## 1 Introduction  
The autonomous driving of unmanned vehicles is in a golden age of development. Trajectory tracking plays a pivotal role in autonomous driving, and tracking controllers are fundamental to the driving performance of unmanned vehicles. In recent years, the number of autonomous controllers has increased. The most widely used of these controllers are proportional-integral-derivative (PID) controllers, linear quadratic regulators (LQR), and model predictive control (MPC) controllers [1; 2; 3; 4; 5]. A PID controller is a negative feedback controller based on error compensation. It does not require complex modeling of the control object, and adjust the controller parameters to obtain a good tracking effect [6; 7]. However, the parameters of the PID controller need to be determined through a large number of simulations, and have to be adjusted dynamically under different working conditions. An LQR is based on state feedback, which can reduce steady-state lateral errors during curve driving [8]. Additionally, LQRs can perform real-time computations to handle emergency situations [9]. However, this depends on an accurate control model and requires a curvature continuity trajectory. An MPC has the basic characteristics of a predictive model, rolling optimization, and feedback correction. It is especially suitable for situations where accurate mathematical models cannot be established and there are multiple constraints [10]. MPCs are more applicable than LQRs, which rely too much on models. In real driving processes, it is necessary to predict driving actions over a future horizon according to the surrounding environment, with complicated road and traffic conditions, even if the vehicle is driving manually. Neither PID nor LQR can realize this. However, MPC could realize the predictions needed for real-time controls with its basic characteristics of predicting several time steps in the future. Linear model predictive controls (LMPCs) was proposed and applied due to its simple linearization model and fast calculation speed [11]. Researchers establish kinematic models for different control object, and use LMPC to obtain good trajectory tracking effects [12; 13]. In [14], a multilayer control system that includes three path tracking controllers was proposed based on LMPC, which can actively adjust the longitudinal velocity and use suitable path tracking controller to improve the real-time performance. In [15], a LMPC is designed to linearize the nonlinear vehicle model locally at each sampling point to reduce the computational complexity. In addition, many MPC algorithms for autonomous driving use linear predictive models based on approximate linearization [16]. A linear time-varying predictive model considering cornering characteristics was designed and optimized in [17]. [PERSON] et al. directly used a simplified vehicle linear kinematics model and verified the algorithm by a single-shift simulation [18]. However, since the control objects are generally nonlinear complex systems, the accuracy of approximate linearization is poor, and more and more system nonlinear factors are added to the consideration of MPCs [19]. [PERSON] et al. compared the control effects of nonlinear model predictive controls (NMPCs) and LMPCs, and the result shows that the LMPC algorithm are not satisfactory at high speeds or when target trajectory curvatures change rapidly [20]. [PERSON] et al. compared results for LMPCs and NMPCs in simulations, showing that LMPC lateral errors fluctuated greatly at different speeds and that the performance of the LMPCs was lower than that of the NMPCs [21]. The solution complexity of NMPCs is closely related to the dimensions of the equations. Even if a simple dynamic model is used, multiple time steps need to be predicted, and the computational load will be large [22]. Thus, NMPCs remain challenging to use in real-time operations.