## 7 Test System
### Design of FOPID Controller  
An SCA-based FOPID controller is proposed for the first order interval system, (46), by considering the frequency domain specifications. In order to establish the efficacy of the SCA-based FOPID controller, other algorithms, namely particle swarm optimization (PSO) [61], Nelder-Mead simplex (NMS) [4] and Luus-Jaakola (LJ) [62; 63] are also used to obtain the parameters of \(PI^{\alpha}D^{\beta}\) controller. A total of 50 candidates are considered in whole population with 100 iterations for all algorithms. Figure 8: Step response of system \(G_{3}(s)\) and model \(\widehat{G_{1}}\left(s\right)\). The key feature of the SCA algorithm [61; 64; 65; 66] is that it is free from algorithm-specific parameters. The algorithm-specific parameters of all other algorithms are listed in Table 3.  
In order to design the FOPID controller, different transfer functions are constructed as shown in Section 4.1. Using (16) and (18), the system transfer functions corresponding to maximum plots of gain and phase of the Bode envelope given as  
\[\widehat{G}_{1b}\ (s)=\frac{15.382}{85.175s+35.06} \tag{47}\]  
\[\widehat{G}_{1d}\ (s)=\frac{13.638}{86.054s+35.06} \tag{48}\]  
The design criteria of the controlled system are considered as: gain crossover frequency (\(\omega_{gc}\))= 50 rad/sec and phase margin (\(\phi_{pm}\))= 85 deg. Figure 9 represents the block diagram of ROCIP with an SCA-based FOPID controller. To obtain the desired specifications, (21) and (22) are solved using SCA. The optimal parameters (\(K_{a}\), \(K_{b}\), \(K_{c}\), \(\alpha\) and \(\beta\)) of \(PI^{\alpha}D^{\beta}\) controller obtained are  
\[K_{a}=324.565,K_{b}=658.081,K_{c}=204.214,\] \[\alpha=0.616,\beta=0.992 \tag{49}\]  
The FOPID controller for the specified design criteria is given as  
\[\overset{\wedge}{C}(s)=324.565+\frac{658.081}{s^{0.616}}+204.214s^{0.992} \tag{50}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & **Parameters** & **Values** \\ \hline \multirow{3}{*}{PSO} & Cognitive parameter, \(c_{1}\) & 2 \\  & Social parameter, \(c_{2}\) & 2 \\  & Inertia weight, \(w\) & 0.5 \\ \hline \multirow{3}{*}{NMS} & Refection coefficient, \(\alpha\) & 1 \\  & Contraction coefficient, \(\beta\) & 1/2 \\  & Expansion coefficient, \(\gamma\) & 2 \\  & Shrinking coefficient, \(\delta\) & 1/2 \\ \hline \hline \multicolumn{2}{c}{LJ} & Contraction factor, \(\gamma\) & 0.95 \\ \hline \multicolumn{2}{c}{SCA} & No parameter & – \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Algorithm-specific parameters for different algorithms. Figure 9: ROCIP with an SCA-based FOPID controller. The FOPID controllers for the above defined specifications obtained due to PSO, NMS and LJ are given as  
\[\overset{\wedge}{C}_{PSO}\left(s\right)=245.496+\frac{335.474}{s^{0.457}}+443.675s ^{0.845} \tag{51}\]  
\[\overset{\wedge}{C}_{NMS}\left(s\right)=832.765+\frac{147.164}{s^{0.329}}+379.85 2s^{0.693} \tag{52}\]  
\[\overset{\wedge}{C}_{LJ}\left(s\right)=473.359+\frac{259.876}{s^{0.478}}+156.95 2s^{0.573} \tag{53}\]  
Figure 10 presents the unit step response of closed-loop control system of Figures 4 and 5 having ROCIP and HOCIP, respectively. From Figure 10, it is clearly observed that the designed FOPID controller \(\overset{\wedge}{C}(s)\) using SCA for ROCIP, \(\overset{\frown}{G}_{1}\left(s\right)\), is also suited for HOCIP, \(G_{3}(s)\). This establishes the usefulness of proposed technique. Figure 11 depicts the Bode plots of the open-loop controlled systems with FOPID controller. It is clear from the Figure 11 that frequency response of the SCA-based FOPID controller is also excellent for both ROCIP and HOCIP. Figures 12 and 13 plots the step responses of ROCIP and HOCIP with FOPID controllers obtained due to SCA, PSO, NMS and LJ. However, Figures 14 and 15 are showing the Bode plots. It is clearly observed that the step responses of the closed-loop system with an SCA-based FOPID controller is better when compared to other optimization algorithms. The same is true for frequency response plotted in Figures 14 and 15.