# A Tube Linear Model Predictive Control Approach for Autonomous Vehicles Subjected to Disturbances
## 5 The Tube MPC Optimization Problem  
In this section, the disturbances are introduced into the MPC framework to formulate the tube MPC optimization problem. In this paper, it was assumed that the control systems are perturbed by a bounded disturbance \(w\).  
\[x_{k+1}=A_{T}x_{k}+B_{T}u_{k}+w_{k}. \tag{34}\]  
The tube MPC optimization problem can be described as follows:  
\[\begin{array}{l}\min\limits_{u}f_{N}=\sum\limits_{k=0}^{N-1}\left(x_{k}^{T} Qx_{k}+u_{k}^{T}Ru_{k}\right)+x_{N}^{T}Px_{N}+\rho^{T}H\rho\\ s.t\;\;x_{k}\in\overline{x}=x_{k}\in\;\varepsilon\;\;\;\;\;k=0,1,\ldots,N-1\\ \mathrm{h}(x_{k},u_{k})\;\leq\delta\;\;\;\;k=0,1,\ldots,N-1\\ u_{k}\in\overline{u}=u\ominus\kappa\epsilon\;\;\;\;k=0,1,\ldots,N-1\end{array}, \tag{35}\]  
where \(\varepsilon\) expresses the robust invariant set for the states; \(N\) is the predicted horizon; \(\rho\) is the slack factor; \(\overline{u}\) and \(\overline{x}\) express the restricted input constraint and state constraint, respectively; and \(Q\), \(R\), \(P\), and \(H\) represent the weights of the appropriate dimension penalizing state tracking error, control action, change rate of control, and violation of the soft constraints, respectively.  
The solving process of the tube MPC optimization problem can be described as follows. Firstly, all states are calculated for all inputs and disturbances; specifically, the robust positively invariant set when under additive disturbance conditions is calculated by Algorithm 1. Then, the constraints are tightened to keep the autonomous system meeting the requirement of the constraints when under additive disturbance conditions. In this paper, the tightened constraints are equal to the Pontryagin difference of the original constraints, and the maximal robust positively invariant set is based on Equation (35). Finally, the tightened constraints are introduced into the MPC framework, and the optimal input sequence is computed by solving a tightened constrained optimal control problem at each sampling time. The computed optimal control input sequence and the corresponding predicted vehicle state are stored as the nominal input and state. In the next step, the optimal control problem is solved, starting with new state measurements. From the above description, it can be obtained that the optimization problem's decision variable is the front tires' steering input sequence. Though the steering input sequence is the 4 \(\times\) 1 column vector, it has only one effective dimension.  
Note that the proposed linear MPC is based on the simplified linear bicycle model, and it is available when the disturbances make the vehicle go into the linear area. The studies in [30] show that most of the existing disturbances did not make the vehicle go into the non-linear area. Therefore, the proposed linear MPC can adapt to most disturbances. When the simplified linear bicycle model cannot precisely describe the vehicle motion, the proposed linear MPC system is not available. For example, the bicycle model cannot describe the vehicle's motion on the icy or slippery road precisely. The proposed linear MPC system is not available for the above two conditions.