# Adaptive Cruise Control Based on Model Predictive Control with Constraints Softening
## 2 Related works  
Improving safety is the basic requirement for ACC systems; this is achieved by keeping a desired vehicle inter-distance. Many researchers have studied the safety of car following when designing an ACC strategy. [PERSON] et al. [5] proposed a continuous car-following model based on the time gap for an ACC system. [PERSON] et al. [6] proposed a nonlinear model predictive control (NMPC) approach to realize distance tracking for ACC with a \"stop&go\" function. Besides safety, fuel consumption and driver comfort are also important for ACC systems. [PERSON] et al. [7] introduced an ACC method in multi-lane traffic conditions with the aim of enhancing driving comfort. The difficulty of the development and application of ACC systems is how to build the multi-target control strategy, which includes safety, economy, and comfort, amongst others. For example, on the basis of model predictive control (MPC), [PERSON] et al. [8] proposed a fuel consumption optimization algorithm for ACC of a hybrid electric vehicle. [PERSON] and [PERSON] [9] presented an ecological adaptive cruise controller based on action-dependent heuristic dynamic programming for parallel hybrid electric vehicles, with the aim of realizing optimal fuel economy and comfortable driving. [PERSON] et al. [10] utilized the adaptive neuro fuzzy inference control strategy to enhance the control of ACC systems, which can improve the robustness and efficiency of the ACC system. They realized this function by predicting the environmental factors in real time, such as the velocity of the wind and the inclination angle of the road. [PERSON] et al. [11] proposed an ACC algorithm with multi-objectives, which was based on MPC with several hard constraints. [PERSON] et al. [12] used MPC with the multiple objectives of safety, desired driver response, fuel consumption, and car-following error, where the constant time headway was utilized to model the car-following system. [PERSON] et al. [13] designed a predictive cruise controller under connected circumstances. They used reinforcement learning to approach the predictive problem, which can regulate the headway, velocity, and the acceleration of each vehicle at the same time. The above researches indicate that the MPC theory is effective for designing the ACC strategy with multi-objectives and constraints. When applying MPC to the design of the controller, it is possible that the entire problem cannot be calculated by a feasible solution due to the constraints; this can result in various problems, such as a weak robustness, the solution being unfeasible, and a high computational complexity [14]. To avoid the divergence problem of the MPC problem, [PERSON] et al. [15] presented a soft-constrained MPC based on the idea that violation can often be tolerated for short periods due to the nature of the state constraints. [PERSON] et al. [16] introduced an efficient and accurate calculation method for continuous MPC problems based on the pseudospectral method, which was introduced to improve the calculation accuracy and reduce the calculation amount of an ACC system. In summary, aimed at the requirements of multi-objectives and the convergence of the MPC problem, the constraint softening method is introduced, which can strengthen the performance of feasible solutions. Enlightened by the above ideas, this paper puts forward a constraint-softening method by adding a penalty function item to the multi-objective functions, which can provide a larger feasible set while improving the closed-loop convergence of the MPC problem. The inter-distance, the relative speed, the acceleration, and the jerk between the host vehicle and the preceding vehicle are integrated into a longitudinal kinematics prediction model. Furthermore, the multi-target requirements are converted into performance indicators and constraints for the ACC system. The constraint softening method is adopted to improve the robustness and to avoid the solution becoming unfeasible because of the overlap feedback correction.