# Adaptive Cruise Control Based on Model Predictive Control with Constraints Softening
## 1 Introduction  
Traffic safety is facing severe challenges with the continuous growth of highway traffic mileage and car ownership. In the face of severe traffic accidents and environmental issues, intelligent transportation systems (ITSs) and advanced driver assistance systems (ADASs) have been proposed worldwide. ADASs are systems aimed at enhancing active safety and providing a better driving experience for drivers; however, they have recently aroused concerns from a wide range of sources [1]. Adaptive cruise control (ACC) is a newly developed ADAS system and has been an important milestone in the history of driving assistance [2; 3]. Aside from the basic functions of a cruise control system, the ACC system has the additional ability to avoid forward collisions and keep a safe distance from the preceding vehicle by considering the relative distance, the relative velocity, and the relative acceleration of the vehicle. [PERSON] and [PERSON] [4] indicated that the success of ACC can be considered a good example for the development of a complete cooperation ADAS system.  
Commonly, ACC systems can reduce the working intensity of the driver and ensure the safety of driving through the automatic control of longitudinal movement, and can provide auxiliary support for drivers through a convenient human-computer interface. At present, most scholars consider car following and safety in the design of ACC strategies. However, comfort and economy are also important performance features for ACC systems. Therefore, the control goal of ACC is to improve the ride comfort and economy on the premise of tracking the expected safe car spacing, with the difficulty being in the construction of multi-objective control strategies involving such features as safety,economy, and comfort. The motivation of this paper is to design an ACC system that can consider the abovementioned multi-objectives, which include safety, comfort, economy, and car following.  
For the above multi-objective optimization problem, linear quadratic (LQ) and model predictive control (MPC) are commonly used solutions. For automotive applications, nonlinearity and constraint problems must be considered. The realization of LQ needs an implicit assumption with no external disturbance. However, the ACC system has external disturbances, with the state of the preceding vehicle causing a standard LQ regulator to lose its optimality to a certain extent. It is important to consider that MPC is capable of optimally controlling constrained nonlinear problems in each step by rolling optimization while LQ cannot. Since the above optimization problem is non-nonlinear and constrained, this paper puts forward an ACC algorithm under the framework of MPC by analyzing the current ACC problems and challenges. The commonly used MPC methods do not consider the infeasibility of the solution as a result of extreme conditions and the use of fixed constraints. To achieve the multi-objectives of ACC strategy and meet the multiple control objectives present during the driving process, an MPC-based ACC strategy with constraint softening is proposed to address the multi-objectives of the ACC system.  
The paper is structured in the following manner. Section 2 discusses related works and tries to assess the challenges and differences as compared to existing works. Section 3 builds the mathematical model for the proposed ACC system. Section 4 presents the MPC-based multi-objective ACC strategy with constraints and the constraints softening method. Section 5 verifies the proposed ACC strategy by case studies and provides comparisons in different scenarios using CarSim and MATLAB/Simulink. Finally, Section 6 concludes the paper and provides details of possible future work.