# Sliding Mode Control of Laterally Interconnected Air Suspensions
## Full Vehicle Modeling and Road Excitations
### Modeling of the Vehicle and Air Suspensions  
Considering that the deformation speed of each air spring is no more than 15 m/s, while the air pressure transfers at a speed of 345 m/s, we assume that the thermodynamic state within each air spring is equal. Meanwhile, during work time, the air pressure of each spring is no more than 1 MPa and the temperature is much higher than 0 K, thus the air can be treated as ideal gas with acceptable error. In addition, the air spring has less heat exchange with outside world, which can be treated as an adiabatic process. On the basis of the assumptions above and thermodynamic theory, air states in air spring can be described as  
\[P_{i}\left(\frac{V_{i}}{m_{i}}\right)^{\kappa}=\text{const}, \tag{1}\]  
where \(I\) = 1,2,3\(A\) represents each air spring in this paper; \(P_{i}\) is the pressure of each air spring; \(V_{i}\) is the volume of each air spring; \(m_{i}\) is the air mass of each air spring; \(\kappa\) is adiabatic exponent and equals to 1.4; and const means a constant. Assuming no mass flow leakage in the system components, the pressure of each air spring at any time can be calculated with Equation (2)  
\[P_{i}=P_{i0}\left(\frac{m_{i}V_{i0}}{m_{i0}V_{i}}\right)^{k}, \tag{2}\]  
where \(P_{i0}\), \(V_{i0}\), and \(m_{i0}\) represent the initial pressure, volume, and air mass of each air spring, respectively. The parameters \(V_{i}\) and \(m_{i}\) can be expressed as  
\[V_{i}=V_{i0}+\frac{dV_{i}}{dh}fd_{i}, \tag{3}\]  
\[m_{i}=m_{i0}+\int_{0}^{t}q_{ai}dt, \tag{4}\]  
where d_V_i/d_t_ is the volume change rate with height of each air spring. \(fd_{i}\) is the suspension travel of each suspension. \(q_{ai}\) represents the air mass flow in and out of each air spring, which can be calculated with the interconnection model in the next section. Moreover, the effect area of the air spring also varies with the suspension travel, thus the effect area \(A_{ei}\) of each air spring can be calculated as  
\[A_{ei}=A_{ei0}+\frac{dA_{ei}}{dh}fd_{i}, \tag{5}\]  
where \(A_{ei0}\) is the initial effect area of each air spring and d_Aei_/d_t_ is the effect area change rate with height of each spring. Hence, the force of air spring can be expressed as  
\[F_{si}=(P_{i}-P_{a})\cdot A_{ei}, \tag{6}\]where \(P_{a}\) is the atmospheric pressure and equals to \(1\times 10^{5}\) Pa.  
For control purposes, this study utilizes a simplified seven-DOF suspension model. The more realistic vehicle suspension model and damper model can be found in [46; 47]. The full-vehicle dynamic model with air suspension feature is established for modeling the entire system, which comprises the vertical, roll, and pitch movements of the body and vertical movements of four wheels, as shown in Figure 1.  
On the basis of [PERSON]'s law and vehicle dynamics, the seven-DOF full vehicle model can be formulated as follows  
\[\left\{ \begin{array}{c}M_{\rm{b}}\ddot{Z}_{b}=F_{1}+F_{2}+F_{3}+F_{4}\\ I_{r}\ddot{\theta}=(F_{1}-F_{2})\frac{B_{r}}{2}+(F_{3}-F_{4})\frac{B_{r}}{2}\\ I_{p}\ddot{\varphi}=(F_{1}+F_{2})I_{f}-(F_{3}+F_{4})I_{r}\\ M_{ti}\ddot{Z}_{ti}=K_{t}(q_{i}-Z_{ti})-F_{i}\end{array} \right., \tag{7}\]  
where  
\[\left\{ \begin{array}{c}F_{i}=(p_{i}-p_{a})A_{ei}+c_{i}\dot{f}d_{i}\\ fd_{1}=Z_{1t}-\left(Z_{b}-l_{f}\theta+0.5\varphi B_{f}\right)\\ fd_{2}=Z_{t2}-\left(Z_{b}-l_{f}\theta-0.5\varphi B_{f}\right)\\ fd_{3}=Z_{t3}-\left(Z_{b}+l_{r}\theta+0.5\varphi B_{r}\right)\\ fd_{4}=Z_{t4}-\left(Z_{b}+l_{r}\theta-0.5\varphi B_{r}\right)\end{array}