## 4 Case Studies to Validate the Self-Tuning Algorithm
The control of a nonlinear plants using a linear controller is a difficult task (and a classical PID controller is a linear controller). Although a PID controller certainly has advantages for linear systems, this controller is not suitable for high-order nonlinear systems [17; 33; 34]. Also, the mechanical torque variation (\(\Delta T_{m}\)) is a measurable external disturbance, which does not directly act on the controlled output (terminal voltage), but rather as an internal perturbation caused by a variation of a process model parameter (\(T_{m}\)-see Appendix A). Therefore, the variation of the mechanical torque (considered as a [PERSON]'s model parameter) leads to a change of the operating point along the nonlinear characteristic of the process. For linear plants, there are many tuning methods for a PID controller (Kessler, Zigler-Nichols, Cohen-Coon, etc.) described by the technical literature [35]. However, an accurate tuning of a PID controller becomes very difficult for a highly nonlinear system. For this case, the tuning was performed through empirical attempts. The next test scenario considers two study cases for a mechanical torque variation: the case of a small perturbation \(\Delta T_{m}=10\%\) (Figure 10a,b) and the case of a high perturbation \(\Delta T_{m}=24\%\) (Figure 11a-d), as the maximum disturbance allowed only by using the proposed self-tuning strategy of control penalty factor (\(\rho\)). For a small perturbation, by comparing the results obtained by using a self-tuning minimum variance controller with constant control penalty (Figure 10a) and a classical PID controller (Figure 10b), there can be noticed relatively comparable performances. The overshoot is smaller for the PID control system, while the settling time is smaller for the classical minimum variance control. It is mentioned that the same response (see Figure 10a) is obtained also by using the new proposed self-tuning mechanism, which does not activate an increase of the control penalty factor for a small disturbance, so \(\rho=0.0001\). Therefore, for small disturbances, the control performances are comparable for all these analyzed control systems. Figure 10: (**a**) Controlled output—minimum variance controller ((\(\rho=0.1,\Delta T_{m}=10\%\)), small disturbance); (**b**) Controlled output—classical PID controller ((\(\Delta T_{m}=10\%\)), small disturbance). By considering a high disturbance, as we already mentioned, even for a high control penalty factor (\(\rho=0.1\)), the classical minimum variance control system (with constant control penalty) stabilizes the system only for a maximum disturbance \(\Delta T_{m}=19\%\) (see Figure 3f) and cannot stabilize the system for a higher disturbance \(T_{m}=24\%\) (Figure 11a). Thus, for this situation, the proposed self-tuning control penalty strategy must be taken into consideration (Figure 11b). In this case, for the same already tuned parameters of the PID controller, the change of the operating point leads to poor performances of the PID control (see comparatively the settling times and the overshoots in Figure 11b,c). Therefore, a retuning of the PID controller is required (and this task is not easy at all) and a new set of PID parameters were tuned corresponding to this new functioning point. The result depicted in Figure 11d proves that a proper retuning can considerably improve the PID control performance. Although the control system response presented in Figure 11d is relatively good, by comparison with the one from Figure 11b, it can be noticed that the settling time is much reduced for the case of the proposed solution. As a conclusion, it can be stated that the performance of a classical PID controller is lower than that of the proposed self-tuning minimum variance controller (with variable control penalty factor), due to the nonlinearity of the controlled process. However, a self-adapting PID controller can be an alternative solution.