting the following three conditions

$$\begin{array}{c}\mathrm{\Psi }=\{{\psi }_{\lambda }{\}}_{\lambda \ge 0}\hfill \\ \forall \lambda \ge 0,X\subset Y\iff {\psi }_{\lambda }(X)\le {\psi }_{\lambda }(Y)\hfill \\ \forall \lambda \ge 0,{\psi }_{\lambda }(X)\le X\hfill \\ \forall \lambda ,\mu \ge 0,{\psi }_{\mu }({\psi }_{\lambda }(X))={\psi }_{\lambda }({\psi }_{\mu }(X))={\psi }_{max(\lambda ,\mu )}(X)\hfill \end{array}$$

(1)

Among the four basic mathematical operators, opening operation is able to meet the above three conditions so that mathematical morphological particle analysis is usually applied based on opening operator. Assuming that g is unit structuring element and structuring element on scale λ is defined as

$$g_{\lambda }=\lambda g=g\oplus \cdots \oplus g$$

(2)

Mathematical morphological particle analysis is defined as

$$\psi (X)=X\circ \lambda g\lambda \ge 0$$

(3)

PS is introduced using the definition above, and it is a form of curve presenting mathematical particle size distribution. The change information for different λ values can be extracted when used in one-dimensional signal analysis. Supposing f is time-domain signal and g is unit structuring element, PS for f on scale λ is defined as follows

$$\mathrm{PS}(f,\lambda ,g)=\{\begin{array}{cc}\hfill -\frac{\mathrm{dA}(f\circ \lambda g)}{d\lambda }\hfill & \hfill \lambda \ge 0\hfill \\ \multicolumn{1}{c}{-\frac{\mathrm{dA}(f·(-\lambda )g)}{d\lambda }}& \hfill \lambda <0\hfill \end{array}$$

(4)

A(f) means the coverage for f in the definition domain. Mathematical opening operator is applied as λ ≥ 0 while mathematical closed operator is introduced as λ < 0, plus part for PS represents the structuring information and minus part represents the background information. The two parts mentioned above are accordant, and the plus part as λ ≥ 0 is usually introduced for analysis, this article is also in this case.

When analyzing one-dimensional discrete signal, λ is defined as continuous integer and the definition above can be simplified as follows. As to the non-epitaxial feature of opening operator, the result below is a set of non-negative number sequences

$$\begin{array}{c}PS+(\lambda ,g)=A[f\circ \lambda g-f\circ (\lambda +1)g]0\le \lambda \le {\lambda }_{max}\hfill \\ P{S}_{-}(\lambda ,g)=A[f·(-\lambda )g-f·(-\lambda +1)g]{\lambda }_{min}\le \lambda \le 0\hfill \end{array}$$

(5)

Proposition of feature extractionbased on IPS

Proposition of IPS

Considering different features of morphological operators, morphological erosion operator is introduced and a definition of IPS proposed. Supposing f is time-domain signal and g is unit structuring element, IPS for f on scale λ can be defined as follows

$$\mathrm{IPS}(f,\lambda ,g)=\{\begin{array}{cc}\hfill -\frac{\mathrm{dA}(f\Theta \lambda g)}{d\lambda }\hfill & \hfill \lambda \ge 0\hfill \\ \multicolumn{1}{c}{-\frac{\mathrm{dA}(f\oplus (-\lambda )g)}{d\lambda }}& \hfill \lambda <0\hfill \end{array}$$

(6)

Mathematical erosion operator is introduced as λ ≥ 0 and dilation operator when λ < 0. Considering the consistency of plus and minus parts, the plus part of IPS is usually used for analysis. When analyzing one-dimensional discrete signal, λ is defined as continuous integer and the definition above can be simplified as follows—the result below is also a set of non-negative number sequences

$$\begin{array}{c}\mathrm{IP}S+(\lambda ,g)=A[f\Theta \lambda g-f\Theta (\lambda +1)g]0\le \lambda \le {\lambda }_{max}\hfill \\ \mathrm{IP}{S}_{-}(\lambda ,g)=A[f\oplus (-\lambda )g-f\oplus (-\lambda +1)g]{\lambda }_{min}\le \lambda \le 0\hfill \end{array}$$

(7)

Operator is the key distinction comparing with definition of PS, mathematical opening operator is the combination of erosion and dilation operators, and mathematical opening operation is able to filter peak noise. As to the definition of PS, a weaker regularity may be got because of the coupled operation in mathematical opening operator. The monotonicity of IPS may be evident owning to the sole erosion operator in formula (7).

Simulation

Roller bearing simulation signal is modeling according to the operation and failure mechanisms ²²

$$y(k)=e^{-\alpha t}\sin 2\pi f_c\mathrm{kT},t=mod(\mathrm{kT},\frac{1}{f_m})$$

(8)

In the formula above, the parameters of α, f_m, f_c, and T represent the exponential frequency, modulation frequency, carrier frequency, and sampling interval, respectively. In this section, α = 800, f_c = 5000 Hz, T = 1/20,000, and the sampling time is 0.1 s. Gaussian white noise is added into simulation signal. f_m is set as 80, 160, and 240 Hz in order to simulate different bearing fault types, respectively. The simulation signals with different f_m values are shown in Figure 1.

Figure 1.

Time domain of simulation signal with different modulation frequencies.

Analyzing simulation signals with IPS proposed, the unit structuring element is set as g = [0 0 0]. According to some analysis on different scales, the maximum analytical scale is set as λ_max = 20 which is considered as a proper value in this article. The results are shown in Figure 2(a) where IPS1–IPS3 correspond three signals with f_m equals 80, 160, and 240 Hz, respectively. It is clear that the curves present a monotone decreasing trend with the increase in the analytical scale, and D-values among the curves are shown in the histogram, in which the values are all minus and present a steady distinction of different fault types.

Figure 2.

IPS and PS curves with D-value of different feature frequencies: (a) IPS curves and (b) PS curves.

Comparing with traditional PS method, three PS curves are shown in Figure 2(b) with the same g and λ_max as IPS. It seems that there is no monotone trend and the D-values are not consistent that presenting a bad distinction of different fault types. In addition, morphological opening operation is defined as the sequential execution of morphological corrosion and dilation operation, and the proposed method based on IPS has a less time-consumption than traditional PS method.

Influence analysis

The unit structuring element g and maximum analytical λ_max are two important parameters in IPS analysis. Triangular and rectilinear structuring elements are analyzed for contrast and then λ_max is especially discussed.

IPS curves with triangular structuring element are shown in Figure 3. The regularity is the same as Figure 2(a), and the D-values become more and more smaller with the increase in the triangular structuring element’s amplitude. For example, there has no distinctions for three IPS curves when λ > 5 in Figure 3(b).

Figure 3.

IPS curves based on triangular structuring element with different amplitudes: (a) g = [0 0.05 0] and (b) g = [0 0.2 0].

IPS curves with rectilinear structuring element are shown in Figure 4, which have no apparent distinction contrast with Figure 2(b); therefore, g =[0 0 0] is a good choice considering the calculation amount and effect.

Figure 4.

IPS curves based on triangular structuring element with different amplitudes: (a) g = [1 1 1] and (b) g = [5 5 5].

According to the basic principle of multi-scale mathematical morphology, the length of rectilinear structuring element has the same influence as the largest analytical scale, so we focus on the largest analytical scale in this part. We set structuring element as g = [0 0 0], and the IPS curves are shown in Figure 5 when λ_max = 80. It is clear that the value of IPS keeps decreasing trend until the curve fells into a named “stable zone,” where λ_max > 66 and IPS value remains constant. In order to improve feature effectiveness, the “stable zone” should be avoided. As introduced above, λ_max is set as 20 for reducing calculation amount.

Figure 5.

IPS curves based on triangular structuring element with different scales: (a) IPS curves when λ_max = 80 and (b) stable zone.

Fault diagnosis flow using IPS and SVM

Fault diagnosis flow using IPS and FOA-SVM is proposed in Figure 6. First, unit structuring element as well as the maximum analytical scale should be set as a proper value. Second, analyze the signal with IPS method. In order to reduce the dimensions of IPS curve, calculate five statistic parameters and constituting feature vectors including the maximum value, the minimum value, the average value, the geometric mean, and the standard deviation. Third, FOA ²³ is introduced in SVM parameters optimization using feature vectors of training samples. The optimization procedure is shown in Figure 7. Finally, the fault types will be diagnosed by the optimized SVM model.

Figure 6.

Flowchart of the proposed method.

Figure 7.

Process of FOA.

Application and discussion

Experimental setup

The bearing vibration data were employed from CWRU.^24,25 The experimental setup is shown in Figure 8. The chosen vibration data contain four fault types under the load of 0.746 kW, and four typical signals are shown in Figure 9.

Figure 8.

The experimental setup for bearing.

Figure 9.

Time-domain waveform of the four fault types.

Descriptions of the bearing data sets are listed in Table 1. The data sets are divided into training and testing samples with 1:2 proportion randomly. Four types of bearing conditions are labeled as 1–4.

Table 1.

Description of bearing data set.

Bearing condition	Number oftraining samples	Number oftesting samples	Label ofclassification
Good condition	6	12	1
Roller element fault	6	12	2
Inner race fault	6	12	3
Outer race fault	6	12	4

Feature extracting based on IPS

IPS is employed in feature extraction in this section, the maximum analytical scale is also set as λ_max = 20, and the unit structuring element is set as g = [0 0 0]. Calculating IPS for training samples and the IPS curves are shown in Figure 10, it can be seen that evident distinction exists among the conditions and the IPS curves contain information for each condition.

Figure 10.

IPS curves distribution for 24 groups of testing sample.

A group of representative curves above are shown in Figure 11. It is clear that IPS curves present a decreasing trend along with the increase in the analytical scale. The curve of outer race fault has a maximum value in the range of scale section. The curve of inner race fault takes the second place. The curve of normal status has the minimum value, which has a less discrimination with curve for ball fault. Thus, feature extraction based on IPS has a good effect compared with PS curves calculated with traditional mathematical morphology spectrum as shown in Figure 12.

Figure 11.

IPS curve of typical testing sample.

Figure 12.

PS curve of typical testing sample.

The IPS calculated above has 19 dimensions that will increase the computational complexity for SVM model. Five statistic parameters are calculated to reduce the dimensions as shown in Figure 13.

Figure 13.

Five statistics over the IPS.

Pattern using FOA-SVM

The five-dimensional features are used to train SVM, and FOA is introduced to get the optimal parameters of SVM including penalty factor and kernel function parameter γ.

In the training process with FOA, the iteration generation is set as 100 and fruit fly group is set as 20. The classification accuracy for training features is set as objective function. In the end of optimization, the classification accuracy reaches the best value as 97.83%. In this situation, the optimal penalty parameter C = 0.5642 and kernel parameter γ = 2.9129.

After training with FOA, the testing feature set is input to the SVM model, and the output is shown in Figure 14. The misclassified samples include three in the training and four in the testing. The whole classification accuracy of the proposed approach reaches 87.5% (21/24) in training and reaches 91.7% (44/48) in testing data sets. The result indicates that the effectiveness and suitability of IPS as feature vectors.

Figure 14.

The classification output using the proposed approach.

A typical backpropagation (BP) neural network is introduced for comparison. The structure is 5 × 15 × 1, and the training and testing samples have no differences with SVM. The recognition result of BP network is shown in Figure 15. Five samples are misclassified in training, and the training classification accuracy reaches 79.2% (19/24). Furthermore, eight samples are misclassified in the testing, and the testing classification accuracy reaches 81.8% (36/44). This comparison presents the advantage of SVM with FOA.

Figure 15.

The classification output using BP network.

Conclusion

In this article, morphological erosion operator is introduced to the mathematical morphological particle analysis, and a fault diagnosis method using IPS and FOA-SVM is proposed. Experiments were conducted, and the proposed method is verified by roller bearing vibration data including different fault types. Some conclusions of the study are as follows:

1. Morphological erosion operator has different operation character compared with morphological opening operator, and IPS based on erosion operator is able to present morphological characters of signals on different scales, leading a better effect in distinguishing fault types.

2. Considering that morphological opening operation is defined as the sequential execution of morphological corrosion and dilation operation, the proposed method based on IPS has a less time-consumption and a better effect than traditional PS method. Analysis on the parameters’ influence approved its feasibility.

3. Considering that morphology operation involves only simple addition and subtraction, the proposed IPS method has a small calculation amount and high efficiency, and it is promising to introduce this method to online monitoring processing system.

4. The method proposed got a 91.7% classification accuracy toward testing data set and a better effect than BP neural network, reflecting a good effect and adaptability when facing small training samples.