## 3 The Self-Tuning Algorithm of Control Penalty Factor
In this context, the \"normal\" state of the system is the stationary regime (when the process is not disturbed or is disturbed only by a low stochastic noise) or a regime in which the system is weakly disturbed by small external disturbances. In both cases, the proposed adjustment mechanism for tuning the control penalty factor is not active, the control penalization being set to lowest value of (\(\rho=\rho_{0}=0.0001\)). Figure 1: The self-tuning mechanism of the control penalty based on a sharp hysteresis loop. Based on simulations results analysis, the calibration procedure that was followed to determine the two thresholds of the error when the control penalty factor (\(\rho\)) switches between the two levels (\(\rho_{0}\), \(\rho_{1}\)), involves the following steps:  
1. The minimum variance control system is started (without enabling the proposed self-tuning switching mechanism), setting a constant maximum penalty of control (\(\rho=0.1\)) and considering the maximum allowed value of the disturbance for which it remains stable and the control performance is good;
2. The maximum value of the system output error (the overshoot) is determined to be used as an initial benchmark for setting the high switching threshold \(e2\);
3. The value of the high switching threshold \(e2\) is set about 10% lower than the maximum value of the output error previously determined. 4. Also, as an initial benchmark, the value for the low threshold \(e1\) is set with an order of magnitude larger than the error produced in stationary regime by the stochastic noise that affects the process (needed to ensure the numerical stability of the parameters estimator). 5. Keeping the same disturbance, the control system is restarted, enabling the proposed self-tuning switching mechanism, initially calibrated with the values \(e1\) and \(e2\) previously established (analyzing the control system stability and its performances: the settling time, the transient regime duration). 6. Small adjustments around these values \(e1\) and \(e2\) can be performed to obtain better results based on the observations that fine changes of the high switching threshold \(e2\) can affect the control stability and fine changes in the low switching threshold e1 can affect the control performance (the settling time or the duration of the transient oscillating regime). 7. Finally, larger perturbations are applied and step 6 is resumed, fine adjustments of these two thresholds (\(e1\), \(e2\)) being carried out successively until the result is considered appropriate (stable system and good control performance). 8. The procedure stops when, for a very high disturbance, an adequate set (\(e1\), \(e2\)) can no longer be determined to ensure the stability of the system (this disturbance being considered the maximum limit for the control system operating range). Comparing the solution described by the authors in [17] with the one presented in this paper, there are some major differences. Thus, when a high measurable disturbance occurs, the algorithm presented in [17] involves one single rectangular pulse variation of the control penalty factor, with a fixed duration (previously well-established) and the method can be applied only for the case of  
Figure 2: General structure of the proposed self-tuning minimum variance control system. measurable disturbances. The presented strategy can involve multiple rectangular pulse changes of the control penalty factor (with a variable duration based on a logic algorithm using a sharp hysteresis loop) and (additionally) can be applied also for unmeasurable disturbances (for example, the unpredictable load/unload regimes), requiring only a measurement of the controlled output. Also, in [10], the authors describe some classical minimum variance control systems (with a constant penalty control factor), analyzing the control performance for different orders of the induction generator linearized model (used only as a starting point to design the control law).