## 2 System Overview and Analysis
### System Model and Analysis  
In this paper, we do not discuss the issue of the manipulation simulation of the booster. Hence, from the above descriptions of the manipulation simulation process, and to facilitate the study, the booster can be regarded as an electromagnetic valve controlled a hydraulic cylinder that is driven by the electromagnetic valve input. The studied equivalent schematic diagram is shown in Figure 2.  
The mathematical model of the electro-hydraulic loading system can be set up by means of theoretical analysis or experiment identification. The theoretical model plays an important role in the design of controllers and the selection of system parameters. As shown in Figure 2, the load force \(F_{\mathrm{L}}\) (N) applied to the booster can be measured by the load cell, and in consideration of \(K_{\mathrm{s}}\) (N/m), the elastic stiffness of the force sensor, it can be expressed as  
\[F_{\mathrm{L}}  
points. When the valve opening is very small, the flowrate-pressure characteristics are determined by the leakage characteristics between the sleeve and the spool of the valve, and then \(K_{\mathrm{c}}\) should be calculated by \(K_{\mathrm{c}}=\pi v_{\mathrm{c}}^{2}w/(32\mu)\) according to the laminar flow formula.  
The flowrate continuity equation of the hydraulic power mechanism can be expressed as  
\[q_{\mathrm{L}}=A_{\mathrm{L}}\frac{dx_{\mathrm{L}}}{dt}+\frac{V_{\mathrm{L}}} {4\beta_{\mathrm{e}}}\frac{dp_{\mathrm{L}}}{dt}+C_{\mathrm{sl}}p_{\mathrm{L}}, \tag{4}\]  
where \(A_{\mathrm{L}}\) is the piston effective area(m\({}^{2}\)); \(\beta_{\mathrm{e}}\) is the fluid bulk modulus (N/m\({}^{2}\)); \(C_{\mathrm{sl}}\) is the total leakage coefficient of the hydraulic cylinder (m\({}^{5}\)/(N\(\cdot\) s)); and \(V_{\mathrm{L}}\) represents the total control volume of the loading system (m\({}^{3}\)). The force equilibrium equation for the loading hydraulic cylinder piston is  
\[A_{\mathrm{L}}p_{\mathrm{L}}=m_{\mathrm{L}}\frac{d^{2}x_{\mathrm{L}}}{dt^{2}} +B_{\mathrm{L}}\frac{dx_{\mathrm{L}}}{dt}+F_{\mathrm{L}}, \tag{5}\]  
where \(m_{\mathrm{L}}\) is the equivalent load mass (kg) including the mass of the piston, rods, force sensor and the load; and \(B_{\mathrm{L}}\) represents the equivalent viscous damping coefficient (N\(\cdot\) s/m). Combining and solving Equations (1) and (3)-(5), the transfer function of the output load force \(F_{\mathrm{L}}\), with the valve spool displacement \(x_{\mathrm{v}}\) as the input and the displacement \(x_{\mathrm{M}}\) of the loaded booster as the disturbance, can be derived as follows  
\[F_{\mathrm{L}}(s)=G_{\mathrm{P}}(s)X_{\mathrm{v}}(s)-G_{\mathrm{d}}(s)sX_{ \mathrm{M}}(s)=\frac{N_{\mathrm{U}}(0)}{P_{\mathrm{D}}(s)}X_{\mathrm{v}}(s)- \frac{N_{\mathrm{D}}(s)}{P_{\mathrm{D}}(s)}sX_{\mathrm{M}}(s), \tag{6}\]  
where \(K_{\mathrm{tm}}=K_{\mathrm{c}}+C_{\mathrm{sl}}\); \(N_{\mathrm{D}}(s)=\frac{V_{\mathrm{L}}m_{\mathrm{L}}}{4\beta_{\mathrm{e}}K_{ \mathrm{tm}}}s^{2}+\left(m_{\mathrm{L}}+\frac{V_{\mathrm{L}}B_{\mathrm{L}}}{4 \beta_{\mathrm{e}}K_{\mathrm{tm}}}\right)s+B_{\mathrm{L}}+\frac{A_{\mathrm{L}} ^{2}}{K_{\mathrm{tm}}}\); \(N_{\mathrm{U}}(0)=A_{\mathrm{L}}K_{\mathrm{q}}/K_{\mathrm{tm}}\); \(P_{\mathrm{D}}(s)=\frac{V_{\mathrm{L}}m_{\mathrm{L}}}{4\beta_{\mathrm{e}}K_{ \mathrm{tm}}K_{\mathrm{s}}}s^{3}+\left(\frac{m_{\mathrm{L}}}{K_{\mathrm{s}}}+ \frac{V_{\mathrm{L}}B_{\mathrm{L}}}{4\beta_{\mathrm{e}}K_{\mathrm{tm}}K_{ \mathrm{s}}}\right)s^{2}+\left(\frac{B_{\mathrm{L}}}{K_{\mathrm{s}}}+\frac{A_ {\mathrm{L}}^{2}}{K_{\mathrm{tm}}K_{\mathrm{s}}}+\frac{V_{\mathrm{L}}}{4\beta _{\mathrm{e}}K_{\mathrm{tm}}}\right)s+1\); \(X_{\mathrm{v}}(s)\) and \(X_{\mathrm{M}}(s)\) denote the Laplace transform of \(x_{\mathrm{v}}\) and \(x_{\mathrm{M}}\), respectively. It can be noticed from Equation (6) that, besides the valve spool displacement \(x_{\mathrm{v}}\), \(F_{\mathrm{L}}\) is related to the booster displacement \(x_{\mathrm{M}}\) as well. The latter motion disturbance exactly results in the extraneous force in the loading system.