CN104596767B - Method for diagnosing and predicating rolling bearing based on grey support vector machine - Google Patents

Abstract

The invention provides a method for diagnosing and predicating a rolling bearing based on a grey support vector machine. The method is characterized in that the rolling bearing is used as a key part of a mechanical device, and the advantages and disadvantages of the operation state influence the operation performances of the whole device. The method is the method for diagnosing and predicating the rolling bearing based on GM (1, 1)-SVM. The method comprises the steps of extracting a vibration signal time domain and frequency domain feature values of the rolling bearing under various fault and normal states; selecting important feature parameters to build a predicating model, namely, grey model; predicating the feature value; training a binary tree supporting vector machine according to various fault feature values and normal state feature values of the bearing; creating a rolling bearing decision making tree for determining the fault as well as classifying the fault type to diagnosis the fault of the bearing; then predicating the fault according to the predicating value and the trained supporting vector machine.

Description

A Method of Rolling Bearing Fault Diagnosis and Prediction Based on Gray Support Vector Machine

technical field

The invention belongs to the field of bearing fault diagnosis, and is a comprehensive fault diagnosis and prediction model GM(1,1)-SVM developed for rolling bearings.

Background technique

Rolling bearings are the most widely used mechanical parts in electric power, petrochemical, metallurgy, machinery, aerospace and some military industries, and are also one of the most vulnerable parts. It has the advantages of high efficiency, small frictional resistance, convenient assembly, and easy lubrication. It is widely used in rotating machinery and plays a key role. Many failures of rotating machinery are closely related to rolling bearings. According to relevant statistics, 70% of mechanical failures are vibration failures, and 30% of vibration failures are caused by rolling bearings. The consequences caused by rolling bearing failure range from reduced and lost some functions of the system to severe or even catastrophic consequences. So the fault diagnosis method of rolling bearing has always been one of the key development technologies in mechanical fault diagnosis. This paper is devoted to the research of monitoring and prediction technology of rolling bearing fault.

In order to solve the problem of bearing fault diagnosis and prediction, various algorithm models have been proposed, but these methods cannot effectively predict bearing faults. Therefore, it is necessary to propose a model that can not only realize the diagnosis of bearing faults but also realize the effective early warning of faults.

Contents of the invention

The present invention is based on the gray support vector machine GM (1, 1)-SVM bearing fault diagnosis and early warning method, not only can realize the fault diagnosis to the rolling bearing, but also can realize the effective early warning to the fault, help to improve the rolling bearing with rolling bearing safe operation of rotating machinery systems.

The technical scheme that the present invention adopts is as follows,

The method for bearing fault diagnosis and early warning based on gray support vector machine GM (1,1)-SVM provided by the present invention at least includes the following parts:

S1 feature variable extraction and correlation analysis. Rolling bearings are typical rotating machinery. The time-domain characteristic variables of the vibration signal include root mean square value, peak-to-peak value, and average value, etc., and the frequency-domain characteristic variables include fundamental frequency, 2-fold frequency, 3-fold frequency, 4-fold frequency, and 8-fold frequency. etc., they contain rich failure information. Compare and analyze the characteristic variables of the time domain and frequency domain of the vibration signal of various faults and normal conditions of rolling bearings, and select the appropriate characteristic variables. The characteristic variables selected in this paper for fault discrimination are:

X = (RMS, peak-to-peak, 1x amplitude, 2x amplitude, 3x amplitude, 4x amplitude, 8x amplitude).

RMS is the root mean square value of the vibration signal, which can best represent the overall characteristics of the signal. The RMS time series is selected as the reference sequence, and the other 6 sequences are used as the comparison sequence, and the gray correlation degree between the reference sequence and each comparison sequence is obtained, and the correlation degree is removed. Low 2 characteristic variables.

Let the reference sequence Y={y(k)|k=1,2,...,n},

Comparison sequence X _i ={X _i (k)|k=1,2,…,n}, i=1,2,…,m

Dimensionlessize variables:

The gray correlation coefficient of the reference series and the comparison series:

Calculate the degree of association:

Sorting relevance, if r _i <r _j , then x _j (k) is closer to reference sequence y(k) than x _i (k).

S2 Establish a prediction model, use the first 10 groups of eigenvalues of each state to establish a gray model and an orthogonal polynomial as a least squares fitting prediction model, and the last two groups are used as the comparison value of the predicted value to calculate the error of the predicted value of the two models, Choose a model with a smaller error—the gray model.

(1) Gray prediction model

The gray system model GM(1,1) performs an accumulation process through the univariate time series { _xi }(i=1,2,3,…), and establishes a first-order differential equation for this generation sequence to reveal its internal development law .

Define characteristic gray system

do: have

Set up the following differential equation for X ⁽¹⁾ :

Record the differential equation of the first-order one variable as GM(1,1)

In the above formula, a and u can be fitted by the least square method:

In formula (5), Y _M is a column vector Y _M =[X ⁰ (2),X ⁰ (3),…,X ⁰ (n)] ^T ,

B is a matrix:

The time response function corresponding to the differential equation (5)

The predicted value of the accumulated sequence generated by formula (6): X ⁽⁰⁾ (t) = X ⁽¹⁾ (t)-X ⁽¹⁾ (t-1) (7)

(2) Use orthogonal polynomials for least square fitting prediction

Data fitting is to determine the type of curve y=s( _x ;a ₀ ,a ₁ ,…,an ) based on the relationship between the measured data, and then according to the principle that the sum of the squares of the errors at a given point reaches the minimum , which solves the unconstrained problem:

Determine the optimal parameters Thus a fitting curve y=s ^* (x) is obtained.

Let φ ₀ ,φ ₁ ,…,φ _n be n+1 functions, and ω _i be the coefficient, satisfying:

That is, φ ₀ , φ ₁ ,…,φ _n are orthogonal on X={x ₁ ,x ₂ ,…,x _m }, where Then the solution of the normal equation (8) is:

S3 support vector machine is a typical neural network based on statistical learning theory, which establishes an optimal classification hyperplane to maximize the distance between the two types of samples on both sides of the plane, thus providing a good generalization ability for classification problems . For samples (x _i , y _i ), i=1,2,...,s, where x _i ∈ R ^m , y _i ∈ {+1,-1}, s is the dimension of the input variable. The hyperplane equation for classification is: ω·x+b=0

Divide the samples into two categories: ω·x+b≥0, (y=+1)

ω·x+b≥0,(y=-1) (10)

The optimal hyperplane of a support vector machine is a hyperplane that maximizes the classification margin, that is, is the largest, so to solve the optimal hyperplane, that is

It should satisfy the constraints: y _i (ω· _xi + b)-1≥0, i=1,2,...,l

Under nonlinear conditions, the maximization function of a linear non-separable support vector machine:

The discriminant objective function is:

Training support vector machine, for the fault classification of rolling bearings, this paper chooses two types of SVM to construct a multi-class classifier. Since the two types of SVM 1-to-many algorithms and 1-to-1 algorithms have their own shortcomings, this paper adopts a binary tree-based support vector machine multi-class classification method. The construction steps of the two-class classifier based on the binary tree are: the i-th classifier separates the i-th class from the i+1, i+2, ..., N classes, and constructs SVMi until the N-1 classifier divides the N- Class 1 is separate from Class N. Combine N-1 SVMs into a multi-class classifier, and construct an SVM decision tree to identify N types of faults.

Compared with the prior art, the method can not only realize the fault diagnosis of the rolling bearing, but also realize the effective early warning of the fault, and help to improve the safe operation of the rotating machinery system with the rolling bearing.

Description of drawings

Figure 1. Gray support vector machine failure prediction flow chart.

Specific implementation method

The implementation of the present invention will be specifically described below in conjunction with the accompanying drawings.

1. Rolling bearing failure experiment of Western Reserve University in the United States. The experimental platform includes a 2-horsepower motor, a torque sensor, a power meter and electronic control equipment. The bearing to be tested supports the motor shaft. The bearing model is SKF bearing, and single-point faults are arranged on the bearings using EDM technology, and the fault diameters are 0.007, 0.014, and 0.028 inches respectively. In the experiment, an acceleration sensor is used to collect vibration signals, and the sensors are respectively installed on the drive end and fan end of the motor housing and the motor support chassis. The vibration signal is collected by a 16-channel DAT recorder, and the sampling frequency of the digital signal is 12000S/s.

The fault data used in this paper are the outer ring fault data, inner ring fault data, and ball fault data of the bearing when the motor load is 0 horsepower and 1 horsepower respectively. The diameters of each fault are 0.007, 0.014, and 0.021 inches respectively. Under normal circumstances, 12 sets of data are collected, with a total of 216 sets of data. A total of 48 sets of data are selected from 4 normal states with motor loads ranging from 0 to 3 horsepower, and the total number of data is 264 sets.

2. Extraction of feature variables and correlation analysis

Rolling bearings are typical rotating machinery. The time-domain characteristic variables of the vibration signal include root mean square value, peak-to-peak value, and average value, etc., and the frequency-domain characteristic variables include fundamental frequency, 2-fold frequency, 3-fold frequency, 4-fold frequency, and 8-fold frequency. Wait. The characteristic variables selected in this paper for fault discrimination are:

X = (RMS, peak-to-peak value, 1x amplitude, 2x amplitude, 3x amplitude, 4x amplitude, 8x amplitude)

Record 12 sets of time-series data of the rolling bearing when the outer ring fault is 0.007 inches, the speed is 1797, and the motor load is 0 horsepower. Fast Fourier transform is performed on each set of data to extract its frequency domain eigenvalues. Extract feature variables to obtain a 12x7 order time series matrix, as shown in Table 1.

Table 1. 12 sets of time-series data of the rolling bearing when the outer ring fault is 0.007 inches, the speed is 1797, and the motor load is 0 horsepower

RMS represents the energy of the entire signal, and the RMS time series is selected as the reference sequence, and the other 6 sequences are used as the comparison sequence, and the gray correlation degree between the reference sequence and each comparison sequence is calculated, as shown in Table 2.

Table 2. The gray correlation degree of the reference sequence and each comparison sequence

Set reference sequence Y={y(k)|k=1,2,...,n}, comparison sequence X _i ={X _i (k)|k=1,2,...,n}, i=1,2 ,...,m

Dimensionlessize variables:

The gray correlation coefficient of the reference series and the comparison series:

Calculate the degree of association:

Sorting relevance, if r _i <r _j , then x _j (k) is closer to reference sequence y(k) than x _i (k).

It is calculated that the peak-to-peak value, fundamental frequency amplitude, 8x amplitude, and 2x amplitude are more correlated with RMS, and the 3x amplitude and 4x amplitude are less correlated with RMS. Remove the 3x and 4x variables and use RMS, The gray model GM (1, 1) was established with 5 variables of peak-to-peak value, fundamental frequency amplitude, 8x amplitude and 2x amplitude.

3. Establishment of prediction model

(1) Gray prediction model

The gray system model GM(1,1) performs an accumulation process through the univariate time series { _xi }(i=1,2,3,…), and establishes a first-order differential equation for this generation sequence to reveal its internal development law .

With characteristic gray system

do: have

Set up the following differential equation for X ⁽¹⁾ :

Record the differential equation of the first-order one variable as GM(1,1)

In the above formula, a and u can be fitted by the least square method:

In formula (5), Y _M is a column vector Y _M =[X ⁰ (2),X ⁰ (3),…,X ⁰ (n)] ^T ,

B is a matrix:

The time response function corresponding to the differential equation (5)

The predicted value of the sequence generated by one accumulation by formula (6): X ⁽⁰⁾ (t) = X ⁽¹⁾ (t)-X ⁽¹⁾ (t-1) (20)

The gray model GM(1,1) was established by using the first 10 groups of eigenvalues of each state, and the last 2 groups were used as the comparison value of the predicted value to calculate the error of the predicted value. The model and prediction accuracy established by the 10 sets of eigenvalues of the 0.007-inch outer ring fault in 1797 revolutions of the bearing are shown in Table 3. The average prediction error of the 10 sets of data is 4.94%.

(3) Use orthogonal polynomials for least square fitting prediction

Data fitting is to determine the type of curve y=s( _x ;a ₀ ,a ₁ ,…,an ) based on the relationship between the measured data, and then according to the principle that the sum of the squares of the errors at a given point reaches the minimum , which solves the unconstrained problem:

Determine the optimal parameters Thus a fitting curve y=s ^* (x) is obtained.

Suppose φ ₀ ,φ ₁ ,…,φ _n are n+1 functions, ω _i is the coefficient, satisfying That is, φ ₀ , φ ₁ ,…,φ _n are orthogonal on X={x ₁ ,x ₂ ,…,x _m }, where

Then the solution of the normal equation (8) is:

Orthogonal polynomials were used to perform least squares fitting on the first 10 groups of eigenvalues of each state, and the last two groups of data were used as the comparison of predicted values to calculate the error of predicted values. The 10 groups of eigenvalue square fitting prediction values and accuracy values of 0.007 inches in 1797 revolutions of the bearing are shown in Table 3. The average prediction error of the 10 groups of data is 7.05%. Through the comparison of the two prediction models, it can be seen that the average error value of the gray model is 4.91% less than the average value of the least squares fitting prediction error of the orthogonal polynomial 7.05%, so the gray model is selected as the prediction model.

Table 3. The model and prediction accuracy established by the 10 sets of eigenvalues of the outer ring fault of 0.007 inches in 1797 revolutions of the bearing

4. Training Support Vector Machines

Support vector machine is a typical neural network constructed based on statistical learning theory. Its main idea is to establish an optimal classification hyperplane so that the distance between the two types of samples on both sides of the plane is maximized, thus providing a good general model for classification problems. ability. For samples (x _i , y _i ), i=1,2,...,s, where x _i ∈ R ^m , y _i ∈ {+1,-1}, s is the dimension of the input variable. The hyperplane equation for classification is: ω·x+b=0

Divide the samples into two categories: ω·x+b≥0, (y=+1)

ω·x+b≥0,(y=-1) (23)

The optimal hyperplane of a support vector machine is a hyperplane that maximizes the classification margin, that is, maximum. So to solve the optimal hyperplane, that is

It should satisfy the constraints: y _i (ω· _xi + b)-1≥0, i=1,2,...,l

Under nonlinear conditions, the maximization function of a linear non-separable support vector machine:

The discriminant objective function is:

For the fault classification of rolling bearings, this paper chooses two types of SVM to construct a multi-class classifier. A multi-class classifier is constructed based on rolling bearing outer ring faults, inner ring faults, ball faults and normal states. Take 60 sets of training samples and 12 sets of test samples for each fault, a total of 200 sets of training samples, 40 sets of test samples; normal state 40 sets of training samples, 8 sets of test samples. Taking the outer ring fault as an example, the outer ring fault SVM1 is constructed, and the other two SVMs are the same. The training set is composed of 60 sets of training samples and 40 sets of normal samples for each type of fault as {(x ₁ ,y ₁ ),(x ₂ ,y ₂ ),…,(x ₂₂₀ ,y ₂₂₀ )}, where x _i ∈ R ⁵ , y _i ={-1,1}, when y=1, it means that the sample has an outer ring fault, and when y=-1, it means that there is no outer ring fault.

This paper uses the LIBSVM toolkit. The main task of training is to select the appropriate classifier parameters according to the sample, select the type of support vector machine (s), the type of kernel function (t), the coef0 setting (c) in the kernel function, and the The gamma function setting (g). Select the vector machine as C-SVC, that is, s is 0, and the kernel function is Gauss kernel function, that is, t is 2, c is 1.2, and g is 2.8.

Taking the outer ring fault as an example, it can be obtained that the recognition rate of the training samples by the support vector machine is 99%. A total of 36 groups of 12 groups of fault test samples of each type and 8 groups of normal state test samples were selected for testing, and the fault identification rate was obtained as 91%. Similarly, the other two support vector machines were trained, and each vector machine's recognition of all test samples Rates are shown in Table 4.

Table 4. The recognition rate of each vector machine for all test samples

Finally, the 12 groups of test samples of single-type faults are identified by using the trained three-type binary tree support vector machine multi-class classification method for fault identification, which is recorded in Table 5. It can be seen from Table 5 that the recognition rate of single-type faults can reach more than 90% by using the support vector machine classification method based on binary tree, so it is feasible and efficient for rolling bearing fault diagnosis.

Table 5. Training results on single-class failure test data

Bring the 44 sets of prediction data (36 sets of fault data and 8 sets of normal data) of the gray model into the trained three-class binary tree support vector machine, and record the training results of each bearing state, as shown in Table 6. It can be seen from Table 6 that by extracting the eigenvalues of the vibration data collected in each state of the rolling bearing, establishing a gray model, predicting the data, and bringing them into the three types of binary tree support vector machines trained by the eigenvalues of each fault state for classification, it can be effectively carried out. The early warning of faults greatly improves the safe operation status of mechanical equipment.

Table 6. Training results on prediction data

Claims (1)

1. A method for rolling bearing fault diagnosis and prediction based on gray support vector machine, characterized in that: the implementation process of the method is as follows, Extraction of S1 characteristic variables and correlation analysis; rolling bearings are typical rotating machinery, the time-domain characteristic variables of its vibration signal include root mean square value, peak-to-peak value, and mean value, and the frequency-domain characteristic variables include fundamental frequency, double frequency, and triple frequency frequency, 4 times frequency, 8 times frequency, they contain a wealth of fault information; compare and analyze the characteristic variables of the time domain and frequency domain of the vibration signal of various faults and normal conditions of rolling bearings, and select the appropriate characteristic variables; The feature variables are: X = (RMS, peak-to-peak value, 1x amplitude, 2x amplitude, 3x amplitude, 4x amplitude, 8x amplitude); RMS is the root mean square value of the vibration signal, which can best represent the overall characteristics of the signal. The RMS time series is selected as the reference sequence, and the other 6 sequences are used as the comparison sequence, and the gray correlation degree between the reference sequence and each comparison sequence is obtained, and the correlation degree is removed. Low 2 characteristic variables; Let the reference sequence Y={y(k)|k=1,2,...,n}, Comparison sequence X _i ={X _i (k)|k=1,2,…,n}, i=1,2,…,m Dimensionlessize variables: The gray correlation coefficient of the reference series and the comparison series: ${\mathrm{<span\; class="notranslate">\; \&xi;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&rho;</span>}\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&rho;</span>}\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$ Calculate the degree of association: Sorting relevance, if r _i <r _j , then x _j (k) is closer than x _i (k) to the reference sequence y(k); S2 Establish a prediction model, use the first 10 groups of eigenvalues of each state to establish a gray model and an orthogonal polynomial as a least squares fitting prediction model, and the last two groups are used as the comparison value of the predicted value to calculate the error of the predicted value of the two models, Select a model with a smaller error - the gray model; (1) Gray prediction model The gray system model GM(1,1) performs an accumulation process through the univariate time series { _xi }(i=1,2,3,…), and establishes a first-order differential equation for this generation sequence to reveal its internal development law ; Define characteristic gray system do: have Set up the following differential equation for X ⁽¹⁾ : Record the differential equation of the first-order one variable as GM(1,1) In the above formula, a and u can be fitted by the least square method: In formula (5), Y _M is a column vector Y _M =[X ⁰ (2),X ⁰ (3),…,X ⁰ (n)] ^T , B is a matrix: The time response function corresponding to the differential equation (5) The predicted value of the accumulated sequence generated by formula (6): X ⁽⁰⁾ (t) = X ⁽¹⁾ (t)-X ⁽¹⁾ (t-1) (7) (2) Use orthogonal polynomials for least square fitting prediction Data fitting is to determine the type of curve y=s( _x ;a ₀ ,a ₁ ,…,an ) based on the relationship between the measured data, and then according to the principle that the sum of the squares of the errors at a given point reaches the minimum , which solves the unconstrained problem: $\mathrm{<span\; class="notranslate">\; min</span>}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; ...</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; m</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ;</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; ...</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$ Determine the optimal parameters Thereby fitting curve y=s ^* (x) is obtained; Let φ ₀ ,φ ₁ ,…,φ _n be n+1 functions, and ω _i be the coefficient, satisfying: That is, φ ₀ , φ ₁ ,…,φ _n are orthogonal on X={x ₁ ,x ₂ ,…,x _m }, in Then the solution of the normal equation (8) is: S3 support vector machine is a typical neural network based on statistical learning theory, which establishes an optimal classification hyperplane to maximize the distance between the two types of samples on both sides of the plane, thus providing a good generalization ability for classification problems ; For samples ( _xi , y _i ), i=1,2,…,s, where x _i ∈ R ^m , y _i ∈ {+1,-1}, s is the input variable dimension; used for classification The hyperplane equation is: ω·x+b=0 Divide the samples into two categories: ω·x+b≥0, (y=+1) ω·x+b≥0,(y=-1) (10) The optimal hyperplane of a support vector machine is a hyperplane that maximizes the classification margin, that is, is the largest, so to solve the optimal hyperplane, that is It should satisfy the constraints: y _i (ω· _xi + b)-1≥0, i=1,2,...,l Under nonlinear conditions, the maximization function of linear non-separable support vector machines: $\mathrm{<span\; class="notranslate">\; max</span>}\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; l</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; l</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; l</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\mathrm{<span\; class="notranslate">\; K</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; )</span>$ The discriminant objective function is: Training support vector machine, for the fault classification of rolling bearings, choose two types of SVM to construct a multi-class classifier; because the two types of SVM 1-to-many algorithm and 1-to-1 algorithm have their own shortcomings, the binary tree-based support vector machine multi-class Classification method; the construction steps of the two-class classifier based on the binary tree are: the i-th classifier separates the i-th class from the i+1, i+2, ..., N classes, constructs SVMi, until the N-1 classifier Separate the N-1 category from the N-th category; combine N-1 SVMs into a multi-class classifier, and construct an SVM decision tree to identify N-type faults.