## 2 Materials and Methods
On the one hand, the presence of adjustable parameters (\(k_{1},k_{2},k_{3},\alpha\)) allows for fine-tuning the controller's gain, where the exponential factor \(e^{-\alpha\cdot t}\) reduces the influence of sharp disturbances, preventing overgain and oscillations, which ensures an optimal balance between the system's speed and stability. On the other hand, empirical data on the gas-generator part of the TV3-117 engine operation's modelling [13] demonstrate that the use of \(g(t)\) allows for reducing the transient process time by half compared to traditional methods and improves the optimization algorithm's convergence, reducing the number iterations and minimizing the loss function to \(MSE\approx 0.005\). Experimental research [8; 13; 25; 26; 27; 28] confirms that the function's selected form ensures the controller's reliable adaptation in real time, the system parameters' correct restoration, and a smoother and more predictable transient process, which is confirmed by the Hessian matrix's positive definiteness and the stability of the system's dynamics. The object operator parameters' fundamental values (2) are taken, taking into account the significant delay condition, i.e., \(\frac{T_{\text{th}}}{T_{\text{th}}^{\infty}}>1\). Taking into account the expression that determines the control action, \(u(t)\), values, similarly to [25], the analytical expression describing \(u(t)\) is presented as follows:  
\[u(t)= \begin{cases}\theta_{1}\cdot\varepsilon\big{(}t,\overline{\theta}\big{)} +\frac{\theta_{2}}{s}\cdot\varepsilon\big{(}t,\overline{\theta}\big{)},g \big{(}t,\varepsilon\big{(}t,\overline{\theta}\big{)},\dot{\varepsilon}(t, \overline{\theta}\big{)}\big{)}=1,\\ \theta_{1}\cdot\varepsilon\big{(}t,\overline{\theta}\big{)}+\frac{\theta_{2 }}{s}\cdot 0(t),g\big{(}t,\varepsilon\big{(}t,\overline{\theta}\big{)},\dot{ \varepsilon}\big{(}t,\overline{\theta}\big{)}\big{)}=0.\end{cases}  \tag{5}\]  
Similarly to [25], this research sets the task of investigating the influence of the different variants of the switching function (4) in the controller (5) on the transient processes' quality. Figure 1 shows the controller's structural diagram under research [25]. As a criterion for optimizing the automatic parametric optimization-generated algorithm, based on [29], the functionality widely used in the practice of automatic regulation was selected:  
\[J=\int\limits_{0}^{\infty}\Big{(}a_{1}\cdot\varepsilon^{2}\big{(}t,\overline{ \theta}\big{)}+a_{2}\cdot u^{2}(t)\Big{)}dt, \tag{6}\]  
where \(a_{1}\) and \(a_{2}\) are weighting coefficients. Figure 1: Structural diagram of the researched PI controller with semi-permanent integration. In the formation of the automatic parametric optimization algorithm in this research, sensitivity theory methods [10, 11] for a system's continuous class were used [18, 19] since the control action, \(u(t)\), discontinuity point's coordinate \(t_{m}=(1,2,\ldots,m)\) does not depend on \(\overline{\theta}\)[21, 25, 26]. Similarly to [25, 26, 27, 28], in this research, a model was obtained for the computation of the controller's adjustable parameters' (5) sensitivity functions:  
\[\xi_{i}(t)=G_{p}(s)\cdot\frac{\partial u(t)}{\partial\theta_{i}},\ i=1,2, \tag{7}\]  
\[\frac{\partial u(t)}{\partial\theta_{i}}=\frac{\partial G_{c}(s,\overline{ \theta})}{\partial\theta_{i}}\cdot\varepsilon\big{(}t,\overline{\theta}\big{)} -\xi_{i}(t)\cdot G_{c}\big{(}s,\overline{\theta}\big{)};\frac{\partial u(t)}{ \partial\theta_{1}}=\varepsilon\big{(}t,\overline{\theta}\big{)}-\xi_{i}(t) \cdot G_{c}\big{(}s,\overline{\theta}\big{)}; \tag{8}\]  
\[\frac{\partial u(t)}{\partial\theta_{2}}=