## 4 Simulation Results and Discussion
### Simulation Results  
The simulation results of sprung mass displacement and acceleration, tracking error of sprung mass displacement, RSD, RTF, and control signals of passive suspension, traditional backstepping, and ESO backstepping, and developed control are displayed in Figures 3-9. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Value** & **Unit** \\ \hline \(m_{s}\) & \(550\pm 100\sin(\pi\mathrm{t})\) & kg \\ \(m_{u}\) & 60 & kg \\ \(k_{s}\) & 16,000 & Nm\({}^{-1}\) \\ \(k_{t}\) & 145,000 & Nm\({}^{-1}\) \\ \(c_{a}\) & 2300 & Nsm\({}^{-1}\) \\ \(c_{t}\) & 1100 & Nsm\({}^{-1}\) \\ \(z_{R}\) & 0.04 & m \\ \(z_{a80}\) & 0.18 & m \\ \(R\) & 287.5 & J-kg\({}^{-1}\).K\({}^{-1}\) \\ \(A_{ss}\) & 0.0047 & m\({}^{2}\) \\ \(\kappa\) & 1.4 & - \\ \(T\) & 293.15 & K \\ \hline \end{tabular}
\end{table} 
Table 1: Pneumatic vehicle suspension parameters. \begin{table}
\begin{tabular}{c c} \hline
**Controller** & **Parameter** \\ \hline Backstepping & \(k_{1}=100\), \(k_{2}=20\), \(k_{3}=5\) \\  & \(k_{1}=100\), \(k_{2}=20\), \(k_{3}=5\) \\ ESO & \(\omega_{0}=150\) \\  & \(k_{1}=100\), \(k_{2}=20\), \(k_{3}=5\) \\ Proposed & \(\omega_{0}=150\), \(n=21\), \(\Lambda_{h}=50\) \\  & \(\sigma=0.01,\zeta=[-2,0.1,2]\) \\ \hline \end{tabular}
\end{table} 
Table 2: Control parameters. With the developed control, the ride comfort is strongly improved because the sprung mass acceleration and displacement have been reduced. By introducing a PPF, the proposed control can guarantee the displacement of sprung mass convergence in a finite time to get the ride comfort as shown in Figure 3. Besides, we can see that the time of zero convergence for the error signals \(e_{1}\) is about \(t=1.2\) (\(s\)) as shown in Figure 4. Moreover, passenger comfort is strongly improved thanks to the developed method because root mean square values of chassis acceleration are decreased by 69.43% as shown in Figure 5.  
Figure 4: Tracking error of sprung mass displacement. Figure 5: Simulation result of sprung mass acceleration. Figure 3: Simulation result of sprung mass displacement. Figure 8: Control signal (V). Figure 6: Relative suspension deflection. Figure 7: Relative tire force. Figure 3.0 Road profile. In comparison with passive suspension, the sprung mass displacement can reduce with traditional backstepping but there have some high peak values because of external disturbance. Moreover, the developed method can guarantee the sprung mass deflection, which is smaller than traditional backstepping and passive suspension. From Figure 6, the proposed control gives the magnitude of the RSD smaller than the conventional backstepping and passive, and this RSD is guaranteed within limits. The proposed control design can improve passenger comfort by selecting the sprung mass vertical displacement under the exogenous road disturbance while keeping the result of RSD within the limit value. It is also possible to see from the RTF values in Figure 7 that dynamic stroke constraints are satisfied because the limit is less than 1. Although both ESO backstepping and proposed control can ensure that dynamic and steady tracking errors are limited in the PPF constraints, only the proposed control can converge the system state \(x_{1}\) in a small neighborhood of the origin. Under the effect of parametric uncertainties, the ESO backstepping needs to consume more than the proposed control to compensate for this model error as shown in Figure 3.1  
Figures 10-12 plotted the estimation results of system states under the effect of external disturbances and their estimation errors. The simulation results show that the ESO can estimate system states accurately. Because the stiffness coefficient of the air spring depends on the rubber behavior under vertical displacement, the disturbance observer can estimate the influent of air spring stiffness and unmodeled dynamic of air pressure to guarantee the suspension performance.