## 4 Simulation Results
### Quantitative Comparison under Critical Conditions  
The performance of the CFF-NFPID controller was compared with that of the CFF-FPID, CFF-PID, and SMC controllers proposed in [15], the SMC proposed in [16], and the cascade PID and conventional PID controllers when there were sudden changes in the desired drill bit velocity to evaluate the benefits of using the suggested control scheme. All of the controllers were simulated on the drill string system with the parameters listed in Table 1. Figure 9 shows the simulated responses of the different controllers when there were sudden variations in the drill bit's angular velocity reference. Furthermore, a quantitative performance analysis in terms of the peak overshoot/undershoot and the settling time of the different controllers is summarized in Table 2. It can be noted that the proposed CFF-NFPID approach provided high-performance control in the dynamic state (fast settling  
Figure 8: Drill string stabilization when using the CFF-NFPID controller with slow changes in the drill bit reference and WOB = 100 kN: (**a**) angular velocities; (**b**) control torque. time and very small overshoot/undershoot) in comparison with the other controllers. In the steady state, all of the controllers succeeded in regulating the velocities of both the drill bit and rotary table to follow the desired velocity. It can also be noted in Table 2 that employing the cascade PID controller improved the control performance in comparison with that of the conventional PID controller. On the other hand, the CFF-PID controller provided enhanced control performance in comparison with that of the SMC algorithms proposed in [15; 16]. In addition, we noticed that, with the exception of the proposed CFF-NFPID controller, the CFF-FPID controller provided better performance than that of the other controllers in all cases of changes in the reference. Figure 9: Simulated waveforms of the angular velocity during variations in the desired angular velocity with different controllers: (**a**) rotary table; (**b**) drill bit. Secondly, another quantitative comparison was conducted when considering a sudden change in the WOB from 10 kN to 8 kN at 150 s and from 8 kN to 12 kN at 300 s. The reference angular velocity was set to 12 rad/s. The simulation waveforms obtained with different controllers are shown in Figure 10. The detailed control performance of all controllers is presented in Table 3. High-performance control in terms of the settling time and peak overshoot was guaranteed by the proposed controller in comparison with the other controllers. On the other hand, the C-PID controller improved the performance obtained with the conventional PID controller. The CFF-PID controller provided a fast settling time and low peak overshoot/undershoot in comparison with the SMC algorithms, as we noted in the previous test. Due to the fact that the design of the feedforward term requires some system parameters to be set, a quantitative comparison in the case of parametric uncertainty was conducted. The purpose of this critical test was to show if the proposed control scheme was able to keep the superior performance that it had in the previous tests. The obtained waveforms are presented in Figure 11, and the detailed control performance is summarized in Table 4. It can be clearly observed that the control performance of all controllers was degraded in this critical scenario. However, the proposed CFF-NFPID controller still provided better performance than that of the other controllers. Generally, the simulation results that were obtained proved the greater ability of the proposed control scheme to suppress stick-slip vibrations. In addition, the CFF-NFPID controller achieved high control performance in terms of the response time overshoot/undershoot in different critical scenarios in comparison with the other controllers.