## 3 Case Study
At the same time, the parameter \(\theta_{1}\) has a more substantial impact on the system's behaviour than \(\theta_{2}\), both in the first-order and second-order sensitivity terms. In turn, the mutual influence of the parameter \(\xi_{12}(t)\) is insignificant, which simplifies the optimization issue since the parameters can be selected almost independently. As a result, it is proven that the output coordinate \({}_{x}(t)\) stabilizes at optimal \(\mathbf{q}^{*}\), and the criterion \(J\) is minimized at \(\mathbf{q}^{*}=[0.45,\,0.12]\). When conducting a computational experiment using the developed neural network (Figure 3) and the developed algorithm for its training (Table 2), a maximum accuracy of 0.993 was achieved on both the training and validation datasets (Figure 10(a)) with 160 training epochs. At the same time, the loss function drops from 2.5 to 0.5% on both the training and validation datasets (Figure 10(b)) with 160 training epochs. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Modelling Step** & **Modelling Step** & **Description** \\
**Number** & **Name** & \\ \hline
1 & Setting system parameters & \(T_{obj}^{(1)}=1.0\), \(T_{obj}^{(2)}=4.0\), \(k_{ob}=1.0\), \(\tau_{ob}=5.0\). \\  & Setting system parameters & Controller parameters: \(\theta_{1}=0.5\), \(\theta_{2}=0.1\). \\  & & Reference action: \(r(t)=\sin(t)\). \\ \hline
2 & System modelling & The system block diagram creation, including the controller \(G_{c}(p,\overline{\theta})\) and the object \(G_{p}(p)\). \\  & & Adding blocks to compute \(\xi_{i}(t)\) and \(\xi_{ij}(t)\). \\ \hline
3 & Sensitivity computation & The implementation of the numerical calculation of the derivatives \\  & & \(\frac{\partial u(t)}{\partial t_{i}}\) and \(\frac{\partial^{2}u(t)}{\partial\overline{\theta}\partial\overline{\theta}}\). \\  & & The finite difference methods or analytical solution application. \\ \hline
4 & The optimality criterion & \(J=\int\limits_{0}^{\infty}\big{(}a_{1}\cdot\varepsilon^{2}\big{(}t,\overline{ \theta}\big{)}+a_{2}\cdot u^{2}(t)\big{)}dt\), \\  & computation & where \(a_{1}=1.0\) and \(a_{2}=0.5\). \\ \hline
5 & Analysis of results & The computation of the first- and second-order sensitivity functions. \\  & & The gradient \(\
abla\overline{g}J\) and Hessian matrix \(H(J)\) computation. \\  & & Testing [PERSON]’s criterion for \(H(J)\) and optimizing the \(\overline{\theta}\) parameters. \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Developed algorithm for conducting simulation modelling. The obtained data confirm that the proposed neural network (Figure 3), which trains 160 epochs using the developed training algorithm (Table 3), is sufficient to achieve the minimum value of the error function (\(MSE_{\text{min}}\) = 0.005). For comparison, similar experiments were performed using standard training algorithms for this neural network (Figure 3), the results of which are presented in Table 7.  
Figure 10: Schemes follow the same formatting. The comparative analysis results showed that the developed training algorithm is the most effective, achieving the error function's minimum value (\(MSE_{min}\) = 0.005) for the minimum number training epochs (160). Compared with traditional and alternative algorithms, the minimum values of \(MSE_{min}\) achieved by them exceed the developed algorithm's similar indicator by 1.86 to 6.02 times, while the optimal number of training epochs increases by 1.38\(\ldots\)2.38 times. The developed method for neural network implementation using the developed neural network (Figure 3) comparison was made with other well-known neural network architectures--the traditional RBF network [20; 56], Jordan neural network [21; 57], Elman neural network [22; 58], Hopfield neural network [23; 59], and Hamming neural network [24; 60]--according to traditional quality metrics: accuracy, precision, recall, F1-score, and AUC-ROC, as well as neural network training time (Table 8). These metrics are calculated according to the following expressions [61; 62; 63]:  
\[