# A Tube Linear Model Predictive Control Approach for Autonomous Vehicles Subjected to Disturbances
## 1 Introduction  
With the increase in the number of vehicles, traffic safety and efficiency have become a severe issue for governments and humans [1; 2; 3; 4; 5]. Autonomous vehicles can provide more excellent safety benefits and improve traffic efficiency, and they are assumed to be an efficient solution for traffic safety and efficiency issues [6; 7; 8]. The control layer for automatic vehicles has a significant effect on traffic safety and efficiency issues, and many scholars have devoted themselves to creating effective control methods for automatic vehicles [9; 10; 11; 12]. The commonly used controllers are LQR (Linear Quadratic Regulator) [13], fuzzy logic control [14], sliding mode control [15], and backstepping [16; 17]. Compared with other controllers, model predictive control has better real-time performance, stability, and robustness [13]. Therefore, it is widely used in AVs. [PERSON] et al. [18] designed a non-linear MPC controller that navigates AVs to avoid obstacles. The non-linear MPC method introduces a computational burden during the iterative calculations of MPC and requires expensive hardware configurations, which makes the non-linear MPC difficult to apply in practical engineering applications. To deal with the issue, many scholars [19; 20; 21] have carried out a great deal of studies. [PERSON] et al. [22] proposed a lower-dimensional MPC approach that entailed a trade-off between the tracking performance and the computational burden. [PERSON] et al. [23] employed the linear time-varying (LTV) approach to decrease the online computational burden of the MPC. However, the tracking performance is degraded under disturbance conditions, like parameter uncertainty and cross-wind [24]. For example, due to the cross-wind disturbance, the actual path of a vehicle deviates from its planned path and the vehicle impacts with the obstacle (see Figure 1). To attenuate the disturbances in the MPC, [PERSON] et al. [25] proposed a tube nonlinear MPC approach to ensure that the states satisfy the constraints when under common disturbance conditions. [PERSON] et al. [26] adopted a tube non-linear MPC for lane keeping and obstacle avoidance when under disturbance conditions. The results showed the effectiveness of the controller in the presence of uncertainty. [PERSON] et al. [27] employed a tube non-linear MPC to develop a safe lane changes for automated driving by considering environmental information, sensor uncertainties, and collision risks. [PERSON] et al. [28, 29] presented a tube non-linear MPC approach for the trajectory tracking of an autonomous race car at the handling limits. The results demonstrated that the tube MPC controller reduced the constraint violations compared with the traditional MPC controller. To consider the effect of disturbances, the tube non-linear MPC is usually employed with a Multi-Parametric Toolbox (MPT) to calculate the most robust positively invariant set to tighten the constraints. The computational burden of the most robust positively invariant set is big, which makes it hard to apply a tube non-linear MPC in engineering applications. To improve autonomous vehicle tracking performances and computational efficiency when under disturbance conditions, this paper proposes a tube linear model predictive controller (MPC) framework. The proposed framework simplifies the complex vehicle dynamic models to a bicycle model with three degrees of freedom. In addition, the complex computation of a robust positively invariant set is transferred to a linear programming (LP) problem instead of an MPT to solve. The computational burden is significantly reduced by the proposed framework. The proposed framework not only considers disturbances, but also operates with high computational efficiency. This paper is structured as follows: Section 2 introduces the tube linear MPC theory. Section 3 establishes a simplified bicycle dynamic model. Section 4 presents the collision avoidance constraints, and Section 5 formulates the tube linear MPC mathematical problem.