# 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
The second condition can be fulfilled by designing the IMC filter \(f\) as:  
\[f=\frac{\beta s+1}{\left(\lambda_{1}s+1\right)^{2}} \tag{43}\]  
where \(\lambda_{1}\) is an adjustable parameter which controls the tradeoff between the performance and robustness; \(\beta\) is an extra degree of freedom to cancel the poles near zero in \(G_{d_{1}}\), and determined via the following equation:  
\[\left(1-\widetilde{G}_{pm}q\right)\Big{|}_{s=-1/\tau_{p}}=\left.\left(1-\frac{ \beta s+1}{\left(\lambda_{1}s+1\right)^{2}}\right)\right|_{s=-1/\tau_{p}}=0 \tag{44}\]  
Thus, the value of \(\beta\) is obtained as:  
\[\beta=2\lambda_{1}-\frac{\lambda_{1}^{2}}{\tau_{p}} \tag{45}\]  
Then, the IMC controller is expressed as follows:  
\[q=p_{m}^{-1}f=\frac{\tau_{p}s+1}{K_{p}}\frac{\beta s+1}{\left(\lambda_{1}s+1 \right)^{2}} \tag{46}\]  
From Figure 5a,b, the ideal feedback controller can be derived in terms of the internal model \(\widetilde{G}_{pm}\) and the IMC controller \(q\):  
\[G_{c_{1}}=\frac{q}{1-\widetilde{G}_{pm}q} \tag{47}\]  
Substituting Equation (46) into Equation (47), the ideal feedback controller is finally obtained:  
\[G_{c_{1}}=\frac{\left(\tau_{p}s+1\right)\left(\beta s+1\right)}{K_{p}\Big{[} \left(\lambda_{1}s+1\right)^{2}-\left(\beta s+1\right)\Big{]}} \tag{48}\]  
It is obvious that the resulting controller, Equation (48), does not have the FOPI-type controller form as in Equation (11) and also not in the frequency domain. Therefore, it is necessary to convert it into the frequency domain by replacing \(s=j\omega\) and rearranging it into the complex form. After several calculations, the final result is as follows:  
\[ \begin{array}{l}G_{c_{1}}(j\omega)=\frac{1}{K_{p}}\frac{\left(1+j\omega\tau_ {p}\right)\left(1+j\omega\beta\right)}{\left(1+j\lambda_{1}\omega\right)^{2} -\left(1+j\omega\beta\right)}\\ =\frac{\left\{-\lambda_{1}^{2}\omega^{2}\left(1-\omega^{2}\tau_{p}\beta \right)+\omega^{2}\left(\tau_{p}+\beta\right)\left(2\lambda_{1}-\beta\right) \right\}}{K_{p}\left[\lambda_{1}\omega^{4}+\omega^{2}\left(2\lambda_{1}-\beta \right)^{2}\right]}-j\frac{\left\{\omega\left(1-\omega^{2}\tau_{p}\beta \right)\left(2\lambda_{1}-\beta\right)+\omega^{3}\left(\tau_{p}+\beta\right) \lambda_{1}{}^{2}\right\}}{K_{p}\left[\lambda_{1}\omega^{4}+\omega^{2}\left(2 \lambda_{1}-\beta\right)^{2}\right]}\end{array}  \tag{49}\]  
By comparing Equation (49) and Equation (11), the analytical tuning rules of the FOPI controller can be obtained as follows:  
\[K_{l_{1}}=\frac{1}{K_{p}}\frac{\omega^{\lambda}\big{\{}\omega\big{(}1-\omega^ {2}\tau_{p}\beta\big{)}(2\lambda_{1}-\beta)+\omega^{3}\big{(}\tau_{p}+\beta \big{)}\lambda_{1}{}^{2}\big{\}}}{\sin\gamma_{I}\Big{[}\lambda_{1}\omega^{4} \omega^{4}+\omega^{2}(2\lambda_{1}-\beta)^{2}\Big{]}} \tag{50}\]  
\[K_{c_{1}}=\frac{\left\{-\lambda_{1}{}^{2}\omega^{2}\big{(}1-\omega^{2}\tau_{p} \beta\big{)}+\omega^{2}\big{(}\tau_{p}+\beta\big{)}(2\lambda_{1}-\beta)\right\} }{K_{p}\Big{[}\lambda_{1}\omega^{4}\omega^{4}+\omega^{2}(2\lambda_{1}-\beta) ^{2}\Big{]}}-\frac{K_{I}\cos\gamma_{I}}{\omega^{\lambda}} \tag{51}\]  
From these above equations, two tuning algorithms, Algorithms 1 and 2, for the primary controllers are proposed for case 2, and 3. It is obvious that the control parameters of case 1 are easily obtained. ```
1: Initialization  Calculate \([K_{d\min},K_{d\max}]\) according to Equations (32) and (41) \(K_{d}=K_{d\min}\); \(\Delta K_{d}=0.01\); assign the value of \(M_{s\_desired}\)
2:while\(|M_{s}-M_{s\_desired}|>0.01\)do
3: Compute \(K_{p_{1}},\,r_{p_{1}}\) according to Equation (31)
4: Determine \(\lambda_{1}\) based on Equation (40)
5: Compute \(K_{c_{1}},\,K_{1}\) according to Equations (50) and (51)
6: Calculate \(M_{s}\) for each set of control parameters
7: Increase the value of \(K_{d}\) by \(\Delta K_{d}\): \(K_{d}=K_{d\min}+\Delta K_{d}\)
8:If (\(K_{d}>K_{d\max}\)): assign \(M_{s}=M_{s\_desired}\) % avoid an infinite loop
9:endwhile
10:end
```  
**Algorithm 1**The proposed tuning algorithm for case 2.