# Sliding Mode Control of Electro-Hydraulic Position Servo System Based on Adaptive Reaching Law
## 2 Problem Formulation and Dynamic Models  
In the electro-hydraulic servo system, compared with the symmetrical hydraulic cylinder, the asymmetrical cylinder has the advantages of occupying a small space, being a simple structure and having a low cost, so it is widely used. In this paper, a four-way three-position (4/3) valve-controlled asymmetric hydraulic cylinder was used as the power mechanism of the electro-hydraulic servo system; the structure diagram of the valve-controlled asymmetric hydraulic cylinder is shown in Figure 1.  
Figure 1: Structure diagram of valve-controlled asymmetric hydraulic cylinder. Let the positive direction of the displacement of the spool of the servo valve be the direction shown by the \(x_{v}\) arrow in Figure 1, at time \(x_{v}>0\). Let the ratio of the effective area of the bore area to the rod area of the asymmetric cylinder be \(n=A_{2}/A_{1}\), where \(A_{1}\) is the effective area of the bore, \(A_{2}\) is the effective area of the rod. When \(x_{v}>0\), the spool moves to the right, the piston rod extends, and the load pressure is  
\[p_{L}=p_{1}-np_{2} \tag{1}\]  
and the load flow is:  
\[q_{L}=q_{1} \tag{2}\]  
When, \(x_{v}<0\), the spool moves to the left, the piston rod retracts, the load pressure is  
\[p_{L}=p_{2}-np_{1} \tag{3}\]  
and the load flow is:  
\[q_{L}=q_{2} \tag{4}\]  
The flow continuity equation of the piston chamber of the valve-controlled asymmetric hydraulic cylinder is:  
\[q_{1}=A_{1}\frac{dx_{p}}{dt}+C_{ip}(p_{1}-p_{2})+C_{ep}p_{1}+\frac{V_{1}}{ \beta_{e}}\frac{dp_{1}}{dt} \tag{5}\]  
The flow continuity equation of the rod chamber of the valve-controlled asymmetric hydraulic cylinder is:  
\[q_{2}=A_{2}\frac{dx_{p}}{dt}+C_{ip}(p_{1}-p_{2})-C_{ep}p_{2}-\frac{V_{2}}{ \beta_{e}}\frac{dp_{2}}{dt} \tag{6}\]  
where \(V_{1}\) is the effective volume of the piston-side chamber; \(V_{2}\) is the effective volume of the rod-side chamber; \(\beta_{e}\) is the effective bulk modulus of the hydraulic oil; \(C_{ip}\) is the internal leakage coefficient of the cylinder; \(C_{ep}\) is the external leakage coefficient of the cylinder; \(x_{p}\) is the displacement of the piston; \(p_{1}\) is the pressure of the piston-side chamber; and \(p_{2}\) is the pressure of the rod-side chamber. The positive direction of piston displacement is defined as the direction shown by the \(x_{p}\) arrow in Figure 1, when the piston extends. Taking \(x_{p}>0\) as an example, the flow equation of the four-way three-position (4/3) valve is established as  
\[\left\{ \begin{array}{l}q_{1}=C_{d}\omega x_{v}\sqrt{\frac{2(p_{s}-p_{1})}{ \rho}}\\ q_{2}=C_{d}\omega x_{v}\sqrt{\frac{2p_{2}}{\rho}}\end{array} \right. \tag{7}\]  
where \(C_{d}\) is the flow coefficient of the valve; \(\omega\) is the area gradient of the valve; and \(\rho\) is the oil density. The flow rate of the piston-side chamber can be expressed by the product of the velocity of the piston and the effective area of the bore area. Similarly, the flow rate of the rod-side chamber can be expressed by the product of the velocity of the piston and the effective area of the rod area. Therefore, the relationship between the flow rate of the piston-side chamber and the flow rate of the rod-side chamber is  
\[\frac{q_{2}}{q_{1}}=\frac{vA_{2}}{vA_{1}}=\frac{A_{2}}{A_{1}}=n \tag{8}\]  
where \(v\) is the velocity of the piston. The pressure of the piston-side chamber can be expressed as  
\[p_{1}=\frac{n^{3}p_{s}+p_{L}}{n^{3}+1} \tag{9}\]  
The pressure of the rod-side chamber can be expressed as  
\[p_{2}=\frac{n^{2}(p_{s}-p_{L})}{n^{3}+1} \tag{10}\]  
where \(p_{s}\) is the system pressure. Equations (1), (2) and (7)-(10) can be combined into the load flow equation:  
\[q_{L}=\eta x_{v}\sqrt{p_{s}-p_{L}} \tag{11}\]  
where \(\eta=C_{d}\omega\sqrt{\frac{2}{p(1+n^{3})}}\) is a constant.