## 4 Case Studies to Validate the Self-Tuning Algorithm
Figure 6a-d present the obtained results for each test and the control performances are concisely shown in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \(\rho\) & \(\Delta R\) & **Control Performance** \\ \hline \(\rho=0.0001\) (constant) & \(\Delta R\leq 7\%\) & Stable system and very good control performances (settling time: 0.2 s.) \\ \hline \(\rho=0.001\) (constant) & \(\Delta R\leq 7\%\) & Stable system and very good control performances (settling time: 0.2 s.) \\ \hline \(\rho=0.01\) (constant) & \(\Delta R\leq 7\%\) & Stable system and very good control performances (settling time: 0.2.) \\ \hline \(\rho=0.1\) (constant) & \(\Delta R\leq 8\%\) & Stable system but poor control performances (settling time: under 2 s.) \\ \hline \(\rho\in\{0.0001,\ 0.1\}\) (variable) & \(\Delta R\leq 11\%\) & Stable system and very good control performances (settling time: 0.5 s.) \\ \hline \end{tabular}
\end{table} 
Table 2: The case studies—results for maximum allowed values of electrical load variation (\(\Delta R\)). Also, for this disturbance (electrical load) and for small control penalizations (\(\rho\in\{0.0001,\ 0.001,\ 0.01\}\)), it can be noticed that the maximum allowed disturbance (electrical resistive variation for which the system still remains stable) is the same in all three cases: \(\Delta R=7\%\) (Figure 6a-c). The best control performance is obtained for a small control penalization (\(\rho=0.0001\) --see Figure 6a). Similar good results are also provided by setting \(\rho=0.001\) or \(\rho=0.001\) (Figure 6b,c). For a higher value \(\rho=0.1\) and for a slightly increased disturbance \(\Delta R=8\%\) (Figure 6d), the system slowly stabilizes but the control performances are poor (long oscillating regime, long settling time--5 s). Although the system is at last stabilized, the control performance is too weak to be acceptable. Over this value of the disturbance (\(\Delta R>8\%\)), the system becomes unstable due to the excessive control (too weakly penalized). Therefore, also for this disturbance (load variation), the conclusion is similar to the one reached for the case of a mechanical torque variation:  
* A small \(\rho\) ensures very good performances, but cannot stabilize a highly disturbed system;
* A high \(\rho\) ensures the system stabilization for a higher disturbance, but with the price of an unacceptable degradation of control performance (long oscillating transient regime, long settling time). Using the proposed tuning mechanism, similar tests were performed for load regimes (generated by connecting/disconnecting an electrical consumer at generator terminals). Thus, taking into consideration a higher electrical load (\(\Delta R=11\%\)), the results are depicted in Figure 7a-d. It is highlighted that for a constant smaller \(\rho\), the maximum value of the disturbance which allowed stabilization of the control system was \(\Delta R=7\%\). For this maximum resistive load \(\Delta R=11\%\), the proposed control strategy is still efficient, stabilizing the control system (Figure 7a). A short oscillating regime and a good settling time (0.5 s) can be noticed for the controlled output (Figure 7a), the tuning mechanism being triggered for a limited time (Figure 7c) in order to penalize the control (Figure 7b) and thus to stabilize the system. Therefore, for a load regime, the new control strategy covers an extended value range of disturbance (electrical resistance variation) up to 50%, from \(\Delta R=7\%\) (the maximum allowable load/unload for which a small constant \(\rho=0.0001\) still provides good control performance and system stabilization--see Figure 6a) up to \(\Delta R=11\%\) (the maximum load for which the proposed tuning mechanism \(\rho\in\{0.0001,\ 0.1\}\) can ensure good performance and can stabilize the system--see Figure 7a).