# Research on Prediction Method of Hydraulic Pump Remaining Useful Life Based on KPCA and JITL
## 5 RUL Prediction Method of Hydraulic Pump Based on KPCA and JITL  
When repairing and maintaining mechanical equipment, if the maintenance personnel can timely and accurately determine their RUL, a scientific and appropriate maintenance plan can be formulated, and the property losses and casualties caused by their untimely maintenance can be avoided. Therefore, it is extremely essential to predict the RUL of mechanical equipment on time and accurately. As the core of a hydraulic system, it is highly important to predict the RUL and manage the health of a hydraulic pump. Therefore, an RUL prediction method of a hydraulic pump based on KPCA and JITL is proposed in this paper. The KPCA method was used to perform the weighted fusion calculation of the selected characteristic parameters, and the first and second principal components were extracted as the evaluation indices representing the performance degradation of a gear pump. Subsequently, the RUL prediction model of a hydraulic pump based on KPCA and JITL was established using the RUL prediction method of JITL based on k-VNN. The KPCA method was used to analyze the time- and frequency-domain characteristic parameters of the gear pump pressure signals. Based on Table 5, when the kernel parameter is \(\sigma=450\), the _CPVs_ of the first and second principal components of the four gear pumps exceed 90%. Therefore, in this study, the first and second principal components of tested gear pump 1-3 were selected as the training samples in the modeling. Moreover, the data of tested gear pump 4 were chosen as the test samples to verify the accuracy of the model. As aforedescribed, the selected minimum neighborhood, \(k_{m}\), and the weight factor, \(\gamma\), in the JITL method have different effects on the predicted results. Among them, the maximum relative error (MRE) is listed in Table 6.  
It can be observed from Table 6 that when the minimum neighborhood, \(k_{m}\), is set as 2, 3, 4, and 5, respectively, the weight factors, \(\gamma\), are 0.9 and 1.0; the corresponding MREs are smaller than the other MREs. To further optimize the values of the above parameters, the average relative errors (AREs) of the predicted results were compared, and the results are tabulated in Table 7.  
Based on Table 7, when minimum neighborhood \(k_{m}=5\) and weight factor \(\gamma=1.0\), the corresponding ARE is minimum. Therefore, in this study, parameters \(k_{m}\) and \(\gamma\) of the model established are 5 and 1.0, respectively. The model was established based on the life-cycle test data of gear pumps 1-3. The prediction results of the RUL of gear pump 4 obtained using the model are shown in Figure 11. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(k_{m}\) & \(\gamma\) & **1.0** & **0.9** & **0.8** & **0.7** & **0.6** & **0.5** \\ \hline
2 & **0.5340** & **0.7438** & 0.8205 & 0.8114 & 0.8114 & 0.8428 \\
3 & **0.6644** & **0.7438** & 0.8205 & 0.8489 & 0.8495 & 0.8476 \\
4 & **0.5681** & **0.7949** & 0.8243 & 0.8489 & 0.8495 & 0.8299 \\
5 & **0.6184** & **0.7949** & 0.8243 & 0.8489 & 0.8495 & 0.8299 \\
6 & 0.6184 & 0.8053 & 0.8243 & 0.8489 & 0.8495 & 0.8312 \\
7 & 0.6192 & 0.8053 & 0.8243 & 0.8489 & 0.8495 & 0.8623 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: MREs of predicted results. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\gamma\) & \(k_{m}\) & **2** & **3** & **4** & **5** \\ \hline
0.9 & 15.8639 & 15.6194 & 15.4281 & 15.2987 \\
1.0 & 15.9545 & 15.7997 & 15.3460 & **15.1547** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: AREs of predicted results. As can be seen from Figure 11, the gear pump is in a healthy state in the initial operating state without any mechanical failure; therefore, its life is not changed significantly. Over time, when the gear pump has operated for approximately 700 h, its RUL begins to decrease gradually. The predicted values of the RUL basically coincides with the real values.