# Nonlinear Adaptive Robust Control of the Electro-Hydraulic Servo System
## 2 Modeling of the Hydraulic System  
The hydraulic system shown in Figure 1 is comprised of a single-rod cylinder, a 4/3- way servo valve, and a mass load. The goal of this paper is to make the mass load track the desired motion trajectory as closely as possible.  
The dynamic of the piston can be described as follows::  
\[P_{1}A_{1}-P_{2}A_{2}=m\ddot{x}+f_{L} \tag{1}\]  
where \(P_{1}\), \(P_{2}\) are the pressure in two chambers of cylinder, respectively. \(A_{1}\), \(A_{2}\) are the piston area of the two chambers, respectively. \(m\) is the mass of load, \(x\) is the displacement of load, and \(f_{L}\) is lumped disturbance acting on the cylinder.  
Assuming that both the internal leakage f of the servo valve and the external leakage flow of the cylinder have been neglected, then the dynamic of oil flow of the cylinder can be written as  
\[\begin{cases}A_{1}\dot{x}+C_{t}(P_{1}-P_{2})+\frac{V_{01}+A_{1}x}{\beta_{e}} \dot{P_{1}}=Q_{1}\\ A_{2}\dot{x}+C_{t}(P_{1}-P_{2})=\frac{V_{02}-A_{2}x}{\beta_{e}}\dot{P_{2}}+Q_{2 }\end{cases} \tag{2}\]  
where \(C_{t}\) is the coefficient of cylinder internal leakage flow, and \(\beta_{e}\) is the effective bulk modulus of oil in the container. \(V_{01}\), \(V_{02}\) are the original total control volumes of two chambers respectively, which include the volume of the servo valve, pipelines, and cylinder chambers, and \(V_{1}=V_{01}+A_{1}x\), \(V_{2}=V_{02}-A_{2}x\). \(Q_{1}\) represents the supply flow rate of the forward chamber, and \(Q_{2}\) represents the return flow rate of the return chamber.  
\(Q_{1}\) and \(Q_{2}\) are given as  
\[\begin{cases}Q_{1}=k_{q}x_{v}[s(x_{v})\sqrt{P_{s}-P_{1}}+s(-x_{v})\sqrt{P_{1}- P_{r}}]\\ Q_{2}=k_{q}x_{v}[s(x_{v})\sqrt{P_{2}-P_{r}}+s(-x_{v})\sqrt{P_{s}-P_{2}}]\end{cases} \tag{3}\]  
with an index function of valve direction \(s(x)\) being defined as  
\[s(x)=\begin{cases}1,&\text{if }x\geq 0\\ 0,&\text{if }x<0\end{cases} \tag{4}\]  
where \(P_{s}\) is the system supply pressure and \(P_{r}\) is the container pressure. \(x_{v}\) is the spool displacement of servo valve, \(k_{q}\) is the valve flow gain coefficient and \(k_{q}=C_{d}w\sqrt{\frac{2}{\rho}}\), in which \(C_{d}\) is the discharge coefficient, \(w\) is the spool area gradient, and \(\rho\) is the oil fluid density.  
A high-performance servo valve having a natural frequency of 80 Hz is used, which is much higher than the frequency of desired motion trajectories. Thus, we can assume that the dynamic of spool displacement is linear with the control input, and the following equation \(x_{v}=yu\) is satisfied, where \(\psi\) is a positive constant, and \(u\) is control input. Meanwhile, \(s(x_{v})=s(u)\) can be obtained by following the above assumption.  
Figure 1: The scheme of the hydraulic system.  
Therefore, Equation (3) can be transformed into  
\[\begin{cases}Q_{1}=gu[s(u)\sqrt{P_{s}-P_{1}}+s(-u)\sqrt{P_{1}-P_{r}}]\\ Q_{2}=gu[s(u)\sqrt{P_{2}-P_{r}}+s(-u)\sqrt{P_{s}-P_{2}}]\end{cases} \tag{5}\]  
where \(g=k_{q}\psi\).  
Defining the system state variables as  
\[x\triangleq(x_{1},x_{2},x_{3},x_{4})^{\mathrm{T}}=(y,\dot{y},P_{1},P_{2})^{ \mathrm{T}},\]  
and  
\[\begin{cases}R_{1}(x)=s(u)\sqrt{P_{s}-x_{3}}+s(-u)\sqrt{x_{3}-P_{r}}\\ R_{2}(x)=s(u)\sqrt{x_{4}-P_{r}}+s(-u)\sqrt{P_{s}-x_{4}}\end{cases}\]  
then the entire state equations of the hydraulic system can be expressed as follows:  
\[\begin{cases}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=\frac{1}{m}\left(A_{1}x_{3}-A_{2}x_{4}-f_{L}\right)\\ \dot{x}_{3}=\frac{\beta_{e}}{V_{01}+A_{1}x_{1}}\left[-A_{1}x_{2}-C_{t}(x_{3}-x _{4})+guR_{1}(x)\right]\\ \dot{x}_{4}=\frac{\beta_{e}}{V_{02}-A_{2}x_{1}}\left[A_{2}x_{2}+C_{t}(x_{3}-x _{4})-guR_{2}(x)\right]\end{cases} \tag{6}\]  
To simplify the system state equations, some variables and parameters are defined as follows:  
\[\begin{cases}\overline{x}_{3}=\frac{A_{1}x_{3}-A_{2}x_{4}}{m}\\ d=-\frac{f_{L}}{m}\\ \beta=\frac{A_{1}\beta_{e}}{m}\\ k_{A}=\frac{A_{2}}{A_{1}}\end{cases}\]  
and the