## 3 Case Study
For \(\varepsilon\big{(}t,\overline{\theta}\big{)}\cdot\dot{\varepsilon}\big{(}t, \overline{\theta}\big{)}\geq 0\), the system demonstrates smoother control due to a decrease in the gain. However, for \(\varepsilon\big{(}t,\overline{\theta}\big{)}\cdot\dot{\varepsilon}\big{(}t, \overline{\theta}\big{)}<0\), the presence of exponential damping, \(e^{-\alpha\cdot t}\), can lead to oscillations if the value chosen for the parameter \(\alpha\) is too small. The optimization functional \(J\) (6) reflects the balance between minimizing the error and control costs. For these parameters of the object, the weighting factors \(a_{1}\) and \(a_{2}\) must be selected in such a way as to take into account the delay's influence on the control action, \(u(t)\). Based on the above, it was found that the method is stable for the given parameters \(T_{ob}^{(1)}=1.0\), \(T_{ob}^{(2)}=4.0\), \(k_{ob}=1.0\), and \(\tau_{ob}=5.0\), provided that the controller's parameters (\(k_{p}\), \(T_{i}\), \(k_{1}\), \(k_{2}\), \(k_{3}\), \(\alpha\)) are selected correctly. The significant delay of the object (\(\frac{\tau_{ob}}{T_{ob}^{(1)}}>1\)) is levelled out by the use of regularization and an adaptive training rate. The Hessian matrix's positive definiteness confirms the system's stability. To further confirm the stability, simulation modelling was carried out using the second-order sensitivity functions according to the developed algorithm (Table 6). The resulting diagram (Figure 10) for the sensitivity functions demonstrates the system parameters' influence on its behaviour time dynamics. The first-order sensitivity function (\(\xi_{1}(t)\) and \(\xi_{2}(t)\)) diagrams show two curves: the red curve, \(\xi_{1}(t)\), is the system's sensitivity to the first controller parameter, \(\theta_{1}\), and the blue curve, \(\xi_{2}(t)\), is the system's sensitivity to the second parameter, \(\theta_{2}\). At the beginning of the process (\(t\approx 0\)), the \(\xi_{1}(t)\) and \(\xi_{2}(t)\) values can be large, indicating the significant influence of parameters \(\theta_{1}\) and \(\theta_{2}\) on the system dynamics at the initial moment in time. As time increases (\(t\rightarrow\infty\)), both functions tend to zero, meaning a decrease in the parameters' influence on the system's output in the steady-state mode. The \(\xi_{1}(t)\) values are higher than \(\xi_{2}(t)\), which may indicate that the parameter \(\theta_{1}\) had a more substantial influence on the system's dynamics, and the time delay and curve amplitudes differ due to each parameter's different contributions. The second-order sensitivity function (\(\xi_{11}(t)\), \(\xi_{12}(t)\), \(\xi_{22}(t)\)) diagrams show three curves: The red curve, \(\xi_{11}(t)\), is the sensitivity's second derivative with respect to \(\theta_{1}\); it shows how \(\theta_{1}\) influences changes over time. The green curve, \(\xi_{12}(t)\), is the second-order joint sensitivity between the parameters \(\theta_{1}\) and \(\theta_{2}\); it reflects the parameter interactions. The blue curve, \(\xi_{22}(t)\), is the sensitivity's second derivative with respect to \(\theta_{2}\); it shows how the influence of \(\theta_{2}\) changes over time. The second-order functions \(\xi_{11}(t)\) and \(\xi_{22}(t)\) have high values at the beginning (\(t\approx 0\)) and then quickly decrease, demonstrating a decrease in the parameters' influence in the steady-state mode. The joint sensitivity, \(\xi_{12}(t)\), has a smaller amplitude, which may indicate a weak relationship between the parameters \(\theta_{1}\) and \(\theta_{2}\). At the same time, the \(\xi_{11}(t)\) value is higher than \(\xi_{22}(t)\), confirming the greater significance of \(\theta_{1}\) compared to \(\theta_{2}\). The \(\xi_{12}(t)\) values remain low, indicating weak cross-dependence between the parameters. Thus, the diagrams show that the parameters \(\theta_{1}\) and \(\theta_{2}\) have the most significant influence on the system's dynamics at the initial moment in time.