# Sliding Mode Control of Laterally Interconnected Air Suspensions
## 4 Simulations and Analysis
### Simulation Results Under Varying Situations  
As the roll movement is mainly caused by the unequal excitation of left and right side of the road, a right-left twisted road input is firstly applied. The left side of the road raises 5 mm at 0 s, while the right side of the road drops 5 mm at the same time. Figure 3a shows the roll angle responses of sprung mass with different interconnection states or controllers under the applied excitations. Here, the \"ON\" state means that the suspension is interconnected air suspension, while the \"OFF\" state means that the suspension is non-interconnected air suspension. The \"imitated skyhook\" and \"sliding mode\" represent that using the imitated skyhook controller or the sliding mode controller to control the on and off states of the interconnection. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value** & **Unit** & **Parameter** & **Value** & **Unit** \\ \hline \(P_{10},P_{20}\) & \(5.8306\times 10^{5}\) & Pa & \(P_{30},P_{40}\) & \(6.1817\times 10^{5}\) & Pa \\ \(m_{10},m_{20}\) & 0.0166 & kg & \(m_{30},m_{40}\) & 0.0176 & kg \\ \(V_{10}\) & 0.0024 & m\({}^{3}\) & \(A_{c0}\) & 0.009 & m\({}^{2}\) \\ \(\mathrm{d}V_{j}/\mathrm{d}h\) & 0.0072 & m\({}^{2}\) & \(\mathrm{d}A_{ei}/\mathrm{d}h\) & 0.008 & m \\ \(L\), \(B_{r}\), \(B_{r}\) & 1.515 & m & \(M_{b}\) & 1839 & kg \\ \(l_{r}\) & 600 & kg\(\cdot\)m\({}^{2}\) & \(l_{p}\) & 3500 & kg\(\cdot\)m\({}^{2}\) \\ \(L_{f}\) & 1.417 & m & \(L_{r}\) & 1.321 & m \\ \(M_{ii}\) & 50 & kg & \(K_{t}\) & 260,000 & N/m \\ \(c_{i}\) (compress) & 1800 & N-s/m & \(c_{i}\) (rebound) & 5400 & N-s/m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters for full car model. When laterally connected, the steady-state value of the roll angle is higher than others under the applied road input condition, but the peak value of the roll angle drops compared with the \"OFF\" state. Figure 3: The response of the vehicle under banked road excitations: (a) roll angles of sprung mass; (b) accelerations of driver position. The imitated skyhook controller can reduce the roll angle obviously, but the sliding mode controller performs the best in reducing the steady-state and peak value of the response. Under the applied road excitations, the sliding mode controller reduces the roll angle by 72% compared with the \"ON\" state and 41% peak value of roll angle response compared with the \"OFF\" state. Figure 3b shows the acceleration responses of driver position. It illustrates that, with the \"OFF\" state of the interconnection, the acceleration response will reach higher peak and valley values under the applied excitation. Moreover, keeping the interconnection in the \"ON\" state will result in the acceleration response coming to the balance position much faster than other methods. The imitated skyhook controller and the sliding mode controller lead to approximately the same acceleration responses. Both increase by 4.9% in RMS of the responses compared with the \"ON\" state and 4.6% in amplitude compared with the \"OFF\" state. In Equation (31), the lateral acceleration is added considering the same road excitations of the vehicle runs along under a straight line. \[I_{r}\ddot{\theta}=(F_{1}-F_{2})\frac{B_{f}}{2}+(F_{3}-F_{4})\frac{B_{r}}{2}+ a_{y}M_{b}H_{r}, \tag{31}\]  
where \(a_{y}\) is the lateral acceleration, and \(H_{r}\) is the distance from the centroid to roll axis and equals 0.3 m.  
An excitation of 3 m/s\({}^{2}\) lateral acceleration is applied at 0 second in the simulation; Figure 4a,b show the performance with each method. The above figures show that, under the lateral acceleration excitation, the imitated skyhook controller makes a balance between the acceleration and roll angle, it reduces the amplitude of acceleration response compared with \"ON\" state, while increasing the steady-state value of roll angle response compared with the \"OFF\" state. Meanwhile, the sliding mode controller chooses to shut the interconnection.