## 1 Introduction  
The control of drill string systems in oil wells to improve extraction has received significant attention in the gas and petroleum industry. A drill string is composed of a rotary table, a series of hollow drill pipes, and a bottom hole assembly (BHA). Numerous drill collars and intermediate stabilizers connecting to a drill bit are the main components of a BHA [1, 2]. The rotary table provides the required torque on the bit (TOB) to the bottom of the borehole through the components of the drill string system [3, 4]. A drill string system is regarded as a complex dynamic system with many indefinite and unstable parameters due to the changes in the characteristics of a drill string that occur during drilling operations. Furthermore, different challenges within the perforation process can lead to failure or an elevation of its overall cost. The interaction between the drill string and the borehole causes many undesired vibrations. Therefore, these vibrations are the most common cause of perforation process failure [4, 5]. A permanently stuck bit, which occurs in cases where the bit is unable to rotate, is the most detrimental phenomenon aside from stick-slip at the bit. The stick-slip phenomenon occurs when the top rotary table system maintains a constant rotary speed while the bit's rotary speed fluctuates between zero and several times the surface rotary speed [1; 2; 3; 4; 5]. Many researchers have suggested various passive control methods for mitigating these undesirable phenomena. The most common passive control approaches discussed in the literature include drill string reconfiguration, bit selection with redesign, optimization of the BHA configuration, and the use of anti-vibration down hole tools [3]. These approaches prevent stick-slip vibrations by changing the weight on the bit (WOB) and/or the bit velocity. However, the effectiveness of these approaches is not perfect under all conditions [3; 4; 5]. As an example of passive approaches, the authors of [6] introduced a zone with optimal operating conditions that ensured the stability of a drill string system during its operation. The zones bordering the optimal zone underwent stick-slip phenomena, backward whirls, forward whirls, and a low rate of penetration (ROP). Nevertheless, these zones are likely to disappear entirely when drilling into hard formations. Therefore, the control of vibration in drill string systems based on the adjustment of the drilling parameters typically fails, which leads to the degradation of the effectiveness of passive control approaches [3; 4; 5]. Because of the improvements in real-time measurement, different closed-loop control strategies have been designed and applied. Initially, a control scheme based on a proportional-integral (PI) regulator was created and implemented in [7; 8]. A cascade control scheme was designed in [9] to improve the performance provided by conventional control schemes. In [10], [PERSON] and [PERSON] proposed an optimum state feedback control mechanism to regulate a drill string's rotational motion. To limit stick-slip vibrations, [PERSON] et al. developed a linear \(H^{\infty}\) control strategy [11]. Moreover, in [12], a linear quadratic controller was designed through a linearized model of a drill string. The previous active controllers were considered linear control methods. The performance of this type of controller is limited due to the high nonlinearity of drill string systems. By taking the modeling errors in the control design process in terms of uncertain WOB into consideration, [PERSON] et al. developed synthesis control strategy for drill string systems [13]. The design of this control strategy requires the linearization of the dynamic model of the drill string system around an operating point. Therefore, the functionality of the controller cannot be confirmed, especially in the case of more degrees of freedom (DOFs). To avoid the limitations of linear control schemes, in [14], [PERSON] et al. investigated a nonlinear dynamic inversion control method.