## 3 Results and Discussion
### Real-Time Experimental Results  
The membership functions shown in Figure 9 were exported to the experimental inverted pendulum on a cart, shown in Figure 7, without any change; meanwhile the control and adaptive gains used in all experiments were: \(\gamma_{1}=20\)\([\frac{\mathrm{rad}}{\mathrm{s^{y}}}]\), \(\gamma_{2}=20\)\([\frac{\mathrm{m^{2}}}{\mathrm{rad}\cdot\mathrm{s^{y}}}]\), \(\gamma_{3}=0.026\)\([\frac{\mathrm{rad}}{\mathrm{kg}\cdot\mathrm{m}\cdot\mathrm{s}}]\), \(\gamma_{4}=0.026\)\([\frac{\mathrm{m}}{\mathrm{rad}\cdot\mathrm{kg}\cdot\mathrm{s}}]\), \(p_{12}=20\)\([\frac{\mathrm{rad}}{\mathrm{m^{2}}}]\), \(p_{22}=25\)\([\frac{\mathrm{rad}\cdot\mathrm{s}}{\mathrm{m^{2}}}]\), \(L=0.25\)\([\frac{\mathrm{m}}{\mathrm{rad}}]\), \(\mathbf{k}=\left[ \begin{array}{cc}173.42\,[\mathrm{s^{-2}}]&25.26\,[\mathrm{s^{ -1}}]\end{array} \right]^{T}\), \(k_{d_{1}}=1122.864\left[\frac{\mathrm{m}}{\mathrm{rad}}\right]\), \(k_{d_{2}}=153.442\left[\frac{\mathrm{kg}\cdot\mathrm{m}}{\mathrm{s}\cdot \mathrm{rad}}\right]\), \(k_{d_{3}}=105\)\([\frac{\mathrm{kg}}{\mathrm{s^{2}}}]\) and \(k_{d_{4}}=397.1\left[\frac{\mathrm{kg}}{\mathrm{s}}\right]\). As it was said before, in this paper we are interested only on the proposed controller part; that is, on the adaptive fuzzy controller. Thereby, the experiments were carried out for a longer time frame and only the steady state responses were evaluated; this happens from 30 to 80 s.  
The comparison of the position errors of the pendulum and the tracking error of the cart as well as the control input for the not optimized and each optimized controller are shown in Figures 11-13, meanwhile the performance indexes of errors and applied force are available in Table 4. Specifically, Figure 11 shows steady state responses of the PSO algorithm vs. the not optimized algorithm, starting from 30 to 80 s. We can observe that the pendulum position error response for PSO fluctuates between \(+2\) and \(-3\) degrees, while for the not optimized varies between \(+3.5\) and \(-3\) degrees. The same comparison can be observed for the position error of the cart, making more evident the superiority of the PSO response (\(+-10\) cm vs. \(+-5\) cm). The control efforts are similar for both approaches. Figures 12 and 13 show similar advantages for the other optimized algorithms. In order to quantify more precisely the control performance, we use the root-square average index, shown in Table 4. It is evident that the evolutionary algorithms improved the performance of the controller in  
Figure 10: Membership functions plotted together for each input. the physical experiments; the improvement is more noticeable on the cart. Also, it can be concluded that PSO gets the best performance index for both the pendulum and the cart. Figure 11: Time response in real-time for the not optimized and optimized controller by PSO. Figure 12: Time response in real-time for the not optimized and optimized controller by FFA. Figure 13: Time response in real-time for the not optimized and optimized controller by DE. On the other hand, it follows that the values of the optimized parameters are not identical, even though they are pretty similar, which can be better appreciated in Figure 9 where it is clear that the fuzzy sets for each input variable are coincident among those obtained by the three different algorithms. For comparison purposes, a Takagi-Sugeno Fuzzy Servo Control (TSFSC) scheme designed in [28] is used. TSFSC is comprised of only two membership functions with one input (position of the pendulum), and designed via local approximation in a fuzzy partition spaces technique, and the feedback gain is obtained via a linear matrix inequalities (LMIs) approach. The performance of the TSFSC scheme applied on the pendulum-cart system is verified and exhibited in Figure 14.  
Table 5 presents the norms for the TSFSC algorithm.