# Comparison of Single Control Loop Performance Monitoring Methods
## 2 Materials and Methods
### Implemented CPM Indices  
CPM methods for demonstration were chosen based on the criteria previously mentioned. The methods should perform without the need for modeling of the process. A priori knowledge of the process should not be needed for estimating the control performance. As such, the methods described in the following section were chosen. Integrated Squared Error (_ISE_, Equation (1)) is calculated as a function of time and in a sliding window. Here, a sliding window of one day is selected as:  
\[ISE=\int_{n-x}^{n}e(t)^{2}dt, \tag{1}\]  
where _ISE_ is comprised of the sum of squared errors (\(e(t)\)) between the current time (\(n\)) and the duration of the sliding window (\(x\)). For the performed step changes, the Integral of Time-weighted Absolute Error (_ITAE_, Equation (2)) is monitored for identifying longer period faults in the process. The longer a fault is present, the larger the time weight grows and directly increases the metric. The time weight used is the time since the last setpoint change:  
\[ITAE=\int_{1}^{w}t|e(t)|dt, \tag{2}\]  
where _ITAE_ is the sum of the absolute values of errors (\(e(t)\)) multiplied by the time since the last setpoint change (\(w\)). Additionally, Amplitude Index (_AMP_, Equation (3)) is used to measure the ratio between the maximum amplitude of process error and the size of the performed step change. The value is obtained from the minimum and maximum values after the rise time period, as in [3]:  
\[AMP=\frac{y_{max}-y_{min}}{\Delta y_{sp}}, \tag{3}\]  
where \(y_{max}\) and \(y_{min}\) are the maximum and minimum values, respectively, of the process value after the rise time and \(\Delta y_{sp}\) is the magnitude of the performed step change. The difference from normal operation in the process can be identified by utilizing the measurement error residuals and comparing the distribution to a selected time period from normal operation. Kullback-Leibler divergence (\(KL\), Equation (4)) [19] provides a measurement for difference between the two datasets. In real processes, obtaining data from the normal operation may prove difficult, especially if the process has been in operation for a while and unknown disturbances may have occurred. Kullback-Leibler divergence has been previously adapted to an index in MIMO controller performance monitoring [20]:  
\[KL=\int\ h_{1}\log\frac{h_{1}}{h_{2}}dx, \tag{4}\]  
where \(h_{1}\) is the reference dataset and \(h_{2}\) is the dataset in the chosen sliding window. Additionally, histogram intersection (\(HI\), Equation (5)) [21] and Euclidean distance (\(ED\), Equation (6)) [22] are used here to estimate the difference between the reference data and the sliding window testing datasets:  
\[HI=\frac{\sum_{j=1}^{m}min(h_{1,m},h_{2,m})}{\sum_{j=1}^{m}h_{2,m}}, \tag{5}\]\[ED=\sqrt{\sum_{j=1}^{m}(h_{1,m}-h_{2,m})^{2}}, \tag{6}\]  
where the datasets \(h_{1}\) and \(h_{2}\) are divided into \(m\) histogram bins. In addition, a commercial CPM tool was applied to the studied process. Overall Controller Efficiency (OCE) is a method developed by Insta Advance. The idea of OCE was inspired from a more general framework called OEE and is an adaptation of it in the context of PID controller performance. OCE defines one number that describes one PID controller's history, present, and future ability to function in a specified task. The main point in OCE is to follow the trend of the OCE value, and a continuous decrease in the OCE value may indicate the need for action. On the contrary, an increase in the OCE value can show the recovery or improvement of PID controller efficiency. OCE was first developed for detecting long-term phenomena (e.g., crawling due to wearing), but in this paper, OCE's suitability to detect short-term phenomena has been examined in detail in the experimental part.