## 4 Simulation Results
### Quantitative Comparison under Critical Conditions
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Stepwise Decrease in the Angular Velocity**} & \multicolumn{4}{c}{**Stepwise Decrease in the Angular Velocity**} & \multicolumn{4}{c}{**Stepwise Increase in the Angular Velocity**} \\  & \multicolumn{4}{c}{**Reference 0\(\rightarrow\)12 rad/s**} & \multicolumn{4}{c}{**Reference 12\(\rightarrow\) 8 rad/s**} & \multicolumn{4}{c}{**Reference 8\(\rightarrow\)14 rad/s**} \\ \cline{2-13}  & \multicolumn{4}{c}{**Angular Velocity of the Rotary Table**} & \multicolumn{4}{c}{**Angular Velocity of the Drill Bit**} & \multicolumn{4}{c}{**Angular Velocity of the Rotary Table**} & \multicolumn{4}{c}{**Angular Velocity of the Drill Bit**} & \multicolumn{4}{c}{**Angular Velocity of the Drill Bit**} \\ \cline{2-13}  & \multicolumn{4}{c}{**T, [s]**} & \multicolumn{1}{c}{**\(\underline{U}_{\textbf{r}}\)**} & \multicolumn{1}{c}{**\(\underline{U}_{\textbf{r}}\)**} & \multicolumn{1}{c}{**\(\underline{U}_{\textbf{r}}\)**} & \multicolumn{1}{c}{**\(\underline{U}_{\textbf{r}}\)**} & \multicolumn{1}{c}{**\(\underline{U}_{\textbf{r}}\)**} & \multicolumn{1}{c}{**\(\underline{U}_{\textbf{r}}\)**} & \multicolumn{1}{c}{**\(\underline{U}_{\textbf{r}}\)**} & \multicolumn{1}{c}{**\(\underline{U}_{\textbf{r}}\)**} & \multicolumn{1}{c}{**\(\underline{U}_{\textbf{r}}\)**} & \multicolumn{1}{c}{**\(\underline{U}_{\textbf{r}}\)**} & \multicolumn{1}{c}{**\(\underline{U}_{\textbf{r}}\)**} \\ \hline PID & 87.87 & 3.86 & 87.94 & 3.35 & 50.05 & 0.82 & 51.02 & 0.74 & 45.65 & 1.21 & 45.89 & 1.09 \\ \hline C-PID & 50.63 & 5.71 & 54.12 & 10.60 & 42.15 & 0.65 & 39.35 & 2.16 & 41.43 & 0.92 & 38.77 & 3.16 \\ \hline CFF-PID & 35.4 & 7.51 & 34.1 & 12.08 & 35.12 & 1.72 & 34.51 & 2.28 & 34.43 & 2.57 & 33.89 & 3.38 \\ \hline SMC [15] & 39.45 & 2.25 & 47.56 & 6.73 & 36.02 & 1.51 & 33.92 & 1.15 & 36.03 & 3.87 & 26.63 & 1.79 \\ \hline SMC [16] & 40.56 & 3.83 & 48.77 & 7.96 & 36.29 & 2.57 & 34.65 & 1.19 & 36.62 & 4.02 & 35.21 & 1.89 \\ \hline CFF-FFID & 28.68 &Figure 11 shows the simulated responses of the OH-HT2 FPID+I controller with parametric uncertainties. In this case, the parameters \(J_{r},k_{t},d_{t}\), and \(d_{r}\) were changed, and the reference velocity was set to 12 rad/s. It is clear that the parametric uncertainties did not degrade the performance of the proposed OH-IT2 FPID+I controller. This led to the achievement of a small overshoot and fast response time. Figure 10: Simulated waveforms of the angular velocities during variations in the WOB and constant desired velocity with different controllers: (**a**) rotary table; (**b**) drill bit. To demonstrate the effectiveness of using the proposed control scheme, the OH-IT2 FPID+I controller was compared with a conventional PID controller, the SMC from [16], the SMC from [15], and the OH-IT1 FPID+I controller. Figure 9 shows the simulation time response of the velocities with variations in the reference angular velocity when using different controllers. The results showed a large undershoot with large decreases in the reference velocity in the SMC algorithms proposed in [15; 16], which could cause instability in a drill string system. Figure 10 shows the simulation time response of the velocities with variations in the WOB when using different controllers. In Figure 11, it is clear that the proposed OH-IT2 FPID+I controller produced high-performance control with parametric uncertainty in terms of the peak overshoot and settling time. Table 4 depicts a quantitative analysis of the performance in terms of the peak overshoot and settling time for all controllers in the different situations. It is clear that the proposed OH-IT2 FPID+I controller was more effective than the other controllers in all of the tested situations. Figure 11: Simulated waveforms of the angular velocities in the case of parametric uncertainty with different controllers: (**a**) rotary table; (**b**) drill bit.