## 4 Results Obtained from The Tests Performed  
In this section the performance, stability and behavior of the robot is presented using the PI control with intelligent adjustment in the variables (\(K_{p}\), \(T_{i}\)). Each experiment has 15 tests performed with the following configuration of parameters in the meta-heuristic ABC: population size = 20, iterations = 50, and search range [\(-\)1 9], using as a reference the results obtained with the Ziegler and Nichols method. The results obtained are shown in the Table 2.  
\begin{table}
\begin{tabular}{c c c c} \hline
**NUM.** & \(T_{i}\) & \(K_{p}\) & \(BEST\) \\ \hline
1 & **3.8538** & **8.5949** & **2.0947** \\
2 & 1.3049 & 3.3470 & 6.5614 \\
3 & **5.5574** & **7.0028** & **3.0410** \\
4 & 6.5373 & 7.3083 & 4.5221 \\
5 & 7.6869 & 8.1065 & 3.1513 \\
6 & 8.6309 & 0.6437 & 4.6822 \\
7 & 3.8538 & 8.5949 & 3.2261 \\
8 & 6.4815 & 7.2582 & 3.7371 \\
9 & 3.1727 & 6.7449 & 3.5709 \\
10 & 4.2885 & 8.6309 & 3.5917 \\ \hline \end{tabular}
\end{table}
Table 2: Results of control PI using ABC optimization.  
Figure 5: Behavior using PI traditional control.  
In Table 2, we can observe the results obtained from the tests performed on the robot. We consider it important to show the values found for the variables (\(K_{p}\), \(T_{i}\)) and the error obtained. The error was calculated using the equation of the root of the mean square error (RMSE); see Equation (8):  
\[RMSE=\sqrt{\sum_{i=1}^{n}\frac{\left(y_{ref}-y_{data}\right)^{2}}{N}} \tag{8}\]  
where \(y_{ref}\) represents the reference to follow, \(y_{data}\) represents the data obtained from the motors of the robot, and \(N\) is the number of data.  
The behavior of the robot motors when following the reference for tests 19, 1, and 3 is shown in Figures 6-8. Observing the fitness value obtained, we determined that they are the best results obtained from the tests carried out with the robot and the meta-heuristics; moreover, the values for the constants of the controller (PI) are shown.  
The yellow signal is the pulse of input to the system (reference); the magenta and cyan signals are the responses of motor one and two. The type of input to the system was a pulse signal with an amplitude of 150 square.  
For greater ease for the readers of this work, in Table 3, we can observe a comparison of the results obtained with the different approaches. It is important to mention that the error obtained with \"Integral proportional control (PI), traditional\" used the values defined by [29; 30; 31], and the error was 16.6874, which is high. When the optimization is used with values 7.5682 in \(\mathrm{K_{P}}\) and 0.2836 in \(\mathrm{T_{i}}\), the error decreases considerably.  
\begin{table}
\begin{tabular}{c c c c} \hline
**NUM.** & \(T_{i}\) & \(K_{p}\) & **BEST** \\ \hline
11 & 4.1859 & 7.2531 & 4.1478 \\
12 & 4.8299 & 8.8266 & 4.1827 \\
13 & 3.2176 & 6.4975 & 3.1795 \\
14 & \(-\)0.2182 & 6.1521 & 3.3044 \\
15 & 8.1057 & 8.339 & 4.1057 \\
16 & 7.2582 & 8.6190 & 4.5323 \\
17 & 7.002 & 6.0428 & 3.8286 \\
18 & 3.1726 & 6.7449 & 4.0684 \\
19 & **0.2836** & **7.5682** & **1.2493** \\
20 & 1.2345 & 6.4258 & 3.1565 \\ \hline \end{tabular}
\end{table}
Table 2: _Cont._  
Figure 6: Behavior using PI with optimization values, test 19.  
The values shown for \(\rm K_{p}\) and \(\rm T_{i}\) with \"Intelligent integral proportional control\" were found in a dynamic and intelligent way by the proposed algorithm (ABC), as we can see in the comparison: the error difference is very significant compared with that obtained using the traditional PI control. As mentioned in the introductory section, bio-inspired algorithms provide a great variety of solutions to different problems in engineering and other areas with a low computational cost.