## 4 Engineering Test Verification
### Analysis of Test Data  
In order to verify the effectiveness of the fusion algorithm of IFOA-VMD and hierarchical dispersion entropy (HDE) in fuel injector fault feature extraction, it is applied to the test data analysis of high-pressure common rail injector.  
Figure 10: High-pressure common rail test system.  
Figure 9: High-pressure common rail fuel injection system.  
First, on the high-pressure common rail experimental platform, the pressure wave signals of the injector under three working conditions are obtained, including normal operation, solenoid valve failure, and nozzle clogging. The data in a working cycle of diesel engine is show in Figure 11.  
Then, the collected fuel pressure wave signal is decomposed, and the pressure wave signals of the three working states input into the IFOA-VMD algorithm. Taking the normal working state as an example, the IFOA algorithm sets sizepop = 10, maxgen = 20, and the VMD algorithm sets the center frequency init = 1 to terminate the condition \(tol\ =\ 1\times 10^{-7}\). The energy growth factor is used as the concentration determination function, and the optimal \(K=6\), \(\alpha=6000\) is obtained by the IFOA algorithm. Next, the pressure wave signal is run with the optimal \(K\) and the combined parameter \(\alpha\) to obtain the IMF component, and the results are shown in Figure 12.  
Then, the state characteristic parameters of the IMF components of the three working state pressure wave signals decomposed by the IFOA-VMD are extracted, and the hierarchical dispersion entropy of the fuel pressure wave IMF component obtained by the IFOA-VMD algorithm is calculated. In this paper, \(m=2\), \(c=3\), \(k=3\), and \(N=1024\) are set. The hierarchical dispersion entropy of the IMF components of the three working states is calculated, and the feature extraction of the fuel pressure wave signals of the three working states is completed.  
Figure 11: Pressure wave original signal in three states of the injector.  
Figure 12: IFOA-VMD decomposition results under normal conditions.  
Finally, the hierarchical dispersion entropy of IMF components in different working conditions is regarded as the feature vector for fault state classification and recognition, and the input support vector machine is used for classification and recognition. After obtaining the HDE characteristic parameters, it is necessary to establish an intelligent classifier to realize the automatic fault diagnosis of the injector. When dealing with multi-class classification problems, it is necessary to construct a suitable multi-class classifier. Currently, indirect methods are mainly used to construct multi-class SVM classifiers, such as one-to-one (OVR), one-to-one (OVO), directed acyclic graphs (DAG), partial binary trees (PBT), etc. [39]. This paper uses PBT to construct a multi-fault classifier. The kernel function is an important part of SVM, including linear kernel function, polynomial kernel function, RBF kernel function, and Sigmoid kernel function. The RBF kernel function can approximate nonlinear functions with faster convergence speed and better generalization ability. Therefore, this paper uses the RBF kernel function for classification, with a penalty factor of \(C=1000\).  
Since there are three categories of injector status classification problems, two SVMs (SVM1 and SVM2) need to be built. The 10 sets of feature vectors of the three states are input into the SVM classifier as the training samples, and the 20 sets of feature vectors are used as the test samples of the input SVM classifier. The classification results are shown in Figures 13 and 14. The classification results of the support vector machine for all training samples and test samples are shown in the Figures 13 and 14. It can be seen that there are obvious boundaries between the three working state feature vectors of the injector, and the prediction classification of the training set and the test set are consistent with the actual classification