# Optimal Fractional PID Controller for Buck Converter Using Cohort Intelligent Algorithm
## 4 Results and Discussion
### Convergence Plots of the CI Algorithm  
The CI algorithm has a self-supervisory learning behavior for the candidates. This can be shown in the convergence plot of Figure 16, which shows the convergence of the candidates towards the required cost function. Similarly, the convergence of the FOPID parameters towards the final value is shown in Figure 17. The fractional controller parameters start converging to the solution after 5 iterations, and reaches optimal value in 21 iterations.  
The error and control signals of the CI optimized plant with FOPID controller for a reduction factor of 0.45 is shown in Figure 18.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{Parameters} & \multicolumn{6}{c}{\(\mathbf{J_{3}=ITAE}\)} \\ \cline{2-6}  & **CI** & **GA** & **PSO** & **ABC** & **SA** \\ \hline Avg. Overshoot \% & 20 & 29 & 26 & 18 & - \\ \hline Rise Time (ms) & 0.036 & 0.037 & 0.039 & 0.036 & - \\ \hline Settling time (ms) & 0.31 & 0.36 & 0.39 & 0.40 & - \\ \hline Cost function value & \(4\times 10^{-4}\) & \(3.44\times 10^{-4}\) & \(3.97\times 10^{-4}\) & \(2.65\times 10^{-5}\) & \(3.97\times 10^{-4}\) \\ \hline Avg. computation Time (s) & 1020 & 1500 & 1200 & 1500 & 1800 \\ \hline Avg. Function evaluations & 84 & 204 & 88 & 100 & 1000 \\ \hline Avg. No. of iterations & 21 & 51 & 21 & 25 & 200 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of different optimization methods for the cost function ITAE.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{Parameters} & \multicolumn{6}{c}{\(\mathbf{J_{4}=IAE}\)} \\ \cline{2-6}  & **CI** & **GA** & **PSO** & **ABC** & **SA** \\ \hline Avg. Overshoot \% & 20 & 21 & 25 & 18 & 25 \\ \hline Rise Time (ms) & 0.036 & 0.038 & 0.039 & 0.036 & 0.037 \\ \hline Settling time (ms) & 0.28 & 0.42 & 0.39 & 0.38 & 0.29 \\ \hline Cost function value & 0.0116 & 0.00548 & 0.0017 & 0.0024 & 0.005 \\ \hline Avg. computation Time (s) & 542 & 1500 & 1200 & 1500 & 1320 \\ \hline Avg. Function evaluations & 84 & 204 & 88 & 100 & 1000 \\ \hline Avg. No. of iterations & 21 & 51 & 21 & 25 & 200 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Comparison of different optimization methods for the cost function IAE.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{Parameters} & \multicolumn{6}{c}{\(\mathbf{J_{2}=ITSE}\)} \\ \cline{2-6}  & **CI** & **GA** & **PSO** & **ABC** & **SA** \\ \hline Avg. Overshoot \% & 9.2 & 16.11 & 5 & 17.3 & 25 \\ \hline Rise Time (ms) & 0.042 & 0.038 & 0.043 & 0.039 & 0.039 \\ \hline Settling time (ms) & 0.39 & 0.42 & 0.37 & 0.38 & 0.27 \\ \hline Cost function value & 0.0001 & 0.00018 & \(1.7\times 10^{-5}\) & \(1.7\times 10^{-5}\) & 0.00039 \\ \hline Avg. computation Time (s) & 762 & 1500 & 1200 & 1500 & 1320 \\ \hline Avg. Function evaluations & 84 & 204 & 88 & 100 & 1000 \\ \hline Avg. No. of iterations & 21 & 51 & 21 & 25 & 200 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of different optimization methods for the cost function ITSE.  
Figure 16: Convergence plot of cost function.  
Figure 17: Convergence plot of the FOPID parameters.  
Figure 18: _Cont._