A Mechanical Equipment Fault Diagnosis Model Based on TSK Fuzzy Broad Learning System

Abstract

In an intelligent manufacturing context, the smooth operations of mechanical equipment in the production process of enterprises and timely fault diagnosis during operations have become increasingly important. However, the effect of traditional fault diagnosis depends on the feature extraction quality and experts’ empirical knowledge, which is inefficient and costly, and cannot match the needs of mechanical equipment fault diagnosis in intelligent manufacturing. The TSK fuzzy system has a strong approximation capability and the ability to interpret expert knowledge. The broad learning system (BLS) has strong feature extraction and fast computation capabilities. In this paper, we present a new model—the TSK fuzzy broad learning system (TSK-BLS). The model integrates the advantages of the BLS and the fuzzy system at the same time, which can be calculated quickly and accurately by pseudo-inverse and symmetry methods. On the other hand, the model is an embedded model-building mechanism, which extends the application scope of BLS theory. The model was tested on a bearing fault data set, provided by Case Western Reserve University, and the model’s fault diagnosis accuracy was as high as 0.9967. The results were compared with those of the convolutional neural network (CNN) and the BLS models, whose fault diagnosis accuracies are 0.6833 and 0.9133, respectively. Comparison showed that the proposed fault diagnosis model—TSK-BLS, achieved significant improvements.

1. Introduction

With the integrated development of artificial intelligence and manufacturing industry, a variety of expert systems and intelligent auxiliary systems have been developed for production control, equipment maintenance and other links in the manufacturing field. The world’s manufacturing powers have put forward different strategic plans to seize the commanding heights of the future manufacturing industry. In addition, against the background of intelligent manufacturing, the manufacturing industry also puts forward higher requirements for the digitalization and intelligence of mechanical equipment; mechanical equipment fault diagnosis is becoming more and more important. However, human-driven mechanical equipment fault diagnosis methods are inefficient and influenced by production conditions, which makes it difficult to match the current development speed of intelligent manufacturing. Most of the current fault diagnosis methods for mechanical equipment are based on using feature extraction [1] to classify faults and thus determine the fault types [2]. Recently, methods such as fuzzy systems and neural networks have gradually thrived in the field of fault diagnosis, and the reliability and maintainability of mechanical equipment operations have been further improved.

The input, output and state variables of fuzzy systems are defined on fuzzy sets, which are a generalization of deterministic systems. Fuzzy systems [3], which can emulate comprehensive human reasoning [4], are very effective for fuzzy information processing problems [5], and have been used extensively for fault diagnosis. Wu [6] proposed a fault detection filter design method for fuzzy stochastic systems. Wang [7] proposed a fault diagnosis and isolation design method for fuzzy affine systems with sensor faults. Although fuzzy systems have great interpretability, their fuzzy rules are predefined, based on expert knowledge, and do not change as the input data are updated, which often leads to poor accuracy. In addition, when encountering high-dimensional data, fuzzy systems need a large number of fuzzy rules to achieve satisfactory precision [8], which leads to an increase in the computational workload.

The neural network is a network structure that can be used to handle practical problems with multiple nodes and multiple outputs and has the advantages of a strong self-learning capability, self-adaptive capability, and fast computation. Yu [9] proposed a model combining wavelet packets with radial basis function (RBF) neural networks, to conduct a fault diagnosis with rolling bearings of metro vehicles, and the experiments showed that the method has a good fault type classification effect. Mandal [10] used the deep belief network to classify online fault data and trained faulty sensors. The performance of the proposed method was verified using the obtained field data. Guo [11] applied continuous wavelet transform to extract image fault features and used the convolutional neural network (CNN) to further extract features of grayscale images. The proposed method is robust and improves the efficiency of fault diagnosis. Although neural networks are developing rapidly and have become one of the current research hotspots, they still have the disadvantages of poor interpretation, and are time-consuming and labor-intensive.

Fuzzy neural networks have the strengths of both fuzzy systems and neural networks, compensating for their drawbacks when they exist in isolation. Jiang [12] proposed a new method for epileptic electroencephalogram (EEG) recognition based on a multi-view learning framework and fuzzy system. The model introduces the classical TSK fuzzy system as an easy-to-understand recognition model and develops a multi-view TSK fuzzy system method for recognizing epileptic EEG signals with good recognition results. Bai [13] used information fusion to extract the fault features of vibration signals, and established a fault diagnosis model based on fuzzy neural networks. Simulation results show that the proposed model is capable of extracting fault features of gearboxes well; the presented model additionally has a comparatively accurate fault identification capability. Jang [14] combined the TSK fuzzy model with a five-layer neural network to form the adaptive-network-based fuzzy inference system (ANFIS), which can quickly complete highly nonlinear mappings. TSK fuzzy systems are easy-to-understand intelligent models that are highly interpretable and have powerful learning capabilities. To date, TSK fuzzy systems have had a wide range of applications in multiple disciplines.

The broad learning system (BLS) [15] is the new model with a simple and shallow network structure proposed by C. L. Philip Chen, which can replace deep learning [16] for data training [17]. Currently, the BLS is widely used in image classification, numerical regression, EEG signal processing and other fields. Yang [18] proposes a fusion early warning model based on the ensemble empirical modal decomposition (EEMD) and broad learning algorithm. Using the gearbox data in a healthy state as the discriminant, we first perform the EEMD decomposition of the gearbox oil temperature time series signal to obtain the time-frequency characteristics. We then use the broad learning algorithm to model the gearbox, using the data acquisition and monitoring control system data to realize the predictive maintenance of the wind turbine gearbox. Zhao [19] proposes a domain-adapted BLS (DABLS) model based on the stream regularization framework and maximum mean difference to achieve cross-domain image classification in the target domain without labels. The DABLS model first constructs feature nodes and enhancement nodes of the BLS to effectively extract features from the source and target domain data; it then uses the stream regularization framework to construct Laplacian matrices to mine the potential information of the target domain data, which solves the problem of label data scarcity in fault diagnosis classification tasks. The TSK fuzzy system [20] is a well-known classical fuzzy system that is widely used for modeling [21] because of its simple output. In this paper, we combine BLS with TSK fuzzy systems, and propose the TSK fuzzy broad learning system (TSK-BLS), which inherits both the better interpretability of TSK fuzzy systems and the advantages of the simple structure and fast operations of the BLS.

The performance of the TSK-BLS model proposed in this paper is compared with that of the BLS and CNN. The TSK-BLS can achieve satisfactory training accuracy with fewer fuzzy rules and less training time, which shows the superiority of the model. In addition, the ensemble model breaks through the limitation of a single theory and promotes the further development of the theoretical framework of the BLS. Its construction process is also universal and can be applied to other similar models.

The remainder of this paper is in the following organization. Section 2 outlines the basic concepts of the BLS and TSK fuzzy systems. Section 3 presents the structural model and learning algorithm of the proposed TSK-BLS. Section 4 verifies the feasibility of the TSK-BLS through experiments. Section 5 summarizes the conclusions of the TSK-BLS.

2. Related Works

2.1. Broad Learning System (BLS)

The simple and shallow network structure of the BLS is an alternative to deep learning for data training, and has great symmetry. Compared with the currently popular deep learning, the BLS has fewer model parameters and a faster computing speed with a slight loss of accuracy. In addition, the BLS can be trained using features extracted from other models and has better adaptability with other machine learning algorithms.

As shown in Figure 1, the BLS is built on the basis of a random vector function chain neural network, but the BLS enhancement nodes are built in a different way from it. Instead of using the raw data as an input layer, the BLS first performs feature extraction on the original data to form the mapping features and store them in feature nodes. The mapping features are then used as the input of the enhancement nodes to perform further transformations. Finally, the defuzzification outputs of the feature nodes are augmented with the enhancement nodes’ outputs to obtain the outputs of the BLS; the connection weights can be quickly calculated by a pseudoinverse.

For given original data $X$, the feature of the $ith$ mapping can be expressed as:

$$Z_i={\varphi }_i\left(X{W}_{e_i}+{\beta }_{e_i}\right),i=1,2\dots n$$

(1)

The output of the feature layer for the BLS formed by $n$ sets of feature nodes in series is:

$$Z^n\triangleq \left(Z_1,Z_2\cdots Z_n\right)$$

(2)

Subsequently, the mapped features are subjected to nonlinear transformations, forming “enhancement nodes” whose weights are randomly generated; and the first $n$ feature mappings are combined to create the $jth$ set of enhancement nodes, denoted as follows:

$$H_j={\xi }_j\left(Z^n{W}_{h_j}+{\beta }_{h_j}\right), j=1,\dots ,m$$

(3)

The output of the enhancement layer of the BLS formed by $m$ sets of enhancement nodes in a series is:

$$H^m\triangleq \left(H_1,H_2\cdots H_m\right)$$

(4)

Finally, feature and enhancement layers are transferred to the output layer directly; the output $Y$ can be expressed as:

$$\begin{array}{cc}\hfill Y=[Z_1,\dots ,Z_n|\xi (Z^n{W}_{h_1}+& {\beta }_{h_{j1}}),\dots ,\xi \left(Z^n{W}_{h_m}+{\beta }_{h_m}\right)]W_n^m\hfill \\ & =\left[Z_1,\dots ,Z_n|H_1,\dots ,H_m\right]W_n^m\hfill \\ & =\left[Z^n|H^m\right]W_n^m\hfill \\ & =A_n^m{W}_n^m\hfill \end{array}$$

(5)

The corresponding output coefficients can be derived from the pseudoinverse calculation, which is given by:

$$W_n^m={\left[Z^n|H^m\right]}^{+}Y={\left(A_n^m\right)}^{+}Y$$

(6)

$X\in R^{N\times M}; Y\in R^{N\times C}$; $N$ denotes the number of input samples; $M \mathrm{and} C$ represent the input and output data dimensions, respectively. $\varphi$ and $\xi$ represent the activation functions; $W_{e_i}$, $W_{h_j}$ and $W_n^m$ denote the weights; ${\beta }_{ei}$ and ${\beta }_{h_j}$ denote the errors; $A_n^m$ denotes the augmentation matrix of the feature layer output $Z^n$ and enhancement layer output $H^m$.

2.2. Takagi-Sugeno-Kang Fuzzy System (TSK Fuzzy System)

Fuzzy systems have the advantage of being able to incorporate expert experiences, while their generalization capability is less affected by the data. Currently, the main fuzzy systems are TSK fuzzy systems, fuzzy logic systems and Mamdani fuzzy systems. The TSK fuzzy system is structurally and computationally simpler and more expressive than the other two types of systems; and due to the simplicity of the output form, TSK fuzzy systems are particularly suitable for data-driven modeling. TSK fuzzy systems have been used in many applications, including fault diagnosis, system identification, pattern recognition, image processing and data mining. TSK fuzzy systems are described by a group of $if-then$ fuzzy rules, which take the general form of:

$$\begin{array}{c}if x_1 is A_{k1}, x_2 is A_{k2} \cdots x_m is A_{km} \\ then y_k=f_k\left(x\right), k=1,2\cdots K\end{array}$$

$x_j \left(j=1,2\cdots m\right)$ is the fuzzy system input, $A_{kj}$ is the fuzzy sets of the $k\mathrm{th}$ fuzzy rule corresponding to the $j\mathrm{th}$ input variable $x_j$, and $K$ is the number of fuzzy rules.

$f_k\left(x\right)$ is generally a linear function, usually expressed as a polynomial. Therefore, the model’s posterior equation is:

$$y_k=f_k\left(x\right)={\alpha }_0^k+{\alpha }_1^k{x}_1+{\alpha }_2^k{x}_2+\cdots +{\alpha }_m^k{x}_m$$

(7)

The membership function then corresponding to the $k\mathrm{th}$ fuzzy rule is:

$$u_k={\prod }_{j=1}^m{u}_{kj}\left(x_j\right)$$

(8)

${\alpha }_j^k$ denotes the coefficients, and $u_{kj}\left(x_j\right)$ denotes the membership function of input $x_j$ corresponding to the fuzzy set $A_{kj}$.

The TSK fuzzy system applies the center of gravity method for defuzzification. Therefore, the defuzzification output of the TSK fuzzy system is:

$$y={\sum }_{k=1}^K\frac{u_k}{{\sum }_{k=1}^K{u}_k}{f}_k\left(x\right)={\sum }_{k=1}^K\frac{{\prod }_{j=1}^m{u}_{kj}\left(x_j\right)}{{\sum }_{k=1}^K{\prod }_{j=1}^m{u}_{kj}\left(x_j\right)}{f}_k\left(x\right)$$

(9)

3. TSK Fuzzy Broad Learning System (TSK-BLS)

BLS has the strengths of simple structure, fast speed and high accuracy. TSK fuzzy system has good interpretability for what was just learned and can fully use expert knowledge. This study integrates the BLS with the TSK fuzzy system to obtain the new model TSK-BLS—which has the advantages of both fast computations and great interpretability. It additionally has excellent results in the field of fault diagnosis.

The TSK-BLS preserves the basic structure of the BLS, but TSK fuzzy subsystems replace the feature nodes of the BLS, whose structure is shown in Figure 2. The TSK-BLS was evaluated in subsequent experiments and proved to have great stability. Compared to some more advanced methods, the results indicate that the TSK-BLS can attain satisfactory test accuracy in less training time.

If we suppose that there are $n$ TSK fuzzy subsystems and $m$ enhancement nodes in the TSK-BLS, all TSK fuzzy subsystems are first-order TSK fuzzy subsystems; that is, $f_k\left(x\right)$ are all first-order polynomial.

The input data are:

$$X={\left(x_1,x_2\cdots x_N\right)}^T, x_s=\left(x_{s1},x_{s2}\cdots x_{sM}\right),s=1,2\cdots N$$

If we assume that the $i\mathrm{th}$ TSK fuzzy subsystem has $k^i$ fuzzy rules, the $k\mathrm{th}$ fuzzy rule is expressed in the following form:

$$\begin{array}{c}if x_{s1} is A_{k1}^i, x_{s2} is A_{k2}^i \cdots x_{sM} is A_{kM}^i\\ then z_{sk}^i=f_k^i\left(x_{s1},x_{s2}\cdots x_{sM}\right),k=1,2\cdots k^i\end{array}$$

Because $f_k^i\left(X\right)$ is a first-order polynomial, it takes the form:

$$z_{sk}^i=f_k^i\left(x_{s1},x_{s2}\cdots x_{sM}\right)={\sum }_{j=1}^M{\alpha }_{kj}^i{x}_{sj}$$

(10)

${\alpha }_{kj}^i$ is the coefficient of the $j\mathrm{th}$ input $x_{sj}$ corresponding to the $k\mathrm{th}$ fuzzy rule in the $i\mathrm{th}$ TSK fuzzy subsystem.

The Gaussian function has rotation symmetry and is chosen as the membership function of the corresponding fuzzy set, which takes the form:

$$u_{kj}^i\left(x\right)=e^{-{\left(\frac{x-c_{kj}^i}{{\sigma }_{kj}^i}\right)}^2}$$

(11)

${\sigma }_{kj}^i$ and $c_{kj}^i$ are the width and center of the Gaussian function, respectively. We apply the k-means algorithm for the training set, choose appropriate clustering centers for each TSK fuzzy subsystem, and use these clustering centers to initialize the centers of the Gaussian function. Additionally, the number of clustering centers determines the number of fuzzy rules in the TSK fuzzy subsystem. Owing to the randomness of the k-means algorithm under the initial conditions, the training data of each TSK fuzzy subsystem will generate different centers, ensuring that different results are produced, thus enabling better extraction of data features.

Therefore, the activation function for the kth fuzzy rule of the $i\mathrm{th}$ TSK fuzzy subsystem is:

$$u_{sk}^i={\prod }_{j=1}^m{u}_{kj}\left(x_{sj}\right)$$

(12)

The weight of each fuzzy rule can be calculated as:

$$w_k^i=\frac{{\prod }_{j=1}^m{u}_{kj}\left(x_{sj}\right)}{{\sum }_{k=1}^{k^i}{\prod }_{j=1}^m{u}_{kj}\left(x_{sj}\right)}$$

(13)

Because the $i\mathrm{th}$ TSK fuzzy subsystem has $k^i$ fuzzy rules:

$$z_s^i=\left(z_{s1}^i,z_{s2}^i\cdots z_{s{k}^i}^i\right)$$

(14)

Therefore, the $i\mathrm{th}$ TSK fuzzy subsystem output is:

$$z^i={\left(z_1^i,z_2^i\cdots z_N^i\right)}^T$$

(15)

To maintain the consistency of the variables, for $n$ TSK fuzzy subsystems, the intermediate output matrix variables are expressed as:

$$z_n=\left(z^1,z^2\cdots z^n\right)\in R^{N\times (k^1+k^2\cdots k^n)}$$

(16)

$k^i\left(i=1,2\cdots n\right)$ denotes the number of fuzzy rules in $i\mathrm{th}$ TSK fuzzy subsystem.

Subsequently, $z_n$ is used as the input to the enhancement layer for nonlinear transformation. For the $l\mathrm{th}$ enhancement node, assuming it has $L_l$ neurons, its output matrix is expressed as:

$$H^l={\xi }_l\left(Z^n{W}_{hl}+{\beta }_{hl}\right), L=1,\dots ,m$$

(17)

$W_{hl}$ and ${\beta }_{hl}$ denote the weights corresponding to the output $z_n$ of $n$ TSK fuzzy systems and the errors, respectively, which are randomly generated in the interval [0, 1].

Thus, the output of the $m$ membership nodes is:

$$H_m=\left(H^1,H^2\cdots H^m\right)\in R^{L_1\times L_2\times \cdots L_m}$$

(18)

The nonlinear transformation of the enhancement layer can take full advantage of the rule output $z_n$ of the TSK fuzzy system, which can fully extract information from the original input data and interpret it well.

The output of the $i\mathrm{th}$ TSK fuzzy subsystem with input training data $x_s$ is expressed as:

$$\begin{array}{c}{F}_s^i=\left({\displaystyle \sum _{k=1}^{k^i}}{w}_{sk}^i{\delta }_{k1}^i{z}_{sk}^i,\cdots ,{\displaystyle \sum _{k=1}^{k^i}}{w}_{sk}^i{\delta }_{kC}^i{z}_{sk}^i\right)\\ =\left({\displaystyle \sum _{k=1}^{k^i}}{w}_{sk}^i{\delta }_{k1}^i\left({\displaystyle \sum _{j=1}^M}{\alpha }_{kj}^i{x}_{sj}\right),\cdots ,{\displaystyle \sum _{k=1}^{k^i}}{w}_{sk}^i{\delta }_{kC}^i\left({\displaystyle \sum _{j=1}^M}{\alpha }_{kj}^i{x}_{sj}\right)\right)\\ ={\sum }_{j=1}^M{\alpha }_{kj}^i{x}_{sj}·\left(w_{s1}^i,w_{s2}^i,\cdots ,w_{s{k}^i}^i\right)\left(\begin{array}{ccc}{\delta }_{11}^i& \cdots & {\delta }_{1C}^i\\ ⋮& ⋮& ⋮\\ {\delta }_{k^{i}1}^i& \cdots & {\delta }_{k^{i}C}^i\end{array}\right)\end{array}$$

(19)

${\delta }_{kc}^i\left(c=1,2,\cdots ,C\right)$ is the posterior parameter of the fuzzy rule in the $i\mathrm{th}$ TSK fuzzy subsystem. We introduced parameter ${\delta }_{kc}^i$ to reduce the amounts of total parameters to be calculated by the TSK fuzzy subsystem. This was done because, in practice, the input data dimension $M$ is much larger than the output data dimension $C$. It is easier and faster to calculate ${\delta }_{kc}^i$ by the pseudoinverse method than to calculate ${\alpha }_{kj}^i$.

The output of the $i\mathrm{th}$ TSK fuzzy subsystem for input data $X$ is:

$$F^i={\left(F_1^i,F_2^i,\cdots ,F_N^i\right)}^T=D{\omega }^i{\delta }^i\in R^{N\times C}$$

(20)

where:

$$\begin{array}{c}D=diag\left\{{\displaystyle \sum _{j=1}^M}{\alpha }_{kj}^i{x}_{1j},{\displaystyle \sum _{j=1}^M}{\alpha }_{kj}^i{x}_{2j},\cdots ,{\displaystyle \sum _{j=1}^M}{\alpha }_{kj}^i{x}_{Nj}\right\}\hfill \\ {\omega }^i=\left(\begin{array}{ccc}{w}_{11}^i& \cdots & w_{1k^i}^i\\ ⋮& \cdots & ⋮\\ w_{N1}^i& \cdots & w_{N{k}^i}^i\end{array}\right), {\delta }^i=\left(\begin{array}{ccc}{\delta }_{11}^i& \cdots & {\delta }_{1C}^i\\ ⋮& ⋮& ⋮\\ {\delta }_{k^{i}1}^i& \cdots & {\delta }_{k^{i}C}^i\end{array}\right)\hfill \end{array}$$

We then have the aggregate output of $n$ TSK fuzzy subsystems as follows:

$$\begin{array}{cc}\hfill F_n=& {\displaystyle \sum _{i=1}^n}{F}^i={\displaystyle \sum _{i=1}^n}D{\omega }^i{\delta }^i=D{\displaystyle \sum _{i=1}^n}{\omega }^i{\delta }^i\hfill \\ & =D·\left({\omega }^1,{\omega }^2,\cdots ,{\omega }^n\right)·\left(\begin{array}{c}{\delta }^1\\ {\delta }^2\\ ⋮\\ {\delta }^n\end{array}\right)\hfill \\ & =D\omega \delta \in R^{N\times C}\hfill \end{array}$$

(21)

where $\omega =\left({\omega }^1,{\omega }^2,\cdots ,{\omega }^n\right)$ is the matrix consisting of $w_{s{k}^i}^i$; $\delta ={\left({\delta }^1,{\delta }^2,\cdots ,{\delta }^n\right)}^T$ consists of the parameters to be calculated later.

The output $F_n$ of all TSK fuzzy subsystems is now transferred to the output layer of the TSK-BLS together with the output $H_m$ of the enhancement layer. The output is:

$$\begin{array}{c}Y=F_n+H_m{w}_h\hfill \\ =D\omega \delta +H_m{w}_h\hfill \\ =\left(D\omega ,H_m\right)\left(\begin{array}{c}\delta \\ w_h\end{array}\right)\hfill \\ =\left( D\omega ,H_m\right)W\hfill \end{array}$$

(22)

where $W$ is the parameter matrix of a TSK-BLS consisting of $\delta$ and $w_h$.

For a given training target $Y$, the matrix coefficients $W$ can be quickly computed using the pseudoinverse and symmetry.

$$W={\left(D\omega ,H_m\right)}^{+}Y$$

(23)

$${\left(D\omega ,H_m\right)}^{+}={\left[{\left(D\omega ,H_m\right)}^T\left(D\omega ,H_m\right)\right]}^{-}{\left(D\omega ,H_m\right)}^T$$

(24)

The TSK-BLS training algorithm is outlined in Algorithm 1.

Algorithm 1: TSK-BLS training process

Input: Training samples $X$ $\in R^{N\times M}$, numbers of fuzzy rules $k^i$, neuron of the enhancement node $L_l$, TSK fuzzy subsystems $n$ and enhancement node $m$

1: initialize the coefficients ${\alpha }_{kj}^i$ in function $f_k^i\left(X\right)$

2: for $i=1;i\le n$ do

3: obtain $k^i$ clustering centers by applying k-means algorithm to training samples $X$.

4:  applying the values of $k^i$ clustering centers to initialize $c_{kj}^i$.

5:  for $s=1 ; s\le N$ do

6:  calculate $z_s^i$, $F_s^i$

7:  end for

8:   calculate $z^i$, $F^i$

9:  end for

10: calculate $z_n$, $H_m$, $F_n$, $W$

4. Experimental Results and Analysis

4.1. Data Description

In this section, the Case Western Reserve University bearing fault dataset [22] was used as experimental data to verify the reliability of the TSK-BLS, which were collected by accelerometers during the operation of the faulty bearing. Faults ranging from 0.007 inch to 0.014 inch, 0.021 inch, and 0.028 inch were introduced into the inner raceway, ball element and outer raceway, respectively. The fault dataset also records vibration data for motor loads of 0 to 3 horsepower.

Owing to the excessive amount of raw data in the bearing fault dataset, it is difficult to validate the TSK-BLS model for all types of fault data. Therefore, to simplify the experiment, the inner ring fault data of the 0.007 inches bearing diameter and 0 HP bearing load from the Case Western Reserve University bearing fault dataset were chosen as the input data for this experiment. The data selected had a collection speed of 1797 rpm and sampling frequency of 12 kHz. Therefore, there are approximately 400 acceleration data points for one rotation of the bearing. This experiment takes the 400 acceleration data points as a sample and stores these data points in a picture, which is subsequently used as the input of the TSK-BLS in the form of pictures, as shown in Figure 3. Such data processing enables a picture data to contain a large number of data points, with the consequence that as many of the original acceleration data of the faulty bearing as possible can be used to fully extract the data features.

4.2. Analysis of the Experimental Results

The accuracy is a statistical calculation of how correct a model is at predicting its effectiveness. The loss rate is a function of the difference between the predicted output and actual output of the model. The learning rate is the magnitude of updating the network weights in the optimization algorithm and determines the speed of updating the weights. Epochs refer to the number of iterations that the model needs to make.

During the experiment, first the data had to be disrupted; the data then had to be scaled to divide the input data. Here, 70% of the data are used as a training set, and the rest are used as a test set. The training set is entered into the TSK-BLS model for training, and is followed by the test set to verify the trained results. To avoid errors that occur due to random chance, multiple duplicate experiments are also conducted for the TSK-BLS. After the model stops training, we calculate metrics such as the accuracy, number of iterations, training time, small batch accuracy, small batch loss, etc., which are used to verify the feasibility of the TSK-BLS model. Subsequently, the TSK-BLS model is tested with a test set to determine the training accuracy of the model.

The experimental equipment is a personal computer with Intel Core i5 CPU, 16 GB RAM and NVIDIA GTX 1080ti GPU. The operating system is Windows 10 Professional, and the programming platform is MATLAB R2021a. The CNN has powerful automatic feature extraction capability. The BLS has the advantage of a simple structure and fast computation. To highlight the advantages of the TSK-BLS model, CNN and BLS models are selected as comparison models to compare with the model proposed in this paper.

The learning rate of this experiment is constant and appropriate and will not be adjusted during the training process. According to the experimental results shown in Figure 4, Figure 5 and Figure 6, it can be seen that the pictures clearly show the relationship between the number of iterations and the training time, small batch accuracy, verification accuracy, small batch loss rate, and verification loss rate. The more iterations there are, the higher the small batch accuracy and verification accuracy, and the lower the small batch loss and verification loss. When the training time is the main evaluation index, the times of the TSK-BLS and CNN are approximately the same and lower than that of the BLS. When the small batch accuracy and verification accuracy are the main evaluation metrics, the TSK-BLS and BLS have high accuracy, and the CNN has relatively lower accuracy compared to them.

Figure 7 shows the comparison of the small batch loss rate as well as the validation loss rate of the CNN, BLS and TSK-BLS models, demonstrating the comparison of the small batch accuracy, validation accuracy and experimental accuracy. The results from the pictures show that the TSK-BLS has a smaller small batch loss and verification loss, a higher small batch accuracy and verification accuracy, and that its fault diagnosis accuracy is as high as 0.9967 compared to the CNN and BLS. The experimental results show that the TSK-BLS has a higher accuracy. The results from the pictures show that the TSK-BLS has a smaller small batch loss and verification loss, a higher small batch accuracy and verification accuracy, and that its fault diagnosis accuracy is as high as 0.9967 compared to the CNN and BLS. The experimental results show that the TSK-BLS has a higher accuracy than the compared models, which indicates its significant advantages.

5. Conclusions

By integrating the BLS with the fuzzy system, a new model TSK-BLS is obtained. To construct this model, the feature nodes of the traditional BLS are replaced by the TSK fuzzy system, and the outputs of all TSK fuzzy subsystems are passed directly to the enhancement nodes for nonlinear transformation without aggregation into intermediate variables in advance, thus reducing the complexity. In addition, this integrated model retains the advantages of the simple structure and fast computation of the BLS, as well as the strong interpretability of the fuzzy system to the output. In the experiments on the faulty bearing dataset, the TSK-BLS showed good results in terms of experimental accuracy and training time, which also confirms the superiority of the model in this paper. The contributions of this paper are as follows: by combining TSK fuzzy systems with the BLS, the new model has the advantages of a single model at the same time and extends the scope of application; it breaks through the limitations of a single theory and promotes the further development of the theoretical framework of the BLS; the construction of the model in this paper is not a direct splicing between multiple models, but an embedded model construction mechanism, which is more meaningful for practical application.

Although the TSK-BLS has the advantages of fast computations and high interpretability, it can be effectively applied in the field of fault diagnosis. However, there are still some limitations. The TSK-BLS preserves the construct of the BLS, which means that we can logically extend the incremental learning algorithm to it. Due to technical limitations, research on the incremental learning of the TSK-BLS has not yet been conducted, which will be an important research point in the future.