## 4 Results
This is also related to the choice of the reference controller as discussed in Section 3.4. At the onset of the bump disturbance at \(t=1\) s, it led to an increase in \(\dot{z}_{1}\), causing it to deviate from the desired zero velocity. Consequently, the CLF constraint demonstrated a gradual increment, resulting in an elevation of the control action (compression). This adjustment led to a subsequent reduction in \(\dot{z}_{1}\), bringing it closer to zero at \(t=1.4\) s, in alignment with the CLF constraint and the consequent control effort. In this period (\(t\in[1-1.4]\) s), there was a slight increase in the suspension stroke (\(S_{t}=0.05\) m) that caused the CBF to exhibit a slight spike, which was enough to drive \(S_{t}\) toward zero, evidencing why the slack variable did not increase much to relax the CLF. On the other hand, after \(t=1.4\) s, the suspension stroke approached the limit (\(S_{t}=0.09\) m), and the CBF exhibited multiple larger spikes that caused the slack variable to increase, relax the CFL, and enforce the suspension stroke to decrease. It is worth mentioning that the LQR controller failed to limit the suspension stroke within the specified limits during the period (\(t\in[1.25-1.5]\) s). Enhancing the slack variable and relaxing the constraints of the CLF demonstrate the capacity of CLF-CBF-QP to prioritize safety over performance as required. It is worth noting that the chattering behavior of the CBF constraint is due to the sign function in the barrier function formulation. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\eta\) & \(\mu\) & \(M\) & \(q\) & \(St_{max}\) (m) & \(\kappa\) & \(u_{max}\) (N) & \(u_{min}\) (N) \\ \hline
5 & 25 & \(2.2222\times 10^{-5}\) & 25 & 0.1 & 200 & 10,000 & \(-10\),000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the CLF-CBF-QP controller. Figure 4 presents the results of the closed-loop simulations of the nonlinear quarter-car model with the CLF-CBF-QP and LQR controller. At \(t=1\) s, the vehicle encounters a ramped pothole with a depth of 0.1 m on the road. The figure showcases significant improvements in various performance metrics, including sprung mass displacement, sprung mass velocity, and suspension stroke compared to the responses obtained with the LQR controller. In contrast, the CLF-CBF-QP controller significantly reduces the settling time for both sprung mass displacement and velocity and the maximum peak of the sprung mass displacement. As the pothole disturbance commenced, it induced a downward trend in \(\dot{z}_{1}\), causing it to deviate from the desired zero velocity. Consequently, the CLF constraint experienced a gradual increase, leading to an upward adjustment in the control action (Tension). This CFL response aimed to counter the deviation and effectively increased \(\dot{z}_{1}\) towards zero by \(t=1.4\) s, aligning it with the CLF constraint and the corresponding control input. In this period, unlike the bump input, there was a larger increase in the suspension stroke (\(S_{t}=0.09\) m), resulting in a larger spike in the CBF constraint. This spike was adequate to steer the stroke toward zero, and consequently, the slack variable did not experience significant augmentation to relax the CLF. Conversely, beyond \(t=1.4\) s, the CLF constraint is reduced to \(\dot{z}_{1}\) towards zero by \(t=1.4\) s, and the CLF constraint is reduced to \(\dot{z}_{1}\) towards zero by \(t=1.4\) s.  
Figure 3: Time-domain closed-loop simulations for the active suspension with CLF-CBF-QP and LQR controllers in response to smooth bump (black dashed-line) road disturbance. the suspension stroke approached its limit (\(S_{t}=-0.09\) m), prompting the CBF constraint to manifest substantial spikes. These spikes, in turn, led to an augmentation of the slack variable, relaxing the CLF, increasing the control input, and enforcing \(S_{t}\) toward zero. The figure shows that the LQR controller proved inadequate in constraining the suspension stroke within the predefined limits during the interval (\(t\in[1.25-1.5]\) s).