## 4 Discussion
### Generalization of Results
The control action, \(u(t)\), is determined by expression (5), which depends on the error \(\varepsilon(t,\overline{\theta})\), parameters \(\theta_{1}\) and \(\theta_{2}\), and the switching function state \(g\big{(}t,\varepsilon(t,\overline{\theta}),\dot{\varepsilon}(t,\overline{ \theta})\big{)}\). The functional expression in (6), including the weighting coefficients \(a_{1}\) and \(a_{2}\), which take into account the control errors and the control action contributions, is selected as the optimization criterion for adjusting the controller's parameters. Expression (7) is used to calculate the sensitivity to the parameters \(\overline{\theta}\), and the update in the parameters' gradient in the optimization process is determined by expression (10). The conditions for stopping the optimization process are formed on the basis of the analysis of changes in the optimality criterion, as indicated in expressions (12) and (13). The process completion is determined by fulfilling the conditions presented in expression (14). To check the criterion's minimum, the Hessian matrix \(H(t)\) is calculated, whose expression is given in (15). The matrix's positive definiteness condition is estimated using [PERSON]'s criterion, presented in expression (16). The developed method differs from the search-free method [25; 26; 27; 28] by the integration of the nonlinear components into the control object operator, the parameterization of the dynamic training rate \(\eta_{I}\), the regularization coefficient \(\lambda\) used to stabilize the optimization, and improved stopping criteria taking into account the permissible error and the number of changes in the gradient sign. The developed method is implemented as a neural network with a specified architecture (see Figure 3), taking into account the system's dynamic features and its behavioural nonlinearities. The basis of this architecture is a multilayer structure, where the input layer receives the system error \(\varepsilon(t)\) and the reference action \(r(t)\) parameters, and the hidden layers with nonlinear activation functions (SmoothReLU, tanh) model complex relations. A unique feature is the first hidden layer used for processing \(\varepsilon(t)\) with the function \(f_{\textit{nonlinear}}(x,\)\(u)\), the second layer for calculating the regulatory action \(u(t)=G_{c}(s)\cdot\varepsilon(t)\), and the third layer for integrating the switching parameters through the functions \(\phi(t)\). Adaptive network parameters such as \(k_{p}\), \(T_{i}\), \(k_{1}\), \(k_{2}\), \(k_{3}\), and \(\alpha\) are trained using the gradient descent method with regularization, where the coefficient \(\lambda\) stabilizes the training, preventing overfitting and noise. The dynamic training rate \(\eta_{I}\), depending on the change in the optimization functional, ensures convergence acceleration and increased model accuracy, and the output layer calculates the coordinate \(x(t)\), described by the control object operator \(G_{p}(s)\), which allows for taking into account delays and system nonlinearities. A neural network training algorithm (see Table 2) is developed that includes a multilayer architecture with adaptive parameters and takes into account the system's nonlinear dynamic features. In the first stage, the network architecture and its parameters (weights and biases) are initialized using the normal distribution or methods such as the Xavier or He methods. In the forward pass stage, the input parameters \(\varepsilon(t)\) and \(r(t)\) are sequentially processed through three hidden layers: the first layer computes the nonlinear error processing using SmoothReLU, the second layer models the control action with the tanh function, and the third layer integrates the switching parameters via SmoothReLU. The output layer forms the coordinate \(x(t)\) defined by the control object operator. The error is calculated using a loss function, for example, \(MSE\), and the gradients are computed taking into account the sensitivity function, \(\Xi(t)\), reflecting the parameters' effect on the system.