# Analytical Design of Fractional-Order PI Controller for Parallel Cascade Control Systems
## 2 Materials and Methods
### FOPI Controller Design for the Primary Control Loop  
The block diagram of the system is reduced in Figure 4 where \(G_{p}\) and \(\widetilde{G}_{p}\) represent the equivalent process and the obtained transfer function from \(r^{\prime}{}_{2}\) to \(y_{1}\) in Figure 2, respectively. \(\widetilde{G}_{pm}\) is \(\widetilde{G}_{p}\) without the delay term. This structure follows the Smith predictor scheme to eliminate the sluggish response at the primary output when the primary process has a long delay time. The closed loop transfer functions for the outer loop (from \(r_{1}\) to \(y_{1}\))  
\[\frac{\gamma_{1}}{R_{1}}=\frac{G_{c_{1}}(s)G_{p}(s)}{1+G_{c_{1}}(s)G_{pm}(s)} \tag{25}\]  
From Equation (25), it is obvious that the characteristic equation of the closed-loop system is the delay-free function. In order to design the primary controller, the equivalent process \(\widetilde{G}_{p}\) first has to be obtained. In this work, depending on the kinds of secondary process transfer functions, several methods are proposed to derive \(\widetilde{G}_{p}\) in the form of FOPTD. Case 1: The primary process is FOPTD  
The transfer function of the primary process is as Equation (16). In this case, the primary transfer function is also stable, so there is no need to use the stabilized controller, \(G_{d}=0\), and from Equation (14):  
\[\widetilde{G}_{p}(s)=G_{p_{1}}(s)=\frac{K_{1}e^{-\theta_{1}s}}{\tau_{1}s+1}= \frac{K_{p}e^{-\theta_{p}}}{\tau_{p}s+1} \tag{26}\]  
Case 2: The primary process is UFOPTD  
\[G_{p_{1}}(s)=\frac{K_{1}e^{-\theta_{1}s}}{\tau_{1}s-1} \tag{27}\]  
In this case, the primary process is unstable, hence, the stabilized controller is suggested as follows:  
\[G_{d}(s)=K_{d}(1+\tau_{d}s) \tag{28}\]  
The equivalent transfer function is obtained:  
\[\widetilde{G}_{p}(s)=\frac{G_{p_{1}}(s)}{1+G_{d}G_{p_{1}}(s)}=\frac{K_{1}e^{- \theta_{1}s}}{\tau_{1}s-1+K_{d}K_{1}(1+\tau_{d}s)e^{-\theta_{1}s}} \tag{29}\]  
Using Pade 1/1 approximation: \(e^{-\theta_{1}s}\approx\frac{1-0.56\theta_{1}s}{1+0.56\theta_{1}s}\) and choosing  
\[\tau_{d}=0.5\theta_{1} \tag{30}\]  
Figure 4: The control structure design of the primary control loop. \[\widetilde{G}_{p}(s)=\frac{K_{1}e^{-\theta_{1}s}}{\tau_{1}s-1+K_{d}K_{1}(1-0.5 \theta_{1}s)}=\frac{K_{1}e^{-\theta_{1}s}}{(\tau_{1}-0.5K_{d}K_{1}\theta_{1})s+K _{d}K_{1}-1}=\frac{K_{p}e^{-\theta_{p}}}{\tau_{p}s+1} \tag{31}\]  
where \(\theta_{p}=\theta_{1}\), \(K_{p}=\frac{K_{1}}{K_{d}K_{1}-1}\) and \(\tau_{p}=\frac{\tau_{1}-0.5K_{d}K_{1}\theta_{1}}{K_{d}K_{1}-1}\). To guarantee the stability criterion, two parameters \(K_{p}\) and \(\tau_{p}\) have to be greater than zero. Therefore, the parameter \(K_{d}\) has to meet the following requirement:  
\[\frac{1}{K_{1}}<K_{d}<\frac{\tau_{1}}{0.5K_{1}\theta_{1}} \tag{32}\]  
Case 3: The primary process is SOIPTD  
\[G_{p_{1}}(s)=\frac{K_{1}e^{-\theta_{1}s}}{s(\tau_{1}s+1)} \tag{33}\]  
The stabilized controller as in the previous case, Equation (28), is also needed in this case; therefore, the equivalent transfer function is derived as follows:  
\[\widetilde{G}_{p}(s)=G_{p_{1}}(s)=\frac{K_{1}e^{-\theta_{1}s}}{\tau_{1}s+1}= \frac{K_{p}e^{-\theta_{p}}}{\tau_{p}s+1} \tag{26}\]0  
Similar to case 2, using Equation (30) to replace into Equation (34):  
\[\widetilde{G}_{p}(s)=G_{p_{1}}(s)=\frac{K_{1}e^{-\theta_{1}s}}{\tau_{1}s+1}= \frac{K_{p}e^{-\theta_{p}}}{\tau_{p}s+1} \tag{26}\]1  
The condition of \(K_{d}\) is obtained:  
\[\widetilde{G}_{p}(s)=G_{p_{1}}(s)=\frac{K_{1}e^{-\theta_{1}s}}{\tau_{1}s+1}= \frac{K_{p}e^{-\theta_{p}}}{\tau_{p}s+1} \tag{26}\]2  
Moreover, in this case, \(\widetilde{G}_{p}\) is the second-order transfer function. To ensure a servo response of the closed-loop system, as well as to simplify the design problem, the system has to be the overdamped one which the damping factor has not to be less than 1.