## 3 Active Disturbance Rejection Control Design
### Observer Design
#### 3.1.2 Matched Disturbance Observer Design  
Since the load pressure \(P_{L}\) can be directly calculated through measured pressures \(P_{1}\) and \(P_{2}\) in the forward chamber and reverse chamber of the HRA, respectively, the LESO is constructed to estimate the lumped disturbance in the hydraulic system as  
\[\begin{split}\dot{\hat{x}}_{3}&=f_{3}(\hat{\mathbf{ x}})+g_{3}(\hat{\mathbf{x}},u)u+\hat{x}_{\varepsilon 2}+2\omega_{2}(x_{3}-\hat{x}_{3})\\ \dot{\hat{x}}_{\varepsilon 2}&=\omega_{2}^{2}(x_{3}-\hat{x}_ {3})\end{split} \tag{24}\]  
where \(x_{\varepsilon 2}=d_{2}(t)\) and \(\hat{x}_{\varepsilon 2}=h_{2}(t)\) with \(h_{2}(t)\) is the first-order derivative of \(d_{2}(t)\).  
Set \(\hat{x}_{3}=x_{3}-\hat{x}_{3}\) and \(\hat{x}_{\varepsilon 2}=x_{\varepsilon 2}-\hat{x}_{\varepsilon 2}\) as the estimation errors, the estimator error dynamics of the LESO are given by  
\[\begin{split}\dot{\hat{x}}_{3}&=\tilde{f}_{3}+\tilde {x}_{\varepsilon 2}-2\omega_{2}\tilde{x}_{3}\\ \dot{\hat{x}}_{\varepsilon 2}&=h_{2}(t)-\omega_{2}^{2} \tilde{x}_{3}\end{split} \tag{25}\]  
where \(\tilde{f}_{3}=f_{3}(\mathbf{x})-f_{3}(\hat{\mathbf{x}})\), and \(\omega_{2}\) is the bandwidth of the constructed LESO.  
Defining \(\eta=[\eta_{1},\eta_{2}]^{T}=[\hat{x}_{3},\hat{x}_{\varepsilon 2}/\omega_{2}]^{T}\) as scaled estimation error vector, the error dynamics (25) can be transformed into  
\[\dot{\eta}=\omega_{2}A_{2}\eta+B_{2}\tilde{f}_{3}+C_{2}\frac{h_{2}(t)}{\omega_ {2}} \tag{26}\]  
where  
\[A_{2}=\left[\begin{array}{cc}-2&1\\ -1&0\end{array}\right];B_{2}=\left[\begin{array}{c}1\\ 0\end{array}\right];C_{2}=\left[\begin{array}{c}0\\ 1\end{array}\right]\]Since the matrix \(A_{2}\) is strictly negative definite, there exists a symmetric positive definite \(P_{2}\) that satisfies the following [PERSON] equality  
\[A_{2}^{T}P_{2}+P_{2}A_{2}=-2I_{2} \tag{27}\]  
where \(I_{2}\) is the identity matrix of size 2.  
Theorem 2: _The established LESO guarantees that the estimation error of the lumped matched disturbance \(d_{2}(t)\) converge to an arbitrarily small bounded region whose boundary depends on the selection of the observer gain \(\omega_{2}\) and the estimation accuracy of \(x_{2}\) of the above ESMO._  
Proof of Theorem 2: Consider the following candidate Lyapunov function  
\[V_{\eta}=\frac{1}{2}\eta^{T}P_{2}\eta \tag{28}\]  
Differentiating (28) with respect to time yields  
\[\dot{V}_{\eta}=-\omega_{2}\eta^{T}\eta+\eta^{T}P_{2}B_{2}\tilde{f}_{3}+\eta^{ T}P_{2}C_{2}\frac{h_{2}(t)}{\omega_{2}} \tag{29}\]  
Applying [PERSON]'s inequality, we have  
\[\dot{V}_{\eta} \leq-\omega_{2}\eta^{T}\eta+\frac{1}{2}\eta^{T}\eta+\frac{1}{2}B _{2}^{T}P_{2}B_{2}\tilde{f}_{3}^{2}+\frac{1}{2}\eta^{T}\eta+\frac{1}{2}C_{2} ^{T}P_{2}^{T}P_{2}C_{2}\frac{h_{2}^{2}(t)}{\omega_{2}^{2}} \tag{30}\] \[\leq-(\omega_{2}-1)\eta^{T}\eta+\frac{1}{2}\lambda_{\max}(X_{2}) k_{f_{3}}\tilde{x}_{2}^{2}+\frac{1}{2}\lambda_{\max}(Y_{2})\frac{\delta_{2}^{2}}{ \omega_{2}^{2}}\]  
As stated in Theorem 1, it can be observed that \(\tilde{x}_{2}\) converges to the arbitrarily small region, hence, a globally ultimately bounded stability of the LESO is ensured. The bound of this region is specified by the designed observer bandwidths \(\omega_{1}\) and \(\omega_{2}\).  
This completes the proof of Theorem 2.  
Remark 2: _By appropriately choosing the switching gain \(\ell_{1}\) and the bandwidth \(\omega_{1}\) of the ESMO, the estimation errors of inmeasurable velocity and lumped mismatched uncertainty reach arbitrarily small bounded regions. The estimation accuracy of the generalized matched disturbance in hydraulic dynamics depends on not only the bandwidth of the LESO, i.e., \(\omega_{2}\) but also the estimation accuracy of the angular velocity._