## 5 Results  
The quality of control is difficult to determine. The error rates obtained depend on the specifics of the process. In processes where gas flow is maintained, the values change rapidly and fluctuate. In the process studied it is not physically possible to stabilize the vacuum at the setpoint value as oscillations are a typical feature of this control loop. This process shows that an absolute performance cannot be obtained. The quality of control must be measured relative to the process conditions. A certain level of nominal indicator values should be taken as the correct state (good quality) and deviations from this level should be interpreted either as a deterioration or an improvement in the quality of regulation. Indicators calculated for individual days and hours are presented in Figures 2-7 and are used to assess the quality of regulation. Figure 4: Median and mean control error values. Figure 5: Skewness and kurtosis factors for the control error value. Figure 3: Mean square control error value. Figure 2: Mean absolute and standard deviation control error values. The best results according to the criteria of absolute error, squared error and standard deviation were obtained on days 9 and 10. The worst results were collected on days 1 and 18. The largest differences were observed in hourly data for the mean squared error. The daily metrics were more stable. In the case of mean absolute error and standard deviation, the differences were 2-3 times, while for the mean squared error metric the differences are 4 times. The differences between the results cannot be explained by changes in the equipment, because, as described earlier, four days from January (before the repair) and 16 days from August to November (after the maintenance) were analysed, accounting for variations in the quality from varying process conditions. Therefore, the index values must be normalized for objectivity. The median and mean value of the control error gave good results in case of limitations, i.e., when the process value was constantly above or below the setpoint, and the error did not change sign. In a situation where the process oscillates around the setpoint, both the median and mean took the target value of 0.0 and did not reflect the large oscillations and poor performance. We used the control error median and mean to illustrate the problems of setpoint tracking in cases where the system is operating in saturation. In a situation where the process oscillates around the setpoint, both the median and mean can take the ideal value of 0.0, not reflecting the large oscillations and low control quality. Skewness illustrates problems with the symmetry of the distribution. Right skewness (positive) indicates that there are more observations below and fewer above the mean. Skewness (both positive and negative) is associated with tails in the times series distribution and indicates rare but significant over- or undershooting. In the example shown, low values of skewness are present and this indicates a symmetrical distribution. The highest values of skewness were recorded for days 6 and 7 and were below 0.6. Skewness, like mean and median, is a good indicator that the control action works within its limits. The kurtosis shows the concentration of the data. In the example above, the kurtosis value was greater than 3 for each day, indicating a relatively small number of outliers in the measurements. Similar to skewness, kurtosis can be a good indicator of saturation  
Figure 6: Stability exponent \(\alpha\) and skewness \(\beta\) factors of \(\alpha\)-stable distributions. Figure 7: Scale factor \(\gamma\) of \(\alpha\)-stable distributions. conditions when there are problems with keeping process values at the setpoint (e.g., due to saturation of manipulated variables). The parameters of the \(\alpha\)-stable density function (\(\alpha\), \(\beta\), \(\gamma\)) confirm that the distribution is close to normal (\(\alpha=2\)), with low skewness (\(\beta\) factor).