# Tube-Based Robust Model Predictive Control for Autonomous Vehicle with Complex Road Scenarios
## 2 Problem Statement and Preliminaries
### Theoretical Demonstration and Analysis  
**Theorem 1** ([37]). : _For the error model (15), if there are matrices \(Y\) and \(Q>0\) that make the following linear matrix inequalities (16) true, then the controller \(K=YQ^{-1}\), and \(e(k)\) is stable._  
\[ \begin{bmatrix}Q&(A_{k,t}Q+B_{k,t}Y)^{T}\\ A_{k,t}Q+B_{k,t}Y&Q\end{bmatrix} >0,\,A_{k,t},B_{k,t}\in\Omega \tag{16}\]  
Proof. : Since the error system (15) is linear and the disturbances are bounded, the effect of the disturbance on system stability has already been considered when defining the disturbance-invariant set. Given this, we can directly analyze the stability of the error system without the disturbance term (17). \[e(k+1)=(A_{k,t}+B_{k,t}K)e(k) \tag{17}\]Define the Lyapunov function as \(V(k)=e(k)^{\mathsf{T}}(k,t)e(k)\), where it is the Lyapunov matrix, and the feedback control gain satisfies \(V((A_{k,t}+B_{k,t}K)e(k))-V(e(k))<0\) for any \(e(k)\in R^{n}\), \(A_{k,t},B_{k,t}\in\Omega\). Then, the stability condition is equivalent to  
\[(k,t)-(A_{k,t}+B_{k,t}K)^{\mathsf{T}}(k,t)(A_{k,t}+B_{k,t}K)>0 \tag{18}\]  
Let \((k,t)=Q^{-1},K=YQ^{-1}\). Then, we have  
\[Q^{-1}-(A_{k,t}+B_{k,t}YQ^{-1})^{\mathsf{T}}Q^{-1}(A_{k,t}+B_{k,t}YQ^{-1})>0 \tag{19}\]  
Multiplying the left and right sides of (19) by Q results in  
\[Q-(A_{k,t}Q+B_{k,t}Y)^{T}Q^{-1}(A_{k,t}Q+B_{k,t}Y)>0 \tag{20}\]  
Applying the [PERSON] complement lemma to (20) yields  
\[ \begin{bmatrix}Q&(A_{k,t}Q+B_{k,t}Y)^{\mathsf{T}}\\ A_{k,t}Q+B_{k,t}Y&Q\end{bmatrix} >0.\]  
Next, we define \(\overline{\xi}-\overline{\xi}^{\prime}\) by calculating the disturbance-invariant set \(Z\), leading to Lemma 1. \(\square\)  
**Lemma 1** ([38]). : _Let \(Z\) be the interference invariant set of the system (15), the actual uncertain system (9), and the corresponding nominal system (10). If, at time \(k\), the state of the actual system \(\overline{\xi}\) satisfies \(\overline{\xi}(k)\in\overline{\xi}^{\prime}(k)\oplus Z\), which is the control input \(u=K(\overline{\xi}-\overline{\xi}^{\prime})+u^{\prime}\), it can be guaranteed that for all \(A_{k,t},B_{k,t}\in\Omega\) and \(\forall w(k)\in W\), \(\overline{\xi}(k+1)\in\overline{\xi}^{\prime}(k+1)\oplus Z\) is true._  
From Lemma 1, to ensure the optimal control performance, the size of \(Z\) should be minimized; here, we adopt the minimum robust positively invariant (RPI) set as \(Z\). In reference [39], the minimum RPI set is defined as \(S_{\infty}=\oplus_{i=0}^{\infty}(A_{k,t}+B_{k,t}K)^{i}W\), which involves the summation of Minkowski sets of infinite terms. However, computing such a set necessitates that the system dynamics exhibit nilpotency [40]. Therefore, to address this challenge, we simplify the expression of the invariant set by defining it in inequality form: \(Z=\{x\in R^{n}:-z\leq x\leq z\},z>0\). With the help of the method proposed in the literature [41], the original optimization problem is cleverly converted into a linear programming problem. Next, for the design of Tube-RMPC, the first and most crucial step is to determine a central trajectory, which is the optimal trajectory. Based on this principle, we adopt the same control law for each vertex. We select Lyapunov functions \(V^{\prime}(k+i|k)=\overline{\xi}^{\prime}(k+i\,|\,k)^{\mathsf{T}}(k_{0},t_{0} )\overline{\xi}^{\prime}(k+i|k)\), \((k_{0},t_{0})>0\) as a calm matrix, and \(\overline{\xi}^{\prime}(k+i|k)\) on behalf of the nominal system forecast, and at time \(k+i\), \(u^{\prime}(k+i|k)\) is the \(k+i\) moment of nominal control input. The Lyapunov function at time \(k\) is defined as \(V^{\prime}(k|k)=\overline{\xi}^{\prime}(k\,|\,k)^{\mathsf{T}}(k_{0},t_{0}) \overline{\xi}^{\prime}(k|k)\) of the upper bound value of \(\gamma_{0}\). The objective function is defined as  
\[J^{\prime}_{\infty}(k)=\sum_{i=0}^{\infty}\big{(}\big{\|}\overline{\xi}^{ \prime}(k+i|k)\big{\|}_{L_{0}}^{2}+\big{\|}u^{\prime}(k+i|k)\big{\|}_{R_{0}}^ {2}\big{)},\]  
where \(L_{0}\) and \(R_{0}\) are positive definite weighted matrices.