## 4 Case Study
### System Failure Prognostics
#### 4.2.1 System Health Prediction Based on Degradation
In the training process, the reducing learning rate was set to 0.1, and the ratio of the training set to the test set was determined as 4:1. H1 was basically consistent with the functional indicator at the initial stage, and the fusion result deviated greatly at the later stage. If H1 was used to produce the failure prediction, the deviation result was large. H2 could accurately fit the attenuation trend in the global attenuation, local self-healing, and interference stages, which was consistent with the target function and which could fully and accurately express the degradation characteristics of the system. The analysis results of Table 2 can be obtained by Equation (10); as shown in Table 2, the SHI constructed by the two methods are linearly correlated with the system function indicators. The SHI extracted by SAE has a stronger linear correlation with the functional indicator (\(r_{s}=0.998\)). Considering all the factors that influenced the SHI, the SHI fused by the SAE method could better express the health condition of the system. Therefore, using SHI to predict will obtain a more accurate system life. (3) SHI decomposition  
Using experience to choose the parameters of the VMD decomposition method, the initial value of the center frequency was set to 1, the update parameter was set to 0, and the termination condition was set to \(10^{-6}\). The penalty factor was determined by experiment. In the study of the ship system, only the separation of the global degradation, local self-healing, and other sources of noise were considered, so 5 was set to 3. \begin{table}
\begin{tabular}{c c c} \hline
**Spearman Correlation** & **SAE Fusion SHI** & **PCA Fusion SHI** \\ \hline Function indicator & 0.998 & 0.994 \\ \hline \end{tabular}
\end{table} 
Table 2: Correlation analysis of SHI and function indicator. Figure 8: Fusion effect comparison. After signal decomposition, the SHI contained a mixture of three degradation phenomena. VMD was used to separate the local self-healing phenomenon \(u_{2}\) from the global attenuation trend \(u_{1}\) and other sources of noise \(u_{3}\) in the SHI, and this could solve the problem of prediction bias caused by the local instability of the HI. The three decomposed curves, \(u_{1}\), \(u_{2}\), and \(u_{3}\), shown in Figure 9 represent the dominant degradation, local self-healing, and local interference. Dominant degradation \(u_{1}\) was affected by the internal uncertainty, and it showed a downward curve. The local self-healing \(u_{2}\) and interference \(u_{3}\) were influenced by the periodic function recovery characteristics of the equipment in the system and showed periodic curves. The characteristics of the decomposed signal could be predicted by different prediction algorithms. According to the characteristics of \(u_{1}\), the RVM model was selected for prediction. There were obvious nonlinear characteristics in \(u_{2}\) and \(u_{3}\), so the LSTM algorithm was selected. (4) Hybrid prediction  
After TTS calculation, when the system ran for 3334 h, the change of the health value met the requirement of the prediction conditions, so the prediction algorithm was started in order to predict the subsequent health condition. The health values before 3334 were used to train the RVM regression model and the LSTM model. The input data for the models were selected as the 1 st-3334 th samples of the health values. The step size of the multi-step prediction was set to 10, and the target data in the training process were the 11-3344 th samples. The main engine fuel oil inlet pressure was taken as the evaluation standard of the system target function. When the pressure was lower than 0.7 MPa, it did not meet the demand of the main engine. At that time, the set health value was \(HI=0.2\), which was the failure threshold. As shown in Figure 10, after TTS, the fusion features had undergone three obvious changes, so the training data of the hybrid model had been updated three times.