## 5 Conclusions  
In this study, a fast and intelligent cascade control scheme based on hybrid adaptive neuro-fuzzy PID and feedforward terms (CFF-NFPID) was presented for an oil well drill string system. The controller's main objective was to mitigate the torsional stick-slip vibration phenomena in oil well drill strings by controlling the velocities of the drill string components to remain at a desired value. The key feature of the CFF-NFPID controller is the estimation of the top drive velocity reference from the drill bit velocity and its reference via the first control loop. Then, a second control loop was employed to regulate the top drive velocity to its reference. Each control loop was designed based on a hybrid adaptive neuro-fuzzy PID controller and feedforward term. The feedforward (FF) term provides an ideal control action that helps to achieve a fast settling time of the measured signal to the reference signal. The effectiveness of the proposed control scheme was evaluated via simulations in Matlab/Simulink. The obtained results showed the outstanding performance of the proposed CFF-NFPID controller in comparison with that of the fuzzy feedforward (CFF-FPID) controller, sliding-mode controller, cascade feedforward PID controller, cascade PID controller, and conventional PID controller when there were variations in the weight on the bit and the desired drill bit rotary speed. Although the design of feedforward terms requires exact system parameters, the proposed CFF-NFPID controller maintained its superiority over other controllers under parametric uncertainties. Future work will involve combining the predictive capabilities of model predictive control with the proposed cascade control scheme to significantly improve the response time, stability, adaptability, operational efficiency, safety, and cost savings.  
Conceptualization, A.L. and M.G.; methodology, A.L.; software, A.L.; validation, A.L.; formal analysis, A.L.; investigation, A.L.; resources, M.G.; data curation, A.L. and M.G.; writing--original draft preparation, A.L.; writing--review and editing, M.G.; visualization, M.G.; supervision, M.G.; project administration, M.G.; funding acquisition, M.G. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Qatar National Research Fund (grant number NPRP10-0101-170081).  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the corresponding author on request.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Stepwise Increase in the Angular Velocity Reference 0\(\rightarrow\)12 rad/s**} \\  & \multicolumn{4}{c}{**in the Case of Parametric Uncertainty**} \\  & \multicolumn{4}{c}{**Angular Velocity of the Rotary Table**} & \multicolumn{2}{c}{**Angular Velocity of the Drill Bit**} \\ \cline{2-5}  & **Settling Time [s]** & **Undershoot [rad/s]** & **Settling Time [s]** & **Undershoot [rad/s]** \\ \hline PID & 95.95 & 5.49 & 96.19 & 4.93 \\ \hline C-PID & 42.23 & 5.58 & 44.31 & 9.19 \\ \hline CFF-PID & 48.97 & 6.23 & 49.61 & 10.37 \\ \hline SMC [15] & 39.62 & 1.95 & 46.59 & 6.56 \\ \hline SMC [16] & 40.76 & 2.47 & 47.98 & 7.51 \\ \hline CFF-FPID & 28.76 & 2.59 & 35.75 & 5.41 \\ \hline CFF-NFPID & 26.63 & 0.95 & 33.84 & 2.29 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Quantitative comparison using a constant desired angular velocity and parametric uncertainties.