# Design and Optimization of a Neuro-Fuzzy System for the Control of an Electromechanical Plant
## 8 Discussion  
As seen, for linear controller the algorithm locates the zero near the pole in such a way that the pole and zero \"almost cancel\" occurs, which is a PI controller design method as presented in [34]. Frequently, it is possible to create a design by placing the zero of the controllers near the pole of origin in a way that both pole and zero of the system in closed loop \"almost cancel\" each other. Considering [33; 34], a pair pole-zero that almost cancels has an insignificant effect on the time response; thus, the transitory response of the closed-loop system with the controller is approximately the same as in the closed-loop system without the controller. As a result, PI controller offers little deterioration in the transitory response with a significant improvement in steady state error. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Simulation** & \multicolumn{2}{c}{**QR1**} & \multicolumn{2}{c}{**QR2**} & \multicolumn{2}{c}{**QR3**} & \multicolumn{2}{c}{**QR4**} \\ \hline
**Run** & **Linear** & **Fuzzy** & **Linear** & **Fuzzy** & **Linear** & **Fuzzy** & **Linear** & **Fuzzy** \\ \hline
1 & 0.0025 & 0.0024 & 0.0038 & 0.0036 & 0.0119 & 0.0116 & 0.0208 & 0.0213 \\ \hline
2 & 0.0024 & 0.0024 & 0.0036 & 0.0035 & 0.0109 & 0.0107 & 0.0189 & 0.0190 \\ \hline
3 & 0.0023 & 0.0023 & 0.0034 & 0.0033 & 0.0101 & 0.0100 & 0.0172 & 0.0178 \\ \hline
4 & 0.0027 & 0.0028 & 0.0043 & 0.0044 & 0.0147 & 0.0153 & 0.0264 & 0.0289 \\ \hline
5 & 0.0023 & 0.0023 & 0.0034 & 0.0034 & 0.0103 & 0.0102 & 0.0176 & 0.0180 \\ \hline
6 & 0.0023 & 0.0023 & 0.0034 & 0.0033 & 0.0099 & 0.0099 & 0.0168 & 0.0178 \\ \hline
7 & 0.0020 & 0.0020 & 0.0029 & 0.0028 & 0.0075 & 0.0072 & 0.0120 & 0.0120 \\ \hline
8 & 0.0024 & 0.0024 & 0.0036 & 0.0036 & 0.0111 & 0.0114 & 0.0192 & 0.0205 \\ \hline
9 & 0.0022 & 0.0022 & 0.0031 & 0.0031 & 0.0088 & 0.0086 & 0.0145 & 0.0147 \\ \hline
10 & 0.0022 & 0.0022 & 0.0032 & 0.0031 & 0.0090 & 0.0090 & 0.0149 & 0.0157 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Disturbance rejection results of \(J_{T}\) for 10 simulations. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Numerical** & \multicolumn{2}{c}{**QR1**} & \multicolumn{2}{c}{**QR2**} & \multicolumn{2}{c}{**QR3**} & \multicolumn{2}{c}{**QR4**} \\ \hline
**Measure** & **Linear** & **Fuzzy** & **Linear** & **Fuzzy** & **Linear** & **Fuzzy** & **Linear** & **Fuzzy** \\ \hline Mean & 0.0023 & 0.0023 & 0.0035 & 0.0034 & 0.0104 & 0.0104 & 0.0178 & 0.0186 \\ \hline STD & 0.0002 & 0.0002 & 0.0004 & 0.0004 & 0.0020 & 0.0022 & 0.0039 & 0.0045 \\ \hline Max & 0.0027 & 0.0028 & 0.0043 & 0.0044 & 0.0147 & 0.0153 & 0.0264 & 0.0289 \\ \hline Min & 0.0020 & 0.0020 & 0.0029 & 0.0028 & 0.0075 & 0.0072 & 0.0120 & 0.0120 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Numerical measures of \(J_{T}\) for 10 simulations considering disturbance rejection. Figure 22: Multiple comparison using Bonferroni procedure. According to [40], a prefilter can be employed to improve the system's dynamic response. In the linear controller, the prefilter allows the cancellation of the zero effect; also, the system overshoot can be reduced with an adequate prefilter design. The effect of the prefilter might be included in the training algorithm for the optimization of the neuro-fuzzy system. The prefilter may have a linear design and then extend it to a fuzzy design. This application shows the possibility of achieving a non-linear fuzzy controller taking as reference the design of a linear controller (linear controllers with excellent performance can be used for the plant). As the plant considered is well known, the article focuses on showing how a linear controller can be extended to a fuzzy controller (starting from a behavior close to the linear controller). After having the first implementation for the fuzzy controller, nonlinearities can be included to improve the behavior of the control system when having more complex plants with nonlinearities, a subject that will be addressed in later work.