# Comparison of Single Control Loop Performance Monitoring Methods
## 3 Results
### Case 2. Robustness of the Methods with Varying Fault Intensities  
The robustness of the demonstrated methods was considered with the second simulation case, where the fault intensity for valve stiction was changed randomly for 500 different simulations. The resulting index values for the histogram intersection are shown in Figure 7.  
It can be seen in Figure 7 that some of the intensities for valve stiction can be identified, as the index value clearly decreases below the normal operation (dashed horizontal line). However, most of the index values are near normal operation limits, suggesting a limited performance of the CPM index in this case. To improve the identification of the fault, the CPM method parameters need to be adjusted. After testing a different number of bins for the histogram intersection method, the best results in terms of the method's robustness to different fault intensities was achieved with a parameter value of 15 bins. Figure 8 depicts \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**CPM Index**} & \multirow{2}{*}{\
\begin{tabular}{c} **Cont.** \\ **Tuning** \\ \end{tabular} } & \multirow{2}{*}{**Ext. Dist.**} & \multirow{2}{*}{**Rate Limit**} & \multirow{2}{*}{**Quant.**} & \multirow{2}{*}{
\begin{tabular}{c} **Value** \\ **Stiction** \\ \end{tabular} } \\ \hline _ISE_ & X & X & X & - & - \\ \hline _ITAE_ & X & - & - & X & X \\ \hline _AMP_ & X & - & X & - & - \\ \hline _KL_ & X & X & X & X & X \\ \hline _ED_ & X & X & X & X & X \\ \hline _HI_ & X & X & X & X & X \\ \hline \(OCE_{p}\) & X & X & X & X & X \\ \hline \(OCE_{q}\) & - & X & X & - & X \\ \hline \(OCE_{total}\) & X & X & X & X & X \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Qualitative performance of CPM indices. The fault situations marked with X showed statistically significant difference in the monitored index between normal and faulty operation. Figure 6: (**a**) Boxplot of \(OCE_{total}\) during a single fault, (**b**) Boxplot of _ISE_ in sliding window during a single fault. the result. It is notable that the resolution (absolute values of the index) of the method was now considerably lower in comparison to the results in Figure 7.  
Figure 8: Boxplot (median and IQR) for adjusted histogram intersection with 500 different valve stiction intensities. The red vertical line shows the value of valve stiction in Case 1. Figure 7: Histogram intersection for 500 different amplitudes for valve stiction. The vertical dotted line marks the time where the fault starts. The horizontal dashed line is the lower quartile of histogram intersection in the fault-free simulation. Valve stiction was identified for different methods with varying degrees of intensity for the fault required. In Figures 8 and 9, the metrics are drawn as a boxplot (median, and lower and upper quartiles) as a function of the fault intensity. _HI_ (Figure 8), _ED_, and _KL_ (Figure 9a) can identify the fault well, as the intensity of the fault increases above 0.002. _AMP_ performed poorly with all chosen intensities, due to the nature of the valve stiction fault. Among the studied CPM methods, _ITAE_ showed the most robust behavior for different intensities of the valve stiction, shifting from the normal operation with even small disturbance values, as shown in Figure 9b. _ISE_ also performed well, as shown in Figure 9c. The sensitivity of the demonstrated indices was compiled in Table 2, where statistically significant difference (95% confidence) for the medians in the fault-free simulation and the simulations with different fault intensities were compared. The results suggest that _KL_, _HI_, and _ED_ can identify small fault intensities with some accuracy, but only reach accurate identification with the highest intensities. Additionally, the absolute values of these metrics were low, and the performance may vary in more  
Figure 9: (**a**) Boxplot (median and IQR) for adjusted Kullback–Leibler divergence with 500 different valve stiction intensities.