## 1 Introduction
\begin{cases}\Omega_{r}&=\ -\ \frac{d_{s}}{\
u}(\Omega_{r}-\Omega_{p})-\frac{k_{ s}}{\
u}(\Omega_{r}-\Omega_{p})-\frac{d_{s}}{\
u}(\Omega_{r})+\frac{T_{m}}{ \
u}\\ \dot{\Omega}_{p}&=\ \frac{d_{s}}{\
u}(\Omega_{r}-\dot{\Omega}_{p})+\frac{k_{ s}}{\
u}(\Omega_{r}-\Omega_{p})-\frac{d_{s}}{\
u}(\Omega_{p}-\dot{\Omega}_{l})- \frac{k_{isl}}{\
u}(\Omega_{p}-\Omega_{l})\\ \dot{\Omega}_{l}&=\ \frac{d_{s}}{\
u}(\Omega_{p}-\Omega_{l})+\frac{k_{isl}}{\
u}( \Omega_{p}-\Omega_{l})-\frac{d_{m}}{\
u}(\Omega_{l}-\Omega_{b})-\frac{k_{isl}} {\
u}(\Omega_{l}-\Omega_{b})\\ \dot{\Omega}_{b}&=\ \frac{d_{m}}{\
u}(\Omega_{l}-\dot{\Omega}_{b})+\frac{k_{isl}}{ \
u}(\Omega_{l}-\Omega_{b})-\frac{T_{m}(x)}{\
u_{b}}\\ \end{cases}  \tag{1}\]  
\[\mathbf{X}\ =\ [\dot{\Omega}_{r},(\Omega_{r}-\Omega_{p}),\dot{\Omega}_{p},( \Omega_{p}-\Omega_{l}),\dot{\Omega}_{l},(\Omega_{l}-\Omega_{b}),\dot{\Omega}_ {b}]^{T} \tag{2}\]  
where \(j\) is the inertia, \(\Omega\) is the angular position, \(\dot{\Omega}\) is the angular velocity, \(k\) is the stiffness, and \(d\) is the damping. The torques on the system are the drive torque \(T_{m}\), the torque on the bit (TOB) \(T_{ob}\), and the rock-crushing torque \(T_{fb}\). Equation (2) is simplified as follows:  
\[\mathbf{X}\ =\ [x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7}]^{T} \tag{3}\]  
By combining Equations (1) and (3), the mathematical model based on the state equations for the system model can be rewritten as follows:  
\[ \begin{cases}\dot{x}_{1}&=\ \frac{1}{\
u}[-(d_{r}+d_{l})x_{1}-k_{l}x_{2}+d_ {l}x_{3}+T_{m}]\\ \dot{x}_{2}&=\ x_{5}-x_{7}\\ \dot{x}_{3}&=\ \frac{1}{\
u_{p}}[d_{l}x_{1}+k_{l}x_{2}-(d_{tl}+d_{t})x_{3}-k_{ ll}x_{4}+d_{tl}x_{5}]\\ \dot{x}_{4}&=\ x_{3}-x_{5}\\ \dot{x}_{5}&=\ \frac{1}{\
u_{l}}[d_{tl}x_{3}+k_{tl}x_{4}-(d_{tl}+d_{tb})x_{5}-k_{ tb}x_{6}+d_{tb}x_{7}]\\ \dot{x}_{6}&=\ x_{5}-x_{7}\\ \dot{x}_{7}&=\ \frac{1}{\
u_{b}}[d_{tb}x_{5}+k_{tb}x_{6}-(d_{tb}+d_{b})x_{7}-T_{ ob}(x)]\\ \end{cases}  \tag{4}\]  
The torque on the bit \(T_{ob}(x)\) is defined as follows:  
\[T_{ob}\ =\ T_{p}\left(x_{7}\right)+T_{fb}\left(X\right) \tag{5}\]  
Figure 1: The studied system: (**a**) a drilling rig with a drill string system; (**b**) mathematical model of the drill string system [15]. where \(T_{v}(x_{7})\) is the reaction torque of the drilling mud on the drill bit and is defined as  
\[T_{v}\left(x_{7}\right)\ =\ d_{b}\ x_{7} \tag{6}\]  
The interaction between the drill bit and the rock dissipates most of the energy generated by the drive motor [5]. The main cause of vibrations in a drill string is rock-bit interaction. Consequently, the creation of an interaction model requires a detailed analysis of the induced vibration [35]. Friction models have been proposed in the literature [16; 36; 37; 38]. The Stribeck model in [16; 39; 36] and [PERSON]'s model in [36] have been the most frequently employed in recent years. However, the Stribeck model suffers from instability and zero-speed discontinuity. To overcome these weaknesses, [PERSON]'s model is exploited to establish a rock-bit interaction model in which cutting and frictional contact are taken into account [36] (see Figure 2). The reaction torque is estimated using the following mathematical model:  
\[T_{fb}(x)= \begin{cases}T_{eb}(x)&\text{if}\quad|x_{7}|<D_{v},\quad|T_{eb}|\leq T _{sb}\\ T_{sb}\,sign(T_{eb}(x))&\text{if}\quad|x_{7}|<D_{v},\quad|T_{eb}|>T_{sb}\\ T_{cb}\,sign(x_{7})&\text{if}\quad|x_{7}|\geq D_{v}\end{cases}  \tag{7}\]  
where  
\[T_{eb}\ =\ d_{tb}\ (x_{5}-x_{7})+k_{tb}\ x_{6}-d_{b}\ x_{7} \tag{8}\]  
\[T_{sb}\ =\ w_{ob}\ R_{b}\,u_{sb} \tag{9}\]  
\[u_{b}(x_{7})\ =\ u_{cb}+(u_{sb}-u_{cb})\ e^{\frac{\gamma_{b}}{\gamma_{f}}|x_{7}|} \tag{10}\]  
\[T_{cb}\ =\ w_{ob}\ R_{b}\,u_{b}(x_{7}) \tag{11}\]  
where \(T_{eb}(X)\), \(T_{cb}\), and \(T_{sb}\) are the breakaway torque, the torque due to Coulomb friction, and the torque due to static friction, respectively. \(w_{ob}\) is the weight on the bit (WOB), and \(R_{b}\) is the radius of the drill bit.