## 5 Interval Type-2 Controller of Motor Angular Position
\[\text{MSE}=\frac{1}{\text{n}}\sum_{\text{i=1}}^{\text{n}}\Bigl{(}\overline{Y}_ {\text{i}}-Y_{\text{i}}\Bigr{)}^{2} \tag{17}\]  
\[\text{RMSE}=\sqrt{\frac{1}{\text{N}}\sum_{\text{t=1}}^{\text{N}}(x_{\text{t }}-\hat{x}_{\text{t}})^{2}} \tag{18}\]  
\[\text{ITAE}=\sum_{\text{t=1}}^{\text{N}}t(x(t)-\hat{x}(t)) \tag{19}\]  
\[\text{IAE}=\sum_{\text{t=1}}^{\text{N}}\lvert x(t)-\hat{x}(t)\rvert \tag{20}\]  
\[\text{ITSE}=\sum_{\text{t=1}}^{\text{N}}t\Bigl{(}\Bigl{(}x(t)-\hat{x}(t))^{2} \Bigr{)} \tag{21}\]  
\[\text{ISE}=\sum_{\text{t=1}}^{\text{N}}\Bigl{\lvert}\Bigl{(}x(t)-\hat{x}(t))^{ 2}\Bigr{\rvert} \tag{22}\]  
The results of the performance index obtained from the 30 simulations are outlined in Table 5. This table displays the values obtained without noise and with noise for the proposed controller. It can be noted that adding uncertainty to the controller significantly improves its performance. Tables 6-8 present the performance index obtained by varying the URN noise level in the proposed controller. The best result is achieved when a noise level of 0.01 is applied to the controller. As the noise level increases, the RMSE results follow the same order. Therefore, introducing uncertainty to the controller enhances its performance. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Performance Index** & **0.01 URN** & **0.02 URN** & **0.03 URN** & **0.04 URN** \\ \hline MSE & \(6.73\times 10^{-1}\) & \(6.69\times 10^{-1}\) & \(6.61\times 10^{-1}\) & \(6.52\times 10^{-1}\) \\ RMSE & \(2.88\times 10^{-2}\) & \(3.10\times 10^{-2}\) & \(3.07\times 10^{-2}\) & \(3.35\times 10^{-2}\) \\ ITAE & \(2.01\times 10^{-2}\) & \(2.01\times 10^{-2}\) & \(2.00\times 10^{-2}\) & \(1.98\times 10^{-2}\) \\ IAE & \(1.74\times 10^{-1}\) & \(1.73\times 10^{-1}\) & \(1.72\times 10^{-1}\) & \(1.71\times 10^{-1}\) \\ ITSE & \(2.07\times 10^{-2}\) & \(2.05\times 10^{-2}\) & \(2.03\times 10^{-2}\) & \(2.01\times 10^{-2}\) \\ ISE & \(1.72\times 10^{-1}\) & \(1.72\times 10^{-1}\) & \(1.70\times 10^{-1}\) & \(1.71\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results with 0.01, 0.02, 0.03, and 0.04 URN noise levels in the IT2 FLC controller. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Performance Index** & **0.05 URN** & **0.06 URN** & **0.07 URN** & **0.08 URN** \\ \hline MSE & \(6.28\times 10^{-1}\) & \(6.33\times 10^{-1}\) & \(6.26\times 10^{-1}\) & \(6.18\times 10^{-1}\) \\ RMSE & \(3.46\times 10^{-2}\) & \(3.57\times 10^{-2}\) & \(3.46\times 10^{-2}\) & \(3.41\times 10^{-2}\) \\ ITAE & \(1.95\times 10^{-2}\) & \(1.96\times 10^{-2}\) & \(1.95\times 10^{-2}\) & \(1.93\times 10^{-2}\) \\ IAE & \(1.68\times 10^{-1}\) & \(1.69\times 10^{-1}\) & \(1.68\times 10^{-1}\) & \(1.67\times 10^{-1}\) \\ ITSE & \(1.95\times 10^{-2}\) & \(1.96\times 10^{-2}\) & \(1.94\times 10^{-2}\) & \(1.91\times 10^{-2}\) \\ ISE & \(1.62\times 10^{-1}\) & \(1.63\times 10^{-1}\) & \(1.62\times 10^{-1}\) & \(1.60\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results with 0.05, 0.06, 0.07, and 0.08 URN noise levels in the IT2 FLC controller. \begin{table}
\begin{tabular}{c c} \hline \hline
**Performance Index** & **0.09 URN** \\ \hline MSE & \(6.16\times 10^{-1}\) \\ RMSE & \(3.42\times 10^{-2}\) \\ ITAE & \(1.93\times 10^{-2}\) \\ IAE & \(1.67\times 10^{-1}\) \\ ITSE & \(1.90\times 10^{-2}\) \\ ISE & \(1.59\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results with 0.09 URN noise level in the IT2 FLC controller. \begin{table}
\begin{tabular}{c c} \hline \hline
**Level of URN Noise** & **Time (Seconds)** \\ \hline Without URN noise & 640.50 \\
0.01 & 661.12 \\
0.02 & 644.44 \\
0.03 & 650.40 \\
0.04 & 645.90 \\
0.05 & 649.50 \\
0.06 & 645.00 \\
0.07 & 429.20 \\
0.08 & 647.40 \\
0.09 & 659.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Experiment time with each URN noise level in the IT2 FLC controller.