## 3 Active Disturbance Rejection Control Design
### Observer Design  
#### 3.1.1 Extended Sliding Mode Observer Design  
Consider the mechanical system which is constituted by the first two equations of (4) as  
\[\dot{x}_{1} =x_{2} \tag{10}\] \[\dot{x}_{2} =x_{3}+f_{2}(\mathbf{x})+d_{1}(t)\]  
By extending \(x_{e1}=d_{1}(t)\) as a new state and \(h_{1}(t)\) as the derivative of \(x_{e1}\), the system dynamics (10) can be rewritten as  
\[\dot{x}_{1} =x_{2} \tag{11}\] \[\dot{x}_{2} =x_{3}+f_{2}(\mathbf{x})+x_{e1}\] \[\dot{x}_{e1} =h_{1}(t)\]  
Figure 2: The proposed control scheme. To estimate the angular velocity \(x_{2}\) of the actuator and lumped disturbance \(d_{1}(t)\), the ESMO is constructed as  
\[ \begin{split}\dot{\hat{x}}_{1}&=\hat{x}_{2}+\ell_{1} \text{sign}(x_{1}-\hat{x}_{1})\\ \dot{\hat{x}}_{2}&=x_{3}+f_{2}(\hat{\mathbf{x}})+ \hat{x}_{e1}+2\omega_{1}\ell_{1}\text{sign}(x_{1}-\hat{x}_{1})\\ \dot{\hat{x}}_{e1}&=\omega_{1}^{2}\ell_{1}\text{ sign}(x_{1}-\hat{x}_{1})\end{split}  \tag{12}\]  
where \(\ell_{1}\) is a positive constant and \(\omega_{1}\) is the observer bandwidth determined later. Let \(\tilde{x}_{i}=x_{i}-\hat{x}_{i}\) being the estimation error with \(i=1,2,e1\). From (11) and (12), error dynamics of the observer are determined by  
\[ \begin{split}\dot{\hat{x}}_{1}&=\tilde{x}_{2}-\ell_ {1}\text{sign}(\tilde{x}_{1})\\ \dot{\hat{x}}_{2}&=\tilde{f}_{2}+\tilde{x}_{e1}-2 \omega_{1}\ell_{1}\text{sign}(\tilde{x}_{1})\\ \dot{\tilde{x}}_{e1}&=h_{1}(t)-\omega_{1}^{2}\ell_ {1}\text{sign}(\tilde{x}_{1})\end{split}  \tag{13}\]  
**Theorem 1**. : _By employing the ESMO (12), the angular velocity of the HRA and lumped mismatched disturbance estimation errors, i.e., \(\hat{x}_{2}\) and \(\hat{x}_{e1}\), approach to arbitrarily small bounded regions whose boundaries depend on the selection of observer bandwidth \(\omega_{1}\)._  
Proof of Theorem 1.: Consider a Lyapunov function as  
\[V_{x_{1}}=\frac{1}{2}\hat{x}_{1}^{2} \tag{14}\]  
Taking time derivative of it and combining with (13) yields  
\[ \begin{split}\dot{V}_{x_{1}}&=\hat{x}_{1}\hat{x}_{1 }\\ &=\hat{x}_{1}(\tilde{x}_{2}-\ell_{1}\text{sign}(\tilde{x}_{1}))\\ &\leq-(\ell_{1}-|\tilde{x}_{2}|)|\tilde{x}_{1}|\end{split}  \tag{15}\]  
Obviously, in the region \(|\tilde{x}_{2}|\leq\ell_{1}-\ell_{0}\) with \(\ell_{0}>0\), the following inequality holds  
\[\dot{V}_{x_{1}}\leq-\ell_{0}|\tilde{x}_{1}| \tag{16}\]  
When sliding mode occurs, the reduced-order system dynamics are equivalent to \(\tilde{x}_{2}=k_{1}\text{sign}(\tilde{x}_{1})\), hence, (13) becomes  
\[ \begin{split}\dot{\hat{x}}_{2}&=\tilde{f}_{2}+ \tilde{x}_{e1}-2\omega_{1}\tilde{x}_{2}\\ \dot{\hat{x}}_{e1}&=h_{1}(t)-\omega_{1}^{2}\tilde{x} _{2}\end{split}  \tag{17}\]  
Define the scale estimation error vector as \(\varepsilon=[\varepsilon_{1},\varepsilon_{2}]^{\text{T}}=[\hat{x}_{2},\tilde {x}_{e1}/\omega_{1}]^{\text{T}}\), (17) is rewritten as  
\[\dot{\varepsilon}=\omega_{1}A_{1}\varepsilon+B_{1}\tilde{f}_{2}+C_{1}\frac{h_{ 1}(t)}{\omega_{1}} \tag{18}\]  
where  
\[A_{1}=\left[ \begin{array}{cc}-2&1\\ -1&0\end{array} \right];B_{1}=\left[ \begin{array}{c}1\\ 0\end{array} \right];C_{1}=\left[ \begin{array}{c}0\\ 1\end{array} \right]\]  
Because the matrix \(A_{1}\) is Hurwitz, there exists a symmetric positive definite matrix \(P_{1}\) that satisfies the following Lyapunov equation  
\[A_{1}^{T}P_{1}+P_{1}A_{1}=-2I_{2} \tag{19}\]  
where \(I_{2}\) is the identity matrix of size 2. Consider the following Lyapunov function  
\[V_{\varepsilon}=\frac{1}{2}\varepsilon^{T}P_{1}\varepsilon \tag{20}\]  
Taking time derivative of (20), one obtains  
\[\dot{V}_{\varepsilon}=-\omega_{1}\varepsilon^{T}\varepsilon+\varepsilon^{T}P_{1 }B_{1}\tilde{f}_{2}+\varepsilon^{T}P_{1}C_{1}\frac{h_{1}(t)}{\omega_{1}} \tag{21}\]  
Applying [PERSON]'s inequalities, we have  
\[ \begin{split}\dot{V}_{\varepsilon}&\leq-(\omega_{1} -\frac{1}{2}\lambda_{\max}(X_{1})k_{f_{2}}^{2}-1)\varepsilon^{T}\varepsilon+ \frac{1}{2\omega_{1}^{2}}\lambda_{\max}(Y_{1})\delta_{1}^{2}\\ &\leq-a_{1}V_{\varepsilon}+b_{1}\end{split}