## 3 Case Study
For example, in the emergency mode, \(n_{TC}=97.4\%\) or \(n_{TC}=15,486\) rpm; in the takeoff mode, \(n_{TC}=96.3\%\) or \(n_{TC}=15,312\) rpm; in the nominal mode, \(n_{TC}=94.7\%\) or \(n_{TC}=15,057\) rpm; in the first cruising mode, \(n_{TC}=93.6\%\) or \(n_{TC}=14,882\) rpm; and in the second cruising mode, \(n_{TC}=91.7\%\) or \(n_{TC}=14,580\) rpm. In this case, a deviation from the specified value of \(\pm 0.5\%\) is allowed. Since the Mi-8 MTV helicopter's flight tests were conducted in the nominal engine operating mode, the reference action is 256 values of \(n_{TC}=15,057\) rpm \(\pm 0.5\%\), where the deviation of \(\pm 0.5\%\) was selected randomly for each \(i\)-th \(n_{TC}\) value. The \(n_{TC}\) values, as a reference action, were also normalized according to (23) and (24). Thus, the training dataset presented in Table 3 is obtained. Based on [32; 33; 34; 35; 36; 37; 38], the training dataset's (Table 3) homogeneity was assessed using the Fisher-Pearson [45; 46; 47] and Fisher-Snedecor [48; 49; 50] tests at a significance level of \(\alpha=0.01\) (Table 4). The significance level of 0.01 was chosen to ensure a statistically valid conclusion about high reliability, which is especially important for helicopter flight safety. This strict criterion minimizes the first type of error probability, which contributes to the helicopter TE's more accurate and safe control under operational conditions [51; 52]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **The System Error \(\epsilon(t)\) Values** & **The Reference Action \(r(t)\) Values** \\ \hline
1 & 0.011 & 0.913 \\... &... &... \\
46 & 0.012 & 0.918 \\... &... &... \\
115 & 0.015 & 0.933 \\... &... &... \\
173 & 0.013 & 0.926 \\... &... &... \\
209 & 0.011 & 0.911 \\... &... &... \\
256 & 0.010 & 0.908 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The training dataset fragment. The training and test datasets' representativeness was assessed using cluster analysis (\(k\)-means method) [53], which involved dividing the training dataset (Table 3) into \(k\) groups (clusters), where each value belongs to the cluster with the highest mean value [49; 50]. The algorithm repeats the point distribution, minimizing the distance between them and their centroids until the clusters stabilize [54; 55]. The cluster analysis of the training dataset (Table 3) identified eight clusters (I... VIII). The training and test datasets were randomly generated in a 2:1 ratio (67 and 33%, respectively), and the same 8 clusters were found in both sets, indicating their structural similarity. The distances between clusters in the training and test datasets were almost identical, confirming their similarity (Figure 5). Thus, the optimal size of the training dataset is 256 elements (100%), that of the control dataset is 172 elements (67% of the training dataset size), and that of the test dataset is 84 elements (33% of the training dataset size). The conducted performance test results include the computation of the \(H(J)\) matrix and the transient process diagrams at the automatic parametric optimization algorithm's initial and final points (Figures 6-9) with the above-described reference action. The research object's (the TV3-117 engine) parameters are as follows: \(T_{ob}^{(1)}=1.0\), \(T_{ob}^{(2)}=4.0\), \(k_{ob}=1.0\), and \(\tau_{ob}=5.0\)[32; 33; 34; 35; 36; 37; 38]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{2}{c}{**The Fisher–Pearson Criterion**} & \multicolumn{2}{c}{**The Fisher–Snedekor Criterion,**} & \multirow{2}{*}{**Description**} \\  & **Calculated** & **Critical** & **Calculated** & **Critical** & \\ \hline \(c(t)\) & 12.996 & 13.3 & 15.402 & 15.98 & The Fisher–Pearson and Fisher–Snedecor criterium-computed values were below the thresholds, which made it possible to confirm the training dataset’s homogeneity. \\ \(r(t)\) & 13.014 & 15.599 & & dataset’s homogeneity. \\ \hline \hline \end{tabular}
\end{table}