## 2 System Overview and Analysis
### Equivalent Linear Model Set  
Normally, we linearize the flow quantity equation of the valve around the neutral position, because the general hydraulic servo system is often working around that, and \(K_{\mathrm{q}}\) and \(K_{\mathrm{c}}\) are obtained as fixed constants. However, for the hydraulic loading system with strong position interference, its operating point varies over a large range. If the loading system is just described by a linear plant with linear parameters around one operating point, it is only possible to approximate the system dynamics over a limited region of operation. Moreover, parameter uncertainty in some hydraulic systems is inevitable and this can degrade the closed-loop performance. In particular, with the requirement of high closed-loop bandwidth (80 Hz), the inaccurately described system leads to unreliable design. Therefore, any single linear time invariant (LTI) model cannot describe the whole system's dynamics, and it is necessary to establish a set of equivalent LTI models that describe the dynamics of the nonlinear hydraulic system over the whole envelope of operation. From the design point of view, we believe that the transfer function \(G_{\mathrm{v}}(s)\) can accurately reflect the dynamic characteristics of the servo valve, and some structural parameters, such as \(V_{\mathrm{L}}\), \(A_{\mathrm{L}}\), \(K_{\mathrm{s}}\), _etc._, can be precisely measured and kept constant. However, the equivalent load mass \(m_{\mathrm{L}}\), which is described by the lumped mass method, is difficult to measure precisely, and the fluid bulk modulus \(\beta_{\mathrm{e}}\) fluctuates with fluid temperature and mixed air. According to the maximum load demand of the booster with some appropriate margin (from \(-11\) to \(22\) kN), the initial boundaries of \(K_{\mathrm{q}}\) and \(K_{\mathrm{c}}\) can be calculated. The electro-hydraulic loading system generally uses the servo valve with zero opening or positive opening, and even the zero-opening servo valve generally also has a very small positive opening (under-lapped). The positive opening and the variation of oil supply pressure \(p_{\mathrm{s}}\) also result in the perturbations of \(K_{\mathrm{q}}\) and \(K_{\mathrm{c}}\). A lumped perturbation range can be obtained by integrating these perturbations into the initial boundaries of \(K_{\mathrm{q}}\) and \(K_{\mathrm{c}}\). Then, taking into account the leakage of the whole system can get the boundary of \(K_{\mathrm{tm}}\). Table 1 lists all the measured and calculated values of the parameters of the loading system. From the model derived above, the loading system is a fifth-order system with motion disturbance and variable parameters. The denominator polynomial \(P_{\mathrm{D}}(s)\) of \(G_{\mathrm{P}}(s)\) can be expressed as follows  
\[P_{\mathrm{D}}\left(s\right)=\left(\frac{s}{\omega_{\mathrm{L}}}+1\right) \left(\frac{s^{2}}{\omega_{\mathrm{h}}^{2}}+\frac{2\xi_{\mathrm{h}}}{\omega_{ \mathrm{h}}}s+1\right) \tag{8}\]  
where \(\omega_{\mathrm{L}}\) is the cutoff frequency of the inertial term and \(\omega_{\mathrm{L}}=K_{\mathrm{tm}}K_{\mathrm{h}}K_{\mathrm{s}}/\left[A_{ \mathrm{L}}^{2}(K_{\mathrm{h}}+K_{\mathrm{s}})\right]\); \(\omega_{\mathrm{h}}\) and \(\xi_{\mathrm{h}}\) are the second-order oscillation frequency and the damping ratio of the force loop, respectively; \(\omega_{\mathrm{h}}=\omega_{0}\sqrt{(K_{\mathrm{h}}+K_{\mathrm{s}})/K_{ \mathrm{h}}}\); \(K_{\mathrm{h}}=4\beta_{\mathrm{e}}A_{\mathrm{L}}^{2}/V_{\mathrm{L}}\) is the liquid spring stiffness, \(\omega_{0}=\sqrt{4\beta_{\mathrm{e}}A_{\mathrm{L}}^{2}/\left(V_{\mathrm{L}}m_{ \mathrm{L}}\right)}\)