# Optimal Design of I-PD and PI-D Industrial Controllers Based on Artificial Intelligence Algorithm
## 4 Optimal Design of Controllers for Distillation Column
For clarification, consider the transfer function of a closed-loop control system with a second-order controller in the form of a transfer function:  
\[T(s)=\frac{b_{0}}{a_{0}\cdot s^{2}+a_{1}\cdot s+a_{2}},\;\;m=0,\;\;n=2. \tag{18}\]  
In this case, the \(ISE\) has the form:  
\[ISE=\frac{1}{2\cdot a_{2}^{2}\cdot\Delta}.(B_{0}\cdot\Delta_{0})\to min. \tag{19}\]  
According to the model (18):  
\[B_{0}=b_{0}^{2},\;\;\;\;\Delta_{0}=det \begin{vmatrix}a_{1}&-a_{0}\\ a_{2}&a_{1}\end{vmatrix} =a_{1}^{2}+a_{0}\cdot a_{2},\;\;\;\Delta=det \begin{vmatrix} a_{2}&-a_{0}\\ 0&a_{1}\end{vmatrix} =a_{1}\cdot a_{2}. \tag{20}\]  
Figure 3: Modeling results of two control loops: (**a**) First control loop; (**b**) Second control loop. Then,  
\[ISE=\frac{b_{0}^{2}\cdot\left(a_{1}^{2}+a_{0}\cdot a_{2}\right)}{2\cdot a_{2}^{2} \cdot a_{1}\cdot a_{2}}\to min. \tag{21}\]  
This performance criterion is effectively utilized in this work as a fitness function, the minimization of which is achieved through a search algorithm based on the AIS algorithm. Therefore, the task can be simplified to the following challenge: identifying the elusive parameters of controllers \(K_{P1}\), \(T_{I1}\), \(K_{P2}\) and \(T_{I2}\), that yield the minimum value for the quality criterion (21) through the use of the AIS algorithm. This approach results in a unique quality criterion for each subsystem:  
\[ISE_{I-P1}=\frac{1+0.0084\cdot K_{P1}+0.0042^{2}\cdot K_{P1}^{2}+1.9588\cdot 0.0042\cdot\frac{1}{T_{I1}}}{0.0084\cdot\frac{1}{T_{I1}}+2\cdot 0.0042^{2}\cdot K_{P1} \cdot\frac{1}{T_{I1}}}, \tag{22}\]  
\[ISE_{I-P2}=\frac{1+0.0144\cdot K_{P2}+0.0072^{2}\cdot K_{P2}^{2}+1.9588\cdot 0.0072\cdot\frac{1}{T_{I2}}}{0.0144\cdot\frac{1}{T_{I2}}+2\cdot 0.0072^{2}\cdot K_{P2} \cdot\frac{1}{T_{I2}}}. \tag{23}\]  
When formulating quality criteria, several constraints must be taken into account. First, simulation time and, consequently, integration range are limited by the simulation time required for the technological process. Second, controller parameters, which are solutions to an optimization problem, must be strictly positive:  
\[K_{P1}>0,\;\;T_{I1}>0,\;\;\;\;\;\;K_{P2}>0,\;\;T_{I2}>0. \tag{24}\]  
The procedures of the AIS algorithm, which are designed to minimize _ISE_ (22) and (23), are implemented within the MATLAB environment. During the implementation of the algorithm, an initial population is generated. This population consists of controller parameter values that are selected from a range of feasible values, ensuring both the stability of the closed-loop control system and its feasibility from a technical perspective. Subsequently, a complete calculation cycle is carried out, following the steps defined in the search algorithm described in Section 3. Table 2 shows the general algorithm settings. During the implementation of algorithms within the MATLAB environment, parameters of controllers for the first and second control loops were obtained: (see Table 3). In the previous works of the authors [36; 37], the results of the development of various bioinspired algorithms to minimize such quality criteria as _ISE_, _ITSE_, and _IAE_ were obtained. Bioinspired algorithms were applied to the PI controller, such as ACO [36], PSO [36], DA [36], AIS [37], GA [37], etc. As the AIS and GA produce the best results for _ISE_, these algorithms are used in this paper for a comparative analysis with the AIS algorithm for the I-P controller. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Value** \\ \hline \(N\) & 100 \\ \(Ab(x_{i})\) & \(K_{P}\)([0.1; 1]; \(1/T_{I}\) ([0.1; 10] \\ _gen_ & 50 \\ (\(p_{m}\), \(d\), \(\beta\)) & (0.5, 0.3, 0.8) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial condition for the algorithm.