# Comparison of Single Control Loop Performance Monitoring Methods
## 2 Materials and Methods
### Simulated Faults  
In Case 1, the performance of the CPM methods for different kinds of faults was studied in a long simulation period. The faults were added to the simulated process, first occurring individually and later simultaneously. From the common faults presented in [5], the following faults were used in this work:  
* Valve stiction, where a certain difference between the previous and new controller output is required in order to have an effect on the actuator position. Nominally, the valve stiction in a faulty situation was set to 0.002.
* Valve change rate limit--simulating a scenario where the motor controlling the valve has a sudden fault limiting the speed of the valve change. In this case, the speed is limited to 0.04 valve rotations/s.
* Sin-wave with a constant amplitude of 75 bar, a frequency of 0.00002 Hz, and a rising amplitude (from 0 to 141.6 bar), with a frequency of 0.0001 Hz representing an external disturbance to the process. This disturbance acts as the second input variable in the state-space model (pressure error), as described in Section 2.2.
* Quantization, where the measured process value fed back for the controller is quantized within an accuracy of 0.08 L/min instead of a floating number. This value was selected to produce a noticeable effect on the process control behavior.
* PID controller tuning error, where the value of the P-parameter is changed from 8 to 0.8 for the duration of the fault. The simulator with the implemented control and faults is displayed in Figure 3.  
The faults are enabled one by one as follows. The first 15 days of the simulation are fault-free. The following 2.3 days of the simulation have only one fault activated. The remaining faults are then enabled one by one, every 1.2 days, until each fault is enabled. The faults are then disabled one by one every 1.2 days, following the order of activation for the faults. Thus, during the period from 20.8 to 22.0 days, all faults are present. Afterwards, a rising sin-wave external fault is enabled during the time period of 26.6 to 30.1 days. The final 9.9 days of the simulation are without faults, where restoration to the normal state can be observed.  
Figure 4 depicts the simulation of 40 days containing the setpoint changes mentioned in Section 2.3 and the faults described above for the case with PID \(p\)-value disturbance. Simulations were repeated with different fault scenarios for a comparison of different measurement metrics with each fault case.  
In Case 2, the robustness of the methods was tested. A simulation dataset with different values for fault intensities was obtained, focusing on the valve stiction fault. First, a simulation with no faults was performed to obtain a reference dataset. A total of 500 different simulations were performed with different fault intensities for valve stiction, chosen randomly for each simulation from an even distribution. For the valve stiction intensity, a required difference from 0 to 0.0036 between new and old actuator values was chosen. To speed up the simulation, 800 s between setpoint changes was used instead of  
Figure 3: Control and fault implementation for the simulated process. The rounded boxes at the top represent the common faults in this work.  
the 1200 s used in Case 1. With the reduction of the simulation time, the OCE method does not perform adequately and is left out for this case.