# Tube-Based Robust Model Predictive Control for Autonomous Vehicle with Complex Road Scenarios
## 2 Problem Statement and Preliminaries
### Theoretical Demonstration and Analysis
Referring to the MPC algorithm mentioned in reference [42], the feedback gain matrix for the nominal system can be derived by solving optimization problem 1. Optimization problem 1:  
\[ \begin{split}&\min_{F}\max_{[A_{p},B_{g}]\in\Omega}\int_{\mathcal{O} \sigma^{\prime}}(k)=\sum\limits_{i=0}^{\infty}\left[\left\|\overline{\xi}^{ \prime}(k+i\mid k)\right\|_{L_{0}}^{2}+\left\|F\overline{\xi}^{\prime}(k+i \mid k)\right\|_{R_{0}}^{2}\right]\\ & s.t.\\ &\left\|\overline{\xi}^{\prime}(k)\right\|_{(k_{0},t_{0})}^{2} \leq\gamma_{0},\overline{\xi}^{\prime}(k)=\overline{\xi}^{\prime}(k|k)\\ &\left\|\overline{\xi}^{\prime}(k+i+1\mid k)\right\|_{(k_{0},t_{ 0})}^{2}-\left\|\overline{\xi}^{\prime}(k+i\mid k)\right\|_{(k_{0},t_{0})}^{2} \leq-\left\|\overline{\xi}^{\prime}(k+i\mid k)\right\|_{L_{0}}^{2}-\left\|F \overline{\xi}^{\prime}(k+i\mid k)\right\|_{R_{0}}^{2}\\ & u^{\prime}(k)\in U\Theta KZ\end{split} .\]  
Consider constraints: \(u\in U:=\{u\in R|u(k)\leq u_{\max}|\}\) where \(U\subset R\) is a compact set containing the origin. To ensure that the actual perturbated system satisfies the input constraint \(u\in U\), the constraints of the nominal system must be tightened through a constraint-tightening procedure, as formalized by \(u^{\prime}(k)\in U\Theta KZ\) in the aforementioned optimization problem 1. Therefore, optimization problem 1 can be transformed into an LMI problem. For a given initial state \(\overline{\xi}^{\prime}\) and all vertices of the vehicle dynamic model system, if there exist symmetric matrices and a constant \(\gamma>0\) such that the following LMIs hold, then the state feedback control law \(u^{\prime}=F\overline{\xi}^{\prime\prime}\) ensures the stability of the nominal system. Optimization problem 2:  
\[ \begin{split}&\min_{\gamma,\gamma^{\prime},Q^{\prime}}\gamma\\ &\text{s.t}\\ &\left[\begin{array}{cccc}1&*\\ \overline{\xi}^{\prime}(k)&Q^{\prime}\end{array}\right]\geq 0\\ &\left[\begin{array}{cccc}Q^{\prime}&*&*&*\\ A_{k,l}Q^{\prime}+B_{k,l}Y^{\prime}&Q^{\prime}&*&*\\ L^{\frac{1}{2}}Q^{\prime}&0&\gamma I&*\\ R^{\frac{1}{2}}Y^{\prime}&0&0&\gamma I\end{array}\right]\geq 0\\ &\left[\begin{array}{cccc}\chi&Y^{\prime}\\ Y^{\prime\prime}&Q^{\prime}\end{array}\right]\geq 0,\chi\leq U\Theta KZ.\end{split} \]  
with \(\forall(A_{k,t},B_{k,t})\in\Omega\), \(F=Y^{\prime}Q^{\prime-1}\), \(P^{\prime}=\gamma Q^{\prime-1}\). Based on the above analysis, the control law \(u=K(\overline{\xi}-\overline{\xi}^{\prime})+u^{\prime}\) can be rewritten as \(u=K(\overline{\xi}-\overline{\xi}^{\prime})+F\overline{\xi}^{\prime\prime}\). It is evident that if \(\overline{\xi}(k)\in\overline{\xi}^{\prime}(k)\oplus Z,u^{\prime}=F\overline{ \xi}^{\prime\prime}\in U\Theta KZ\), then for \(\forall(A_{k,t},B_{k,t})\in\Omega\) and \(w(k)\in W\), the actual uncertain system's control input satisfies the constraint \(u\in U\). Furthermore, if the local feedback gain \(F\) can robustly stabilize the nominal system \(\overline{\xi}^{\prime}\) and its nominal state, it can effectively track reference trajectories while ensuring that \(\overline{\xi}(k)\in\overline{\xi}^{\prime}(k)\oplus Z\). Consequently, the actual system's state \(\overline{\xi}\) will converge to a tube centered around the nominal trajectory, thereby guaranteeing the robust stability of the actual system.