# Pneumatic Rotary Actuator Position Servo System Based on ADE-PD Control
## 3 Modeling the System
### Mass Flow Throttling Equation  
According to the basic theories of gas dynamics and thermodynamics, the mass flow passing through the port of proportional directional control valve is a function of independent variables which are the displacement \(x\) of proportional valve spool and the gas pressures \(p_{a}\), \(p_{b}\) of two chambers of the actuator [27]. The function can be written as:  
\[x\geq 0,\left\{ \begin{array}{l}\dot{M}_{a}=f_{a}(x,p_{a})\\ M_{b}=f_{b}(x,p_{b})\\ \dot{M}_{a}=f_{a}(x,p_{a})\\ M_{b}=f_{b}t(x,p_{b})\end{array} \right.. \tag{1}\]  
where \(\dot{M}_{a}\) denotes mass flow flowing into chamber **a**, \(\dot{M}_{b}\) is mass flow flowing out from chamber **b**, \(f_{a}\), \(f_{a}\prime\), \(f_{b}\), and \(f_{b}\prime\) are symbols of different equations. Figure 3: Schematic diagram of the valve-controlled actuator. When the proportional valve spool moves in the positive direction, \(x\geq 0\), and chamber **a** is in charge state. The air supply is upflow, and chamber **a** is downflow. While chamber **b** is in discharge state, in which chamber **b** is upflow, and the environment is downflow. The mass flow flowing through chamber **a** and chamber **b** can be expressed by Equations (2) and (3) [28; 29]. \[\dot{M_{a}}=\left\{ \begin{array}{ll}\frac{CWRy_{p}}{\sqrt{RT_{s}}}\sqrt{ \frac{2k}{k-1}\left[\left(\frac{p_{a}}{p_{s}}\right)^{\frac{2}{k}}-\left(\frac {p_{a}}{p_{s}}\right)^{\frac{k+1}{k}}\right]}\left(\frac{p_{a}}{p_{s}}\geq c_{0 }\right)\\ \frac{CWRy_{p}}{\sqrt{RT_{s}}}\sqrt{\frac{2k}{k+1}\left(\frac{2}{k+1}\right)^{ \frac{2}{k-1}}}\left(\frac{p_{a}}{p_{s}}<c_{0}\right)\end{array} \right. \tag{2}\]  
\[\dot{M_{b}}=\left\{ \begin{array}{ll}\frac{CWRy_{p}}{\sqrt{RT_{b}}}\sqrt{ \frac{2k}{k-1}\left[\left(\frac{p_{a}}{p_{s}}\right)^{\frac{2}{k}}-\left(\frac {p_{a}}{p_{s}}\right)^{\frac{k+1}{k}}\right]}\left(\frac{p_{a}}{p_{s}}\geq c_{ 0}\right)\\ \frac{CWRy_{p}}{\sqrt{RT_{b}}}\sqrt{\frac{2k}{k+1}\left(\frac{2}{k+1}\right)^{ \frac{2}{k-1}}}\left(\frac{p_{a}}{p_{s}}<c_{0}\right)\end{array} \right. \tag{3}\]  
where \(C\) represents flow coefficient of orifice in proportional valve; \(W\) is geometric cross-sectional area gradient of orifice in proportional valve; and \(p_{a}\) and \(p_{b}\) are pressures of chamber **a** and **b** respectively; \(p_{s}\) is supply pressure; \(p_{e}\) is atmospheric pressure; \(R\) is gas constant; \(T_{s}\) is ambient temperature; \(T_{b}\) is gas temperature in chamber **b**; \(k\) is adiabatic index of ideal gas; and \(c_{0}\) is critical pressure ratio. According to [PERSON]'s conclusion that the dynamic response of the system is independent of the initial position [30], the position \(i\) of the piston in proportional valve can be set as the initial position. At this point, the Equations (2) and (3) are linearized as Equation (4) by Taylor Formula. \[\left\{ \begin{array}{ll}\Delta\dot{M_{a}}=K_{ma}\Delta x-K_{ca}\Delta p_{a }\\ \Delta\dot{M_{b}}=K_{mb}\Delta x-K_{cb}\Delta p_{b}\end{array} \right. \tag{4}\]  
where \(K_{ma}\) and \(K_{mb}\) are flow gains flowing through **a** port and **b** port of proportional valve, while \(K_{ca}\) and \(K_{cb}\) are flow-pressure coefficients correspondingly. \[K_{ma}=\left.\frac{\partial\dot{M_{a}}}{\partial x}\right|_{i}=\left\{ \begin{array}{ll}\frac{CWRy_{p}}{\sqrt{RT_{s}}}\sqrt{\frac{2k}{k-1}\left[ \left(\frac{p_{ai}}{p_{s}}\right)^{\frac{2}{k}}-\left(\frac{p_{ai}}{p_{s}} \right)^{\frac{k+1}{k}}\right]}\left(\frac{p_{a  
where \(p_{ai}\) and \(p_{bi}\) stand for pressures of chamber \(\mathbf{a}\) and \(\mathbf{b}\) in working position \(\boldsymbol{i}\); \(T_{bi}\) stands for temperature of chamber \(\mathbf{b}\) in working position \(\boldsymbol{i}\); \(x_{i}\) is displacement of proportional valve spool in working position \(\boldsymbol{i}\). When the system is in the initial state, valve spool is in the middle position. The displacement of valve spool at this time is the same as the gap between spool and inwall of the valve, that is \(x_{i}=x_{0}\).