# A Tube Linear Model Predictive Control Approach for Autonomous Vehicles Subjected to Disturbances
## 6 Simulation Results and Discussion
### Comparison of the Robust Invariant Set by the MPT and LP Method  
The predicted horizon has a significant effect on the efficiency and accuracy of model predictive control. When the predicted horizon adopts 20 steps, it can trade-off the accuracy and efficiency of the model predictive control well [32]. Therefore, the predicted horizon was set to 20 steps in this paper. Figure 4 shows the 20 steps of the robust invariant sets that were obtained by the LP and MPT methods. In Figure 4, 20-LP and 20-MPT represent the robust invariant sets by the LP and MPT method, respectively. From Figure 4, it can be observed that the robust invariant sets obtained by the LP method is an approximate of that by the MPT method, and the robust invariant sets obtained by the MPT method is a subset of that obtained by the LP method. Based on Definition 3, the robust invariant sets by the LP method is the maximal robust invariant set. Table 2 shows the computational burden of the 20 steps of the robust reachable sets by the LP and MPT methods. Evidently, the computational burdens of the 20 steps of the robust reachable set obtained by the LP method were much less than those obtained by the MPT method. In general, the robust invariant sets obtained by the LP method were an approximate of that obtained by the MPT method, and the computational burden of the robust invariant sets obtained by the LP method were much less than those obtained by the MPT. The reason can be explained as follows. According to the algorithm of the minimal robust positively invariant set, Algorithm 1 [31], and parameters in Table 1, the minimal robust positively invariant set was obtained by recursively adding the Minkowski sum 80 times (where the \(s\) in (17) is 80). However, it must be noted that 29,030 Minkowski sums were employed in the MPT in [34]. The number of the Minkowski sums obtained by the LP method was much less than that obtained by the MPT method.  
The proposed tube linear MPC is appropriate for real-time applications. The reasons for this can be explained as follows: (1) The computing power in engineering contexts is stronger than power of the authors' laptop. (2) Recently, the MPC has been employed for real-time applications. For example, [PERSON] et al. [35] proposed a robust-to-early termination MPC, and simulations were carried out on a F-16 aircraft to demonstrate the effectiveness of the proposed scheme. [PERSON] et al. [36] presented and analyzed a novel class of stabilizing and numerically efficient model predictive control (MPC) algorithms for discrete-time linear systems, and they demonstrated the effectiveness of the proposed approach. Therefore, it is appropriate for real-time applications.  
\begin{table}
\begin{tabular}{c c} \hline \hline Mass & 1723 kg \\ Inertia moment & 2175 kg/m\({}^{2}\) \\ Distance between the front axle and center mass & 1.232 m \\ Distance between the rear axle and center mass & 1.468 m \\ The cornering stiffness of the front tire & 66,900 N/Rad \\ The cornering stiffness of the rear tire & 62,700 N/Rad \\ The initial speed & 50 km/h \\ Friction coefficients \(\mu\) & 0.5 \\ Predictive horizon & 20 \\ Control horizon & 10 \\ Sample time & 0.02 s \\ State disturbance & \(|x|\leq[0.05,0.2,0.005,0.02]^{T}\) \\ \(Q\) & [1, 1; 0, 5] \\ \(R\) & [1] \\ \(P\) & [1, 1; 1, 1] \\ \(H\) & 10,000 \\ Limited slip angle of vehicle \(\beta_{d}\) & 12\({}^{\circ}\) \\ Limited slip angle of the front and rear tires \(a_{dr}\) & 0.25 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Vehicle parameters.