## 3 Case Study
The comparison of the curves for parameters with different sets (indicated in Table 5) allows us to estimate the algorithm's convergence sensitivity to the choice of initial values. Figure 8 shows the changes' trajectories in the controller's adjustable parameter values (e.g., parameters \(\theta_{1}\), \(\theta_{2}\)) during the iterative optimization process. Each curve (black, blue, and green) corresponds to one of the three sets of initial parameters, which allows us to trace how the parameters are gradually adjusted and approach the optimal values, ensuring an improvement in the system's characteristics. According to Figure 8, the parameter's observed convergence confirms that the selected update method (based on gradient descent with a dynamic training coefficient) effectively tunes the controller. Figure 9 demonstrates how the value of the optimization criterion J changes as the algorithm's iterations are performed. Figure 9 shows that the J value decreases with each iteration and converges to a minimum in about 20 iterations, indicating the algorithm's rapid convergence. In comparison, in the \"search-free\" method [25; 26; 27; 28], the criterion's optimization is achieved in 24 iterations, which emphasizes the proposed method's advantage (the number of iterations reducing by about 1.2 times). This result confirms the effectiveness of the approach used: a rapid decrease in the J value demonstrates that the algorithm successfully minimizes the optimization function, thereby improving the control characteristics. According to Table 5, the matrix \(H(J)\) is positively defined, which confirms, together with the technique given in [25], that the optimal parameters for controller (5) were found by the automatic parametric optimization-generated algorithm, based on the minimum value of criterion (6). Figure 9: Diagrams of the optimization criterion values during the automatic parametric optimization algorithm’s operation: (**black curve**)—parameter set 1; (**blue curve**)—parameter set 2; (**green curve**)—parameter set 3 (note: the values of parameter sets 1, 2, and 3 are presented in Table 5). The presented stability analysis method was carried out under conditions where the object parameters take pre-specified values: \(T_{ob}^{(1)}=1.0\), \(T_{ob}^{(2)}=4.0\), \(k_{ob}=1.0\), and \(\tau_{ob}=5.0\). The controlled object \(G_{p}(p)\) operator takes into account the system's inertia, delay and possible nonlinearities. For these parameters, the object is characterized by a significant delay (\(\frac{\tau_{ob}}{T_{ob}^{(1)}}>1\)), which increases the control and raises the complexity of the requirements for the controller's algorithm. The modified PI controller with the switching function \(g\big{(}t,\overline{\theta}\big{)}\) (4) is adaptive and allows for taking into account the transient process's features. In the stability analysis, the choice of parameters \(k_{1}\), \(k_{2}\), \(k_{3}\), and \(\alpha\) plays an important role since they determine the controller's response to error deviations and its change rate. The first- and second-order sensitivity functions (\(\xi_{i}(t)\) and \(\xi_{ij}(t)\)) show how a change in the controller parameters (\(\theta_{1}\), \(\theta_{2}\)) affects the system's dynamics. For the given object parameters, the sensitivity function takes into account a significant delay and inertia, which leads to the need to adjust the training rate, \(\eta_{l}\), in the gradient procedure (10). Sylvester's condition (16) is used to check the Hessian matrix's positive definiteness, which is necessary to confirm that the minimum optimization criterion \(J\) was found. For this object, given its parameters, the Hessian matrix determinant and its diagonal elements require additional regularization (\(\lambda>0\)) (15) to compensate for the significant delay effect. The switching function, \(g\big{(}t,\overline{\theta}\big{)}\), significantly affects the transient processes' stability.