# Modified Grey Wolf Optimizer and Application in Parameter Optimization of PI Controller
## 5 Conclusions
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Function** & **Dim** & **Index** & **M-GWO** & **IGWO** & **GWO** & **WOA** & **DBO** & **HHO** \\ \hline F21 & 30 & Min & \(2.35\times 10^{3}\) & \(2.37\times 10^{3}\) & \(2.41\times 10^{3}\) & \(2.41\times 10^{3}\) & \(2.42\times 10^{3}\) & \(2.23\times 10^{3}\) \\  & & Mean & \(2.40\times 10^{3}\) & \(2.43\times 10^{3}\) & \(2.47\times 10^{3}\) & \(2.51\times 10^{3}\) & \(2.48\times 10^{3}\) & \(2.57\times 10^{3}\) \\  & & SD & \(2.88\times 10^{1}\) & \(3.23\times 10^{1}\) & \(3.81\times 10^{1}\) & \(6.71\times 10^{1}\) & \(2.92\times 10^{1}\) & \(8.64\times 10^{1}\) \\  & & Min & \(2.47\times 10^{3}\) & \(2.53\times 10^{3}\) & \(2.46\times 10^{3}\) & \(2.57\times 10^{3}\) & \(2.71\times 10^{3}\) & \(2.73\times 10^{3}\) \\  & & Mean & \(2.54\times 10^{3}\) & \(2.64\times 10^{1}\) & \(2.57\times 10^{3}\) & \(2.27\times 10^{3}\) & \(2.82\times 10^{3}\) & \(2.87\times 10^{3}\) \\  & & SD & \(5.61\times 10^{1}\) & \(6.02\times 10^{1}\) & \(7.51\times 10^{1}\) & \(1.08\times 10^{2}\) & \(3.41\times 10^{1}\) & \(7.49\times 10^{1}\) \\ F22 & 30 & Min & \(2.30\times 10^{3}\) & \(2.30\times 10^{3}\) & \(2.39\times 10^{3}\) & \(2.30\times 10^{3}\) & \(3.28\times 10^{3}\) & \(2.33\times 10^{3}\) \\  & & Mean & \(3.83\times 10^{3}\) & \(4.40\times 10^{3}\) & \(4.56\times 10^{3}\) & \(5.55\times 10^{3}\) & \(4.12\times 10^{3}\) & \(6.76\times 10^{3}\) \\  & & SD & \(2.08\times 10^{3}\) & \(1.89\times 10^{3}\) & \(1.84\times 10^{3}\) & \(2.22\times 10^{3}\) & \(2.78\times 10^{2}\) & \(1.77\times 10^{3}\) \\  & & Min & \(7.53\times 10^{3}\) & \(8.53\times 10^{3}\) & \(7.45\times 10^{3}\) & \(8.18\times 10^{3}\) & \(1.03\times 10^{4}\) & \(9.96\times 10^{3}\) \\  & & Mean & \(9.24\times 10^{3}\) & \(9.76\times 10^{3}\) & \(9.81\times 10^{3}\) & \(1.02\times 10^{4}\) & \(1.51\times 10^{4}\) & \(1.19\times 10^{4}\) \\  & & SD & \(1.05\times 10^{3}\) & \(8.78\times 10^{2}\) & \(2.05\times 10^{3}\) & \(9.64\times 10^{2}\) & \(6.33\times 10^{2}\) & \(1.01\times 10^{3}\) \\ F23 & 30 & Min & \(2.71\times 10^{3}\) & \(2.77\times 10^{3}\) & \(2.78\times 10^{3}\) & \(2.87\times 10^{3}\) & \(3.08\times 10^{3}\) \\  & & Mean & \(2.78\times 10^{3}\) & \(2.86\times 10^{3}\) & \(2.93\times 10^{3}\) & \(2.92\times 10^{3}\) & \(2.96\times 10^{3}\) & \(3.21\times 10^{3}\) \\  & & SD & \(4.87\times 10^{1}\) & \(6.20\times 10^{1}\) & \(8.75\times 10^{1}\) & \(9.08\times 10^{1}\) & \(2.63\times 10^{1}\) & \(1.21\times 10^{2}\) \\  & & Min & \(2.94\times 10^{3}\) & \(2.92\times 10^{3}\) & \(2.96\times 10^{3}\) & \(3.08\times 10^{3}\) & \(3.32\times 10^{3}\) & \(3.51\times 10^{3}\) \\  & & Mean & \(3.03\times 10^{3}\) & \(3.25\times 10^{3}\) & \(3.31\times 10^{3}\) & \(3.38\times 10^{3}\) & \(3.40\times 10^{3}\) & \(3.79\times 10^{3}\) \\  & & SD & \(8.21\times 10^{1}\) & \(1.52\times 10^{2}\) & \(1.49\times 10^{2}\) & \(1.62\times 10^{2}\) & \(3.22\times 10^{1}\) & \(1.52\times 10^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The results of the M-GWO and competitor algorithms on composite functions. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Function** & **Dim** & **M-GWO** & **IGWO** & **GWO** & **WOA** & **DBO** & **HHO** \\ \hline F1 & 30 & 1.207 s & 1.337 s & 1.115 s & 1.357 s & 1.088 s & 1.447 s \\  & 50 & 1.886 s & 2.016 s & 1.769 s & 2.224 s & 1.629 s & 2.301 s \\ F3 & 30 & 1.124 s & 1.284 s & 1.054 s & 1.292 s & 1.044 s & 1.331 s \\  & 50 & 1.771 s & 1.904 s & 1.692 s & 1.885 s & 1.706 s & 1.928 s \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The average time required for different algorithms to perform a single optimization on CEC2017.