## 2 Model Predictive Control Methods
### Optimization Problem and Establishment of Performance INDEX Function  
According to Equation (3), we establish the expression of its performance index function:  
\[ \begin{array}{l}J=\sum\limits_{k=0}^{\infty}\left\{(y(k+1)-r(k+1))^{T}(y(k+ 1)-r(k+1))+\Delta u(k)^{T}\Delta u(k)\right\}\\ =\sum\limits_{k=0}^{\infty}\left\{e(k+1)^{T}e(k+1)+\Delta u(k)^{T}\Delta u(k) \right\}\end{array}  \tag{7}\]  
According to Formulas (5) and (6),  
\[ \begin{array}{l}e(k+1)=e(k)+\Delta y(k+1)\\ e(k+2)=e(k+1)+\Delta y(k+2)=e(k)+\Delta y(k+1)+\Delta y(k+2)\\ e(k+N)=e(k)+\Delta y(k+1)+\cdots\cdots+\Delta y(k+N)\end{array}  \tag{8}\]  
When \(C=I\), according to Formulas (6) and (8),\[\left[ \begin{array}{c}e(k+1)\\ e(k+2)\\ \cdots\cdots\\ e(k+N)\end{array} \right]=\left[ \begin{array}{c}e(k)\\ e(k)\\ \cdots\cdots\\ e(k)\end{array} \right]+\left[ \begin{array}{c}A\\ A^{2}+A\\ \cdots\cdots\\ A^{N}+\cdots\cdots+A\end{array} \right]\Delta x(k)\] \[+\left[ \begin{array}{c}B\\ AB+B\\ \cdots\cdots\\ A^{N-1}B+A^{N-2}B+\cdots\cdots+B\end{array} \right. \begin{array}{c}0\\ B\\ A^{N-2}B+A^{N-3}B+\cdots\cdots+B\end{array} \right]\left[ \begin{array}{c}\Delta u (k)\\ \Delta u(k+1)\\ \cdots\cdots\\ \Delta u(k+N-1)\end{array} \right]\] \[+\left[ \begin{array}{cccc}D&0&\cdots\cdots&0\\ AD+D&D&\cdots\cdots&0\\ \cdots\cdots&\cdots\cdots&0\\ A^{N-1}D+A^{N-2}D+\cdots\cdots+D&A^{N-2}D+A^{N-3}D+\cdots\cdots+D&\cdots\cdots&D \end{array} \right]\left[ \begin{array}{c}\Delta w(k)\\ \Delta w(k+1)\\ \cdots\cdots\\ \Delta w(k+N-1)\end{array} \right] \tag{9}\] \[+\left[ \begin{array}{c}e(k+1)\\ e(k+2)\\ \cdots\cdots\\ e(k+N)\end{array} \right]=\left[ \begin{array}{c}e(k)\\ e(k)\\ \cdots\cdots\\ e(k)\end{array} \right]+\left[ \begin{array}{c}A\\ A^{2}+A\\ \cdots\cdots\\ A^{N}+\cdots\cdots+A\end{array} \right]\Delta x(k)\] \[+\left[ \begin{array}{cccc}B&0&\cdots\cdots&0\\ AB+B&B&\cdots\cdots&0\\ \cdots\cdots&\cdots\cdots&\cdots\cdots\\ A^{N-1}B+A^{N-2}B+\cdots\cdots+B&A^{N-2}B+A^{N-3}B+\cdots\cdots+B&\cdots\cdots&B \end{array} \right]\left[ \begin{array}{c}\Delta u(k)\\ \Delta u(k+1)\\ \cdots\cdots\\ \Delta u(k+N-1)\end{array} \right]\] \[+\left[ \begin{array}{cccc}D&0&\cdots\cdots&0\\ AD+D&D&\cdots\cdots&0\\ \cdots\cdots&\cdots\cdots&\cdots\cdots&0\\ A^{N-1}D+A^{N-2}D+\cdots\cdots+D&A^{N-2}D+A^{N-3}D+\cdots\cdots+D&\cdots\cdots&D \end{array} \right]\left[ \begin{array}{c}\Delta w(k)\\ \Delta w(k+1)\\ \cdots\cdots\\ \Delta w(k+N-1)\end{array} \right]\]  
According to Formula (9),  
\[E=E(k)+\widetilde{A}\Delta X(k)+\widetilde{B}\Delta U(k)+\widetilde{D}\Delta W (k)\] \[\left[ \begin{array}{c}e(k+1)\\ e(k+2)\\ \cdots\cdots\\ e(k+N)\end{array} \right]=E,\ \left[ \begin{array}{c}e(k)\\ e(k)\\ \cdots\cdots\\ e(k)\end{array} \right]=E(k),\ \left[ \begin{array}{c}\Delta u(k)\\ \Delta u(k+1)\\ \cdots\cdots\\ \Delta u(k+N-1)\end{array} \right]=\Delta U(k)\] \[\left[ \begin{array}{c}\Delta w(k)\\ \Delta w(k+1)\\ \cdots\cdots\\ \Delta w(k+N-1)\end{array} \right]=\Delta W(k),\ \left[ \begin{array}{c}A\\ A^{2}+A\\ \cdots\cdots\\ A^{N}+\cdots\cdots+A\end{array} \right]=\widetilde{A} \tag{10}\] \[\left[ \begin{array}{cccc}B&0&\cdots\cdots&0\\ AB+B&B&\cdots\cdots&0\\ \cdots\cdots&\cdots\cdots&\cdots\cdots\\ A^{N-1}B+A^{N-2}B+\cdots\cdots+B&A^{N-2}B+A^{N-3}B+\cdots\cdots+B&\cdots\cdots \cdots&B\end{array} \right]=\widetilde{B}\] \[\left[ \begin{array}{cccc}D&0&\cdots\cdots&0\\ AD+D&D&\cdots\cdots&0\\ \cdots\cdots&\cdots\cdots&\cdots\cdots&0\\ A^{N-1}D+A^{N-2}D+\cdots\cdots+D&A^{N-2}D+A^{N-3}D+\cdots\cdots+D&\cdots\cdots&D \end{array}