# Cooperative Control of Interconnected Air Suspension Based on Model Predictive Control
## 2 The Modeling of the System
### Full-Vehicle Vertical Model  
The vehicle is modeled based on the moving vehicle coordinate system, \(XYZ\), which is attached at the center of gravity (CG) and aligned in principal directions. The full-vehicle vehicle model is simplified to seven degrees of freedom (7-DOF): vertical motions of four wheels, the roll motion, pitch motion, and vertical motion of sprung mass. The 7-DOF structure of the full-vehicle physical model shown in Figure 2 has been verified by testing [6].  
Here, \(M_{\text{s}}\) is the sprung mass; \(Z_{\text{s}}\) is vertical displacement at the center of gravity; \(I_{\text{r}}\) and \(I_{\text{p}}\) denote roll and pitch moments of inertia, respectively; \(\theta\) and \(\varphi\) are the roll angle and pitch angle of the vehicle body, respectively; \(I_{\text{r}}\) and \(I_{\text{r}}\) are the distances between the center of gravity to the front and the rear axis; \(B_{\text{f}}\) and \(B_{r}\) are treads of front and rear wheels, respectively; the front and rear unsprung masses on the right and left sides, respectively, are denoted by \(M_{\text{ufl}}\), \(M_{\text{uflr}}\), \(M_{\text{uflr}}\), and \(M_{\text{uflr}}\); \(Z_{\text{uflr}}\), \(Z_{\text{uflr}}\), and \(Z_{\text{uflr}}\) are the vertical displacements of the four wheels; \(K_{\text{t}}\) denotes the tire stiffness of the wheels; \(q_{\text{flr}}\), \(q_{\text{flr}}\), \(q_{\text{flr}}\) and \(q_{\text{rr}}\) are road roughness displacements; \(A_{\text{efl}}\), \(A_{\text{eflr}}\), \(A_{\text{eflr}}\), and \(A_{\text{ermr}}\) are effective areas of four air suspensions; \(c_{\text{uflr}}\), \(c_{\text{uflr}}\), \(c_{\text{uflr}}\), and \(c_{\text{urmr}}\) are the damping coefficients of four suspensions.  
Thus, the 7-DOF model of the full-vehicle dynamic model is written as follows  
\[\left\{\begin{aligned} & M_{\text{s}}\ddot{Z}_{\text{s}}=F_{\text{fl}}+F_{ \text{fr}}+F_{\text{rl}}+F_{\text{rr}}\\ & I_{\text{r}}\ddot{\theta}=(F_{\text{fl}}-F_{\text{fr}})\frac{B_ {\text{r}}}{2}+(F_{\text{rl}}-F_{\text{rr}})\frac{B_{\text{r}}}{2}\\ & I_{\text{p}}\ddot{\varphi}=(F_{\text{fl}}+F_{\text{rr}})I_{ \text{f}}-(F_{\text{rl}}+F_{\text{rr}})I_{\text{r}}\\ & M_{\text{uflr}}\ddot{Z}_{\text{uflr}}=K_{\text{t}}(Z_{\text{rfl }}-q_{\text{fl}})-F_{\text{fl}}\\ & M_{\text{uflr}}\ddot{Z}_{\text{uflr}}=K_{\text{t}}(Z_{\text{uflr }}-q_{\text{fr}})-F_{\text{fr}}\\ & M_{\text{uflr}}\ddot{Z}_{\text{l}}=K_{\text{t}}(Z_{\text{rfl}}- q_{\text{rl}})-F_{\text{rl}}\\ & M_{\text{urmr}}\ddot{Z}_{\text{urmr}}=K_{\text{t}}(Z_{\text{uflr }}-q_{\text{rr}})-F_{\text{rr}}\\ \end{aligned}\right. \tag{1}\]  
Figure 2: Seven-degree-of-freedom physical model of the full vehicle.  
in which  
\[\begin{cases}fd_{\text{fl}}=Z_{\text{s}}-Z_{\text{uff}}-l_{\text{f}}\varphi+0.5 \theta B_{\text{f}}\\ fd_{\text{fr}}=Z_{\text{s}}-Z_{\text{uff}}-l_{\text{f}}\varphi-0.5\theta B_{ \text{f}}\\ fd_{\text{rl}}=Z_{\text{s}}-Z_{\text{url}}+l_{\text{r}}\varphi+0.5\theta B_{ \text{r}}\\ fd_{\text{rr}}=Z_{\text{s}}-Z_{\text{urr}}+l_{\text{r}}\varphi-0.5\theta B_{ \text{r}}\end{cases} \tag{2}\]  
where, \(fd_{\text{fl}}\), \(fd_{\text{fr}}\), \(fd_{\text{rl}}\), and \(fd_{\text{rr}}\) are the suspension dynamic travels of the four suspensions. \(F_{\text{fl}}\), \(F_{\text{fr}}\), \(F_{\text{rl}}\), and \(F_{\text{rr}}\) are suspension forces which consist of damping force and air suspension force.  
The parameters of these equations are listed in Table 1.