## 1 Introduction
The main objective of this method was to suppress lateral and torsional vibrations. In [15], [PERSON] and [PERSON] introduced the dynamic sliding-mode theory to design an effective controller (SMC) that avoided different bit-sticking problems. [PERSON] modified the SMC to achieve a tolerance for parameter uncertainties and robustness to variations in the WOB [16]. Nevertheless, the design of nonlinear control schemes is based on a system model in addition to much information about the system, which degrades the control performance. Recently, artificial intelligence techniques have been increasingly utilized in industrial control applications due to the rapid advancement of signal processing capabilities and the progression of industrial informatics [17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. The ability to develop effective control schemes without depending on a mathematical model of the considered system is a key benefit of artificial intelligence. Fuzzy logic control (FLC) [17; 18; 19; 20; 21], artificial neural networks (ANNs) [22; 23], and adaptive neuro-fuzzy inference systems (ANFISs) [24; 25; 26] have been widely employed in the literature to develop intelligent controllers. These techniques are appropriate for developing robust regulators of unknown or uncertain systems. [PERSON] et al. designed an optimal hybrid interval Type-2 fuzzy PID+I logic controller (OH-IT2 FPID+I) using a practical swarm optimization algorithm for a multi-degree-of-freedom oil well drill string system [27]. The proposed OH-IT2 FPID+I provided high control performance under several critical conditions. However, the FLC's efficacy was restricted, since it depended on expert knowledge to determine the controller's optimal gains. For these reasons, ANFISs have recently been employed instead of FLC to improve control performance because they provide more DOFs to deal with nonlinearities [24; 25; 26]. ANFIS is a type of ANN that exploits the learning ability of neural networks with the performance of fuzzy linguistics [26]. On the other hand, numerous researchers have attempted to combine FLC with conventional PID control to enhance the control performance [28; 29; 30]. To improve the control performance for multi-DOF oil well drill string systems and eliminate stick-slip phenomena, a cascade control scheme for drill string systems was investigated and modified using hybrid neuro-fuzzy PID and feedforward terms (CFF-NFPID). The suggested control configuration involved two control loops. The first controller estimated the top drive velocity reference from the drill bit velocity and its reference. The second controller aimed to regulate the top drive velocity to its reference by generating the required torque. Each control loop was designed based on hybrid adaptive neuro-fuzzy PID and feedforward terms. Adaptive neuro-fuzzy control was combined with conventional PID control to develop a hybrid neuro-fuzzy PID control scheme (H-NFPID). Furthermore, a feedforward term for each control loop was designed based on a mathematical model of a material characteristics of the drilling system. In addition, the bit is considered free, while the rotary table remains fixed. The drill string system's dynamics are represented by Equation (1) with the aforementioned characteristics of a drill string. The state vector of the drill string system is defined in Equation (2) [34]. \[