# Analytical Design of Fractional-Order PI Controller for Parallel Cascade Control Systems
## 2 Materials and Methods
### The General Proposed Design Method for the Above Three Cases of the Primary Controller  
After deriving the equivalent transfer functions of primary processes in the three above cases to the general form of FOPTD, Equation (38), the proposed FOPI controller will be designed based on the IMC approach. However, the frequency domain mentioned above is addressed here. \[\widetilde{G}_{p}=\frac{K_{p}e^{-\theta_{p}s}}{\tau_{p}s+1} \tag{38}\]  
In order to choose the fractional order of the controller, the relative dead time parameter is considered [10]:  
\[\Delta=\frac{\theta_{p}}{\tau_{p}+\theta_{p}} \tag{39}\]In this work, the fractional-order \(\lambda\) is determined according to the value of \(\Delta\) as Equation (40). It is based on the guideline in [10], however, only the fractional order is considered in this study  
\[\lambda=\left\{ \begin{array}{ll}1.1,&\text{if }\Delta\geq 0.6\\ 0.9,&\text{if }0.1\leq\Delta<0.6\\ 0.7,&\text{if }\Delta<0.1\end{array} \right. \tag{40}\]  
Step 1: Design the stabilized controller  
Case 1: The primary process transfer function is stable. Therefore, the stabilized loop will not be used in this case. Hence, \(G_{d}\) is assigned to zero (\(G_{d}=0\))  
Case 2: Note that when the order is greater than 1, system responses will create an overshoot. Therefore, in this work, to guarantee a good servo tracking of the primary control loop, the relative dead time parameter \(\Delta\) will be chosen in the range of \([0.1,0.6]\), it means that \(\lambda=0.9\). From Equations (31) and (39), the following condition is obtained:  
\[0.1\leq\frac{\theta_{1}}{\frac{\tau_{1}-0.5\xi_{K}\xi_{1}\theta_{1}}{K_{d}K_{1 }-1}+\theta_{1}}<0.6\Rightarrow 0.1\leq\frac{\theta_{1}(K_{d}K_{1}-1)}{\tau_{1}- \theta_{1}+0.5\theta_{1}K_{d}K_{1}}<0.6 \tag{41}\]  
Case 3: For this case, the condition (36) and (37) are used to choose \(K_{d}\) for the stabilized controller, remember that \(\tau_{d}=0.5\theta_{1}\) as in Equation (29). Step 2: Analytical design of the FOPI controller  
The ideal feedback controller that is equivalent to the IMC controller can be expressed in terms of the internal model \(\widetilde{G}_{p}\) and the IMC controller \(q\). Due to the Smith predictor structure, the delay term will be removed in the closed-loop transfer function of the primary loop as in Equation (25). Therefore, in this step, the FOPI controller will be designed for the delay-free process \(G_{pm}\). Figure 5a,b shows the block diagrams of the IMC scheme and equivalent classical feedback control structures, respectively, where \(G_{pm}\) is the process, \(\widetilde{G}_{pm}\) the process model, \(q\) the IMC controller, and \(G_{c_{1}}\) the equivalent feedback controller. For the nominal case (\(G_{pm}=\widetilde{G}_{pm}\)), in Figure 5a, the system response can be calculated by the set-point and disturbance signals in the following simplified equation:  
\[y=G_{pm}qr+(1-\widetilde{G}_{pm}q)G_{d_{1}}d \tag{42}\]  
where \(\widetilde{G}_{pm}=\frac{K_{p}}{\tau_{p}+1}=p_{m}\), which is the portion of the model inverted by the controller, and in this case, \(p_{a}=1\) (IMC approach is already mentioned in the previous section). To obtain a good response for processes with poles near zero, the IMC controller \(q\) should be designed to satisfy the following conditions. 1. If the process \(G_{pm}\) has poles near zero at \(z_{1}\), \(z_{2},\cdots,z_{m}\) then \(q\) should have zeros at \(z_{1}\), \(z_{2},\cdots,z_{m}\)  
2. If the process \(G_{d_{1}}\) has poles near zero, \(z_{d1}\), \(z_{d2},\cdots,z_{dm}\) then \(1-\widetilde{G}_{pm}q\) should have zeros at \(z_{d1}\), \(z_{d2},\cdots,z_{dm}\). Figure 5: (a) The internal model control scheme; (b) The classical feedback control structure. Since the IMC controller \(q\) is designed as \(q=p_{m}^{-1}f\), the first condition is satisfied automatically.