## 5 Conclusions  
The purpose of the proposed CLF-CBF-QP controller is to emphasize the effectiveness of safety-critical controllers in guaranteeing safety for active suspension systems, more precisely, limiting suspension stroke. Limiting suspension travel enables the suspension to absorb severe shocks caused by road disturbances without bottoming out or becoming excessively stiff, which is critical for human comfort, vehicle control, and safety. In this study, we employed a safety-critical CLF-CBF-QP controller on a nonlinear quarter-car model featuring a suspension spring with cubic nonlinearity. The design process of the CLF-CBF-QP controller involves the design of a reference controller, Lyapunov function, and barrier function. The reference controller is designed generally in the CLF-CBF-QP controller design framework using steady-state analysis, which does not  
Figure 4: Time-domain closed-loop simulations for the active suspension with CLF-CBF-QP and LQR controllers in response to ramped pothole (black dashed-line) road disturbance. suit second-order problems, such as the suspension systems. Hence, we established the reference controller as an LQR controller based on the linearized quarter-car model. This reference state-feedback controller facilitated the design of the CLF based on the nonlinear model as a function of the sprung mass velocity only, establishing a relative degree of one. This CLF aims to enhance performance by accommodating nonlinearity and employing online optimization. Further, we constructed a CBF to prevent the suspension stroke from exceeding its defined limits. We conduct closed-loop simulations to demonstrate the effectiveness of the CLF-CBF-QP controller in active suspension systems. The two distinct road profiles under investigation offer various complexities. The first road profile has a smooth sinusoidal bump with a peak height of 0.1 m, whereas the second has a ramped pothole with a depth of 0.1 m. The results from both simulations demonstrate substantial enhancements across various performance aspects, including sprung mass displacement, velocity, and suspension stroke, compared to the responses obtained with the reference LQR controller. Furthermore, it demonstrates the CLF-CBF-QP controller's safety-critical capabilities in limiting the suspension stroke under challenging driving conditions. These results establish that the CLF-CBF-QP controller is very successful in improving performance and guaranteeing safety even though the designed reference controller is not capable of dealing with difficult driving conditions in terms of performance or safety. It is worth noting that the suggested controller is not restricted to a single type of suspension nonlinearity, such as cubic. It can handle a broad range of nonlinearities, including quadratic springs, hysteretic springs, and nonlinear dampers, as long as the nonlinear system is a control affine system. Future work may consider expanding the safety-critical CLF-CBF-QP controller to involve additional constraints, including sprung-mass acceleration and road-holding considerations. These safety constraints are crucial for conventional vehicles, especially in the context of autonomous vehicles. Moreover, employing the recently introduced fast global optimizers [43] could facilitate the extension of the CLF-CBF-QP method to encompass non-affine systems, thereby enhancing its applicability and effectiveness across diverse engineering problems. Conceptualization, T.S. and B.R.N. ; Methodology, T.S. ; Software, T.S. ; Validation, T.S. ; Formal analysis, T.S. ; Investigation, T.S. ; Resources, T.S. ; Data curation, T.S. ; Writing--original draft, T.S. ; Writing--review & editing, T.S. and B.R.N. ; Visualization, T.S. ; Supervision, B.R.N. ; Project administration, B.R.N. ; Funding acquisition, B.R.N. All authors have read and agreed to the published version of the manuscript.