## 3 Explicit Model Predictive Control  
To ensure robustness and meet complex constraints, this study employs MPC [27; 28] as a crucial optimal control scheme. At each sampling time k, the optimal control law of the inner loop is solved by formulating and solving a control problem that is then transformed into an online quadratic programming (QP) problem. The cost function used in this control problem is defined as follows:\[\begin{split}\underset{\Delta u}{min}\Bigg{\{}J(U,x(k))& =\chi_{k+N_{p}|k}^{T}Px_{k+N_{p}|k}\\ &+\sum_{i=0}^{N_{p}-1}\big{[}(y_{k+i|k}-y_{k+i|k}^{ref})^{T}Q_{y} \big{(}y_{k+i|k}-y_{k+i|k}^{ref}\big{)}+x_{k+i|k}^{T}Qx_{k+i|k}\\ &+u_{k+i|k}^{T}Ru_{k+i|k}\Bigg{\}}\\ \text{s.t. }x_{k|k}=x(k)\\ \text{x}_{k+i+1|k}=A_{d}x_{k+i|k}+B_{d}u_{k+i|k}\\ y_{k+i|k}=C_{d}x_{k+i|k}\\ u_{min}\leq u_{k+i|k}\leq u_{max},i=0,1,\cdots,N_{c}\\ x_{min}\leq x_{k+i|k}\leq x_{max},i=0,1,\ldots,N_{p}\\ u_{k+i|k}=Kx_{k+i|k},\ N_{c}<i\leq N_{p}\end{split} \tag{19}\]  
where \(A_{d}\), \(B_{d}\), and \(C_{d}\) are the discrete-time versions of the system, input, and output matrices, respectively. \(N_{p}\) is the prediction horizon and \(N_{c}\) is the control horizon. The notation \(x_{k+i|k}\) represents the predicted value of \(x\) at \(i\) steps ahead of \(k\). Here, \(Q\), \(R\), and \(P\) are the weighting matrices for the state, input, and terminal state, respectively. \(Q_{y}\) is the output weighting matrix used to measure tracking error.  
In addition, the output can be obtained with corresponding dimensions \(K\) and \(P\) by solving the discrete-time algebraic Riccati equation [29], given by  
\[\begin{split} P=A_{d}^{T}PA_{d}-\big{(}A_{d}^{T}PB_{d}\big{)}\big{(} R+B_{d}^{T}PB_{d}\big{)}^{-1}\big{(}B_{d}^{T}PA_{d}\big{)}+Q\\ K=\big{(}R+B_{d}^{T}PB_{d}\big{)}^{-1}B_{d}^{T}PA_{d}\end{split} \tag{20}\]  
Using these weighting matrices, the MPC controller computes a sequence of optimal vectors \(U\) that minimize the cost function \(J\).  
Although MPC offers several advantages, such as optimal control and handling of constraints, the online optimization process can lead to a significant computational burden, which is a major drawback. To address this issue, [PERSON] proposed a new MPC scheme that can reduce the computational load [30].  
The equations for predicting the state vector \(X(k)\) can be obtained through the following derivation:  
\[X(k)=\Theta x(k)+\Lambda U(k) \tag{21}\]  
where  
\[X(k)=\begin{bmatrix}x(k+1|k)\\ \vdots\\ x(k+N_{p}|k)\end{bmatrix},U(k)=\begin{bmatrix}u(k|k)\\ \vdots\\ u(k+N_{p}-1|k)\end{bmatrix},\]  
\[\Theta=\begin{bmatrix}A_{d}\\ A_{d}^{2}\\ \vdots\\ A_{d}^{N_{p}}\end{bmatrix},\ \Lambda=\begin{bmatrix}B_{d}&0&0&\cdots&0\\ A_{d}B_{d}&B_{d}&0&\cdots&0\\ A_{d}^{2}B_{d}&A_{d}B_{d}&B_{d}&\cdots&0\\ \vdots&\vdots&\vdots&\vdots\\ A_{d}^{N_{p}-1}B_{d}&A_{d}^{N_{p}-2}B&A_{d}^{N_{p}-3}B_{d}&\cdots&B_{d}\end{bmatrix}.\]  
Equations (19) and (21) can be reformulated as follows:  
\[\begin{split} V(x(k))=U(k)^{T}&\Big{(}\Lambda^{T}Q_{N_{p}} \Lambda+R_{N_{c}}\Big{)}U(k)+2x(k)^{T}\Theta^{T}Q_{N_{p}}\Lambda U(k)\\ &+x(k)^{T}\Big{(}\Theta^{T}Q_{N_{p}}\Theta+Q\Big{)}x(k)\end{split} \tag{22}\]  
By applying different constraints, more critical regions are formed for different groups of states. Therefore, a map of states to optimal control inputs is eventually created.