## 4 Simulations
### Application to Electro Hydrostatic Actuator System  
As presented in this section, the proposed control algorithm developed previously was applied to stabilize an Electro Hydrostatic Actuator (EHA) system. The EHA studied in this section is illustrated in Figure 7, and the major components are listed in Table 2. In the hydraulic circuit, cylinder (1) is driven by the valve system and the hydraulic pump (7), which is actuated by the AC motor (8). The flow discrepancy between the bore chamber and the rod chamber of the cylinder is automatically compensated by the check valves (4.1) and (4.2), with the support of the pilot-operated directional valve (5). In cooperation with two pilot-operated check valves (6), the hydraulic pump regulates the fluid flow inside the hydraulic circuit and consequently controls the movement of the cylinder. Figure 6: Influence of design parameters on observer performance. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Device** & **No.** & **Device** \\ \hline
1 & Hydraulic cylinder & 5 & Directional valve \\
2 & Relief valves & 6 & Pilot check valve \\
3 & Reservoir & 7 & Hydraulic pump \\
4.1 & Check valve 4.1 & 8 & AC motor \\
4.2 & Check valve 4.2 & 9 & AC motor driver \\ \hline \hline \end{tabular} Parameters used for mathematical modeling the EHA system are defined in Table 3 below.  
\end{table} 
Table 3: Major components in the considered EHA system. Figure 7: Schematic diagram of the studied EHA system. Referring to [52], the dynamic model of the EHA system studied here consists of force dynamics and pressure dynamics, as follows. The force dynamics of the actuator is expressed as:  
\[m\ddot{x}=P_{1}A_{1}-P_{2}A_{2}+f_{Fr}+\delta_{2} \tag{42}\]  
where \(\delta_{2}\) represents a force disturbance that may include external forces and unmodeled force dynamics; \(f_{Fr}\) denotes a lumped friction force [53; 54] and is approximated as a differential function of the form:  
\[\left\{ \begin{array}{l}f_{Fr}=-b_{1}\dot{x}-b_{2}\text{tanh}\big{(}\dot{x} \big{)}\\ \text{tanh}\big{(}\dot{x}\big{)}=\frac{1-e^{-\gamma_{2}\dot{x}}}{1+e^{-\gamma_ {2}\dot{x}}}\end{array} \right. \tag{43}\]  
Governed by the continuity law [55], the pressure dynamics of the actuator are:  
\[\left\{ \begin{array}{l}V_{1t}\dot{P}_{1}=\beta_{e}\big{(}\eta_{V}DK_{dr}u-A_ {1}\dot{x}-C_{Li}(P_{1}-P_{2})-Q_{cv1}\big{)}+\delta_{31}\\ V_{2t}\dot{P}_{2}=\beta_{e}(-\eta_{V}DK_{dr}u+A_{2}\dot{x}+C_{Li}(P_{1}-P_{2})+ Q_{cv2})+\delta_{32}\end{array} \right. \tag{44}\]  
where \(Q_{cv1}\) and \(Q_{cv2}\) are, respectively, the flow in the check valves (4.1) and (4.2), and described as:  
\[\left\{ \begin{array}{l}Q_{cv1}=-\dot{x}(A_{1}-A_{2})\text{sm}\big{(}-\dot{x} \big{)}\\ Q_{cv2}=\dot{x}(A_{1}-A_{2})\text{sm}\big{(}\dot{x}\big{)}\\ \text{sm}(*)=(1+e^{-\gamma_{3}*})^{-1}\end{array} \right. \tag{45}\]  
By defining the system state as \([x_{1},x_{2},x_{3}]^{T}=\big{[}x,\dot{x},(P_{1}A_{1}-P_{2}A_{2})/m\big{]}^{T}\) and combining equations from (42) to (45), the total dynamic system can be expressed in a canonical form as follows:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=x_{3}+d_{1}\\ \dot{x}_{3}=f+g\,u+d_{2}\end{array} \right. \tag{46}\]  
where the dynamic functions are specifically defined as:  
\[\left\{ \begin{array}{l}f=\frac{\beta_{e}}{m}\Big{[}\frac{A_{1}}{V_{1t}}(-A_ {1}+(A_{1}-A_{2})sm(-x_{2}))-\frac{A_{2}}{V_{2t}}(A_{2}+(A_{1}-A_{2})sm(x_{2}) )\Big{]}x_{2}\\ g=\frac{\beta_{e}\gamma_{V}DK_{dr}}{m}\big{(}\frac{A_{1}}{V_{1t}}+\frac{A_{2}}{ V_{2t}}\big{)}\\ d_{1}=\frac{1}{m}\big{(}f_{Fr}+\delta_{2}\big{)}\\ d_{2}=-\frac{\beta_{e}}{m}C_{Li}(P_{1}-P_{2})\Big{(}\frac{A_{1}}{V_{1t}}+\frac{ A_{2}}{V_{2t}}\Big{)}+\frac{A_{1}\delta_{11}}{mV_{1t}}-\frac{A_{2}\delta_{23}}{mV_{2t}} \end{array} \right.