# A Hybrid Degradation Modeling and Prognostic Method for the Multi-Modal System
## 3 Experiment and Discussion
### Experimental Simulation  
#### Benchmark Data Description  
The commercial modular aero-propulsion system simulation (C-MAPSS) turbofan aero-engine dataset [39] is widely used in prognostic studies. This dataset simulated the degradation of aero-engine in four working environments to get the corresponding sub-datasets FD001, FD002, FD003 and FD004. Each subset consists of a training set, a test set and corresponding remaining useful life values. The training samples degrade until the system failure, and the last time is regarded as the failure time  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(a_{exp},b_{exp}\) & Parameter & No. 1 & No. 2 & No. 3 & No. 4 & No. 5 & No. 6 & No. 7 & No. 8 & No. 9 & No. 10 \\ \hline (0.01, 0.25) & \(\mu_{g}\) & 1.58 & 1.93 & 2.94 & 3.45 & 3.62 & 3.88 & 4.22 & 4.60 & 4.80 & 4.84 \\  & \(\sigma^{2}\) & 0.04 & 0.06 & 0.08 & 0.10 & 0.10 & 0.11 & 0.12 & 0.13 & 0.14 & 0.14 \\ (0.02, 0.5) & \(\mu_{g}\) & 4.91 & 4.99 & 5.36 & 6.86 & 12.21 & 12.52 & 17.20 & 17.84 & 18.44 & 19.98 \\  & \(\sigma^{2}\) & 0.14 & 0.15 & 0.16 & 0.20 & 0.37 & 0.379 & 0.52 & 0.54 & 0.55 & 0.60 \\ (\(-0.04, 1.3\)) & \(\mu_{g}\) & 20.02 & 22.43 & 23.27 & 24.67 & 25.81 & 25.92 & 28.11 & 28.45 & 29.07 & 33.13 \\  & \(\sigma^{2}\) & 0.60 & 0.67 & 0.70 & 0.74 & 0.78 & 0.79 & 0.85 & 0.86 & 0.88 & 1.00 \\ (\(-0.06, 1.6\)) & \(\mu_{g}\) & 33.80 & 33.84 & 33.91 & 34.04 & 34.08 & 35.59 & 35.86 & 37.51 & 37.68 & 39.32 \\  & \(\sigma^{2}\) & 1.02 & 1.02 & 1.02 & 1.03 & 1.03 & 1.07 & 1.08 & 1.13 & 1.14 & 1.19 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter settings of synthetic data samples.  
of the engine unit. For test samples, the sensor data recording is stopped before the system fails and the failure period is recorded for verification. The details and statistical data of each subset are listed in Table 2. The monitoring data in each sample consist of 21 sensors (e.g., total temperature, pressure at fan inlet, physical core speed, etc.) and three operational settings which jointly determine the system's working mode [40].  
The sample of FD001 suffered high-pressure compressor failure under a single operating condition. For FD002, the sample suffered the high pressure compressor failure under six operating conditions. In FD003, the sample suffered high pressure compressor and fan failure under a single operating condition, while in FD004 the sample suffered high pressure compressor and fan failure under six operating conditions. The proposed method was applied to four sub-datasets which have different working environments to verify the generality of the proposed model.  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Subset** & **Fault Type** & **Operation Mode** & **Training Scale** & **Testing Scale** & **Max-Lifespan** & **Selected Sensor** \\ \hline FD001 & 1 & 1 & 100 & 100 & 362 & 13 \\ FD002 & 1 & 6 & 260 & 259 & 378 & - \\ FD003 & 2 & 1 & 100 & 100 & 525 & 11 \\ FD004 & 2 & 6 & 249 & 248 & 543 & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: Detailed information of C-MAPSS dataset.  
Figure 4: The prediction results of the proposed model for synthetic data. (**a**) The sample of exponential function with \((a_{exp}=0.01,b_{exp}=0.25)\). (**b**) The sample of exponential function with \((a_{exp}=0.02,\ b_{exp}=0.5)\). (**c**) The sample of exponential function with \((a_{exp}=-0.04,\ b_{exp}=1.3)\). (**d**) The sample of exponential function with \((a_{exp}=-0.06,\ b_{exp}=1.6)\).