## 3 Electro-Hydraulic Servo System Modeling
### Modeling of the Position Servo System  
In the position control loop, a servo valve is used to control the trajectory of the hydraulic cylinder \(x_{L}\) to follow a given signal \(x_{d}\). The force on the hydraulic cylinder can be given using Equation (1):  
\[\ddot{x}_{L}=\frac{1}{m}(p_{1}A_{1}-p_{2}A_{2}-b\dot{x}_{L}-f_{L})+\Delta e_{2}, \tag{1}\]  
where \(m\) is the total mass of the piston, \(f_{L}\) is the load force, which is related to the thickness and the material of the steel plate and the shear speed and time, \(b\) is the viscous damping coefficient and \(\Delta e_{2}\) is the uncertainty due to parameter perturbation, modeling errors, and disturbances. Since the natural frequency of the valve is much higher than the natural frequency of the system, the valve's dynamics can be regarded as a proportional component. The relationship between the spool displacement and the input signal is given by Equation (2):  
\[x_{v1}=k_{s1}u_{1}, \tag{2}\]  
where \(x_{v1}\) is the spool's displacement, \(k_{s1}\) is the valve gain and \(u_{1}\) is the signal input to the valves; \(q_{1}\) represents the flow at the inlet, which is related to the opening of the valve orifice \(x_{v1}\) and the pressure drop \(\Delta p_{1}\)[27], and is obtained using Equation (3):  
\[q_{1}=k_{v1}x_{v1}\sqrt{\Delta p_{1}}, \tag{3}\]  
where \(k_{v1}\) is the flow coefficient and \(\Delta p_{1}\) is the pressure drop of the valve, which is given by Equation (4):  
\[\Delta p_{1}=\left\{\begin{array}{l}p_{s}-p_{1},\;x_{v1}\geq 0\\ p_{1}-p_{o},\;x_{v1}<0\end{array}\right., \tag{4}\]  
where \(p_{s}\) is the supply pressure and \(p_{o}\) is the tank pressure.  
For the chamber at the inlet, according to the flow continuity equation, Equation (5) is obtained and shown as following:  
\[\dot{p}_{1}=[q_{1}-A_{1}\dot{x}_{L}-C_{v1}p_{1}-C_{v1}(p_{1}-p_{2})]\frac{ \beta_{e}}{V_{1}+A_{1}x_{L}}+\Delta e_{3}, \tag{5}\]  
where \(\beta_{e}\) is the effective oil bulk modulus, \(p_{1}\) is the pressure of the piston chamber, \(p_{2}\) is the pressure of the rod chamber, \(A_{1}\) is the area of the piston, \(A_{2}\) is the effective area of the rod, \(x_{L}\) is the displacement of the rod, \(V_{1}\) and \(V_{2}\) are the initial volumes of the cylinder chambers, including the chamber volume of the connected pipeline, \(C_{v1}\) is the external leakage coefficient, \(C_{ip1}\) is the internal leakage coefficient and \(\Delta e_{3}\) is a variable arising from parameter perturbation, modeling errors, and disturbance.  
Considering that \(u_{1}\) is the system input and \(x_{L}\) is the system output, the system state variable is defined using Equation (6):  
\[x_{i}=[x_{i1},x_{i2},x_{i3}]^{T}=[x_{L},\dot{x}_{L},p_{1}]^{T}. \tag{6}\]  
According to Equations (1)-(5), the model of the position servo system can be illustrated using Equation (7):  
\[\left\{\begin{array}{l}\dot{x}_{i1}=x_{i2}\\ \dot{x}_{i2}=c_{1}x_{i3}+c_{2}x_{i2}+\delta_{i2}+\theta_{i2}\\ \dot{x}_{i3}=c_{3}u_{1}+c_{4}x_{i2}+\theta_{i3}\end{array}\right., \tag{7}\]where \(\delta_{i2}=\frac{-p_{2}A_{2}-f_{L}}{m}\), \(c_{1}=\frac{A_{1}}{m}\), \(c_{2}=-\frac{b}{m}\), \(c_{3}=\frac{\beta_{e}}{V_{1}}k_{v1}k_{s1}\sqrt{\Delta p_{1}}\), \(c_{4}=-\frac{\beta_{e}}{V_{1}}A_{1}\) and \(\theta_{i2}\) and \(\theta_{i3}\) are the uncertainties due to modeling errors, parameter uncertainties, and disturbance caused by the varying supply pressure. An NDO is designed to compensate for \(\theta_{i2}\) and \(\theta_{i3}\) (as explained in Section 4).