# Optimal Design of I-PD and PI-D Industrial Controllers Based on Artificial Intelligence Algorithm
## 4 Optimal Design of Controllers for Distillation Column
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Controllers** & \(K_{P1}\) & \(1/T_{I1}\) & \(K_{P2}\) & \(1/T_{I2}\) \\ \hline I-P controller (AIS) & 0.5314 & 9.9961 & 0.0463 & 9.9928 \\ PI controller (AIS) & 25.34 & 20.48 & 20.16 & 20.24 \\ PI controller (AIS-GA) & 19.54 & 48.12 & 22.73 & 23.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the controllers. To demonstrate the effectiveness of the proposed approach, the results obtained with AIS were compared with the GA and hybrid GA methods. We present the results of comparing AIS for I-P with algorithms: AIS for PI controller, GA for PI controller, and hybrid GA with AIS for PID controller. The authors developed these algorithms in their articles [37] and obtained the parameters for the controllers (see Table 3). Figure 4 shows a comparison of the convergence characteristics of AIS and GA in minimizing _ISE_. To validate the performance of the selected algorithm, a statistical analysis was conducted using the numerical performance index _ISE_, as shown in Table 4.  
Figure 5 shows the results of a comparative analysis of  
Upon examining the step responses of the control system using the controllers, the time response specifications can be tabulated in Table 5.  
Based on the simulation results obtained, it is possible to conclude the effectiveness of intelligent controllers that were tuned using the AIS algorithm:  
* The settling time for processes using an I-P controller is significantly shorter compared to those using a PID controller;
* The overshoot observed in processes with an I-P controller is lower compared to processes with a PID controller;
* The rise time for the primary circuit's process using an I-P controller is faster than that with a PID controller. As a result, the control laws of the two industrial regulators are as follows: \[ \begin{split} CV_{I-P}(t)&=0.5314\cdot PV(t)+9.9961 \int\limits_{t=0}^{t_{1}}E(t)dt,\\ CV_{I-P}(t)&=0.0463\cdot PV(t)+9.9928\int\limits_{t=0 }^{t_{1}}E(t)dt\end{split} \] (25)  
The simulation results of the two control loops using these designated signals are presented in Figure 6. Based on the analysis of the graphs, it can be concluded that the desired dynamics have been achieved. Over time, the values of \(\chi_{\text{D}}\) and \(\chi_{\text{B}}\) have converged towards the predefined values of 0.98 and 0.02, respectively. In addition, Figure 6 illustrates the control action values of \(L\) and \(V\) required to achieve the desired system performance. Figure 6: Simulation results of two control loops with reference signals: (**a**) First control loop; (**b**) Second control loop. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Controllers** & **Settling Time (T\({}_{\text{SET}}\))** & **Overshoot (P\({}_{\text{OV}}\), \%)** & **Rise Time (T\({}_{\text{R}}\))** \\ \hline First control loop & & & \\ PI (AIS-GA) & 9.53 & 0.261 & 6.18 \\ PI (AIS) & 12.9 & 0.508 & 8.37 \\ PI (GA) & 13.8 & 3.25 & 5.41 \\ I-P (AIS) & 7.01 & 0 & 3.85 \\ Second control loop & & & \\ PI (AIS-GA) & 6.25 & 0.342 & 3.99 \\ PI (AIS) & 18.7 & 0 & 11.1 \\ PI (GA) & 11 & 8.46 & 3.53 \\ I-P (AIS) & 3.02 & 0 & 1.74 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Step response specification. However, the achieved results indicate the output values generated by individual circuits. Figure 2 provides a representation of a diagram to account for the effects of interactions between components. The simulated transfer functions from the mathematical model (14) are depicted in this diagram. The issue of addressing and compensating for interconnections arising from the presence of coupling devices \(H1\) and \(H2\) has been thoroughly explored in [36]. While this current work focuses on the development of controllers based on AIS, a comprehensive explanation of the decoupling process and the calculation of matrices \(H1\) and \(H2\) will not be presented herein.