CN112560914A - Rolling bearing fault diagnosis method based on improved LSSVM - Google Patents

Rolling bearing fault diagnosis method based on improved LSSVM Download PDF

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CN112560914A
CN112560914A CN202011420849.6A CN202011420849A CN112560914A CN 112560914 A CN112560914 A CN 112560914A CN 202011420849 A CN202011420849 A CN 202011420849A CN 112560914 A CN112560914 A CN 112560914A
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whale
lssvm
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王海瑞
罗源睿
李卓漫
燕志星
李众
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Kunming University of Science and Technology
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Abstract

The invention relates to a rolling bearing fault diagnosis method based on an improved LSSVM, and belongs to the technical field of mechanical engineering automation. Firstly, a variable modal decomposition VMD method is adopted to carry out feature extraction on fault data of a rolling bearing, multi-scale permutation entropy MPE of each modal component after decomposition is calculated, a feature vector of a signal is constructed by the VMD-MPE feature extraction method, the feature vector is input into a least square support vector machine optimized by a whale algorithm WOA to carry out identification and classification, and finally a diagnosis result is output. A series of comparison experiments such as fitness comparison, diagnosis accuracy comparison, training time and test time comparison prove the effectiveness and stability of the model, and the model has a good practical effect in a rolling bearing fault diagnosis system. The invention has the advantages of clear and accurate expression of the fault diagnosis process, reasonable and effective fault diagnosis model and improved classification prediction effect.

Description

Rolling bearing fault diagnosis method based on improved LSSVM
Technical Field
The invention relates to a rolling bearing fault diagnosis method based on an improved LSSVM, and belongs to the technical field of mechanical engineering automation.
Background
With the rapid development of scientific technology, the system and its components of large mechanical equipment are also developed toward larger, more complicated and more intelligent. Therefore, the relationships between the mechanical equipment components are becoming more and more complex and intimate, and a series of production accidents are caused once some key components are broken down. Rolling bearings are key components in large mechanical equipment, and are one of the most prone to failure due to the fact that rolling bearings are often under complicated working conditions. The purpose of fault diagnosis of the rolling bearing is to kill various faults of the bearing in the cradle as early as possible.
Disclosure of Invention
The invention aims to solve the technical problem of obtaining fault classification with high accuracy at the highest possible speed, and provides a rolling bearing fault diagnosis method based on an improved LSSVM.
The technical scheme adopted by the invention is as follows: a rolling bearing fault diagnosis method based on improved LSSVM comprises the steps of firstly, adopting a Variational Mode Decomposition (VMD) method to conduct feature extraction on fault data of a rolling bearing, calculating multi-scale arrangement entropy (MPE) of each decomposed modal component, constructing feature vectors of signals through the VMD-MPE feature extraction method, then utilizing Whale Optimization Algorithm (WOA) to conduct Optimization processing on penalty factor c and kernel function parameter sigma parameters of a Least Square Support Vector Machine (LSSVM) Algorithm model to obtain an optimized LSSVM model, namely a WOA-LSSVM fault diagnosis model, and finally utilizing the fault diagnosis model to conduct fault diagnosis on the rolling bearing.
The method comprises the following specific steps:
(1) the rolling bearing is taken as a research object in the experiment, and the rolling bearing vibration signal data from the university of Kaiser storage in the United states is adopted to verify the real effectiveness of the experiment. The drive end bearing failure sampling frequency was 12 kHz. The rotating speed of the rolling bearing is set to 1979r/min, and vibration signals of the bearing in four working states of a normal state, an inner ring fault, a rolling body fault and an outer ring fault are selected. Extracting the characteristics of fault data of the rolling bearing by a Variational Modal Decomposition (VMD) method, calculating the multi-scale permutation entropy (MPE) of each modal component after decomposition, constructing the characteristic vector of the signal by a VMD-MPE characteristic extraction method,
(2) after the characteristics are extracted, a punishment factor c and a kernel function parameter sigma parameter of a Least Square Support Vector Machine (LSSVM) algorithm model are optimized by utilizing a whale algorithm (WOA) to obtain an optimized LSSVM model, namely a WOA-LSSVM fault diagnosis model, and finally fault diagnosis is carried out on the rolling bearing by utilizing the fault diagnosis model. The specific diagnosis method comprises the following steps: after the characteristics of each 90 groups of training data are extracted, inputting the characteristic vectors into a least square support vector machine optimized by a whale algorithm for training, finally inputting each 30 groups of test data into a trained model, finally obtaining the fault type of the test data, and calculating the accuracy of the WOA-LSSVM diagnostic model to be 91.67%.
The predation process of whales is divided into the following three stages:
first, surround the prey
The whale continuously updates the position of the whale to be close to the nearest whale position and surrounds the whale in a surrounding way by judging the position of the prey, and the process is expressed by a mathematical formula:
D＝|CX*(t)-X(t)| (1)
X(t+1)＝X*(t)-AD (2)
in the formula: x*(t) is the current optimal solution; x (t) is the current solution; a is a convergence coefficient vector, C is a wobble coefficient vector, and is defined as:
A＝2ar-a (3)
C＝2r (4)
in the formula: a is a random factor, the value of which decreases linearly from 2 to 0; r is a random vector within [0,1 ].
Second, bubble net attack
The following two methods were designed to simulate this process:
1. shrink wrapping Mechanism (SEM): this process is implemented using a gradually decreasing value of a in equation (3), where A also decreases as the value of a decreases, and A ∈ [ -a, a ]. When a <1, the whale initiates an attack on the prey, and when a >1, the whale forces to abandon the prey and re-search.
2. Spiral Update Position (SUP): whales continuously reduce the range of prey enclosure by spiral bubbles, and the process is expressed mathematically as:
X(t+1)＝D′eblcos(2πl)+X*(t) (5)
in the formula: d' ═ X*(t) -X (t) l is whale X (t) and local optimization X*(t) distance between; b is a constant defining a helix; l is [ -1,1 [ ]]The random number in (c).
In the whale hunting process, the two modes are comprehensively considered according to 50% probability, and the mathematical formula is represented as follows:
Figure BDA0002822294180000031
In the formula: p is a random number within [0,1 ].
Third, search for prey
The final stage is global search, the optimal whale position is not searched and updated locally any more, but random search is carried out in a large range, and the problem that local optimization is easy to fall into is avoided. The specific process at this stage is as follows:
D＝|CXrand-X(t)| (7)
X(t+1)＝Xrand-AD (8)
in the formula: xrandA randomly selected position vector in the current population.
The WOA optimizes two parameters of σ and c of LSSVM as follows:
(1) initializing WOA parameters including whale colony size S, upper and lower limits of whale colony position and maximum iteration number Tmax。
(2) And randomly initializing whale search positions, wherein the search positions contain two parameter information of sigma and c in the LSSVM.
(3) Calculating the fitness value of each individual in the group, comparing the fitness values, and finding out the current global optimal solution X*。
(4) When T is less than or equal to TmaxWhen a, A, C, l and p are updated.
(5) When p is less than 0.5, if A is less than 1, updating the current whale colony individual position through a formula (5), and if A is more than or equal to 1, updating the current whale colony individual position through a formula (8); when p is more than or equal to 0.5, the position of the current whale colony individual is updated through the formula (6).
(6) Calculating the fitness value of each individual in the whale population, comparing the fitness values, and finding out the current global optimal solution X*And (4) judging whether a termination condition is met, if so, jumping to the step (7), and if not, returning to the step (4) to continue optimizing.
(7) The globally optimal whale position and the optimal parameter combination (c, sigma) of the LSSVM are output.
The least square support vector machine method used by the invention is improved on the basis of a Support Vector Machine (SVM), and on the basis of inheriting the basic principles of a structure risk minimization principle, a kernel function and the like of an SVM model, the quadratic programming problem is converted into the problem of solving a linear equation set, so that the computational complexity is reduced, the method is simpler, the convergence speed is higher, and higher precision can be obtained. The LSSVM is realized as follows:
(1) let the training set sequence { (x)i,yi),i＝1,2,...,n}，xi∈RnFor n-dimensional input samples, yiAnd e.R is an output one-dimensional variable group.
(2) First using a non-linear mapping function
Figure BDA0002822294180000041
Mapping the input samples to a high-dimensional feature space, constructing a regression function in the high-dimensional feature space, and according to a structural risk minimization principle, optimizing an objective function by using the LSSVM to obtain the following parameters:
Figure BDA0002822294180000042
Figure BDA0002822294180000043
in the formula: omega is a weight vector; b is a bias vector; xi is a relaxation variable; and c is a penalty factor.
(3) Solving the optimization problem by Lagrange's method, where αiIs a Lagrange multiplier, solved by the KKT condition. According to the Mercer condition, the invention selects a radial basis kernel function (RBF) as a kernel function, and the expression is as follows:
k(xi,xj)＝exp[-||xi-xj||2/σ2] (11)
in the formula: σ is the kernel width. The final decision function of the LSSVM is:
Figure BDA0002822294180000044
the main optimization parameters in the LSSVM model are a kernel function parameter sigma and a punishment parameter c, and the selection of the kernel function parameter sigma and the punishment parameter c can not influence the recognition rate and the generalization capability of the model easily, so that the two parameters of the LSSVM are optimized by adopting a WOA algorithm to obtain the optimal combination (c, sigma).
The invention has the beneficial effects that:
1. the fault diagnosis model and the optimization method of the rolling bearing are provided, so that the fault diagnosis process is expressed clearly and accurately, and the fault diagnosis model is reasonable and effective;
2. and optimizing a penalty factor c and a kernel function parameter sigma of the support vector machine by adopting a whale algorithm so as to optimize the classification prediction effect of the model.
Drawings
FIG. 1 is a flow chart of a diagnostic model of the present invention;
FIG. 2 is MPE diagram of rolling bearing in four states;
FIG. 3 is a fitness curve diagram of three models of WOA-LSSVM, GA-LSSVM and PSO-LSSVM;
FIG. 4 is a classification diagram of a WOA-LSSVM model test set.
Detailed Description
The invention is further described below with reference to the accompanying drawings and specific embodiments.
Example 1: as shown in FIGS. 1-4, the invention provides a fault diagnosis model for a rolling bearing based on whale algorithm optimization least square support vector machine. Firstly, the acquired signals are processed by adopting variational modal decomposition, so that the characteristic vectors of the signals are extracted, then, related parameters of a least square support vector machine are optimized by using a whale algorithm, a network structure of the optimized least square support vector machine is built, and finally, the extracted characteristic vectors are input into the optimized least square support vector machine for training and testing to obtain a final fault diagnosis result. The effectiveness and the stability of the model are proved by setting a series of comparison experiments such as fitness comparison, diagnosis accuracy comparison, comparison between training time and testing time and the like. And the model has good practical effect in a rolling bearing fault diagnosis system.
The method comprises the following specific steps: a rolling bearing fault diagnosis method based on improved LSSVM,
(1) the rolling bearing is taken as a research object in the experiment, and the rolling bearing vibration signal data from the university of Kaiser storage in the United states is adopted to verify the real effectiveness of the experiment. The drive end bearing failure sampling frequency was 12 kHz. The rotating speed of the rolling bearing is set to 1979r/min, and vibration signals of the bearing in four working states of a normal state, an inner ring fault, a rolling body fault and an outer ring fault are selected. And (3) carrying out feature extraction on the fault data of the rolling bearing by a Variation Modal Decomposition (VMD) method, calculating the multi-scale permutation entropy (MPE) of each modal component after decomposition, and constructing the feature vector of the signal by a VMD-MPE feature extraction method.
(2) After the characteristics are extracted, a punishment factor c and a kernel function parameter sigma parameter of a Least Square Support Vector Machine (LSSVM) algorithm model are optimized by utilizing a whale algorithm (WOA) to obtain an optimized LSSVM model, namely a WOA-LSSVM fault diagnosis model, and finally fault diagnosis is carried out on the rolling bearing by utilizing the fault diagnosis model. The specific diagnosis method comprises the following steps: after the characteristics of each 90 groups of training data are extracted, inputting the characteristic vectors into a least square support vector machine optimized by a whale algorithm for training, finally inputting each 30 groups of test data into a trained model, finally obtaining the fault type of the test data, and calculating the accuracy of the WOA-LSSVM diagnostic model to be 91.67%.
Whale Optimization Algorithm (WOA) is a group intelligent optimization algorithm proposed by simulating whale predation behavior. When a whale finds a prey, the whale firstly submerges under the prey, then ascends along a circular path to spit out spiral air bubbles, and finally the prey is swallowed in a small range through the air bubbles. The WOA has the advantages of simple structure, less parameter adjustment, high convergence speed, strong global optimization capability and the like. The predation process of whales is divided into the following three stages:
first, surround the prey
The whale continuously updates the position of the whale to be close to the nearest whale position and surrounds the whale in a surrounding way by judging the position of the prey, and the process is expressed by a mathematical formula:
D＝|CX*(t)-X(t)| (1)
X(t+1)＝X*(t)-AD (2)
in the formula: x*(t) is the current optimal solution; x (t) is the current solution; a is a convergence coefficient vector, C is a wobble coefficient vector, and is defined as:
A＝2ar-a (3)
C＝2r (4)
in the formula: a is a random factor, the value of which decreases linearly from 2 to 0; r is a random vector within [0,1 ].
Second, bubble net attack
The following two methods were designed to simulate this process:
1. shrink wrapping Mechanism (SEM): this process is implemented using a gradually decreasing value of a in equation (3), where A also decreases as the value of a decreases, and A ∈ [ -a, a ]. When a <1, the whale initiates an attack on the prey, and when a >1, the whale forces to abandon the prey and re-search.
2. Spiral Update Position (SUP): whales continuously reduce the range of prey enclosure by spiral bubbles, and the process is expressed mathematically as:
X(t+1)＝D′eblcos(2πl)+X*(t) (5)
in the formula: d' ═ X*(t) -X (t) l is whale X (t) and local optimization X*(t) distance between; b is a constant defining a helix; l is [ -1,1 [ ]]The random number in (c).
In the whale hunting process, the two modes are comprehensively considered according to 50% probability, and the mathematical formula is represented as follows:
Figure BDA0002822294180000061
In the formula: p is a random number within [0,1 ].
Third, search for prey
The final stage is global search, the optimal whale position is not searched and updated locally any more, but random search is carried out in a large range, and the problem that local optimization is easy to fall into is avoided. The specific process at this stage is as follows:
D＝|CXrand-X(t)| (7)
X(t+1)＝Xrand-AD (8)
in the formula: xrandA randomly selected position vector in the current population.
The WOA optimizes two parameters of σ and c of LSSVM as follows:
(1) initializing WOA parameters including whale colony size S, upper and lower limits of whale colony position and maximum iteration number Tmax。
(2) And randomly initializing whale search positions, wherein the search positions contain two parameter information of sigma and c in the LSSVM.
(3) Calculating the fitness value of each individual in the group, comparing the fitness values, and finding out the current global optimal solution X*。
(4) When T is less than or equal to TmaxWhen a, A, C, l and p are updated.
(5) When p is less than 0.5, if A is less than 1, updating the current whale colony individual position through a formula (5), and if A is more than or equal to 1, updating the current whale colony individual position through a formula (8); when p is more than or equal to 0.5, the position of the current whale colony individual is updated through the formula (6).
(6) In counting whale groupsThe fitness value of each individual is compared to find out the current global optimal solution X*And (4) judging whether a termination condition is met, if so, jumping to the step (7), and if not, returning to the step (4) to continue optimizing.
(7) The globally optimal whale position and the optimal parameter combination (c, sigma) of the LSSVM are output.
The least square support vector machine method used by the invention is improved on the basis of a Support Vector Machine (SVM), and on the basis of inheriting the basic principles of a structure risk minimization principle, a kernel function and the like of an SVM model, the quadratic programming problem is converted into the problem of solving a linear equation set, so that the computational complexity is reduced, the method is simpler, the convergence speed is higher, and higher precision can be obtained. The LSSVM is realized as follows:
(1) let the training set sequence { (x)i,yi),i＝1,2,...,n}，xi∈RnFor n-dimensional input samples, yiAnd e.R is an output one-dimensional variable group.
(2) First using a non-linear mapping function
Figure BDA0002822294180000071
Mapping the input samples to a high-dimensional feature space, constructing a regression function in the high-dimensional feature space, and according to a structural risk minimization principle, optimizing an objective function by using the LSSVM to obtain the following parameters:
Figure BDA0002822294180000072
Figure BDA0002822294180000073
in the formula: omega is a weight vector; b is a bias vector; xi is a relaxation variable; and c is a penalty factor.
(3) Solving the optimization problem by Lagrange's method, where αiIs a Lagrange multiplier, solved by the KKT condition. According to the Mercer condition, the invention selects a radial basis kernel function (RBF) as a kernel function, and the expression is as follows:
k(xi,xj)＝exp[-||xi-xj||2/σ2] (11)
in the formula: σ is the kernel width. The final decision function of the LSSVM is:
Figure BDA0002822294180000081
the main optimization parameters in the LSSVM model are a kernel function parameter sigma and a punishment parameter c, and the selection of the kernel function parameter sigma and the punishment parameter c can not influence the recognition rate and the generalization capability of the model easily, so that the two parameters of the LSSVM are optimized by adopting a WOA algorithm to obtain the optimal combination (c, sigma).
The following description will be made with reference to specific examples.
Case one: extracting a specific rolling bearing fault signal, specifically, extracting the characteristics of fault data of the rolling bearing by using a Variational Modal Decomposition (VMD) method, calculating the multi-scale permutation entropy (MPE) of each modal component after decomposition, and constructing the characteristic vector of the signal by using the VMD-MPE characteristic extraction method, so that the VMD-MPE characteristic extraction method is adopted, the MPE structural characteristic vector is used as subsequent input, and the result is shown in a table 1 (only part of samples are listed under each state due to limited space).
TABLE 1 partial MPE feature vectors
Figure BDA0002822294180000082
FIG. 2 shows MPE analysis of each modal component of the bearing in four states intuitively, and MPE in different states is different because the randomness of vibration signals changes when the bearing fails, so that MPE changes. MPE can effectively detect the dynamic change of the vibration signal and reflect the fault characteristics of the vibration signal under different scales, so that MPE is introduced to quantize the fault characteristics of the bearing.
In order to verify the optimizing performance of a Whale Optimization Algorithm (WOA), comparison experiments of three experimental models, namely WOA-LSSVM, GA-LSSVM and PSO-LSSVM are set. The comparative experimental adaptation curve is shown in fig. 3.
When the fitness curve is stable, it shows that the fitness curve tends to converge, and at this time, the optimal parameters can be obtained. As can be seen from the fitness curve in the graph, the GA optimized LSSVM algorithm tends to converge about 30 generations, the PSO optimized LSSVM algorithm tends to converge about 33 generations, the WOA optimized LSSVM algorithm searches an optimal value after 19 iterations, the convergence accuracy is high, and the result obtained by optimizing has a more remarkable effect than other two optimization algorithms.
In order to test the classification accuracy of the rolling bearing fault vibration diagnosis model of the WOA-LSSVM, the test sample set which is left after the model is trained is input into the formed WOA-LSSVM model for prediction, and the prediction result is compared with the actual output truth value. And obtaining a classification chart of the WOA-LSSVM model test set, as shown in FIG. 4. When 120 prediction sample sets are used for prediction output, the number of misjudgment groups of the WOA algorithm is 10, which indicates that the number of correct diagnosis groups is 110, the prediction precision is 91.67%, the misjudgment rate is low, the prediction precision is high, and the fault type of the rolling bearing fault system can be accurately predicted.
While the present invention has been described in detail with reference to the embodiments shown in the drawings, the present invention is not limited to the embodiments, and various changes can be made without departing from the spirit of the present invention within the knowledge of those skilled in the art.

Claims (3)

1. A rolling bearing fault diagnosis method based on an improved LSSVM is characterized in that: the method comprises the following steps: selecting vibration signals of a bearing in four working states of a normal state, an inner ring fault, a rolling body fault and an outer ring fault as fault data, performing feature extraction on the fault data of the rolling bearing by using a variational modal decomposition VMD method, calculating a multi-scale arrangement entropy MPE of each modal component after decomposition, constructing a feature vector of the signal by using a VMD-MPE feature extraction method, then performing optimization processing on a penalty factor c and a kernel function parameter sigma parameter of a Least Squares Support Vector Machine (LSSVM) algorithm model by using a whale algorithm WOA to obtain an optimized LSSVM model, namely a WOA-LSSVM fault diagnosis model, and finally performing fault diagnosis on the rolling bearing by using the fault diagnosis model.
2. The rolling bearing fault diagnosis method based on the improved LSSVM of claim 1, wherein: the method for carrying out fault diagnosis on the rolling bearing by utilizing the WOA-LSSVM fault diagnosis model specifically comprises the following steps: after the characteristics of each 90 groups of training data are extracted, inputting the characteristic vectors into a least square support vector machine optimized by a whale algorithm for training, finally inputting each 30 groups of test data into a trained model, finally obtaining the fault type of the test data, and calculating the accuracy of the WOA-LSSVM diagnostic model.
3. The rolling bearing fault diagnosis method based on the improved LSSVM of claim 1, wherein: the specific process of optimizing the penalty factor c and the kernel function parameter sigma of the LSSVM model by using the whale algorithm WOA is as follows:
the predation process of whales is divided into the following three stages:
first, surround the prey
The whale continuously updates the position of the whale to be close to the nearest whale position and surrounds the whale in a surrounding way by judging the position of the prey, and the process is expressed by a mathematical formula:
D＝|CX*(t)-X(t)| (1)
X(t+1)＝X*(t)-AD (2)
in the formula: x*(t) is the current optimal solution; x (t) is the current solution; a is a convergence coefficient vector, C is a wobble coefficient vector, and is defined as:
A＝2ar-a (3)
C＝2r (4)
in the formula: a is a random factor, the value of which decreases linearly from 2 to 0; r is a random vector within [0,1 ];
second, bubble net attack
The following two methods were designed to simulate this process:
1. shrinkage and wrapping mechanism SEM: the process is realized by gradually decreasing the value a in the formula (3), wherein A also decreases with the decrease of the value a, A is [ -a, a ], when A <1, the whale attacks the prey, and when A >1, the whale forces to abandon the prey and search again;
2. helical position update SUP: whales continuously reduce the range of prey enclosure by spiral bubbles, and the process is expressed mathematically as:
X(t+1)＝D′eblcos(2πl)+X*(t) (5)
in the formula: d' ═ X*(t) -X (t) l is whale X (t) and local optimization X*(t) distance between; b is a constant defining a helix; l is [ -1,1 [ ]]A random number within;
in the whale hunting process, the two modes are comprehensively considered according to 50% probability, and the mathematical formula is represented as follows:
Figure FDA0002822294170000021
in the formula: p is a random number within [0,1 ];
third, search for prey
The final stage is global search, the optimal whale position is not searched and updated locally any more, but random search is carried out in a large range, the problem of easy falling into local optimal is avoided, and the specific process of the stage is as follows:
D＝|CXrand-X(t)| (7)
X(t+1)＝Xrand-AD (8)
in the formula: xrandRandomly selecting a position vector for the current population;
the WOA optimizes two parameters of σ and c of LSSVM as follows:
(1) initializing WOA parameters including whale colony size S, upper and lower limits of whale colony position and maximum iteration number Tmax；
(2) Randomly initializing whale search positions, wherein the whale search positions comprise two parameter information of sigma and c in the LSSVM;
(3) calculating the fitness value of each individual in the group, comparing the fitness values, and finding out the current global optimal solution X*；
(4) When T is less than or equal to TmaxUpdating a, A, C, l and p;
(5) when p is less than 0.5, if A is less than 1, updating the current whale colony individual position through a formula (5), and if A is more than or equal to 1, updating the current whale colony individual position through a formula (8); when p is more than or equal to 0.5, updating the individual position of the current whale colony through a formula (6);
(6) calculating the fitness value of each individual in the whale population, comparing the fitness values, and finding out the current global optimal solution X*Judging whether a termination condition is met, if so, jumping to the step (7), and if not, returning to the step (4) to continue optimizing;
(7) outputting the globally optimal whale position and the optimal parameter combination (c, sigma) of the LSSVM;
the least square support vector machine method is improved on the basis of a Support Vector Machine (SVM), and a quadratic programming problem is converted into a problem for solving a linear equation set on the basis of inheriting basic principles such as a structure risk minimization principle, a kernel function and the like of an SVM model;
the LSSVM is realized as follows:
(1) let the training set sequence { (x)i,yi),i＝1,2,...,n}，xi∈RnFor n-dimensional input samples, yiE is R as an output one-dimensional variable group;
(2) first using a non-linear mapping function
Figure FDA0002822294170000034
Mapping the input samples to a high-dimensional feature space, constructing a regression function in the high-dimensional feature space, and according to a structural risk minimization principle, optimizing an objective function by using the LSSVM to obtain the following parameters:
Figure FDA0002822294170000031
Figure FDA0002822294170000032
in the formula: omega is a weight vector; b is a bias vector; xi is a relaxation variable; c is a penalty factor;
(3) solving the optimization problem by Lagrange's method, where αiThe Lagrange multiplier is solved by a KKT condition, and according to a Mercer condition, a radial basis kernel function (RBF) is selected as a kernel function, and the expression is as follows:
k(xi,xj)＝exp[-||xi-xj||2/σ2] (11)
in the formula: σ is the kernel function width, and the final decision function of the LSSVM is as follows:
Figure FDA0002822294170000033
the main optimization parameters in the LSSVM model are a kernel function parameter sigma and a punishment parameter c, and the selection of the kernel function parameter sigma and the punishment parameter c can not influence the recognition rate and the generalization capability of the model easily, so that the two parameters of the LSSVM are optimized by adopting a WOA algorithm to obtain the optimal combination (c, sigma).
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CN116561520A (en) * 2023-07-06 2023-08-08 昆明理工大学 Blasting vibration signal noise reduction method based on WOA-VMD-MPE
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CN113014170A (en) * 2021-04-21 2021-06-22 昆明理工大学 Permanent magnet synchronous motor minimum loss control method based on whale optimization algorithm
CN113014170B (en) * 2021-04-21 2022-08-12 昆明理工大学 Permanent magnet synchronous motor minimum loss control method based on whale optimization algorithm
CN113358357A (en) * 2021-06-25 2021-09-07 上海应用技术大学 Fault diagnosis method for rolling bearing of numerical control machine tool
CN113358357B (en) * 2021-06-25 2023-08-04 上海应用技术大学 Fault diagnosis method for rolling bearing of numerical control machine tool
CN114325491A (en) * 2021-12-01 2022-04-12 西南交通大学 Fault positioning method for transformer bushing insulation
CN114325491B (en) * 2021-12-01 2022-07-19 西南交通大学 Fault positioning method for transformer bushing insulation
CN114511926A (en) * 2022-01-17 2022-05-17 江苏大学 Pig feeding behavior identification method based on combination of improved support vector machine and optical flow method
CN114511926B (en) * 2022-01-17 2024-05-14 江苏大学 Pig feeding behavior identification method based on combination of improved support vector machine and optical flow method
CN115689767A (en) * 2022-09-23 2023-02-03 国网冀北电力有限公司经济技术研究院 Transformer substation construction cost interval prediction method based on WOA-LS-SVM
CN115876476A (en) * 2023-02-16 2023-03-31 山东科技大学 Rolling bearing fault diagnosis method, system, computer device and storage medium
CN116561520A (en) * 2023-07-06 2023-08-08 昆明理工大学 Blasting vibration signal noise reduction method based on WOA-VMD-MPE
CN117609768A (en) * 2024-01-23 2024-02-27 昆明理工大学 Vertical water pump unit fault diagnosis method based on improved whale algorithm

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