## 5 Results and Discussion  
Tuning procedure of fuzzy design variables using the CS algorithm presented in this paper permitted not only to eliminate the requirement of expertise needed for setting these variables but also to obtain an accurate FLC that enables MLS to perform according to a sort of pre-defined parameters. The CS algorithm used MLS modeling to simulate the real performance of the system with an 8 mm height set point and tuned the FLC according to MLS desired performance parameters. In this regard, the typical parameters of a second-order linear system are overshoot, settling time and steady-state output value. Overshoot represents the maximum response value (extremum), whereas settling time is the time elapsed until the system response remains bounded around the set point tolerance usually between \(\pm\) 5%. The objective function and a minimum tolerance value as a finishing condition of the optimization process, were defined as:  
\[O=\alpha_{1}\frac{\left|\%M_{p,D}-\%M_{p}\right|}{100}+\alpha_{2}\frac{ \left|t_{s,D}-t_{s}\right|}{10}+\alpha_{3}\left|r_{ss,D}-r_{ss}\right|\cdot 100 \tag{18}\]  
\[tolerance=1\%\cdot\left(\%M_{p,D}+t_{s,D}+r_{ss,D}\right) \tag{19}\]  
where \(\%M_{p,D}\) is the desired percentage overshoot, \(\%M_{p}\) is the actual percentage overshoot, \(t_{s,D}\) is the desired settling time, \(t_{s}\) is the actual settling time, \(r_{ss,D}\) is the desired final position, \(e_{ss}\) is the actual final position, and \(\alpha_{x}\) is the weight of each component. Note that \(\alpha_{1}=\alpha_{2}=\alpha_{3}=1\) are used in this paper to prioritize the minimization of every objective equally. In this regard, the aim is to minimize the deviation of every performance parameter from its respective goal. For purposes of comparison, both optimization procedures: For MF and RB, and with only MF have been implemented. On one hand, the optimization process of MF and RB took approximately 8 h on an Intel(R) Core(TM) i7-4510U CPU (2.00 GHz) 12 GB of RAM computer using Matlab(r) program (Matlab 2016b, MathWorks(r), Natick, MA, USA, 2016), whereas most of the computation time is spent on solving the 40 iterations required to reach the optimum solution that minimizes the objective function according to the problem restrictions. On the other hand, the optimization process of only MFtook approximately 12 h on in the same simulation platform, whereas most of the computation time is spent for solving the 60 iterations required to reach the optimum solution that minimizes the objective function according to the problem restrictions. In this regard, Figure 5 illustrates the average objective function convergence for ten runs of each algorithm showing that optimizing MF and RB may lead to a faster convergence speed of the optimization process. In Figure 6, the optimal fuzzy MF encountered by the optimization algorithm are superimposed over non-optimized MF. The optimized MF maintain order and coherency with regards to the initial distribution illustrated in Figure 3, which was one of the primary objectives. Furthermore, as is shown in Table 2, the optimized RB distribution has been displaced in comparison to the initial RB shown in Table 1.  
Figure 5: Objective function convergence. Figure 6: _Cont._  
As aforementioned, FLC is tuned using CS to enable MLS to accomplish a specific settling time, overshoot, and steady-state vertical position. Moreover, MLS performance with optimized FLC can be compared with linear controllers designed from the system's linearized model around the desired steady-state. For this purpose, and in order to fulfill as close as possible the required performance parameters, full-state feedback controller referred to as \"state-space (SS) controller\" has also been designed. In this concern, it should be pointed out that, instead of the FLC which only needs to sense the position of the sphere, the SS controller should also require sensing both the velocity and the electrical current.