## 5 Simulation Results
### Simulations Results for the FHS  
Table 10 summarizes the results obtained for the water tank controller using FHS algorithm and Table 11 shows the results for the inverted pendulum on a cart controller. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Water Tank Controller**} \\ \hline \multirow{2}{*}{**RMSE**} & **HS FLC without Noise** & **HS FLC with Noise** & **FHS** & **FHS** \\  & **Noise** & **Noise** & **FLC without Noise** & **FLC with Noise** \\ \hline Best & \(4.98\times 10^{-1}\) & \(1.01\times 10^{-2}\) & \(4.86\times 10^{-1}\) & \(2.53\times 10^{-2}\) \\ Worst & \(6.24\times 10^{-1}\) & \(6.99\times 10^{-1}\) & \(6.25\times 10^{-1}\) & \(2.01\times 10^{-1}\) \\ Average & \(5.66\times 10^{-1}\) & \(1.40\times 10^{-1}\) & \(5.54\times 10^{-1}\) & \(8.36\times 10^{-2}\) \\ Standard Deviation & \(3.26\times 10^{-2}\) & \(1.68\times 10^{-1}\) & \(2.92\times 10^{-2}\) & \(3.52\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Results obtained from the water tank controller by HS and FHS algorithms. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Inverted Pendulum on a Cart**} \\ \hline \multirow{2}{*}{**RMSE**} & **BCO** & **BCO** & **FBCO** & **FBCO** \\  & **FLC without Noise** & **FLC without Noise** & **FLC with Noise** \\ \hline Best & \(4.37\times 10^{-1}\) & \(3.78\times 10^{-1}\) & \(3.79\times 10^{-1}\) & \(3.80\times 10^{-1}\) \\ Worst & \(5.61\times 10^{-1}\) & \(8.64\times 10^{-1}\) & \(8.77\times 10^{-1}\) & \(9.47\times 10^{-1}\) \\ Average & \(5.07\times 10^{-2}\) & \(6.02\times 10^{-1}\) & \(8.21\times 10^{-2}\) & \(4.82\times 10^{-1}\) \\ Standard Deviation & \(3.23\times 10^{-1}\) & \(3.45\times 10^{-1}\) & \(4.66\times 10^{-1}\) & \(9.19\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results obtained from the inverted pendulum controller by BCO and FBCO algorithms. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Water Tank Controller**} \\ \hline \multirow{2}{*}{**RMSE**} & **De** & **DE FLC With** & **FDE** & **FDE** \\  & **FLC Without Noise** & **Noise** & **FLC Without Noise** & **FLC with Noise** \\ \hline Best & \(1.65\times 10^{-2}\) & \(1.08\times 10^{-1}\) & \(4.82\times 10^{-2}\) & \(1.27\times 10^{-3}\) \\ Worst & \(2.36\times 10^{-1}\) & \(4.87\times 10^{-1}\) & \(1.82\times 10^{-1}\) & \(2.32\times 10^{-1}\) \\ Average & \(9.37\times 10^{-2}\) & \(4.48\times 10^{-1}\) & \(9.44\times 10^{-2}\) & \(3.67\times 10^{-2}\) \\ Standard Deviation & \(4.81\times 10^{-2}\) & \(7.43\times 10^{-2}\) & \(3.34\times 10^{-2}\) & \(5.60\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results obtained from the water tank controller by DE and FDE algorithm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Inverted Pendulum on a Cart**} \\ \hline \multirow{2}{*}{**RMSE**} & **DE** & **DE FLC with** & **FDE FLC without Noise** & **FDE FLC with Noise** \\  & **FLC without Noise** & **Noise** & **Noise** \\ \hline Best & \(8.45\times 10^{-2}\) & \(2.96\times 10^{-1}\) & \(1.83\times 10^{-2}\) & \(1.40\times 10^{-2}\) \\ Worst & \(1.46\times 10^{0}\) & \(2.99\times 10^{-1}\) & \(1.40\times 10^{0}\) & \(7.92\times 10^{-1}\) \\ Average & \(8.99\times 10^{-1}\) & \(2.97\times 10^{-1}\) & \(2.15\times 10^{-1}\) & \(3.78\times 10^{-1}\) \\ Standard Deviation & \(4.15\times 10^{-1}\) & \(6.28\times 10^{-4}\) & \(2.75\times 10^{-1}\) & \(1.90\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Results obtained from the inverted pendulum controller by DE and FDE algorithms. Figure 17 shows that the DE found the better results compared to BCO and HS with the original and fuzzy algorithms for the water tank controller. Figure 18 shows that DE found the better results compared to BCO and HS with the original and fuzzy algorithms for the inverted pendulum on a cart controller.