## 4 Results and Discussion
### Second Application: Control of Tilt, Sidelobes and Nulls
#### 4.3.2 Analysis of the Simulations  
The results of 10 simulations for each operating condition are summarized in Table 4, which, together with parameters in Table 1, contains the number of _SIR_ errors, NE\({}_{\text{SIR}}\), the average value of _SIR_, SIR\({}_{\text{Ave}}\), and the maximum value of _SIR_, SIR\({}_{\text{Max}}\)[13]. Each simulation only registered the worst _SIR_ case, that is, the smallest null value for the two interferences. The overall result was satisfactory, since all simulations converged properly (according to Section 3.2.9) and there was only one tilt error in 80 simulations, no RSLL errors (NE\({}_{\text{RSLL}}\) = 0% in all cases) and most of the tests showed no _SIR_ error (NE\({}_{\text{SIR}}\) = 0% in seven of the eight tests). Hence, the operating conditions were met in 92% of simulations of the tests' total, indicating promising results for a probabilistic method and far greater than the minimum established (70% according to Section 4.1) to validate the GA-MMC effectiveness. As occurred in Section 4.2.2, RSLL showed high values (RSLL\({}_{\text{Max}}\) between 22.2 and 29.6 dB) compared with parabolic reflector radars (PRRs) typically used in rocket tracking systems (RTS), with a typical RSLL range 15-20 dB. Furthermore, there was little difference between RSLL\({}_{\text{Ave}}\) and RSLL\({}_{\text{Max}}\), which shows stability in results, considering the difficulty of the application with rigorous tilt and null restrictions. The sidelobes level could still be lower, but the objective function prioritizes \(\theta_{0}\), \(\theta_{11}\) and \(\theta_{12}\), so it reduces the chance of increasing more RSLL. With respect to the _SIR_, very high values of the nulls of the radiation pattern were obtained in all tests (SIR\({}_{\text{Max}}\) between 55.1 and 77.0 dB). The greatest difficulty occurred in test 06 (\(\theta_{0}=45\,^{\circ}\), \(\theta_{11}=55\,^{\circ}\)and \(\theta_{12}=35\,^{\circ}\)), because, in this case, control leads to high deformation of radiation pattern below the main lobe, \(\theta_{0}<45\,^{\circ}\). This reduces null availability in this region and, consequently, makes it very difficult to eliminate an interfering signal only \(10\,^{\circ}\) below the main lobe (\(\theta_{12}=35\,^{\circ}\)). Even so, GA-MMC got an optimum solution in 40% of simulations. In Table 5, the values of \(A_{n}\) and \(\beta\) for individuals with best fitness obtained in each test simulations are presented [13]. As in Section 4.2.2, GA-MMC reduced the value of \(A_{n}\) in the side elements, in order to raise RSLL, while increased \(\beta\) to decrease \(\theta_{0}\) (\(90\,^{\circ}\)-\(45\,^{\circ}\)). This, once more, confirms that \(\beta\) controls the main lobe direction and \(A_{n}\) controls the sidelobes level [7,13]. Regarding the _SIR_, several combinations of \(A_{n}\) meet the null level specified for the application (50 dB), indicating the highly multimodal nature of the problem. GA-MMC searches one of these combinations, as long as they enable optimizing \(\theta_{0}\) and RSLL. To illustrate, Figure 15 contains polar and Cartesian forms in the E-plane of the radiation pattern optimized by GA-MMC for \(\theta_{0}=45\,^{\circ}\), \(\theta_{11}=10\,^{\circ}\)and \(\theta_{12}=135\,^{\circ}\)in Table 5.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Test** & **NE\({}_{\text{\text{\text{\text{\text{\text{\textasci  
As it was done in Section 4.2.2, in order to validate the optimization performed by GA-MMC, simulation results are compared with data from references. Accordingly, there are similar examples with the proposed application, using GAs for the optimization of multiple objectives (such as null positioning) in controlling the radiation pattern of linear arrays with isotropic elements [53, 54]. Once again, the results of these references demonstrate the GA-MCC success, since it shows null values comparable or even better than those obtained in previous works.