## 3 Methodology
### Artificial Data Generation  
The required data for training the neural network were generated using a Choudhury Stiction data drive model, as shown in Figure 5. It is a data-driven model that has parameters that can be related to plant data, and it produces behavior similar to a physical model. The model only needs the input signal and the specifications of S and J. The stiction model receives the controller's current output, which ranges from 4 to 20 mA, and converts it to a percentage of valve movement, ranging from 0 to 100%. The valve is said to be fully saturated (i.e., fully open or fully closed) if the valve movement is greater than 100%. The stiction model calculates the slope of the input or control signal when the valve movement lies between 0 and 100%. The model output varies with respect to the sign of the slope of the control signal. Figure 5: Signal and logic flowchart for the data-driven stiction model. Figure 5 shows the flowchart of the process of generating stiction data. The stiction block input is the control signal \(x(k)\) at 4-20 mA, and the output is the control valve movement \(y(k)\) in terms of the percentage of full-scale movement. For increasing control input, the sign function gives a positive output, whereas it gives a negative output for a decreasing control signal. The sign function returns a '0' when there is no change in the control signal from its previous value; thus, the valve is stuck. If a sign change from positive to negative is observed in the control signal, it represents the start of the stick phase, and the corresponding valve position is considered to be 'xss'. On observing the change in the sign of the slope, the valve slips and starts moving if the total change in input beats the stick band and/or deadband. This stick-slip action may repeat in the same direction, thus indicating that 'I' is '1' for the stick phase and '0' for the jump phase. The different kinds of stiction that can be obtained based on the values of 'S' and 'J' are specified below:  
* _Deadband:_ The response of the controller output to the valve position creates a deadband without any sudden jumps, as this case is simulated when J is a null value, J = 0, as presented in Figure 6a for S = 6. * _Undershoot:_ This region is created when the value of 'J' is less than the deadband, J < S, as represented in Figure 6b for S = 6 and J = 4. * _No offset:_ When the values of 'S' and 'J' are the same, J = S, this region is created, and it produces pure stick-slip behavior, as shown in Figure 6c for S = J = 6. * _Overshoot:_ If the 'J' value exceeds the 'S' value, J > S; this leads to an overshoot region of stiction as the jump amplitude is greater than the deadband, as depicted in Figure 6d for S = 4 and J = 6. Figure 6: Open-loop simulation results of the data-driven stiction model. The Choudhury stiction model depicted in Figure 5 was implemented in SIMULINK and MATLAB. The open loop stiction behavior was analyzed by applying a sinusoidal control input to the stiction model to simulate the stiction behavior of the valve. The obtained valve position was plotted with the controller output for various combinations of 'S' and 'J' values, as represented in Figure 6. To analyze the response of the flow process for the stiction fault, a flow process model G(S) was considered and placed in series with the plant model. The flow process model G(S) considered for this study is given in Equation (6) [54]. \[G(S)=\frac{0.615}{20s+1}e^{-10s} \tag{6}\]  
Using the Choudhury model, open loop data were generated for stiction behavior for various combinations of 'S' and 'J' values [55]. The responses shown in Figure 6 are from an open-loop system where the input is a sine wave with an amplitude of 50 and a frequency of 0.02 \(\pi\) rad/s. The responses obtained were normalized before plotting the controller output versus the valve stem movement.