## 4 Experiments and Results
### Proof of Concept  
In this verification test (also called _proof of concept_), we compare the performances of both controllers used in our study, i.e., the OAC-DES and linear PI-controller, according to a Root Mean Squared Error (RMSE) indicator. The RMSE indicator is frequently used in control theory for measuring the quality of a control system, where the difference between desired and actual values is observed. The lower the RMSE value, the better the response of the control system. Using this test, we would like to reveal the adaptive nature of the OAC-DES during online control that is its major advance over the linear PI-controller.  
The RMSE indicator is expressed as follows:  
\[\text{RMSE}=\sqrt{\frac{\sum_{i=1}^{N}\left(\omega_{r_{i}}-\omega_{a_{i}} \right)^{2}}{N}}, \tag{6}\]  
where \(N\) represents the number of samples (every velocity sampling time \(T_{sv}=3\) ms) in one cycle of duration \(T_{cycle}=4.5\) s.  
Using the RMSE indicator, the test was conducted similarly for both controllers. At first, the reference generator for velocity control for duration \(T_{cycle}=4.5\) s acts as a repeatable driving cycle. For each velocity sampling time \(T_{sv}=3\) ms, the squared difference between reference and actual velocity \((\omega_{r_{i}}-\omega_{a_{i}})^{2}\) is calculated. The differences are then summed up for the duration of the whole cycle \(T_{cycle}=4.5\) s. For instance, there are approximately \(N=(3\) ms\()^{-1}=333.33\) samples, when \(T_{cycle}=1\) s, and \(N=1500\) samples, when \(T_{cycle}=4.5\) s. The driving cycle is depicted in detail in Section 4.3.1.  
The results of the comparison between OAC-DES and linear PI-controller during the proof of concept test are shown in Figure 8, where the behavior of the OAC-DES with ANN-learning is compared with the behavior of the linear PI-controller without learning. The figure shows that at the beginning, the linear PI-controller controls the non-linear robotic mechanism more effectively. However, the RMSE indicator for the linear PI-controller remains constant, while it decreases significantly for the OAC-DES during online control, and in fact becomes equal to the linear PI-controller after 5 cycles (i.e., 22.5 s). Furthermore, when the introductory ANN-learning continues, the RMSE indicator decreases steeply for the OAC-DES up to the 11-th cycle, when it starts to decrease gradually.  
\begin{table}
\begin{tabular}{l l c} \hline \hline
**Variable** & **Variable Name** & **Value** \\ \hline Initial mutation step size & \(\sigma_{0}\) & 30 \\ Minimum mutation step size & \(\epsilon_{0}\) & 0.0001 \\ Learning rate & \(\tau\) & 0.1 \\ Number of generations & \(MAX\_GEN\) & 10 \\ Initial reference value & \(x_{0}\) & 0 A \\ Maximum reference current & \(i_{r}\) & \(\pm 1.5\) A \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameter setup of (1+1)-DES.  
According to Figure 8, the behavior of the linear PI-controller is much better than that of the OAC-DES during earlier cycles, but falls very far behind after. Obviously, for optimal control, the OAC-DES requires some time to recognize the non-linearity of the system more precisely. This is the crucial limitation, and should be taken into account for the tests that follow.