# Tube-Based Robust Model Predictive Control for Autonomous Vehicle with Complex Road Scenarios
## 2 Problem Statement and Preliminaries
### Model Integration  
Although nonlinear MPC can more accurately represent vehicle dynamics, its high online computational burden limits its applicability in high-frequency trajectory tracking [33]. Through dynamic modeling, a nonlinear system is formulated, which cannot be directly applied to linear time-varying model predictive control. Linear approximation is required to obtain a suitable linear time-varying representation. Taylor series expansion [34] is performed around specific operating points, retaining only the first-order terms and neglecting higher-order components, resulting in  
\[\dot{\xi}=f(\xi_{dyn},u_{dyn})+\frac{\partial f}{\partial\xi}\big{|}_{\xi}= \xi_{dyn}\quad\left(\xi-\xi_{dyn}\right)+\frac{\partial f}{\partial u}\big{|} _{\xi}=\xi_{dyn}\quad\left(u-u_{dyn}\right) \tag{6}\]  
The nonlinear dynamic system is changed as follows:  
\[\dot{\overline{\xi}}=A(t)\overline{\xi}+B(t)\overline{u} \tag{7}\]  
with \(\overline{\xi}=\xi-\xi_{dyn}\), \(A(t)=\frac{\partial f}{\partial t}|_{(\xi_{r},u_{r})}\), \(B(t)=\frac{\partial f}{\partial u}|_{(\xi_{r},u_{r})}\). Figure 2: The framework of the Tube-RMPC strategy. The derived equation is continuous, and continuous-time models are theoretically applicable to predictive control, but they often result in excessive computational effort in real-time scenarios. Therefore, it is a reasonable and effective approach to discretize the continuous linear system. The continuous linear system is converted into its discrete-time form using the Forward Euler method [35] as follows:  
\[\overline{\xi}(k+1)=A_{k,t}\overline{\xi}(k)+B_{k,t}\overline{u}(k), \tag{8}\]  
with \(A_{k,t}=I+TA(k)\), \(B_{k,t}=TB(k)\). \[A_{k,t} = \begin{bmatrix}\hat{A}_{1,1}(k)&\hat{A}_{1,2}(k)&0&\hat{A}_{1,4}( k)&0&0\\ 0&1&0&0&0&0\\ 0&0&1&T_{s}&0&0\\ \hat{A}_{4,1}(k)&\hat{A}_{4,2}(k)&0&\hat{A}_{4,4}(k)&0&0\\ \hat{A}_{5,1}(k)&\hat{A}_{5,2}(k)&\hat{A}_{5,3}(k)&0&1&0\\ \hat{A}_{6,1}(k)&\hat{A}_{6,2}(k)&\hat{A}_{6,3}(k)&0&0&1\end{bmatrix} ,\;\hat{A} _{1,1}(k)\;=\;\;1-\frac{T(2C_{cf}+2C_{cr})}{m\bar{x}},\] \[\hat{A}_{1,2}(k)=\frac{T(2C_{cf}+2C_{cr})\bar{y}}{m\bar{x}^{2}}- \frac{T(2C_{cf}-2 aC_{cf})\bar{\varphi}}{m\bar{x}^{2}},\hat{A}_{1,4}(k)=T \Big{(}-\dot{x}+\frac{2\mathbb{C}_{cr}-2 aC_{cf}}{m\bar{x}}\Big{)},\hat{A}_{4, 1}(k)=T_{s}\frac{2C_{cr}-2 aC_{cf}}{l_{z}\bar{x}(k)},\] \[\hat{A}_{4,2}(k)\;=\;\;\;-\frac{T(2\mathbb{C}_{cf}-2 aC_{cf})}{l_ {z}\bar{x}^{2}}\dot{y}\;+\;\frac{T(2a^{2}C_{cf}+2b^{2}C_{cr})}{l_{z}\bar{x}^{2 }}\dot{\varphi},\;\;\hat{A}_{4,4}(k)\;=\;\;1-T\frac{2a^{2}C_{cf}+2b^{2}C_{cr}}{ l_{z}\bar{x}^{2}},\] \[\hat{A}_{5,1}(k)\;=\;T\cos\varphi,\;\hat{A}_{5,2}(k)\;=\;T\sin\varphi, \;\hat{A}_{5,3}(k)\;=\;T\dot{x}\cos\varphi-T\dot{y}\sin\varphi,\;\hat{A}_{6,1} (k)\;=\;-T\sin\varphi,\] \[\hat{A}_{6,2}(k)\;=\;T\cos\varphi,\;\hat{A}_{6,3}(k)\;=\;-T\dot{y }\sin\varphi-T\dot{x}\cos\varphi,\;B_{k,t}=\Big{[}\frac{2 TC_{cf}}{m}\;\;\;0 \;\;\;0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Firstly, the error system is defined by taking the difference between Equations (9) and (10):  
\[ \begin{array}{l}\overline{\xi}(k+1)-\overline{\xi}^{\prime}(k+1)=A_{k,t}( \overline{\xi}(k)-\overline{\xi}^{\prime}(k))\\ +B_{k,t}(\overline{u}(k)-\overline{u}^{\prime}(k))+w(k)\end{array}  \tag{11}\]  
Let \(e(k)=\overline{\xi}(k)-\overline{\xi}^{\prime}(k),\overline{u}_{e}(k)= \overline{u}(k)-\overline{u}^{\prime}(k)\).