###### Abstract  
This article presents a method for researching processes in automatic control systems based on the operator approach for modelling the control object and the controller. Within the method framework, a system of equations has been developed that describes the relations between the control error, the reference and control action, the output coordinate and the controller and the control object operators. The traditional PI controller modification, including a switching function for adaptation to operating conditions, allows for the system's effective control in real time. The controller optimization algorithm is based on a functional expression with weighting coefficients that take into account control errors and the control action. To train the neural network through implementing the proposed method, a multilayer architecture was used, including nonlinear activation functions and a dynamic training rate, which ensure high accuracy and accelerated convergence. The TV3-117 turboshaft engine was chosen as the research object, which allows the method to be demonstrated in practical applications in aviation technology. The experimental results showed a significant improvement in control characteristics, including a reduction in the gas-generator rotor speed parameter transient time to \(\approx\)1, which is two times faster than the traditional method, where the transient process reaches \(\approx\)0.5. The model achieved a maximum accuracy of 0.993 with 160 training epochs, minimizing the error function to 0.005. In comparison with similar approaches, the proposed method demonstrated better results in accuracy and training speed, which was confirmed by a reduction in the number of iterations by 1.36 times and an improvement in the mean square error by 1.86-6.02 times. s 1 2025 15 286 2025 1 2025 1  
tasks is to optimize the controllers' parameters to ensure the transient processes' stability under varying external conditions. Of particular interest are controllers with a variable structure and semi-permanent integration [3; 4], which allow the system parameters' flexible adaptation depending on its current state. Modern approaches [5; 6; 7] to the parametric optimization of variable-structure controllers are based on the analysis and numerical calculations of the mathematical methods used. Key areas of research are gradient optimization methods [8; 9], approaches based on sensitivity theory [10; 11] and the optimal control methods used [12; 13]. For example, in [14; 15], the research is devoted to the use of gradient algorithms, and attention is primarily paid to the computation of the quality functional gradient in ACSs using first-order sensitivity. The key drawbacks of modern approaches to parametric optimization are the first-order sensitivity limiting its applications, which reduces the tuning accuracy in systems with pronounced nonlinearity [14; 15]. In addition, gradient methods often require significant computational resources, especially when working with high-dimensional systems, which limits their practical implementation [8; 16]. Optimal control and sensitivity theory methods also face scalability problems and difficulties in being integrated with real-time adaptive controllers [10; 11; 12; 13]. The indicated shortcomings [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16] are eliminated in methods based on the computation of second-order sensitivity functions. Methods based on the computation of second-order sensitivity functions [16; 17; 18] allow for increasing the parametric tuning accuracy. Methods based on the computation of second-order sensitivity provide more accurate parametric tuning by taking into account the interaction between parameters and their influence on system changes [16; 17]. These methods take into account changes in the system, not only depending on the current state but also on the influence of parameter interactions [16; 17]. However, such approaches are often difficult to implement due to the high computational load and the need to use specialized software [16; 18].