# Development of Automatic Electric Drive Drilling System for Core Drilling
## 2 Modeling and Theory
\] (5) where the state vector is \(x(t)=[x_{1},x_{2},x_{3}]^{T}=[\dot{\varphi}_{b},\varphi,\dot{\varphi}_{r}]^{T }\in R^{3}\), and the measured output variable is \(y(t)\in R\). \[f(x)= \begin{bmatrix}-b_{1}\dot{\varphi}_{b}+c_{1}\varphi-d_{1}T_{tob}(\dot{ \varphi}_{b})\\ \dot{\varphi}_{r}-\dot{\varphi}_{b}\\ -b_{2}\dot{\varphi}_{r}-c_{2}\varphi+d_{2}\Omega_{ref}\end{bmatrix} \] (6) \[g(x)=[0\quad 0\quad d_{2}]^{T}\] \[ \begin{array}{l}b_{1}=C_{1}/J_{b},c_{1}=K/J_{b},d_{1}=1/J_{b}\\ b_{2}=C_{2}/J_{r},c_{2}=K/J_{r},d_{2}=1/J_{r}\\ \varphi=\varphi_{r}-\varphi_{b}\end{array} \]  
If the control object of the control system is nonlinear, linearity control techniques generally fail to produce the desired performance, especially for the control system with high precision requirements. The exact linearization method is a nonlinear control design method [25; 26]. Exact linearization can turn an affine nonlinear system to a linear system by choosing adequate diffeomorphisms and nonlinear feedback control. The basic principle of this method is that the dynamic characteristics of the nonlinear system can be converted into linear dynamic characteristics (in whole or in part) through appropriate feedback transformation and nonlinear transformation, so that the exact linearization of input-state and input-output can be realized [27; 28; 29]. Finally, the problem of complex nonlinear system is transformed into a linearized problem, so the familiar linear control method can be used for design [30; 31; 32]. An output must be selected to make the system feasible feedback linearization, so the output turns out to be the angular velocity of the downhole part, which is \(h(x)=\dot{\varphi}_{b}\). If the conditions are satisfied, it is possible to find a scalar function \(\phi(x):U\in R^{n}\) to satisfy Equation (8), as follows:  
\[ \begin{array}{l}x(t)=f[x]+g(x)u\\ y(t)=h[x]\end{array}  \tag{7}\]  
A control input is then obtained through  
\[u=\alpha(x)+\beta(x)\
u \tag{8}\]  
such that the closed-loop system has a new set of variables in linear state:  
\[\dot{z}=Az+bv \tag{9}\]  
where  
\[A=\left[ \begin{array}{ccccc}0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 0&0&0&\cdots&1\\ 0&0&0&\cdots&0\end{array} \right]\quad,\quad B=\left[ \begin{array}{c}0\\ 0\\ \cdots\\ 0\\ 1\end{array} \right] \tag{10}\]  
\(A\) and \(B\) are in the canonical form of Brunovski, however, there is no loss of generality. According to the theoretical basis of input-state linearization control, the linearization method can be used for the mathematical model of the drilling rig rotation system, and the complex nonlinear system can be converted into a linear system [31]. The input-state linearization controller of the 2-DOF drilling rig rotation system is designed through the familiar control method of linear system. It can compensate the harmful vibration caused by the nonlinear characteristics of the rig. According to the nonlinear motion equation state equation and output equation of the drilling rig rotation system given by Equation (6), the bit angular velocity can be obtained by linearization of the state equation. The first component of state transformation \(z\) is  
\[z_{1}=\dot{\varphi}_{b} \tag{11}\]  
Select other state variables as  
\[z_{2}=L_{f}z_{1}=-a_{1}\dot{\varphi}_{b}+b_{1}\varphi-c_{1}T_{tob}(\dot{\varphi }_{b}) \tag{12}\]  
\[z_{3}=L_{f}^{2}z_{2}=Hz_{2}+b_{1}(\dot{\varphi}_{r}-\dot{\varphi}_{b}) \tag{13}\]  
where  
\[H=-b_{1}-d_{1}[\frac{2T_{tob(dyn)}}{\pi}(\alpha_{1}e^{-b_{2}z_{1}}-\alpha_{1} \alpha_{2}z_{1}e^{-a_{2}z_{1}}+\frac{\alpha_{3}}{1+\left(\alpha_{3}z_{1} \right)^{2}})] \tag{14}\]  
resulting in the new state equation:  
\[\left\{ \begin{array}{c}\dot{z}_{1}=z_{2}\\ \dot{z}_{2}=z_{3}\\ \dot{z}_{3}=Mz_{2}^{2}+H^{2}z_{2}+H(z_{3}-Hz_{2})+c_{1}Q+c_{1}d_{2}u\end{array} \right.