## 5 Simulation Experiment  
In the above sections, the outer loop controller and the disturbance compensator have been designed based on the established linear mathematical model with perturbations. The influence of some factors has been neglected, and part of the system nonlinearity has been treated by equivalent linearization. In order to validate the effectiveness of the designed hybrid controller, a simulation model that is close to the practical situation is needed. So, in the simulation experiment section, the model of the loading system is built in AMSEim (LMS Imagine, Leuven, Belgium, 2010), as depicted in Figure 15a; it contains the nonlinear characteristics of the load flowrate of the servo valve and the un-modeled dynamics, and it is closer to the actual plant than the linear model. The developed algorithm is conducted in MATLAB/Simulink (MathWork, New York, NY, USA, 2010). Through co-simulation using AMSEim and MATLAB/Simulink as shown in Figure 15b, the performance of the designed hybrid controller can be evaluated. The schematic diagrams of co-simulation are shown in Figure 15.  
In Figure 15a, the gain block with a value of 1000 is used to convert the unit of the control current of the servo valve from amperes (A) in MATLAB to milliamperes (mA) in AMSEim, and the gain for the signal output of the force sensor is set to \(-1\) to make the output direction of the force sensor consistent with Equation (1).  
In the AMSEim model, besides the nonlinear dynamic characteristics of the servo valve, the following factors are considered: the supply pressure; the internal leakage, static friction, Coulomb friction and viscous friction of the loading cylinder; the equivalent mass of the load; and the oil bulk modulus. Parameters related to the loading cylinder are chosen according to the characteristics of the servo hydraulic cylinder in the industrial environment. A friction model with the Stribeck effect is designed. The used parameters in AMSEim are listed in Table 2 and are consistent with those in Table 1.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Supply pressure \(p_{\text{s}}\) & 180–210 bar & Clearance on diameter & 0.02 mm \\ Maximum flow rate & 40 L/min & Coefficient of viscous friction & 800 N/(m/s) \\ Rated pressure drop & 70 bar & Maximum static force & 150 N \\ Valve rated current & 20 mA & Maximum Coulomb friction & 100 N \\ Valve natural frequency & 200 Hz & Force sensor stiffness & \(1\times 10^{8}\) N/m \\ Valve damping ratio & 0.7 & Dead volume & 40 cm\({}^{3}\) \\ Piston diameter & 70 mm & Mass \(m_{\text{L}}\) & 8–12 kg \\ Rod diameter & 42 mm & Oil modulus \(\beta_{\text{e}}\) & 7000–11,000 bar \\ Full stroke & 150 mm & Oil density & 870 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters in AMSEim.  
Figure 15: Schematic diagrams of co-simulation: (**a**) AMSEim block diagram; and (**b**) co-simulation in MATLAB/Simulink.  
In the course of co-simulation, all experiments are performed under the typical operating condition, _i.e._, in the presence of motion disturbance, and some unified settings are made: (1) The motion disturbance input is set as \(x_{\text{M}}=7\text{sin}14t\) (mm/s) unless otherwise stated to produce the largest motion disturbance. Its magnitude is higher than the maximum magnitude when the motion disturbance is at its maximum frequency. (2) The \(\dot{x}_{\text{M}}\) is acquired by the direct differential method. (3) The nominal values of the perturbed parameters \(p_{\text{s}}\), \(\beta_{\text{e}}\), and \(m_{\text{L}}\) are 21, 900 MPa, and 10 kg, respectively. (4) The AMESim print interval is set as 0.0001 s.