## 1 Introduction  
The need for safety-critical active suspension control is on the rise within the automotive sector [1; 2; 3]. These control systems must factor in the parameter uncertainty and the nonlinear characteristics of suspension elements [4; 5]. An efficient way to unravel this issue is employing data-driven optimization-based control such as safety critical controllers [6; 7; 8]. The safety requirements of a well-designed suspension system should provide a comfortable ride by isolating the passenger from the road roughness. It should also ensure vehicle handling and road holding by maintaining proper tire contact with the road and that the suspension stroke falls within acceptable limits. One of the primary objectives of safety-critical active suspension control is to limit the suspension stroke from reaching its physical limits. Restricting suspension stroke in suspension systems is critical for vehicle safety, stability, ride comfort, and overall performance. It contributes to the reliability of suspension components, maintains consistent handling, and protects the vehicle and its occupants during normal and difficult driving styles. Active suspension system modeling often relies on linear car suspension models to simplify the analysis, albeit with significantly deteriorating model accuracy. For active suspension systems, many researchers have developed a variety of linear control techniques during the last few decades. [PERSON], [PERSON], [PERSON] et al., [PERSON] et al., and [PERSON] [9; 10; 11; 12; 13] employed the optimal control theory including Linear Quadratic Gaussian (LQG) and Linear Quadratic Regulator (LQR) to active suspension systems. On the other hand, [PERSON] et al., [PERSON] et al., [PERSON] et al. and [PERSON] et al. [14; 15; 16; 17] applied robust control theory for active suspension systems due to their capability to address model and parameter uncertainties. However, when vehicles experience substantial suspension stroke, the linear approximation degrades significantly. This motivates the employment of nonlinear car models for active suspension design. More precisely, [PERSON] et al. [18] studied various types of nonlinearities in suspension systems and demonstrated that cubic nonlinearity in the suspension spring is essential for accurate suspension system modeling. Nonlinear control strategies are also applied in active suspension systems featuring nonlinearity, including Linear Parameter Varying Control (LPV) [19; 20; 21], backstepping control [22; 23; 24], Sliding Mode Control [25; 26; 27], and feedback linearization [28; 29; 30]. Nevertheless, none of the aforementioned techniques considers safety-critical control techniques to ensure safety constraints, such as restricting suspension stroke to a specific range, though they consider minimizing the effect of the road disturbance on the suspension stroke. Conventional control strategies lack explicit mechanisms to online monitor safety constraints and enforce them where necessary, such as limiting suspension stroke within a specific range to prevent vehicle instability or component failure. Safety-critical control approaches emerge as a significant extension to the current control strategies to address this issue. Safety-critical controllers prioritize safety by incorporating optimization-based strategies customized to enforce critical safety constraints, ensuring the system operates within specified limits even under challenging conditions. However, safety-critical controllers are optimization-based and generally require large online computations. With increases in hardware-computing capacity, many researchers are using optimization-based approaches for active suspension systems. A well-received optimization-based control approach in the realm of active suspensions is Model Predictive Control (MPC) [7; 8; 31]. MPC aims at each time step to minimize a cost function, which defines the desired system performance while adhering to constraints on inputs, outputs, and states.