## 4 Simulation Results
### Effect of Applying the Proposed CFF-NFPID Approach to a Drill String System  
Figure 6 shows the behavior of a drill string system with a WOB of 97 kN and without a closed control loop. It can be observed that the drill string system suffered from stick-slip phenomena in this situation. The velocities of the rotary table and drill bit periodically oscillated from zero to a value that surpassed the rotary table's velocity in addition to the variation in the drill bit velocity. These oscillations negatively affected the drill string components and could cause failure or complete damage. For this reason, the utilization of closed-loop control in drill string systems is imposed as a compulsory solution for dampening these oscillations. As described in this section, the proposed CFF-NFPID control scheme was implemented to eliminate stick-slip vibrations. Furthermore, the performance of the proposed CFF-NFPID strategy was tested when undergoing sudden and slow changes in the drill bit reference when the WOB was set to 100 kN, as shown in Figures 7 and 8.  
As illustrated in Figure 7a, for the time range from 0 s to 150 s, the reference angular velocity was set to 12 rad/s. The proposed controller dampened the stick-slip vibration and guaranteed a quick stabilization of both the rotary table and the drill bit to the reference angular velocity at t = 25.33 s and t = 24.73 s, respectively, with a very small peak overshoot. Then, a stepwise decrease in the change in the drill bit reference from 12 rad/s to 8 rad/s was applied at 150 s. The angular velocities of both the rotary table and the drill bit were quickly stabilized to the desired drill bit reference without undershoots at t = 25.33 s and t = 24.73 s, respectively. Furthermore, a large and sudden increase in the change in the drill bit reference from 8 rad/s to 14 rad/s occurred at 300 s. In addition, the CFF-NFPID strategy provided a fast stilling time for the angular velocities of both the rotary table and the drill bit at t = 15.40 s and t = 12.50 s, respectively, without overshoots. The simulation response of the drive torque with sudden changes in the drill bit reference is presented in Figure 7b. It can be clearly observed that the drive torque increased or decreased depending on the change in the angular velocity reference with a small overshoot/undershoot (less  
\begin{table}
\begin{tabular}{l c c c} \hline
**Parameter** & **Element(s)** & **Symbol** & **Value** & **Units** \\ \hline Coefficient of static friction & \(\mu_{sb}\) & 0.8 & – \\ Coefficient of Coulomb friction & \(\mu_{cb}\) & 0.5 & – \\ Coefficient of viscous damping & \(d_{r}\) & 425 & N\(\cdot\)m\(\cdot\)s/rad \\ \hline \end{tabular}
\end{table}
Table 1: _Cont._  
Figure 6: Simulation results with a WOB of 97 kN and without a closed control loop.  
than 18%). As presented in Figure 8a, a slow decrease in the variation in the reference angular velocity from 12 rad/s to 8 rad/s occurred from 75 s to 175 s. Moreover, a slow increase in the variation in the reference angular velocity from 8 rad/s to 12 rad/s occurred from 250 s to 350 s. Despite these changes, the proposed CFF-NFPID controller guaranteed accurate angular velocity tracking. On the other hand, the drive torque increased or decreased depending on the slow variation in the angular velocity reference with very small deviations, as presented in Figure 8b.  
Figure 7: Drill string stabilization when using the CFF-NFPID controller with sudden change in the drill bit reference and WOB = 100 kN: (**a**) angular velocities; (**b**) control torque.