# Analytical Design of Fractional-Order PI Controller for Parallel Cascade Control Systems
## 3 Results
### Example 3  
In this example, the process models include a SOIPTD as a primary process and a stable FOPTD secondary process, Equation (57). These models were also investigated by several researchers [3; 8; 9].  
\[G_{p_{1}}(s)=G_{d_{1}}(s)=\frac{e^{-6.5672s}}{s(3.4945s+1)} \tag{57}\] \[G_{p_{2}}(s)=G_{d_{2}}(s)=\frac{2e^{-2s}}{s+1}\]  
The secondary controller is obtained similarly to two previous examples:  
\[G_{c2}(s)=\frac{s+1}{2(0.6s+1)} \tag{58}\]  
The primary controller, in this example, is derived according to Algorithm 2. \(K_{d}\) is chosen from the range of Equations (36) and (37), and in this example \(K_{d}=0.02\). From Equation (35) the equivalent transfer function \(\widetilde{G}_{p}\) is obtained:  
\[\widetilde{G}_{p}=\frac{e^{-6.5672s}}{3.494s^{2}+0.9343s+0.02} \tag{59}\]  
Figure 12: The perturbed primary closed-loop responses for example 2.  
Figure 13: The control signals for example 2 in the perturbed case.  
Using the PSO algorithm proposed in [29] to reduce \(\widetilde{G}_{p}\) to the FOPTD. Figure 14 illustrates the unit step response of the original Equation (59) and the approximated transfer function Equation (60). It can be seen that the reduced-order transfer function approximates well to the second-order one, and it could be adopted for the next step of the design procedure.  
\[\widetilde{G}_{p}\simeq\frac{50.237e^{-6.5672s}}{47.5566s+1} \tag{60}\]  
The control parameters are obtained and tabulated in Table 5. In this example, the proposed method is compared with other methods suggested by [PERSON] et al. [3] (2016) and [8] (2017). Figures 15 and 16 show the nominal closed-loop responses as well as the corresponding control signals. The proposed method yields a fast response accompanied by an overshoot compared to other methods. However, the methods proposed by [PERSON] have an issue in the case of the regulator problem in which the steady-state has a deviation after a load disturbance. In contrast, the proposed tuning  
In order to evaluate the robustness of the proposed controller to uncertainties in process dynamics, the gains and time delays of \(G_{p_{1}},G_{p_{2}},G_{d_{1}}\) and \(G_{d_{2}}\) are perturbed by \(+20\%\) whereas the time constants of those transfer functions are changed by \(-20\%\). Figures 17 and 18 show the process responses and control signals respectively in the perturbed case and prove that the proposed controller still keeps robust stability and performs a good steady-state response due to the load disturbance. The performance indexes in the perturbed case listed in Table 6 state the effectiveness of the proposed settings.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Nominal**} & \multicolumn{3}{c}{**Perturbed (\(\pm 20\%\))**} \\ \cline{2-7}  & **IAE** & **ISE** & **TV** & **IAE** & **ISE** & **TV** \\ \hline Proposed & 17.9520 & 7.5790 & 0.1845 & 18.5801 & 7.9852 & 0.2038 \\ Raja (2016) & 39.9687 & 21.848 & 0.0819 & 39.8605 & 22.166 & 0.0851 \\ [PERSON] (2017) & 17.9977 & 8.1274 & 0.1591 & 17.9531 & 8.3991 & 0.1703 \\ \hline \hline \end{tabular}
\end{table}
Table 6: The performance index in nominal and perturbed models for example 3.  
Figure 16: The control signals for example 3 in the nominal case.  
Figure 17: The perturbed primary closed-loop responses for example 3.