## 2 Plant Description  
The EHS is composed of a servo valve, a symmetrical cylinder, a fixed displacement pump, a motor, and a relief valve as shown in Figure 1. The external load on this EHS is a driven force or torque of any mechatronic plant. The pump outputs the supply pressure \(p_{s}\), which is also the pressure threshold of the relief valve.  
**Hypothesis 1**.: _Since the cut-off frequency of servo valve is far greater than the control system bandwidth, the valve dynamics can be neglected in EHS model construction as \(x_{v}=K_{sv}u\), where \(x_{v}\) is the pool position of servo valve, \(u\) is the control voltage of servo valve, \(K_{sv}\) is the gain of the servo valve [37]._According to Hypothesis 1, if the three state variables are defined as \([x_{1},x_{2},x_{3}]^{T}=[y,\dot{y},p_{L}]^{T}\) where \(y\) and \(\dot{y}\) are the cylinder position and velocity, \(p_{L}\) is the load pressure of the hydraulic cylinder, then the state space model of the EHS is given by  
\[\left\{\begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=\dfrac{1}{m}(A_{p}x_{3}-Kx_{1}-bx_{2}-F_{L})\\ \dot{x}_{3}=-\dfrac{4\beta_{e}A_{p}}{V_{t}}x_{2}-\dfrac{4\beta_{e}C_{ tl}}{V_{t}}x_{3}+\dfrac{4\beta_{e}C_{d}wK_{sv}u}{V_{t}\sqrt{\rho}}\sqrt{p_{s}- \text{sgn}(u)x_{3}}\end{array}\right. \tag{1}\]  
where \(C_{d}\) is the discharge coefficient, \(w\) is the area gradient of the servo valve, \(\rho\) is the density of the hydraulic oil, \(C_{tl}\) is the coefficient of the total leakage of the cylinder, \(\beta_{e}\) is the effective bulk modulus, \(A_{p}\) is the annulus area of the cylinder chamber, \(V_{t}\) is the half-volume of cylinder, \(m\) is the load mass, \(K\) is load spring constant, \(b\) is the viscous damping coefficient of the hydraulic oil, \(F_{L}\) is the external load on the EHS, \(\text{sgn}(\cdot)\) is the sign function.  
**Remark 1**.: _In practice, the hydraulic parameters \(C_{d}\), \(\rho\), \(w\), \(b\), \(\beta_{e}\), \(C_{tl}\) are usually uncertain constants, but the other parameters are known [38; 39]._  
**Remark 2**.: _The external load \(F_{L}\) is unknown dynamic variable, which is caused by the driving force of someone mechatronic plant. Although the dynamic value of \(F_{L}\) depends on the variables \(y,\dot{y},\dot{y}\), \(F_{L}\) is bounded by \(|F_{L}(t)|\leq F_{L\max}\), where \(F_{L\max}\) is an unknown bounded constant [40; 41]._  
Thus from Remarks 1 and 2, the state space model (1) is rewritten as follow  
\[\left\{\begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=\bar{f}_{2}(x_{1},x_{2})+\bar{g}_{2}x_{3}+\Delta_{2}(x_{1},x_{2}) \\ \dot{x}_{3}=f_{3}(x_{2},x_{3})+\bar{g}_{3}(x_{3},u)u+\Delta_{3}(x_{1},x_{2},x_ {3})\end{array}\right., \tag{2}\]  
Figure 1: The EHS control mechanism.  
where \(\bar{C}_{d}\), \(\bar{\rho}\), \(\bar{K}\), \(\bar{b}\), \(\bar{\beta}_{e}\), \(\bar{C}_{tl}\) are nominal values of these uncertain parameters respectively,  
\[\bar{f}_{2}(x_{1},x_{2}) = -\frac{\bar{K}x_{1}+\bar{b}x_{2}}{m},\quad\bar{g}_{2}=\frac{A_{p} }{m}\] \[\bar{f}_{3}(x_{2},x_{3}) = -\frac{4\bar{\beta}_{e}A_{p}}{V_{t}}x_{2}-\frac{4\bar{\beta}_{e} \bar{C}_{tl}}{V_{t}}x_{3} \tag{3}\] \[\bar{g}_{3}(x_{3},u) = \frac{4\bar{\beta}_{e}\bar{C}_{d}wK_{sp}}{V_{t}\sqrt{\bar{\rho}}} \sqrt{p_{s}-\text{sgn}(u)x_{3}}\]  
and \(\Delta_{2}(x_{1},x_{2})=f_{2}(x_{1},x_{2})-\bar{f}_{2}(x_{1},x_{2})-F_{L}(t)/m\), \(\Delta_{3}(x_{1},x_{2},x_{3})=f_{3}(x_{2},x_{3})-\bar{f}_{3}(x_{2},x_{3})+g_{3 }(x_{1},x_{2},x_{3})-\bar{g}_{3}(x_{1},x_{2},x_{3})\) are the integrated elements of parametric uncertainties and the external load disturbance.  
Due to limited boundaries of the parametric uncertainties and the external load mentioned in Remarks 1 and 2, the two uncertain nonlinearities \(\Delta_{2}\), \(\Delta_{3}\) are bounded by \(|\Delta_{2}|<\Delta_{2\max}\), \(|\Delta_{3}|<\Delta_{3\max}\), where \(\Delta_{2\max}\), \(\Delta_{3\max}\) are unknown bounded constants [28].