# Optimal Design of I-PD and PI-D Industrial Controllers Based on Artificial Intelligence Algorithm
## 4 Optimal Design of Controllers for Distillation Column
At the end of Section 4, the results of the modeling of a control system with controllers in the TIA Portal environment are presented. The distillation column can be described as a system with two input and output variables, making it a two-input, two-output system (TTTO). This type of system is commonly referred to as a two-dimensional control system [32]:  
\[ \begin{cases}\frac{d\chi_{D}(t)}{dt}=-\frac{1}{1.9588}\,\chi_{D}(t)+\frac{0.004 2}{1.9588}\,L(t)-\frac{0.0062}{1.9588}\,V(t),\\ \frac{d\chi_{B}(t)}{dt}=-\frac{1}{1.9588}\,\chi_{B}(t)+\frac{0.0072}{1.9588}\, V(t)-\frac{0.0052}{1.9588}\,L(t),\end{cases}  \tag{12}\]  
where \(L\) is the reflux rate; \(V\) is the vapor boil-up rate; \(\chi_{D}\) is the distillating composition; \(\chi_{B}\) is the bottom composition. Let us introduce two column vectors of input, \(CV_{1}\), \(CV_{2}\) and output, \(PV_{1}\), \(PV_{2}\), signals:  
\[ \begin{vmatrix}PV_{1}(s)\\ PV_{2}(s)\end{vmatrix} = \begin{vmatrix}\chi_{D}(s)\\ \chi_{B}(s)\end{vmatrix} ^{\prime} \begin{vmatrix}CV_{1}(s)\\ CV_{2}(s)\end{vmatrix} = \begin{vmatrix}L(s)\\ V(s)\end{vmatrix} . \tag{13}\]  
By the introduced vectors, the expression for the model of a TITO process control system for a distillation column has the form:  
\[ \begin{vmatrix}PV_{1}(s)\\ PV_{2}(s)\end{vmatrix} = \begin{vmatrix}G_{11}(s)&G_{12}(s)\\ G_{21}(s)&G_{22}(s)\end{vmatrix}\begin{vmatrix}CV_{1}(s)\\ CV_{2}(s)\end{vmatrix} , \tag{14}\]  
where \(G_{11}\) and \(G_{22}\) are transfer functions of isolated subsystems; \(G_{12}\) and \(G_{21}\) are transfer functions of interconnections between subsystems (see Figure 2):  
\[ \begin{vmatrix}G_{11}(s)=\frac{0.0042}{1.95886+1}&G_{12}(s)=\frac{-0.0052}{1. 95886+1}\\ G_{21}(s)=\frac{-0.0062}{1.95886+1}&G_{22}(s)=\frac{0.0072}{1.95886+1}\end{vmatrix} . \tag{15}\]  
Figure 2 shows the structural diagram of the TITO system. The structural diagram consists of three modules: controller, decoupling unit and TITO control plant. To achieve stabilization of the distillation process, specific target values need to be set. These values  
Figure 2: Scheme for modeling a TITO system with controllers. include the purity of the distillate, defined as the percentage of isobutane and butane in the product, which is \(SV_{1}=0.98\), and the purity of material from the bottom of the column, also defined in terms of the percentage of these components, at \(SV_{2}=0.02\). We will now derive the results from simulating the transfer functions of isolated circuits by applying a decoupling technique, ensuring that there are no interconnections. To determine the necessary system dynamics, transient simulations were performed (see Figure 3), with \(PV_{1}\) and \(PV_{2}\) representing the current output values of the two circuits. In these simulations the signals \(SV_{1}=0.98\) and \(SV_{2}=0.02\) was used as the desired input. From the analysis of the step responses, it can be concluded that they are monotone. Consequently, for the two circuits described by Equation (14), the control law (1), incorporating the time constant, is selected, thus opting for the I-P controller. By expression (2), the transfer functions for the closed-loop system with I-P controllers can be expressed as:  
\[T_{I-P1}(s)=\frac{0.0042\cdot\frac{1}{T_{I1}}}{\left(\frac{1}{T_{I1}}+0.0042 \cdot T_{D1}\right)\cdot s^{2}+(K_{P1}+1)\cdot s+K_{P1}\cdot\frac{1}{T_{I1}}}, \tag{16}\]  
\[T_{I-P1}(s)=\frac{0.0072\cdot\frac{1}{T_{D2}}}{\left(\frac{1}{T_{I2}}+0.0072 \cdot T_{D2}\right)\cdot s^{2}+(K_{P2}+1)\cdot s+K_{P2}\cdot\frac{1}{T_{I2}}}. \tag{17}\]  
To achieve the desired dynamics described as \((SV)\), we introduce a fitness function designed for optimization purposes. To solve this problem, this study proposes a methodology that is similar to the approach described in [31]. This methodology is based on deriving a quality criterion without an integration function from the Formula (6).