# A Control-Oriented Linear Parameter-Varying Model of a Commercial Vehicle Air Brake System
## 2 Nonlinear Mathematical Modeling
### Pilot Chamber and Spool Valve Displacement  
Assuming that the flow in the pilot chamber is ideal gas and isentropic, the temperature and pressure in the pilot chamber can be expressed by Equation (6). \[T_{p}=T_{0}(\frac{P_{p}}{P_{0}})^{\frac{\gamma-1}{\gamma}} \tag{6}\]  
where \(\gamma\) is the ratio of specific heats and is assumed to be constant; \(T_{p}\) and \(P_{p}\) are the temperature and pressure of the pilot chamber, respectively; and \(P_{0}\) and \(T_{0}\) are the temperature and pressure of supply air, respectively. The relation among the temperature, pressure, volume, and air mass in the pilot chamber can be achieved based on the ideal gas state as Equation (7). \[m_{p}=\frac{V_{p}P_{p}}{RT_{p}} \tag{7}\]  
The pilot chamber is considered as a control volume as a function of the spool displacement with an initial volume of \(V_{0}\). The volume defined as Equation (8) is  
\[V_{p}=V_{0}+A_{p}y \tag{8}\]  
Substituting Equations (6) and (8) into Equation (7) and differentiating the resulting equation yield Equation (9). \[q_{m}=\frac{1}{\gamma}\frac{V_{p}\dot{P}_{p}}{RT_{p}}+\frac{P_{p}\dot{V}_{p}}{ RT_{p}} \tag{9}\]  
or equivalently the air mass flow rate into the pilot chamber can be determined by Equation (10)  
\[q_{m}=\frac{(V_{0}+A_{p}y)\dot{P}_{p}}{\gamma RT_{0}}(\frac{P_{p}}{P_{0}})^{ \frac{1-\gamma}{\gamma}}+\frac{P_{0}A_{p}\dot{y}}{RT_{0}}(\frac{P_{p}}{P_{0}})^{ \frac{1}{\gamma}} \tag{10}\]Then, the pressure change in the pilot chamber is defined as Equation (11). \[\dot{P}_{p}=\frac{\delta q_{m1}-\frac{P_{0}A_{p}y}{RT_{0}}(\frac{P_{p}}{P_{0}})^{ \frac{1}{\gamma}}}{\frac{(V_{0}+A_{p}y)}{\gamma RT_{0}}(\frac{P_{p}}{P_{0}})^{ \frac{1-\gamma}{\gamma}}} \tag{11}\]  
or equivalently  
\[\dot{P}_{p}=\frac{\gamma RT_{0}\delta q_{m1}-\gamma P_{0}A_{p}\dot{y}(\frac{P_ {p}}{P_{0}})^{\frac{1}{\gamma}}}{(V_{0}+A_{p}y)(\frac{P_{p}}{P_{0}})^{\frac{1- \gamma}{\gamma}}} \tag{12}\]  
Spool valve displacement is a function of pilot chamber pressure, outlet pressure, spring and damping forces applied to the spool. Furthermore, the spring forces applied to the spool is a function of displacement \(y\) described below as Equation (13):  
\[f_{spool}=\left\{ \begin{array}{ll}k_{1}y+k_{3}y+f_{30},&y\geq 0\\ k_{1}y,&-y_{0}\leq y<0\\ k_{1}y+k_{2}y-f_{20},&y<-y_{0}\end{array} \right. \tag{13}\]  
where \(k_{1}\), \(k_{2}\), and \(k_{3}\) are equivalent spring stiffness of springs \(s_{1}\), \(s_{2}\) and \(s_{3}\) (see Figure 2), respectively; and \(f_{20}\) and \(f_{30}\) are preload spring forces assiated with springs \(s_{2}\) and \(s_{3}\), respectively. Then the spool motion is analyzed. The equation of motion can be derived by [PERSON]'s second law as Equation (14):  
\[m\ddot{y}+B\dot{y}+f_{spool}=P_{p}a_{1}-P_{b}(a_{1}-a_{2}) \tag{14}\]  
or equivalently Equation (15) below. \[P_{p\ddot{p}i}a_{1}-P_{out}(a_{1}-a_{2})=\left\{ \begin{array}{ll}(m_{1}+m_{3 })\ddot{y}+(B_{1}+B_{3})\dot{y}+k_{1}y+k_{3}y+f_{30},&y\geq 0\\ m_{1}\ddot{y}+B_{1}\dot{y}+k_{1}y,&-y_{0}\leq y<0\\ (m_{1}+m_{2})\ddot{y}+(B_{1}+B_{2})\dot{y}+k_{1}y+k_{2}y-f_{20},&y<-y_{0}\end{array} \right. \tag{15}\]  
In Equation (15), \(m_{1}\) is the mass of spring 1 and piston; \(m_{2}\) is the mass of spring 2; and \(m_{3}\) is the mass of spring 3; \(B_{i}\) (\(i=1,2,3\)) are damping coefficient associated with spring 1, 2, and 3, respectively; \(k_{i}\) (\(i=1,2,3\)) are spring stiffness associated with spring 1, 2, and 3, respectively; \(a_{1}\) is the contact area of pilot chamber and supply air; \(a_{2}\) is the contact area of diagram and push rod; Since the preload of spring is different under the charging or discharging process. To avoid step load change due to preload, the preload is approximated by a continuous function. Note that the mass flow rate is controlled by the equivalent area as a function of spool displacement. Based on the mass flow rate Equation (3), the mass flow through the supply valve can be determined, where the output pressure is changed from the pilot chamber pressure \(P_{p}\) to the brake chamber pressure \(P_{b}\).