## 3 Dynamic Prediction of Servo Valve Performance Degradation
### Dynamic Prediction of Performance Indicators  
In Section 2.2.1, we explained the theory of the exponential smoothing model, and in this section, we use the pressure gain or internal leakage of the servo valve as a one-dimensional time series for dynamic prediction, wherein the main process is as follows:  
Step 1: Confirmation of primary time points  
As described in dynamic prediction, the time nodes to be confirmed include the training set start time, the first training set end time, the first test set end time, and the last test set end time. For example, in this study, the training and prediction selected the pressure gain from 30,000 s to 1,400,000 s of working condition 1 as the training set, and for the prediction of the servo valve pressure gain after 1,400,000 s, the training set start time was 1,400,500 s, the first training set end time was 1,400,500 s, and the first test set end time was 1,401,000 s.  
Step 2: Slice and dice the training set and test set  
The data set was split according to the training set start time, the training set end time, and the test set end time. For example, in this study, the first training and prediction in the training set start time was 30,000 th s, the training set end time was 1,400,000 th s, and the test set end time was 1,401,000 th s. Then, the pressure gain from the 30,000 th to the 1,400,000 th s of condition 1 was intercepted as the training set, and the 1,400,500 th s pressure gain was intercepted as the test set. It should be noted that the time series method training set and test set are one-dimensional data because the pressure gain or and internal leakage units do not coincide. Step 3: Parameter estimation  
Based on the training set data, parameters such as the smoothing index and damping trend were estimated. Depending on the model being selected, the parameters being predicted in this step varied. For example, when simple exponential smoothing was chosen, only the smoothing index was estimated; when double exponential smoothing was chosen, only the smoothing index and the damping trend were estimated; when triple exponential smoothing was chosen, the smoothing index, the damping trend, and the seasonal parameters needed to be estimated. Step 4: Prediction \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Internal Leakage (L/min)** & **Pressure Gain (MPa/mm)** \\ \hline Average value & 1.49 & 971.53 \\ Variance & 0.13 & 49.04 \\ Minimum value & 1.24 & 878.10 \\
25\% quantile & 1.39 & 934.59 \\
50\% quantile & 1.51 & 967.49 \\
75\% quantile & 1.57 & 1000.09 \\ Maximum value & 1.87 & 1153.53 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Leakage and pressure gain statistics table. Prediction values for the specified prediction length were calculated from the estimates of the smoothed parameters and initial values in the training set from step 2 the end values of the training set, as well as the parameter estimates obtained in step 3. Step 5: Advancing the sliding time window  
After completing a single parameter estimation and prediction, the sliding time window needs to be pushed for the next training and prediction to achieve the effect of dynamic prediction. In this study, the prediction length was fixed at 500 s, so each time, the end time of the training set and the end time of the test set were pushed forward by 500 s. At the same time, the training set start time was kept constant in the time series model. For example, in this study, the training set started at 30,000 th s, the first training set ended at 1400,000 th s, and the first test set ended at 1400,500 th s. After the first parameter estimation and prediction, the training set start time was kept constant, the training set end time was advanced by 500 s to 1,400,500 s, and the test set end time was also advanced by 500 s to 1,401,000 s.  
Step 6: Loop  
Loop through steps 2 to 5 until the end time of the test set advances to the end time of the last test set, which was 1,800,000 s at the end of the time series for this data set.