A novel fault diagnosis method based on variational mode decomposition (VMD) and multikernel support vector machine (MKSVM) optimized by Immune Genetic Algorithm (IGA) is proposed to accurately and adaptively diagnose mechanical faults. First, mechanical fault vibration signals are decomposed into multiple Intrinsic Mode Functions (IMFs) by VMD. Then the features in timefrequency domain are extracted from IMFs to construct the feature sets of mixed domain. Next, Semisupervised Locally Linear Embedding (SSLLE) is adopted for fusion and dimension reduction. The feature sets with reduced dimension are inputted to the IGA optimized MKSVM for failure mode identification. Theoretical analysis demonstrates that MKSVM can approximate any multivariable function. The global optimal parameter vector of MKSVM can be rapidly identified by IGA parameter optimization. The experiments of mechanical faults show that, compared to traditional fault diagnosis models, the proposed method significantly increases the diagnosis accuracy of mechanical faults and enhances the generalization of its application.
To ensure the safe and reliable operation of mechanical equipments, vibrations are usually analyzed to diagnose mechanical faults [
As an adaptive method which processes signal in timefrequency domain, empirical mode decomposition (EMD) [
In pattern recognition, support vector machine (SVM) [
The rest of this paper is organized as follows. In Section
IMF components obtained by EMD should meet the following criteria: (1) In a data sequence, the number of extreme points and the number of zero crossing points are equal, or up to a difference of 1; (2) at any data point, the average of the local maximum envelope and the local minimum envelope is 0. The basic algorithm of EMD is as follows.
First, determine the local extremes of signal
Repeat Step
Define
Since the residual component
The original signal
In VMD algorithm [
To obtain IMF components, VMD algorithm does not use the cycled screening stripping signal processing mode of EMD. Instead, VMD moves the signal decomposition process into the variational framework. It realizes adaptive signal decomposition by searching the optimal solution of the constrained variational model. The frequency center and bandwidth of each IMF component are updated in the iterative solving process of the variation model. The signal band is adaptively split according to the frequency domain features of the signal. Finally the narrow band IMF components are obtained.
Assuming the original signal
To obtain the optimal solution of the above constrained variational problem, the following augmented Lagrange function is introduced:
The optimal solution of the constrained variational model is derived by using alternating direction multiplier algorithm, which solves the saddle point of the above augmented Lagrange function. The decomposition of the original signal
Initialize
Perform the first inner cycle and update
Perform the second inner cycle and update
Update
Repeat steps (2) to (7) until the criterion
Assume two linearly separable sample sets,
Normalizing the decision equation so that the samples of both classes satisfy
Support vectors are samples satisfying (
Schematic of the optimal surface.
Under the constraint of (
An optimal hyper surface is found such that the average classification error for the entire training sample set reaches minimum. Introducing a nonnegative relaxation factor
A penalty term
Using Lagrange optimization method, the above optimal classification surface problem is converted into the following dual problem of convex quadratic programming optimization:
By solving the above problem, the optimal decision function can be described as follows:
Since different kernel functions correspond to different decision functions, the selection of kernel functions is very important in fault identification using SVM, and it directly affects the identification accuracy of SVM.
The kernel functions of SVM mainly include local kernel functions and global kernel functions. The Gaussian kernel function is a typical kernel function, which is described as follows:
Polynomial function is a typical global kernel function, which is described as follows:
Local kernel functions have strong learning ability but weak generalization ability; while global kernel functions have strong generalization ability but weak learning ability. In order to achieve better learning and generalization abilities of SVM, MKSVM is constructed based on local kernels and global kernels:
IGA treats the object problem to be solved as biological invasion antigen and the feasible solution of the problem as antibody. The searching process of the optimal solution can be viewed as the process of seeking maximum antigen affinity antibodies by biological systems. The inhibition and promotion of antibodies can ensure the diversity of antibodies in the population and improve the local searching ability of GA. Crossover and mutation of antibodies can ensure that the antibody population evolves towards the direction of high fitness and maintain the diversity of the population. The memory unit accelerates searching by constantly updating with better solutions, which improves the global searching capability of the algorithm. The flowchart of IGA algorithm is shown in Figure
Flowchart of IGA.
In IGAMKSVM, IGA algorithm is used to optimize the weight factor, penalty parameter, and kernel parameter. First, an antibody gene vector
To minimize the square error between the actual output and the expected output of MKSVM, the fitness function
The flowchart of IGAMKSVM algorithm is shown in Figure
Initialize the population and determine the population size, fitness threshold, and maximum iteration number. Determine the initial vector of each antibody within the ranges of penalty parameter and kernel parameters.
Compute the fitness value of each antibody according to (
For the current population, choose the antibody with the highest fitness value as the elite antibody, and save this antibody in a special variable.
If it is the first generation of antibody population, go to step (7); otherwise, go to the next step.
Determine the fitness value of each antibody. If no antibody in the current antibody generation has the same fitness value as the elite antibody, then replace the antibody having the smallest fitness value in the current antibody population by the elite antibody saved in a special variable; otherwise, go to the next step.
If the maximum fitness value in the current antibody population is larger than that of the elite antibody, then copy the antibody with the maximum fitness value into the special variable to replace the current elite antibody; otherwise, go to the next step.
According to the similarity definition, compute the density and selection probability of each antibody; perform selection and copy operations for the antibody population according to the selection probability.
Perform crossover and mutation operation for the antibody population.
Judge whether the criterion of termination is satisfied. If yes, output the results and the algorithm ends; otherwise, return to step (2) and continue the cycle.
The flowchart of fault diagnosis based on VMD and IGAMKSVM is shown in Figure
First, decompose the obtained fault vibration signals of rotating machinery by VMD, and generate
Extract 5 statistical features in time domain and 7 statistical features in frequency domain. These features constitute a
Use the SSLLE algorithm to merge the combined highdimensional feature set and reduce its dimension and then input the feature set to IGAMKSVM as feature vectors.
Use training samples to train the IGAMKSVM and obtain the optimal weight factor
The dimensionless time domain characteristic index.
Number  Feature expression 

1 

2 

3 

4 

5 

The characteristic parameters in frequency domain.
Number  Feature expression 

6 

7 

8 

9 

10 

11 

12 

The process of fault diagnosis based on VMD and IGAMSVM.
The experimental test system is composed of a speed motor, a driving belt, a coupling, a test bearing, a magnetic brake, an acceleration sensor, and a signal record analyzer. The bearing model is N205. The experimental devices are shown in Figure
Experimental devices.
Three N205 bearings were used to simulate different damages, including the outer ring damage, rolling damage, and inner ring damage. Bearing damages were implemented by processing a slot in different parts of bearings with a laser cutting machine. The width of the slot is 0.3 mm, and the depth is 0.1 mm. Different bearing faults are shown in Figure
Experimental parameters of rolling bearing.
Experimental parameter  Value 

Outer ring diameter  52 mm 
Inner ring diameter  25 mm 
Rolling ring diameter  7.5 mm 
The number of rolling elements  12 
Bearing contact angle 

Driving speed  1000 r/min 
Bearing type  N205 
Sampling frequency  10 KHz 
Rolling bearing fault.
ICP accelerometers and data acquisition equipment DP/INV306U were used to collect the vibration signals of bearings. The sampling frequency is 10 kHz. In the input terminal, a low pass filter was used for antialiasing. For each operating status, 120 sample groups were collected, with a total of 10000 points for each sample group. Thirty sample groups were randomly selected as training samples, and the other 90 sample groups were test samples. Figure
The original signal waveforms in different status of rolling bearing.
Normal operating status
Bearing outer race fault
Bearing inner race fault
Bearing rolling element fault
VMD decomposition was performed for the four modes of vibration signals. The decomposition results of VMD and EMD for normal vibration signals of bearings are shown in Figure
The decomposition results of VMD and EMD for a normal vibration signal.
VMD decomposition
EMD decomposition
Comparison of dimension reduction after the decomposition of VMD and EMD.
The fault samples after dimension reduction were inputted into the IGAMKSVM diagnosis model for classification recognition. The recognition results are shown in Figure
Average accuracy of classification by two different decomposition methods.
To further verify the advantage and stability of the IGAMKSVM fault diagnosis model, the standard MKSVM, SVM with Gaussian kernel (GSVM), and SVM with polynomial kernel (PSVM) were choose for comparison. We used fold cross validation [
Comparison of classification accuracy with four different parameters.
Diagnostic model  Average percentage of correct recognition (%)  

Normal  Outer ring fault  Inner ring fault  Rolling element fault  
IGAMSVM  100  96.67  93.33  90.00 
MSVM  100  86.67  91.11  85.56 
GSVM  100  81.11  84.44  74.44 
PSVM  100  78.89  82.22  68.89 
To verify the robustness and generalization ability of IGAMKSVM diagnostic model, stochastic noise signals with
Comparison of the noise immunity by using four different models for fault diagnosis.
Diagnostic model  Average correct ratio of recognition (%)  





IGAMSVM  95.56  91.11  80.00 
MSVM  88.89  83.33  71.11 
GSVM  81.11  78.89  63.33 
PSVM  77.78  68.89  48.89 
To achieve accurate and adaptive identification of mechanical faults, a new fault diagnosis method based on VMD and IGA optimized MSVM is proposed in this paper. IGA overcomes the immature convergence problem of traditional algorithms and is able to solve the initial parameter selection problem of MSVM. From this aspect, it makes MSVM more applicable, robust, and accurate. Experimental results demonstrate that, compared to traditional methods, the combination of VMD method and IGAMSVM for fault diagnosis produces more repeatable results, with stronger generalization ability as well as antiinterference ability, providing a new effective method for mechanical fault diagnosis.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by National Key Basic Research and Development Program (973 Program) (2015CB057702), the National Natural Science Foundation of China (Project no. 51205431), Chongqing Commission of Science and Technology Research Projects (Project no. KJ1401303), and the Research Foundation of Chongqing University of Science & Technology (Project no. CK2015Z19). Finally, the authors are very grateful to the anonymous reviewers for their helpful comments and constructive suggestions.