## 3 Controller Design  
### Pressure Observer  
The following observer is proposed to estimate the pressure of the actuator chamber \(A\) and the average air pressure in the transmission line:  
\[\left\{\begin{array}{ll}\dot{\hat{p}}_{\rm a}=\frac{RT_{\rm s}}{V_{\rm a}} \hat{\hat{m}}_{\rm a}-\frac{A_{\rm a}\dot{x}}{V_{\rm a}}\hat{p}_{\rm a}\\ \hat{p}_{l}=\frac{RT_{\rm s}}{V_{\rm l}}(\hat{m}_{\rm v}-\hat{m}_{\rm a})\\ \end{array}\right., \tag{7}\]  
where \(\hat{p}_{\rm a}\) and \(\hat{p}_{l}\) represent the estimates of \(p_{\rm a}\) and \(p_{l}\), \(\hat{\hat{m}}_{\rm a}\) and \(\hat{m}_{\rm v}\) denote the estimated mass flow rates according to Equation (3) and Equation (5) based on \(\hat{p}_{\rm a}\) and \(\hat{p}_{l}\). It should be noted that the proposed observers are closed loop because of the relationship between the estimated mass flow rates and the estimated pressures. Proof of the convergence between the actual pressures and the estimated ones will be given later.  
To verify the convergence between the actual pressures and the estimated pressures, the following Lyapunov function candidate was considered:  
\[V_{\rm o}=V_{\rm o1}+V_{\rm o2},V_{\rm o1}=\frac{1}{2}(\overline{p}_{\rm a}V_{ \rm a})^{2},V_{\rm o2}=\frac{1}{2}(\overline{p}_{l}V_{l})^{2}, \tag{8}\]  
where \(\overline{p}_{\rm a}=p_{\rm a}-\hat{p}_{\rm a}\), and \(\overline{p}_{l}=p_{l}-\hat{p}_{l}\).  
The time derivative of \(V_{\rm o1}\) is:  
\[\begin{array}{ll}\dot{V}_{\rm o1}&=(p_{\rm a}V_{\rm a}-\hat{p}_{\rm a}V_{ \rm a})(\dot{p}_{\rm a}V_{\rm a}+p_{\rm a}\dot{V}_{\rm a}-\dot{\hat{p}}_{\rm a} V_{\rm a}-\hat{p}_{\rm a}\dot{V}_{\rm a})\\ &=RT_{\rm s}V_{\rm a}(p_{\rm a}-\hat{p}_{\rm a})(\dot{m}_{\rm a}-\dot{m}_{\rm a })\end{array} \tag{9}\]  
According the Equation (3), one can obtain:  
1. \(p_{\rm a}>\hat{p}_{\rm a}\) implies that \(\dot{m}_{\rm a}<\hat{\hat{m}}_{\rm a}\), yielding \((p_{\rm a}-\hat{p}_{\rm a})(\dot{m}_{\rm a}-\hat{\hat{m}}_{\rm a})<0\).
2. \(p_{\rm a}<\hat{p}_{\rm a}\) implies that \(\dot{m}_{\rm a}>\hat{\hat{m}}_{\rm a}\), yielding \((p_{\rm a}-\hat{p}_{\rm a})(\dot{m}_{\rm a}-\hat{\hat{m}}_{\rm a})<0\).
3. \(p_{\rm a}=\hat{p}_{\rm a}\) implies that \(\dot{m}_{\rm a}<\hat{\hat{m}}_{\rm a}\), yielding \((p_{\rm a}-\hat{p}_{\rm a})(\dot{m}_{\rm a}-\hat{\hat{m}}_{\rm a})=0\).  
Therefore, \(\dot{V}_{\rm o1}\) is negative semi-definite.  
The time derivative of \(V_{\rm o2}\) is:  
\[\begin{array}{ll}\dot{V}_{\rm o2}&=(p_{l}V_{l}-\hat{p}_{l}V_{l})(\dot{p}_{l }V_{l}-\dot{\hat{p}}_{l}V_{l})\\ &=RT_{\rm s}V_{l}(p_{l}-\hat{p}_{l})(\dot{m}_{\rm v}-\dot{m}_{\rm v})-RT_{\rm s }V_{l}(p_{l}-\hat{p}_{l})(\dot{m}_{\rm a}-\hat{\hat{m}}_{\rm a})\end{array} \tag{10}\]  
According the Equation (3), one can obtain:  
1. \(p_{l}>\hat{p}_{l}\) implies that \(\dot{m}_{\rm a}>\hat{\hat{m}}_{\rm a}\), yielding \((p_{l}-\hat{p}_{l})(\dot{m}_{\rm a}-\hat{\hat{m}}_{\rm a})>0\).
2. \(p_{l}<\hat{p}_{l}\) implies that \(\dot{m}_{\rm a}<\hat{\hat{m}}_{\rm a}\), yielding \((p_{l}-\hat{p}_{l})(\dot{m}_{\rm a}-\hat{\hat{m}}_{\rm a})>0\).
3. \(p_{l}=\hat{p}_{l}\) implies that \(\dot{m}_{\rm a}=\hat{\hat{m}}_{\rm a}\), yielding \((p_{l}-\hat{p}_{l})(\dot{m}_{\rm a}-\hat{\hat{m}}_{\rm a})=0\).  
For the mass flow rate relationship given by Equation (5), it has proven in [25] that the term \((p_{l}-\hat{p}_{l})(\dot{m}_{\rm v}-\hat{\hat{m}}_{\rm v})\) is always non-positive. Thus, \(\dot{V}_{\rm o2}\) and \(V_{\rm o}\) are negative semi-definite, one can conclude that the proposed pressure observers are globally Lyapunov stable with regard to the errors in the estimated pressures.