## 5 Case Study  
Taking the trainer aircraft CESSA172R of the Civil Aviation Flight University of China as the research object, the engine of CESSA172R is produced by the American Textron Lycoming engine manufacturer. The engine model is I0-360-L2A, its health management system is the GARMIN1000 system, and the data is obtained from the sensors on the aircraft. Generated, and the GARMIN1000 system is responsible for collecting and summarizing. The collection of engine data is mainly based on reading, saving, and backup by an on-board download after the flight over. There are 12 parameters related to the engine, and the meaning of each parameter is shown in Table 1. Random captures 20 partial raw flight data engine revolutions per minute (RPM) data and exhaust gas temperature (EGT) of a general aviation aircraft in cruise state shown in Table 2, the engine revolutions per minute (RPM) data as the monitoring data without trend item shows the normal flight training. According to those data, five fitting curves are drawn in Figures 2-5. Herein we set Y1, Y2, Y3, Y4 to represent the value of EGT1, EGT2, EGT3, EGT4, respectively, and X represents the value of RPM, for increasing the conciseness of multi-order polynomial, Z was employed here Z = (X - 1.1 \(\times\) 103)/20. According to the Polynomial Least Squares Analysis fitting method, the multi-order fitting equation relationship between EGT (Y) and RPM (X) was established, respectively. In order to make the equation expression between Y and X more concise, the parameter Z is introduced here, where Z = (X - 1.1 \(\times\) 103)/20; in fact, Z still represents RPM (X). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Saserial Number** &
\begin{tabular}{c} **Parameter** \\ **Abbreviation** \\ \end{tabular} & **Parameter Explanation** \\ \hline
1 & E1 FFlow & fuel flow \\
2 & E1 OilT & lubricating oil temperature \\
3 & E1 Oilp & Oil pressure \\
4 & E1 RPM & Engine speed \\
5 & E1 CHT1 & No. 1 cylinder temperature \\
6 & E1 CHT2 & No. 2 cylinder temperature \\
7 & E1 CHT3 & No. 3 cylinder temperature \\
8 & E1 CHT4 & No. 4 cylinder temperature \\
9 & E1 EGT1 & No. 1 cylinder exhaust gas temperature \\
10 & E1 EGT2 & No. 2 cylinder exhaust gas temperature \\
11 & E1 EGT3 & No. 3 cylinder exhaust gas temperature \\
12 & E1 EGT4 & No.4 cylinder exhaust gas temperature \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Flight data engine data structure. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Flight Data** & **RPM** & **EGT1** & **EGT2** & **EGT3** & **EGT4** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 2: Partial raw flight data RPM and four different EGT. Figure 3: Five different polynomial fitting curves for RPM (Z) and EGT2 (Y2). Figure 2: Five different polynomial fitting curves for RPM(Z) and EGT1(Y1). We use the squares polynomial fitting method to fit these known numerical scatter plots, and the fitting multi-order polynomial results are listed following:  
Y1 = 13 \(\times\) Z\({}^{2}\) + 58 \(\times\) Z + 9.3 \(\times\) 102 Y1 = 1.1 \(\times\) Z\({}^{3}\) +15 \(\times\) Z\({}^{2}\) + 57 \(\times\) Z + 9.3 \(\times\) 102 Y1 = -5.8 \(\times\) Z\({}^{4}\) - 15 \(\times\) Z\({}^{3}\) + 17 \(\times\) Z\({}^{2}\) + 95 \(\times\) Z + 9.1 \(\times\) 102 Y1 = -15 \(\times\) Z\({}^{5}\)-61 \(\times\) Z\({}^{4}\) - 40 \(\times\) Z\({}^{3}\) + 77 \(\times\) Z\({}^{2}\) +95Z + 9.1 \(\times\) 102 Y1 = -5.3 \(\times\) Z\({}^{6}\) - 37 \(\times\) Z\({}^{5}\) - 75 \(\times\) Z\({}^{4}\) - 15 \(\times\) Z\({}^{3}\) + 94 \(\times\) Z\({}^{2}\) + 87 \(\times\) Z +9.1 \(\times\) 102  
To verify those multi-order polynomial results and find the optimum order, we select a real value (1069, 906.1), where Z = \(-\)3.66. According to the error results from Table 3, we can find four-order polynomial is the best. Figure 4: Five different polynomial fitting curves for RPM(Z) and EGT3(Y3). Figure 5: Five different polynomial fitting curves for RPM (Z) and EGT4(Y4).