## 3 Case Study

Table 4: The training dataset’s homogeneity, assessing results using the Fisher–Pearson and Fisher–Snedecor criteria. Figure 5: Cluster analysis results: (a) training dataset, (b) test dataset. Figure 6 demonstrates that the automatic parametric optimization-developed algorithm of the system with controller (5) ensures \(\overline{\theta}^{*}\) computation from the initial values \(\overline{\theta}^{0}\), at which point the transient process in the automatic system has both oscillatory and monotonic characteristics. Table 5 shows that the matrix \(H(J)\) is positive-definite at the algorithm's endpoints \(\overline{\theta}^{*}\). Figures 6-9 illustrate the algorithm's (10) convergence patterns. Figure 6 shows that the proposed method's application allowed the transient process time for the parameter \(n_{\text{TC}}\) to be reduced by 2 times, while the \(n_{\text{TC}}\) values were set at the level of \(\approx\)1 (i.e., \(n_{\text{TC}}\) values were restored to the initial ones). The search-free method's [25; 26; 27; 28] application also allows for a reduction in the transient process time for parameter \(n_{\text{TC}}\) by 2 times. However, the \(n_{\text{TC}}\) values were set at the level of \(\approx\) 0.5 (i.e., \(n_{\text{TC}}\) values were restored to half of the initial ones). The \(n_{\text{TC}}=0.5\cdot n_{\text{TC}}\) values used in the real helicopter TE's operation conditions are impossible. Figure 6: The transient processes’ \(n_{\text{TC}}(t)\) diagrams, reflecting the dynamics in the initial (**black curves**) and final (**blue curves**) states of the automatic parametric optimization algorithm’s operation: 1—parameter set 1; 2—parameter set 2; 3—parameter set 3 (note: the values of parameter sets 1, 2, and 3 are presented in Table 5). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Number** & \(\mathbf{\theta_{1}^{0}}\) & \(\mathbf{\theta_{2}^{0}}\) & \(\mathbf{\theta_{1}^{*}}\) & \(\mathbf{\theta_{2}^{*}}\) & \(\mathbf{J^{0}}\) & \(\mathbf{J^{*}}\) & \(\mathbf{H(J)}\) & **det(H(J))** \\ \hline
1 & 0.1 & 0.01 & 0.253 & 0.016 & 38.325 & 20.974 & \
\begin{tabular}{c}
\begin{tabular}{c} \(1.049\) \\ 0.659 \\ \end{tabular} & 0.889 \\ \end{tabular} & 0.011 \\ \hline
2 & 0.25 & 0.025 & 0.347 & 0.016 & 38.261 & 20.659 &
\begin{tabular}{c} \(1.013\) \\ 0.611 \\ \end{tabular} & 0.805 \\ \hline
3 & 0.35 & 0.001 & 0.415 & 0.012 & 59.939 & 19.993 &
\begin{tabular}{c} \(1.001\) \\ 0.602 \\ \end{tabular} & 0.738 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The automatic parametric optimization algorithm results. Figure 8: Diagrams of the controller-adjustable parameters \(\theta_{1}^{(l)}\) (**a**) and \(\theta_{2}^{(l)}\) (**b**) during the automatic parametric optimization algorithm’s operation: (**black curve**)—parameter set 1; (**blue curve**)—parameter set 2; (**green curve**)—parameter set 3 (note: the values of parameter sets 1, 2, and 3 are presented in Table 5). Figure 7: Diagrams of the values of the gradient components, \(\frac{\partial I}{\partial\theta_{1}^{(l)}}\) (**a**) and \(\frac{\partial I}{\partial\theta_{2}^{(l)}}\) (**b**) of the optimization criterion \(J\) during the automatic parametric optimization algorithm’s operation: (**black curve**)—parameter set 1; (**blue curve**)—parameter set 2; (**green curve**)—parameter set 3 (note: the values of parameter sets 1, 2, and 3 are presented in Table 5). Figure 7 illustrates the change in the two components of the gradient of the optimization function J during the automatic parametric optimization algorithm's operation. Figure 7 displays the changes in trajectories in the corresponding gradient components for each of the initial parameters (\"set 1\" is the black curve, \"set 2\" is the blue curve, and \"set 3\" is the green curve) in three sets. Figure 7 shows that, as the iterations progress, the gradient component values gradually decrease, indicating that the algorithm converges to the optimization criterion's minimum point.