## 2 Materials and Methods  
According to [25; 26; 27; 28], delays are often encountered in industrial processes such as transportation and substance mixing. Provided that the ratio \(\frac{T_{ab}}{T_{ob}^{(m)}}>1\) (where \(\tau_{ob}\) is the delay time and \(T_{ob}^{(m)}\) is the object's maximum time constant), standard controllers such as P, PI and PID do not provide the required quality in transient processes. One of the control methods involves PI controllers with a delay, which can be used in the structure. Semi-constant integration in the PI controller is also used, which allows for avoiding an increase in the number of adjustable parameters. These controllers with a variable structure are difficult to adjust, which requires approximate methods, reducing their efficiency. In this regard, algorithmic tuning methods are necessary. Positive results from the gradient algorithm applied based on sensitivity theory [10; 11] to calculate the proportional controller's adjustable parameters confirm its applicability for solving the parametric optimization issue, which consists of minimizing a given optimization criterion. The parametric optimization issue is to find values for the controller's adjustable parameters, vector \(\theta=(\theta_{1},\theta_{2})\), so that the minimum optimization criterion is met. Thus, according to [10; 11; 25; 26; 27; 28], the processes occurring in the system are described by the following equation:  
\[\varepsilon\big{(}t,\,\overline{\theta}\big{)}=r(t)-x(t);\,u(t)=G_{c}\big{(}s, \,\overline{\theta}\big{)}\cdot\varepsilon\big{(}t,\,\overline{\theta}\big{)}; \,x(t)=G_{p}(p)\cdot u(t), \tag{1}\]  
where \(\varepsilon(t,\,\overline{\theta})\) is the control system error; \(r(t)\) is the reference action; \(u(t)\) is the control action; \(x(t)\) is the ACS output coordinate; \(G_{c}(s,\,\overline{\theta})\) is the controller operator; \(G_{p}(s)\) is the control object operator; and \(s=\frac{d}{dt}\) is the differentiation operator (according to [25; 28]). The control object operator, \(G_{p}(s)\), in this research is chosen in such a form that it is possible to describe the most complex dynamic object processes:  
\[G_{p}(s)=\frac{k_{ob}}{\Big{(}T_{ob}^{(1)}\!\cdot\!s+1\Big{)}\!\cdot\!\Big{(}T _{ob}^{(2)}\!\cdot\!s+1\Big{)}}\!\cdot\!e^{-\tau_{ob}\cdot s}+f_{nonline}(x,u), \tag{2}\]  
where \(k_{ob}\) is the static gain; \(T_{ob}^{(1)}\) and \(T_{ob}^{(2)}\) are the time constants; \(\tau_{ob}\) is the delay time, and \(f_{nonline}(x,u)\) is the nonlinear part depending on the system's state. The proposed controller operator \(G_{c}(s,\,\overline{\theta})\), which is a traditional PI controller extension including a switching function, has the form:  
\[G_{c}(s)=k_{p}\cdot\left(1+\frac{1}{T_{i}\cdot s}\right)\cdot g\big{(}t, \overline{\theta}\big{)}, \tag{3}\]  
where \(k_{p}\) is the proportionality coefficient that determines the controller's response intensity to a deviation from the set value, \(T_{i}\) is the integration time that characterizes the influence of previous errors on the controller's action, and \(g(t,\,\overline{\theta})\) is the switching function that depends on time \(t\) and the parameter \(\overline{\theta}\), which is responsible for the controller's dynamic adaptation. The function \(g(t,\overline{\theta})\) regulates changes in the controller's behaviour depending on the system's current operating conditions, based on [25], and it is represented by one of the following possible expressions:  
\[g(t,\overline{\theta})= \begin{cases}k_{1},\varepsilon(t,\overline{\theta})\cdot \dot{\varepsilon}(t,\overline{\theta})\geq 0,\\ k_{2}\cdot e^{-\alpha\cdot t}+k_{3},\varepsilon(t,\overline{\theta})\cdot\dot{ \varepsilon}(t,\overline{\theta})<0,\end{cases}  \tag{4}\]  
where \(k_{1},k_{2},k_{3},\) and \(\alpha\) are adjustable parameters depending on \(\theta\). The switching function \(g(t)\) in (4) is chosen to ensure the adaptive behaviour of the controller in a dynamically changing control environment [8], which is confirmed by both theoretical and empirical considerations.