## 1 Introduction
### Mathematical Model of the APRV  
The mathematical model of the APRV includes a solenoid valve subsystem and relay valve subsystem. The key points of the model are the movement of the valve core and piston and the relationship between the pressure and mass flow rate. The movement of the valve core in the solenoid valve is determined by electromagnetic force, spring force, damping force and flow force. The dynamic equation of the valve core of the solenoid valve is as follows:  
\[m\frac{d^{2}x_{1}}{dt^{2}}=F_{e}-k_{1}(x_{0}+x_{1})-sgn(\frac{dx_{1}}{dt}) \cdot c(\frac{dx_{1}}{dt})-(P_{m}-P_{c})\pi{r_{1}}^{2} \tag{1}\]  
where \(F_{e}\) is the electromagnetic force (N), \(m\) is the mass of the valve core in the solenoid valve, \(k_{1}\) is the stiffness of the spring (N/m), \(c\) is the coefficient of the viscous damping (N\(\cdot\)m\({}^{-1}\cdot\)s\({}^{-1}\)), \(P_{in}\) is the pressure of the inlet port (Pa), \(P_{c}\) is the pressure of the control cavity (Pa), \(r_{1}\) is the effective radius of the flow area (m), \(x_{1}\) is the displacement of the valve core in the solenoid valve (m) and \(x_{0}\) is the initial shape variable of the spring (m).  
The movement of the piston and valve core in the relay valve includes the processes of pressurization and decompression. In the pressurization process of the APRV, the piston and the valve core stay in contact and, hence, can be considered as a whole. The dynamic equation of the piston and valve core is as follows:  
\[(m_{p}+m_{c})\frac{d^{2}x}{dt^{2}}=P_{c}A_{1}-PA_{2}-(k_{x2}x+F_{0})-sgn( \frac{dx}{dt})\cdot(c_{1}+c_{2})\frac{dx}{dt}-sgn(\frac{dx}{dt})\cdot F_{f}+( m_{p}+m_{c})g\ (x\geq 0) \tag{2}\]  
In the process of decompression, the piston and the valve core detach. The dynamic equation of the piston is as follows:  
\[m_{p}\frac{d^{2}x}{dt^{2}}=P_{c}A_{1}-PA_{2}-sgn(\frac{dx}{dt})\cdot c_{1} \frac{dx}{dt}-sgn(\frac{dx}{dt})\cdot F_{f}+m_{p}g\ (x<0) \tag{3}\]  
where \(x\) is the displacement of the piston (m), \(m_{p}\) is the mass of the piston in the relay valve (kg), \(m_{c}\) is the mass of the valve core in the relay valve (kg), \(A_{1}\) is the upper surface area of the piston (m\({}^{2}\)), \(A_{2}\) is the lower surface area of the piston (m\({}^{2}\)), \(k_{x2}\) is the stiffness of the spring (N/m), \(F_{0}\) is the preload force of the spring (m), \(c_{1}\) is the viscous damping coefficient of the piston (N\(\cdot\)m\({}^{-1}\cdot\)s\({}^{-1}\)), \(c_{2}\) is the viscous damping coefficient of the valve core (N\(\cdot\)m\({}^{-1}\cdot\)s\({}^{-1}\)) and \(F_{f}\) is the friction force (N).  
The processes of pressurization and decompression in the APRV can be considered the inflation and the deflation processes of a variable-volume chamber. These processeshappen in an extremely short time and, therefore, can be regarded as adiabatic processes, so the relationship between the pressure and mass flow rate [6] is as follows:  
\[\frac{dp}{dt}=\frac{R}{V}k\theta_{in}q_{in}-\frac{R\theta}{V}kq_{out}-(k-1)\frac {P}{V}\frac{dV}{dt}-\frac{P}{V}\frac{dV}{dt} \tag{4}\]  
where \(V\) is the volume of the chamber (m\({}^{3}\)), \(q_{in}\) is the inlet mass flow rate (kg/s), \(q_{out}\) is the outlet mass flow rate (kg/s), \(k\) is the adiabatic index and \(\theta_{in}\) is the temperature of the air that flows through the orifice (K).