## 2 Mathematical Model of Stepless Capacity Control System  
### Mathematical Model of Actuator  
A stepless capacity control system mainly includes hydraulic system, actuator, control system. The composition of the system is shown in Figure 1. The compressor transmits the pressure signal to the DCS (Distributed Control System) system. The DCS system calculates the load signal according to the pressure signal and transmits the load signal to the intelligent controller. The intelligent controller outputs the electric signal to the solenoid valve according to the load signal and the speed signal. The hydraulic force provided by the hydraulic system drives the actuator to control the opening and closing time of the suction valve, realizing the stepless regulation of the capacity. The actuator mainly includes a hydraulic cylinder, unloader and suction valve. As can be seen from Figure 2, status1 is the initial installation state. Since the actuator is not driven by hydraulic pressure, the spring force makes the actuator at the upper limit, and the suction valve is in the closed state. Status2 is the opening process of the suction valve plate. After the suction process of the compressor begins, the valve plate opens automatically. The actuator is still not driven by the hydraulic pressure and is at the upper limit. Status3 is the opening process of the actuator. The actuator is driven by hydraulic pressure to overcome the spring force and air force, and move downward. Status4 is the backflow process. Since the suction valve plate is forced to open, the gas reflux, when the hydraulic force is small, in the backflow process. During the backflow phase, the lower surface of the valve plate has a clearance of \(h_{2}\) from the valve lift limiter. Status5 is the closing process of the actuator and the suction valve. When the capacity meets the technical requirements, the hydraulic force does not work, and the valve plate and the actuator withdraw at the same time. Figure 1: Composition of stepless capacity control system. During the process of ejection, the actuator is subjected to hydraulic force, spring force, gas force, friction force, etc. The rebound during the impact of the actuator is taken into consideration. The rebound coefficient \(\gamma\) is measured by experiment. The differential equations of ejection and withdrawal of the actuator are shown in Equation (1):  
\[m\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}=\left\{ \begin{array}{l}P_{h}-F_{i}+mg \cos\alpha-f-F_{s},status3-ejection\\ P_{h}-F_{i}+mg\cos\alpha+f-F_{s},status3-rebound\\ F_{s}+F_{i}+F_{cy}-f-mg\cos\alpha,status5-withdrawal\\ F_{s}+F_{i}+F_{cy}+f-mg\cos\alpha,status5-rebound\end{array} \right. \tag{1}\]  
where, \(m\) represents the quality of the moving parts of the actuator, \(x\) represents the actuator displacement, \(P_{h}\) represents hydraulic force, \(F_{i}\) represents gas force of suction chamber, \(\alpha\) represents mounting angle of the unloader, \(f\) represents total friction of actuator, \(F_{s}\) represents spring force of unloader, \(F_{cy}\) represents The resultant gas force of the valve plate is related to the backflow clearance \(h_{2}\), in the process of backflow, the suction valve plate is in contact with the actuator claw due to the increased pressure in the cylinder, and the valve plate exerts a force on the actuator, that is \(F_{cy}\). The hydraulic force and spring force in Equation (1), initial conditions and constraint conditions are as follows:  
(1) Hydraulic force \(P_{h}\):  
\[P_{h}=p_{1}A_{unloader},\alpha_{2}\leq\theta\leq\alpha_{3} \tag{2}\]  
where, \(p_{1}\) represents the oil pressure in the ejection process of the actuator, \(A_{unloader}\) represents the sectional area of the hydraulic cylinder plunger\(\alpha_{2}\) represents the start angle of the ejection,\(\alpha_{3}\) represents the end angle of the ejection. (2) Spring force \(F_{s}\):  
\[F_{s}=\left\{ \begin{array}{l}k_{unloader}(x_{0}+x),\alpha_{2}\leq\theta\leq \alpha_{3}\\ k_{unloader}(x_{max}-x),\alpha_{5}\leq\theta\leq\alpha_{6}\end{array} \right.