## 5 Conclusions  
A method for researching processes in automatic control systems using an operator approach and a modification of the PI controller has been developed, which allows for the control and regulation of the efficient modelling of object dynamics. The control object operator includes parameters such as static gain and time constants (e.g., \(T_{\phi b}^{(1)}\) = 0.5, \(T_{\phi b}^{(2)}\) = 0.3) and a nonlinear part \(f_{nonline}(x,\,u)\), which makes it possible to take into account various aspects of the system. The PI controller modification, which includes a switching function, adapts the system to changing conditions, increasing the regulation flexibility and accuracy. The algorithm for optimizing the controller's parameters using gradient descent and regularization (\(\lambda\) = 0.01) stabilizes the training process, minimizing overtraining and data errors. The developed method's application to the TV3-117 engine control system as an example demonstrated high efficiency. The algorithm significantly improved transient processes, reducing the gas-generator rotor speed parameter transition time from 2.0 to 1.0 and increasing the control accuracy. At the same time, the neural network, with the use of adaptive parameters (\(k_{p}\) = 1.2, \(T_{i}\) = 0.8, \(k_{1}\) = 0.5, \(k_{2}\) = 0.3, \(k_{3}\) = 0.7, \(\alpha\) = 0.2) and a special architecture, ensured the error function's minimization to the value \(MSE_{\min}\) = 0.005 for 160 epochs, which confirms the method's high efficiency and stability for the current operating conditions. Comparative analysis showed that the proposed method outperforms alternative approaches in terms of accuracy (accuracy = 0.993), convergence rate (reduction in the number of epochs from 380 to 160) and the number of training epochs, which makes it promising for use in aviation and other complex dynamic systems. The developed method has a number of key limitations, including high dependence on the precise identification of the controlled object's model parameters, such as static gain, time constants and nonlinear components, which can reduce the accuracy with errors in these parameters. The method also requires careful tuning of the controller's adaptation \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Research Title** & **Aim** & **Actions** & **Expected Results** \\ \hline The significant delays in modelling optimization [81] & Develop methods for more accurate modelling of significant delays in the system. & Research and implementation of methods that take into account dynamic delays and significant delays in the system. & Improve model accuracy for systems with long delays. \\  & & & Development of prediction methods to take into account long delays timing models’ & \\  & & & integration to take into account phase shifts. & \\ \hline Simplifying the implementation of accurate systems [82]. & Simplify the implementing process of the method on real systems with limited computing power. & Developing lighter and faster versions of the algorithm for implementation on systems with limited computing power. & Increase the availability and practicality of the method for implementation in real systems with limited resources. \\  & & & & \\ \hline Preventing overfitting [83]. & Develop methods to prevent overfitting on small or heterogeneous training datasets. & Use cross-validation and regularization methods to prevent overfitting. & Reduce risk of overfitting and improve generalization properties of the model. \\ \hline \hline \end{tabular}
\end{table} 
Table 9: _Cont_. parameters and optimization coefficients, which complicates its implementation in practical conditions. High computational complexity and requirements for the neural network architecture limit its applicability, especially under limited computing resource conditions.