# Design and Optimization of a Neuro-Fuzzy System for the Control of an Electromechanical Plant
## 7 Results
On the other hand, Table 3 presents the values obtained for \(J_{T}\) in the initial and the final iteration (30 iterations) when performing the optimization process. These results show that the value of the objective function \(J_{T}\) increases when increasing the value of \(R\). In addition, Table 3 shows that linear and fuzzy controls obtain similar values of \(J_{T}\). These results highlight that the design and choice of the initial configuration of the fuzzy controller were satisfactory since it presents a similar result to the linear controller. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{Fuzzy controller parameters for QR1 case} \\ \hline \multicolumn{1}{c}{\(v_{ij}\)} & \multicolumn{2}{c}{\(\rho_{ij}\)} & \multicolumn{2}{c}{\(\chi_{ij}\)} \\ \hline \(-1.3358\times 10^{4}\) & \(1.3358\times 10^{4}\) & \(-0.0101\) & \(0.0101\) & \(-0.3382\times 10^{-5}\) & \(-0.3382\times 10^{-5}\) \\ \hline \(1.3334\times 10^{4}\) & \(-1.3334\times 10^{4}\) & \(-0.0099\) & \(0.0099\) & \(0.3446\times 10^{-5}\) & \(0.3446\times 10^{-5}\) \\ \hline \(-0.0200\times 10^{4}\) & \(0.0200\times 10^{4}\) & \(-0.0100\) & \(0.0100\) & \(0\) & \(0\) \\ \hline \multicolumn{5}{c}{Fuzzy controller parameters for QR2 case} \\ \hline \multicolumn{1}{c}{\(v_{ij}\)} & \multicolumn{2}{c}{\(\rho_{ij}\)} & \multicolumn{2}{c}{\(\chi_{ij}\)} \\ \hline \(-9.6960\times 10^{3}\) & \(9.6960\times 10^{3}\) & \(-0.0100\) & \(0.0100\) & \(-0.1066\times 10^{-5}\) & \(-0.1066\times 10^{-5}\) \\ \hline \(9.6780\times 10^{3}\) & \(-9.6780\times 10^{3}\) & \(-0.0100\) & \(0.0100\) & \(0.1064\times 10^{-5}\) & \(0.1064\times 10^{-5}\) \\ \hline \(-0.2000\times 10^{3}\) & \(0.2000\times 10^{3}\) & \(-0.0100\) & \(0.0100\) & \(0\) & \(0\) \\ \hline \multicolumn{5}{c}{Fuzzy controller parameters for QR3 case} \\ \hline \multicolumn{1}{c}{\(v_{ij}\)} & \multicolumn{2}{c}{\(\rho_{ij}\)} & \multicolumn{2}{c}{\(\chi_{ij}\)} \\ \hline \(-4.5320\times 10^{3}\) & \(4.5320\times 10^{3}\) & \(-0.0100\) & \(0.0100\) & \(0.1086\times 10^{-6}\) & \(0.1086\times 10^{-6}\) \\ \hline \(4.5240\  
Regarding the statistical tests application [39], the Kruskal-Wallis test is used to determine statistically significant differences between two or more experimental groups, obtaining a \(p\) value of \(3.7814\times 10^{-7}\), which indicates differences, then it the Bonferroni procedure is employed to perform multiple comparisons, the results are presented in Figure 19, where we observe no significant difference between the linear and fuzzy implementations for the \(Q\) and \(R\) configurations. We also observed a significant difference between cases QR1 and QR4. Meanwhile, Figure 20 shows the system behavior for \(\theta(t)\) when there are disturbances \(p(t)\). This figure also displays the responses for different configurations in \(Q\) and \(R\). It can be observed that the disturbance effect is more noticeable for the controllers that employ a small signal control, as in the cases QR3 and QR4 (according to Figures 17 and 18). In Figure 21, the behavior comparison of the linear and fuzzy controllers is presented considering different perturbation values. For the disturbance values considered (Figure 18), some fuzzy controller configurations decrease the disturbance rejection capacity. In order to statistically observe the performance of the linear and fuzzy controllers for disturbance rejection (considering different values of \(Q\) and \(R\)) 10 simulations were performed with randomly generated \(p(t)\) disturbance values (six variations in each simulation). Results of \(I_{T}\) for 10 simulations are shown in Table 6, in this way, Table 7 displays the mean, standard deviation, and maximum and minimum values. Figure 19: Multiple comparison using Bonferroni procedure. Figure 20: Response of the system for different disturbance values. Considering Tables 6 and 7, to determine statistically significant differences among the groups, we used the Kruskal-Wallis test, having a \(p\)-value of \(2.8463\times 10^{-13}\), which indicates differences.