## 5 Results and Discussion
It is worth pointing out that the main reason to optimize both the RB and the MF's mapping is to reach the optimal minimum of the objective function given by the overall FLC processing in comparison to an only MF's parameter optimization procedure, as in [8; 11; 42; 43]. In this regard, both optimization procedures (i.e., MFs and RB optimization and only MFs optimization) have been applied to the FLC of the MLS (MLS-FLC). The obtained solutions of the aforementioned optimization procedures are compared with the MLS using a non-optimized FLC and the MLS with a full state feedback controller in terms of MLS' performance parameters. MLS real model representation described in Section 2 is used for simulations. The objective of optimization was to accomplish a specific set of performance parameters for MLS (i.e., \(\%M_{p,D}=5\%\), \(t_{s,D}=0.5\)\(s\), \(r_{\text{ss,D}}=0.008\)\(m\)) by minimizing the objective function defined by (18). MF and RB optimization optimum solution shows a smaller deviation value of every defined goal in comparison with an only MF optimization. Both optimization procedures' results are also compared with the performance of MLS with a SS controller. In this regard, according to resulting values listed in Table 3, MF and RB optimization proposed in this paper converges to a better solution which results in a set of slightly better performance parameters in comparison with only MF optimization and the results are close to the ones obtained using an analytic design approach (i.e., full state feedback controller). Furthermore, simulation results comparison is shown in Figure 7. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Control Signal**} & \multicolumn{6}{c}{**Position Error**} \\ \cline{3-6}  & & **NB** & **NS** & **ZE** & **PS** & **PB** \\ \hline \multirow{6}{*}{Derivative of error} & NB & PB & PB & PB & PS & PS \\  & NS & PB & ZE & PS & NS & NB \\ \cline{1-1}  & ZE & ZE & NS & NS & NS & NB \\ \cline{1-1}  & PS & NB & NS & NB & NB & NB \\ \cline{1-1}  & PB & NB & NS & NB & NB & NB \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fuzzy Logic optimized rule base (RB) for MLS control. Figure 6: Optimized and non-optimized MF used for inputs and output data: (**a**) Position error (input), (**b**) derivative of error (input), (**c**) control signal (output). Table 4 presents the objective function values in terms of MLS performance parameters for the MF and RB optimized FLC, the MF optimized FLC, and the SS controller. Notice that the objective global optimum value obtained with the MF and RB optimization is approximately half the obtained value with only MF optimization and closer to the SS controller approach, being a better solution the one with its objective function value closer to zero. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Objective Function Value**} \\ \hline MF optimized & 0.0047 \\ \hline MF and RB optimized & 0.0005 \\ \hline SS Controller & 0.0040 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Objective function values comparison. Figure 7: Fuzzy logic control of the MLS (MLS-FLC) with only MF’s positions optimized, with membership function (MF) and rule base (RB) optimized, non-optimized and MLS with state-space (SS) controller step response: (**a**) Steady-state response, (**b**) dynamic response. It is worth pointing out that the analytic design of any classic or modern controller for a non-linear system (i.e., MLS) either by a linearization approach, thus not dealing with large signal perturbations, or eventually by using complex nonlinear techniques, requires a certain level of expertise for system parameter identification and the development of complex system modeling, resulting in a long implementation time. Consequently, by using the presented approach, these requirements are not completely necessary to obtain similar results than other complex control techniques.