## 4 Results  
In this section, we present the results of our simulation investigations into the performance of the proposed CLF-CBF-QP controller with the suspension system. We begin by discussing the introduced road disturbances to the suspension system. Subsequently, we present the detailed design parameters of the LQR and CLF-CBF-QP controller. Furthermore, we present comprehensive analyses to validate the robustness and significance of our closed-loop results in response to challenging road profiles. This section delves into two typical simulations of the nonlinear quarter-car model's closed-loop response when employing CLF-CBF-QP and LQR controllers. It is worth mentioning that all simulations were carried out in the MATLAB environment on an Intel six-core PC with a 2.2 GHz CPU and 16 GB of RAM. The purpose of these simulations is to assess the CLF-CBF-QP controller performance in guaranteeing safety and stability requirements across various road profiles. Two distinct road profiles are under consideration that offer diverse complexities. The first road profile features a smooth sinusoidal bump of 0.1 m peak height, while the second involves a ramped pothole of 0.1 m depth. The vehicle model's parameters are detailed in Table 1, and it is worth noting that these parameters closely resemble those used by [42]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(m_{1}\) (Kg) & \(m_{2}\) (Kg) & \(k_{1}^{L}\) (N/m) & \(k_{2}\) (N/m) & \(\varepsilon\) (Ns/m) & \(k_{1}^{NL}\) (N/m) \\ \hline
300 & 45 & 40,000 & 150,000 & 2000 & 150,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the nonlinear quarter-car model featuring cubic nonlinearity. Figure 2: Block diagram of the CLF-CBF-QP control strategy with LQR reference controller applied to suspension system. The online optimizer (QP) computes the optimal control input (\(u^{*}\)) for the suspension system considering the CLF, CBF, and input constraints. First, the reference controller is designed as an LQR controller as discussed in Section 3.4 using the following weights:  
\[Q=\left[ \begin{array}{cccc}10&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right],\quad R=5\times 10^{-8}.\]  
Then, the CLF-CBF-QP controller is designed by solving quadratic programs formulated in Equation (10), subject to Equation (11). The CLF-CBF-QP controller's parameters are listed in Table 2. This controller computes the optimum control law (\(u\)) every 0.01 s.  
Figure 3 illustrates the outcomes of the closed-loop simulations involving the quarter-car model featuring cubic nonlinearity when utilizing CLF-CBF-QP controller. In this scenario, the vehicle encounters a road disturbance of 0.1 m high smooth bump. Further, the figure provides a comparative analysis of the closed-loop outcomes achieved with the suggested controller and those obtained with the LQR controller. The figure undoubtedly demonstrates substantial enhancements in various performance aspects, including sprung mass displacement, sprung mass velocity, and suspension stroke, when compared to the responses obtained from active suspension system with LQR controller (\(u_{ref}\)). For instance, the settling time for both sprung mass displacement and velocity is around 1.7 s for the closed-loop response with the LQR controller, while the CLF-CBF-QP controller successfully reduced the settling time to less than 1 s. Moreover, the maximum peak of the sprung mass displacement (\(z_{1}\)) is approximately three times the bump height for the closed-loop response with the LQR controller, while the CLF-CBF-QP controller successfully managed to cope with the road disturbance with a maximum sprung mass displacement peak of almost the same of bump height. Note that the CLF objective is to keep the sprung mass velocity (\(\dot{z}_{1}\)) as close to zero as possible. Interestingly, even though the CLF did not include the sprung mass displacement in the Lyapunov function, the CLF managed to improve it indirectly by observing only the sprung mass velocity.