## 1 Introduction
### Background  
Adaptive cruise control is one of the most widely adopted ADAS technologies. It extends traditional cruise control by automatically adjusting the vehicle's speed to maintain a safe distance from the vehicle ahead [9]. This capability not only enhances driving comfort by reducing the need for manual speed adjustments and reducing frequent changes in acceleration but also has the potential to improve fuel efficiency by promoting smoother driving behavior. Moreover, full-range ACC can take care of the longitudinal control of the vehicle in urban, highway, and mixed scenarios. Several studies have sought to improve ACC's performance by optimizing fuel consumption while ensuring safety and comfort for both conventional and hybrid vehicles [10]. In some works, pollutant emissions were also considered as control objectives [11]. Various optimization approaches, such as dynamic programming (DP) [12; 13; 14], model predictive control (MPC) [15; 16; 17; 18], linear-quadratic regulators [19; 20], and reinforcement learning solutions [21; 22; 23] have been proposed. Each approach has its merits and disadvantages. Dynamic programming is effectively the only technique among these that can guarantee optimal solutions, albeit with high computational costs. In short, DP works by breaking down a complex decision-making process into simpler sub-problems and solving them sequentially. [PERSON]'s principle of optimality is central to DP, enabling the formulation of an optimal policy accounting for the cumulative cost of future actions [24]. In the context of ACC, DP can be used to determine the optimal acceleration and deceleration patterns that minimize fuel consumption over a given driving cycle. However, its high computational complexity makes it impractical for real-time applications. Because of this limitation, DP is best suited for offline applications, where it remains valuable as a benchmark for evaluating the performance of other, more computationally efficient methods. Model predictive control is advantageous for online applications due to its ability to anticipate future events and optimize control actions accordingly. MPC's capability to handle multi-variable interactions and constraints explicitly makes it a promising approach for real-time ACC optimization. Unlike DP, MPC operates in a receding horizon manner, continuously updating its predictions based on the latest state measurements. MPC involves solving an optimization problem at each time step, where the objective is to minimize a cost function. By incorporating constraints directly into the optimization process, MPC ensures that the resulting control actions are feasible and safe. The primary challenge with MPC is maintaining real-time feasibility, especially when dealing with complex vehicle models and long prediction horizons. In a recent study [25], deep neural networks were used to generate an approximated policy that reproduces an MPC-based ACC system in order to reduce execution time and memory footprint. Nonetheless, explicit MPC implementations have been proven to be capable in real-time, such as the robust MPC algorithm developed by [26]. Linear quadratic regulators (LQRs) can be seen as a special case of model predictive control under specific conditions. Firstly, LQRs always assume that the system is linear and the cost function is quadratic [27]. In MPC, this is also assumed to be the standard problem structure, but other variations can be treated. For example, nonlinear costs can be considered, although in this case, the term nonlinear MPC is commonly used. Secondly, LQR does not handle constraints, while one of the primary strengths of MPC is its ability to explicitly incorporate constraints on inputs, states, and outputs. Lastly, LQR is inherently an infinite-horizon controller, while MPC typically operates over a finite horizon in the receding horizon technique that was previously described. Despite all its limitations, LQR presents a notable advantage with respect to MPC in terms of computational costs.