# Optimal Design of I-PD and PI-D Industrial Controllers Based on Artificial Intelligence Algorithm
## 4 Optimal Design of Controllers for Distillation Column  
In this section, we describe our computation experiments. In the context of applying AIS algorithms to optimize _ISE_ (6), the primary fitness function typically focuses on a cumulative assessment of the difference between \(SV\) and \(PV\). In essence, the discrete formulation of criterion (6) is employed. Consequently, the operation of these intelligent search algorithms unfolds as follows: within a predefined timeframe, the current \(PV\) values are input and compared against the \(SV\). The resulting disparity is assessed using the _ISE_. However, these algorithms tend to identify and halt at a relative local minimum, contingent upon the initial population provided. This section shows the results of solving this problem based on the transformation of the _ISE_ criterion [31]:  
\[ISE=\frac{1}{2a_{n}^{2}\Delta}(B_{m}\Delta_{m}+B_{m-1}\Delta_{m-1}+\ldots+B_{ 0}\Delta_{0})-\frac{b_{m}b_{m-1}}{a_{n}^{2}} \tag{11}\]  
where \(a\), \(b\), \(B\), \(\Delta\) are the parameters of the transfer function of a closed-loop control system, \(T(s)\). To practically apply the method, a two-loop control system is employed for the operations of a tray-type distillation column. This type of column holds paramount importance in industries such as oil, gas, and chemicals, as it is designed to purify liquids and gases. To implement the method in practice, a two-loop control system is used for the operation of a tray-type distillation column. These columns are of great importance in industries such as oil and gas, where they are used to purify liquids and gases. A typical distillation column consists of a vertical structure that contains trays or packing material. It includes a reboiler that provides heat in the form of steam to the bottom of the column, a condenser, and a liquid reflux tank that stores the condensed vapor returning from the top of the column. The input material entering the column consists of primary components (for example, butane and pentane for raw gas), along with impurities from other components [32]. It is important to minimize or eliminate these impurities from the final product. Key components can be categorized into heavier and lighter fractions. The heavier components accumulate at the bottom of the column, while the lighter components are collected as the distillate at the top. The internal structures of the column facilitate the flow of liquids and vapors. These structures can be plates, nozzles, or other devices. Currently, a significant body of the literature explores the modeling and control of distillation columns, taking into account internal structures, specific process conditions, and requirements. Numerous studies focus on binary control structures that govern changes in reflux flow rates and reboiler heat loads to regulate the composition of components in the distillate and bottoms. Within this context, liquid flow rates (\(L\)) and vapor flow rates (\(V\)) are inputs, while product concentrations \(\chi_{\rm D}\) and \(\chi_{\rm B}\) are outputs. The \(L-V\) configuration is a typical control structure for distillation with composition control. In this setting, the control system's objective is to maintain the desired product concentrations \(\chi_{\rm D}\) and \(\chi_{\rm B}\) at the output of the unit using industrial controllers. Various mathematical models have been developed to describe the \(L-V\) configuration, including the Wood-Berry distillation column model [33], a simplified model proposed by [PERSON] and [PERSON] [34], and a detailed model for ethanol-water separation presented in [PERSON] [35]. For this study, we utilize the model proposed by [PERSON] et al. [32] for a tray-type distillation column with 14 trays. To validate this model, experiments were conducted in the TIA Portal environment using real data. These experiments allowed us to confirm the accuracy of the model and its relevance to real industrial processes.