# A Shadowed Type-2 Fuzzy Approach for Crossover Parameter Adaptation in Differential Evolution
## 5 Interval Type-2 Fuzzy Systems Controller
Figure 8 shows the minimum RMSE of the simulation of the 30 experiments performed with DEST2-FS without noise, where the value obtained was \(5.48\times 10^{-1}\). In Figure 6, when starting the simulation, it exceeds what is desired, but later it stabilizes in the second 0.12 to 0.14 and later it exceeds again from the second 0.14 to 0.2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Level of Confidence & 95\% \\ Alpha \(\alpha\) & 5\% \\ \(H_{4}\) & \(\mu_{1}<\mu_{2}\) \\ \(H_{o}\) & \(\mu_{1}\geq\mu_{2}\) \\ Critical Value & \(-1.645\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: \(Z\)-test parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **DE-ST2-FS** & **DE-ST2-FS** & **DE-ST2-FS** & **DE-ST2-FS** \\  & **without Noise** & **with Noise 0.5** & **with Noise 0.7** & **with Noise 0.9** \\ \hline Min. & \(5.48\times 10^{-1}\) & \(5.28\times 10^{-1}\) & \(9.38\times 10^{-2}\) & \(1.31\times 10^{-2}\) \\ \hline Max. & \(6.08\times 10^{-1}\) & \(6.01\times 10^{-1}\) & \(9.78\times 10^{-2}\) & \(7.63\times 10^{-2}\) \\ \hline Average & \(5.99\times 10^{-1}\) & \(5.63\times 10^{-1}\) & \(9.58\times 10^{-2}\) & \(5.92\times 10^{-2}\) \\ \hline Standard D. & \(1.23\times 10^{-2}\) & \(1.55\times 10^{-2}\) & \(6.22\times 10^{-4}\) & \(2.15\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: DE-ST2-FS fuzzy system. Finally, Figure 11 shows the RMSE of the simulation of the 30 experiments carried out with DE-ST2-FS with noise 0.9, where the value obtained was \(1.31\times 10^{-2}\). In the simulation obtained, it is observed that from second 0.1 the output of the plant has stability, but in between seconds 0.16 and 0.2 it is possible to stabilize. In order to demonstrate that better results are obtained when using DE-ST2-FS with a high noise level, a statistical test was performed. The statistic used in this work is the  
Figure 11: Best result DE-ST2-FS with noise 0.9. Figure 10: Best result DE-ST2-FS with noise 0.7. Figure 9: Best result DE-ST2-FS with noise 0.5. \(Z\)-test, the parameters used are shown in Table 4, and Equation (24) was used to calculate the \(Z\) value. \[Z=\frac{\left(\overline{X}_{1}-\overline{X}_{2}\right)-\left(\mu_{1}-\mu_{2} \right)}{\sigma_{\overline{X}_{1}-\overline{X}_{2}}} \tag{24}\]  
The objective of the statistical test is to show that DE-ST2-FS with noise 0.9 has a lower average than the other three variants of experimentation carried out: DE-ST2-FS without noise, DE-ST2-FS with noise 0.5, and DE-ST2-FS with noise 0.7. Three statistical tests were performed, where \(\mu_{1}\) is equal to DE-ST2-FS with noise 0.9 and \(\mu_{2}\) represents each of the statistical tests carried out. The results of the tests are outlined in Table 5.  
The results of the tests show that the greater the uncertainty, the better results are obtained. The three statistical tests carried out show that DE-ST2-FS with noise 0.9 has a lower average than the other three proposals carried out. Next, a comparison was made with other works found in the literature, in order to be able to validate the proposal to make CR dynamic in the DE and having an interval type-2 controller. We use as reference \"Shadowed Type-2 Fuzzy Systems for Dynamic Parameter Adaptation in Harmony Search and Differential Evolution\" [51]. In this previous work, two metaheuristics are used, where one of the parameters is dynamically adapted with a ST2-FS, and the difference with the reference is that the controller used is a T1-FS compared to ours, which is an IT2-FS. Table 6 exhibits the results from the literature and the results of the best experiments carried out in our work with ST2-FS with noise 0.9, reporting the averages and standard deviations. Using the data from Table 6, two statistical tests were performed to check if our proposal has a lower average than other methods in the literature. The previously shown \(Z\)-test was used, according to Equation (23), and the same parameters used in Table 4 were employed.