## 4 Results and Discussion  
### Preliminaries  
The modeling step is the most difficult in the control design of the radiation pattern of phased array radars (PARs), because it involves an optimization problem with various restrictions, requiring the use of computing tools, such as genetic algorithms (GAs). In this context, Genetic Algorithm with Maximum-Minimum Crossover (GA-MMC) was developed to calculate offline the optimum values for excitation signals of the phased array (PA) elements, which are used to control the radiation pattern. GA-MMC is quite generic and flexible, requiring only changes in input parameters and the objective function to suit various applications. The algorithm was developed in Scilab software, since it is an easy-to-use computing environment and it allows creating functions on instruction files compatible with different hardware and operating systems [67]. The tests were run on multiple computers, taking advantage of Scilab portability. Each GA-MMC test lasted for a reasonably short amount of time on a PC with processor Intel Pentium 4 CPU 2.8 GHz, but with the use of most modern PCs, this processing time is reduced by about 60%. In addition, tests indicate that the use of a programming language, instead of Scilab, drastically decreases the processing time.  
To test GA-MMC validity, a linear array was used, mainly because it represents a more challenging task for the optimization method, since it has less freedom to format the radiation pattern than a planar array does. Hence, a linear setting requires a rigorous control from the elements excitation signals. Moreover, as the GA-MMC is generic and allows adaptation to any type of element, isotropic radiators were chosen, for they reduce calculations during tests, and can be easily extended to other types of elements, such as dipoles or microstrip antennas [7]. They also allow a comparison with other studies [20,22].  
The tests included two applications, in which the first one has two optimization objectives (optimize tilt and RSLL), and the second one is a continuation of the first, with three optimization objectives (optimize tilt, RSLL and null). In both applications, a PAR with similar structure to that outlined in  
Figure 11: Evolution of population fitness in GA-MMC for 10 simulations (tilt angle equal to 30 ) [13]. (**a**) Maximum Fitness. (**b**) Average Fitness.  
Figure 7b is used. In this case, the angle of interest for the tilt varies from \(0\,^{\circ}\)-\(45\,^{\circ}\), since each face covers a \(90\,^{\circ}\) scanning area. Nevertheless, to test GA-MMC, tilt angles between \(0\,^{\circ}\) and \(90\,^{\circ}\) (\(180\,^{\circ}\) scanning area) were considered.  
In each test, 10 simulations were performed to verify the ability of convergence and repeatability of GA results [55]. The algorithm was considered approved on a test when converged in all simulations and reached operating conditions in at least 70% of the simulations. This percentage is used because GA has several probabilistic operators, progressing to an optimal solution by different paths each time it runs. Hence, tests should be analyzed in terms of success of the average or most of the results [22].