# Nonlinear Model Predictive Control with Terminal Cost for Autonomous Vehicles Trajectory Follow
## 1 Introduction  
In recent years, with the development of lidar, millimeter-wave radar, depth camera and AI technology, self-driving technology is also constantly improving. At the same time, a considerable number of domestic and foreign companies and research institutions have invested huge research and development funds and manpower in self-driving technology to conduct research. There is thus reason to believe that self-driving vehicles will gradually enter our lives in the near future with the advancement of self-driving technology and the gradual improvement of laws, regulations and user acceptance. In addition, the superiority of self-driving vehicles over traditional vehicles is also obvious, with one of the most obvious superiorities being that they can reduce traffic accidents and ease traffic congestion [1]. In order to achieve self-driving, self-driving vehicles often have a set of decision-making systems, which are mainly composed of the following function modules: route planning, behavioral decision-making, motion planning, and path tracking [2]. First of all, the vehicle obtains an optimal route from the starting point to the end point according to the current position and road network information. Secondly, the vehicle makes specific behavior decisions such as lane changing, parking, overtaking and so on in accordance with the current environment. Then the vehicle plans a rational path. Finally, the vehicle gives a reasonable control signal according to the current position and the reference path to achieve path tracking. Unmanned path tracking and vehicle control are the last and indispensable part of self-driving. The crux of them can be summarized as the establishment of vehicle models and the design of control algorithms. Vehicle models include kinematic bicycle models, dynamic bicycle models [3] and so on. And some typical control algorithms include model predictive control [4; 5], linear quadratic optimal control [6], PID control [7] and adaptive control [8], etc. In recent years, nonlinear model predictive control has been widely used in path tracking of self-driving vehicles, nevertheless, most of the current researches use the method of linear expansion at the trajectory reference point, without considering the stability of nonlinear model predictive control. [PERSON] et al. [4] adopted the kinematics model of self-driving vehicles, and linearly expanded the nonlinear vehicle kinematics model at the trajectory reference point, established a linearized model predictive controller, and finally converted it into a Quadratic Programming (QP) solution. [PERSON] and [PERSON] [9; 10; 11] pointed out that nonlinear model predictive control (NMPC) is a variant of model predictive control (MPC), which is mainly applicable to the nonlinear control system or the case that is inconvenient to linearize. In common with linearized MPC, NMPC needs to iteratively solve the optimal control problem in a finite time domain. However, due to the nonlinearity of the system, these problems are not necessarily non-convex. [PERSON] et al. [12] proposed a Particle Swarm Optimization (PSO) algorithm to change the MPC predictive control step size in real time to study the influence of the MPC predictive step size on vehicle path tracking. [PERSON], [PERSON] et al. [13] proposed a nonlinear model predictive control algorithm, which verifies the feasibility of nonlinear model predictive control in vehicle path tracking. [PERSON], [PERSON] et al. [14] proposed a model predictive controller with a PID feedback controller to improve the tracking accuracy of the model predictive control in the path tracking process, so that the state error tended to zero during the path tracking process. [PERSON], [PERSON] et al. [15] gave a summary of MPC theoretical research, and it is pointed out that the stability of the MPC controller can be enhanced with tail-end contraction constraints, while lead to an accompanying increase in the computing time. [PERSON], [PERSON] et al.