## 4 Discussion  
### Generalization of Results  
The developed method for studying processes in ACSs (see Figure 2) is based on the operator approach used for modelling both the control object and the controller. The processes in the automatic control system are described by the system of equations given in (1), which determines the relationship between the control error, \(\varepsilon(t,\,\overline{\theta})\), the reference action, \(r(t)\), the control action, \(u(t)\), the output coordinate, \(x(t)\), and the controller, \(G_{c}(s,\,\overline{\theta})\), and control object, \(G_{p}(s)\), operators. The control object operator, presented in the form of (2), includes parameters that allow for describing the most complex object dynamics, including the static gain, \(k_{ob}\); time constants \(T_{ob}^{(1)}\) and \(T_{ob}^{(2)}\); the delay time, \(\tau_{ob}\); and the nonlinear part, \(f_{nonlinear}(x,\,u)\). A modification of the traditional PI controller is used for the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number** & **Training Algorithm** & **Achieved \(MSE_{min}\) Value** & **Minimum Number of Epochs** & **Conclusions** \\ \hline
1 & Developed algorithm & 0.0050 & 160 & The most efficient learning algorithm is characterized by the minimum \(MSE_{min}\) value achieved in the minimum number of training epochs. \\ \hline
2 & Traditional backpropagation algorithm & 0.0093 & 220 & The minimum \(MSE_{min}\) value achieved exceeds the developed algorithm’s similar indicator by 1.86 times, with an increase in the optimal number of training epochs by 1.38 times. \\ \hline
3 & Genetic algorithm & 0.0108 & 240 & The minimum \(MSE_{min}\) value achieved exceeds the developed algorithm’s similar indicator by 2.16 times, with an increase in the optimal number of training epochs by 1.5 times. \\ \hline
4 & Modified inverse gradient descending method [51] & 0.0163 & 250 & The minimum \(MSE_{min}\) value achieved exceeds the developed algorithm’s similar indicator by 3.26 times, with an increase in the optimal number of training epochs by 1.56 times. \\ \hline
5 & Traditional inverse gradient descending method & 0.0237 & 330 & The minimum \(MSE_{min}\) value achieved exceeds the developed algorithm’s similar indicator by 4.74 times with an increase in the optimal number of training epochs by 2.07 times. \\ \hline
6 & Hybrid algorithm & 0.0301 & 380 & The minimum \(MSE_{min}\) value achieved exceeds the developed algorithm’s similar indicator by 6.02 times, with an increase in the optimal number of training epochs by 2.38 times. \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparative analysis results for the proposed neural network training. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Neural Network Architecture** & **Accuracy** & \multicolumn{4}{c}{**Metrics for Evaluating Methods**} & **Training Time** \\ \hline Developed neural network & 0.993 & 0.991 & 1.0 & 0.995 & 0.825 & 1 min 12 s \\ Traditional RBF network [20; 56] & 0.992 & 0.989 & 1.0 & 0.994 & 0.811 & 2 min 25 s \\ Jordan neural network [21; 57] & 0.993 & 0.989 & 1.0 & 0.994 & 0.781 & 4 min 17 s \\ Elman neural network [22; 58] & 0.993 & 0.989 & 1.0 & 0.994 & 0.780 & 4 min 12 s \\ Hopfield neural network [23; 59] & 0.837 & 0.832 & 0.835 & 0.833 & 0.635 & 0 min 47 s \\ Hamming neural network [24; 60] & 0.811 & 0.806 & 0.831 & 0.819 & 0.602 & 0 min 33 s \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparative analysis results. controller, presented in the form of (3). It includes the switching function \(g\big{(}t,\overline{\theta}\big{)}\), which describes the controller's adaptation to the current operating conditions. This function is regulated by the parameters \(k_{1}\), \(k_{2}\), \(k_{3}\), and \(\alpha\) and is determined according to (4).