## 2 Modeling of Electro-Hydraulic Servo System for Valve-Controlled Cylinder
### System Modeling  
Servo valve dynamic characteristics equation: The deviation voltage input signal of a servo valve is derived from the amplifier, the spool displacement is the output signal, and the second-order oscillation element is usually used to simplify the description of the dynamic characteristics of the servo valve [37], whose ideal transfer function is:  
\[\frac{X_{v}(s)}{\Delta U(s)}=\frac{K_{sv}}{\frac{s^{2}}{\omega_{sv}^{2}}+ \frac{2\xi_{sv}}{\omega_{sv}}}, \tag{1}\]  
Figure 1: Configuration of electro-hydraulic position servo system. Figure 2: System of four-way slide valve-controlled asymmetric cylinder. where \(\Delta u\) is the deviation voltage input signal of the servo valve in volts (V); \(K_{sp}\) is the servo valve amplification factor in meters per volt (m/V); \(\omega_{sv}\) is the servo valve natural frequency in radians per second (rad/s); \(\xi_{sD}\) is the servo valve damping ratio; \(x_{p}\) is the valve spool displacement in meters (m). The system deviation input equation: the input voltage is \(u_{g}\), the displacement sensor can be regarded as a proportional element, its feedback voltage is \(u_{f}\), and the gain is \(K_{f}\); assume that the system controller is a proportional controller, and the amplification factor is \(K_{p}\). From the feedback control principle, the relationship between the deviation voltage \(\Delta u\) and the system input voltage \(u_{g}\) is:  
\[\Delta u=K_{p}(u_{g}-u_{f})=K_{p}(u_{g}-K_{f}x_{p}), \tag{2}\]  
where \(u_{g}\) is the input voltage in V; \(u_{f}\) is the hydraulic cylinder piston rod displacement feedback voltage in V; \(K_{f}\) is the displacement sensor gain in volts per meter (V/m); \(K_{p}\) is the proportional controller amplification factor; \(x_{p}\) is the piston rod displacement in m.  
Servo valve flow equation: assuming that the rightward movement of the spool is positive, that is, \(x_{p}\geq 0\), the piston rod is pushed to the right, then based on the valve port flow-pressure equation, the flow rate \(q_{1}\) into the left chamber of the hydraulic cylinder and the flow rate \(q_{2}\) out of the right chamber of the hydraulic cylinder are:  
\[q_{1} = C_{d}Wx_{v}\sqrt{\frac{2}{\rho}|p_{s}-p_{1}|}\text{sign}(p_{s}- p_{1}), \tag{3}\] \[q_{2} = C_{d}Wx_{v}\sqrt{\frac{2}{\rho}|p_{2}|}, \tag{4}\]  
where sign indicates the symbolic function. When the valve spool moves left, that is, \(x_{v}<0\), the piston rod is pushed to the left, the flow rate \(q_{1}\) out of the left chamber of the hydraulic cylinder and the flow rate \(q_{2}\) into the right chamber of the hydraulic cylinder are:  
\[q_{1} = C_{d}Wx_{v}\sqrt{\frac{2}{\rho}|p_{1}|}, \tag{5}\] \[q_{2} = C_{d}Wx_{v}\sqrt{\frac{2}{\rho}|p_{s}-p_{2}|}\text{sign}(p_{s}- p_{2}), \tag{6}\]  
where \(q_{1}\) is the hydraulic cylinder left chamber flow rate in cubic meters per second (m\({}^{3}\)/s); \(q_{2}\) is the hydraulic cylinder right chamber flow rate in m\({}^{3}\)/s; \(C_{d}\) is the orifice flow factor; \(W\) is the orifice area gradient in m; \(\rho\) is the oil density in kilograms per cubic meter (kg/m\({}^{3}\)); \(p_{s}\) is the system oil supply pressure in mega pascals (MPa); \(p_{1}\) is the hydraulic cylinder left chamber pressure in MPa; \(p_{2}\) is the hydraulic cylinder right chamber pressure in MPa.