# A New Life Prediction Scheme for Mechanical System with Considering the Mission Profile Switching
## 3 The Proposed Method
### The Construction of the Reference Degradation Model  
In the proposed method, the reference degradation models are constructed based on the prediction results of each constant operating condition of the reference samples instead of the historical lifecycle data of the reference samples. By this way, not only can the final life prediction results be more accurate, but also the life prediction under different mission profiles can be achieved. After obtaining the partitioned operating conditions \(\mathbf{OT}_{j}^{i}\) of similar reference samples by Step 2, the reference degradation models \(DM_{j}^{i}\) can be built as follows. Where \(i=1,2,\ldots,m\), \(j=1,2,\ldots,C\), \(m\) is the number of similar reference samples, and \(C\) is the number of operating conditions.  
Step 3.1: Calculate the standardized samples. The standardized samples \(\mathbf{NT}_{j}^{i}\) of the partitioned operating conditions are calculated as follows.  
\[\mathbf{NT}_{j}^{i}=\frac{\mathbf{OT}_{j}^{i}-\text{mean}(\mathbf{OT}_{j}^{i})}{\text{std} (\mathbf{OT}_{j}^{i})} \tag{1}\]  
Step 3.2: Extract the maximum value of the standardized samples. The maximum value of the standardized samples is used as the life terminal point of each constant operating condition in one reference sample and is obtained as follows.  
\[NM^{i}=\max_{0\leq j\leq C}(\max(\mathbf{NT}_{j}^{i})) \tag{2}\]  
where, for the standardized sample, the maximum value only needed to be calculated once. However, for the standardized samples \(\mathbf{NT}_{j}^{i}\) of the partitioned operating conditions, the maximum value of each partitioned operating condition should be calculated first, then the maximum of the maximum value of each partitioned operating condition is the maximum value of the standardized samples.  
Step 3.3: Calculate the life thresholds of each constant operating condition in one reference sample. According to the maximum value of the standardized samples and the partitioned operating conditions, the life threshold \(OM_{j}^{i}\) of each constant operating condition can be obtained.  
\[OM_{j}^{i}=NM^{i}\cdot\text{std}(\mathbf{OT}_{j}^{i})+\text{mean}(\mathbf{OT}_{j}^{i}) \tag{3}\]  
As in Step 4.2, we should calculate the life of each operating condition of the sample to be predicted according to STM. Therefore, it is necessary to calculate the similarity between each operating condition of the sample to be predicted and that of similar reference samples. However, according to Equation (1), if we want to calculate the standardized sample of each operating condition, we must know the mean and variance of the operating condition. For the similar reference samples, each operating condition is the data under the whole life. So the mean and variance of each operating condition can be easily obtained. However, for the sample to be predicted, each operating condition is not the data under the whole life. Therefore, it is impossible to standardize the operating condition of the sample to be predicted reasonably and effectively. Thus, it is necessary to restore the endpoint of similar reference samples to values before standardization according to Equation (3).  
Step 3.4: Predict the lifecycle degradation curve of each constant operating condition. According to the life thresholds of each constant operating condition, the lifecycle degradation curve of each constant operating condition is predicted by RVM, respectively. The specific process of RVM refers to [34; 35].  
Step 3.5: Build the reference degradation models. According to the prediction results of the lifecycle degradation curve in Step 3.4, the trend \(\mathbf{RT}_{j}^{i}\) and the lifecycle \(LT_{j}^{i}\) of each constant operating condition are used to construct the reference degradation models \(DM_{j}^{i}\).  
Figure 3 is the process of the reference degradation models construction.