## 2 Materials and Methods
\begin{array}{l}-\xi_{2}(t)-\frac{1}{s}\cdot\xi_{1}(t)-G_{c}\big{(}s, \overline{\theta}\big{)}\cdot\xi_{21}(t),g\big{(}t,\varepsilon(t,\overline{ \theta}),\dot{\varepsilon}(t,\overline{\theta})\big{)}=1,\\ -\xi_{2}(t)-\frac{1}{s}\cdot 0(t)-G_{c}\big{(}s,\overline{\theta}\big{)}\cdot\xi_{21}(t ),g\big{(}t,\varepsilon(t,\overline{\theta}),\dot{\varepsilon}(t,\overline{ \theta})\big{)}=0,\end{array} \right.\]  
The matrix \(H(J)\) for model (17) according to criteria (6) is presented similarly to in [25]:  
\[H(J)=\left[ \begin{array}{cc}\lambda\cdot J+2\cdot\int\limits_{0}^{\infty} \big{(}\xi_{1}^{2}(t)-\xi_{11}(t)\cdot\varepsilon(t,\overline{\theta})\big{)} dt&\lambda\cdot J+2\cdot\int\limits_{0}^{\infty}\big{(}\xi_{1}(t) \cdot\xi_{2}(t)-\xi_{12}(t)\cdot\varepsilon(t,\overline{\theta})\big{)}dt\\ \lambda\cdot J+2\cdot\int\limits_{0}^{\infty}\big{(}\xi_{1}(t)\cdot\xi_{2}(t) -\xi_{21}(t)\cdot\varepsilon(t,\overline{\theta})\big{)}dt&\lambda\cdot J+2 \cdot\int\limits_{0}^{\infty}\big{(}\xi_{2}^{2}(t)-\xi_{22}(t)\cdot\varepsilon (t,\overline{\theta})\big{)}dt\end{array} \right] \tag{19}\]  
The optimization process is complete if the following conditions are met:  
\[\left|I^{(l)}-I^{(l-1)}\right|<\varepsilon\ \text{and}\ n_{SI}\leq N, \tag{20}\]  
where \(\varepsilon\) is the permissible error and \(n_{SI}\) is the gradient sign change number. As noted above, in addition to testing the automatic parametric optimization-generated algorithm's performance and, accordingly, testing the computation (19) correctness, a technique oriented towards a computational experiment was used. The idea behind this technique is that the automatic parametric optimization algorithm is launched with at least three different initial values for the adjustable parameters \(\overline{\theta}_{k}^{0}=\big{(}\theta_{1k}^{0},\theta_{2k}^{0}\big{)}\) and \(k=\overline{1,2,3}\); then, at the adjustable parameters' space points \(\overline{\theta}_{k}^{*}=\big{(}\theta_{1k}^{*},\theta_{2k}^{*}\big{)}\), the necessary condition for the optimality criterion extremum for \(\overline{\theta}^{*}\) is checked:  
\[\frac{\partial J\Big{(}\overline{\theta}^{*}\Big{)}}{\partial\overline{\theta }}=(0\pm\Delta), \tag{21}\]  
where \(\Delta\) is the calculation error. Additionally, to check the automatic parametric optimization-generated algorithm's results reliability, the following condition is used:  
\[\left|J\Big{(}\overline{\theta}^{*}\Big{)}-J\big{(}\overline{\theta}\big{)} \right|<\delta,\ \forall i,j \tag{22}\]  
for \(\overline{\theta}^{*}\) within the range of adjustable parameters used in this research, where \(\delta\) is the permissible error. Based on the above, we summarize that, in this research, the automatic parametric optimization-generated algorithm's performance indicator comprises criteria (16), (21), and (22) for the fulfilment for \(\overline{\theta}^{*}\). In the proposed algorithm, the controller's parameters are updated according to (10): \(\theta_{(l+1)}=\theta_{(l)}-\eta_{l}\cdot(W\cdot\
abla J)\), where the dynamic training rate, \(\eta_{l}\), is adjusted based on the change in the optimization criterion, which allows for adapting the optimization step as it approaches the minimum of the function \(J\) (see (10)-(12)). The stopping conditions specified in (12)-(14) ensure that the iterative process is completed when the difference between the \(J\) values at successive steps becomes lower than a given threshold, which prevents overtraining. Regularization is used to stabilize the algorithm: adding the coefficient \(\lambda\) to the Hessian matrix, \(H(J)\) in (15), and checking for positive definiteness using [PERSON]'s criterion (16) minimizes the influence of noise and nonlinearities. In addition, the choice of switching function, \(g(t)\) in (4), where the parameter \(\alpha\) determines the exponential decay rate and the coefficients \(k_{1}\), \(k_{2}\), and \(k_{3}\) regulate the gain level, allows one to precisely balance the controller's speed and stability under dynamically changing control conditions.