## 3 Design of Control System
### Mpc
#### 3.1.2 Design of Objective Function  
Define \(kT\) as \(k\), The state functions are:  
\[\left\{ \begin{array}{c}\chi(k{+}1)=\overset{\wedge}{A}_{k}\chi(k)+\overset{ \wedge}{B}_{k}\Delta u(k)\\ \eta(k)=\overset{\wedge}{C}_{k}\chi(k)\end{array} \right. \tag{12}\]  
\[\text{where}\quad\chi(k)=\left[ \begin{array}{c}\overset{\wedge}{\zeta}(k)\\ u(k{-}1)\end{array} \right]_{(m+n)\times 1},\quad\overset{\wedge}{A}_{k}=\left[ \begin{array}{cc}\widetilde{A}_{n\times n}&\widetilde{B}_{n\times m}\\ 0_{m\times n}&\mathrm{I}_{m}\end{array} \right]_{(m+n)\times(m+n)},\]  
\[\overset{\wedge}{B}_{k}=\left[ \begin{array}{c}\widetilde{B}_{n\times m}\\ 1_{m}\end{array} \right]_{(m+n)\times m},\overset{\wedge}{C}_{k}=\left[ \begin{array}{cc}\mathrm{I}_{n\times n}&0_{n\times m}\end{array} \right]_{n \times(m+n)},\]  
\(m\) is the number of control quantity, and \(n\) is the number of status quantity. At time \(k+2\), Equation (12) can be written as:  
\[\left\{ \begin{array}{c}\chi(k{+}2)=\overset{\wedge}{A}_{k+1}\chi(k{+}1)+ \overset{\wedge}{B}_{k+1}\Delta u(k{+}1)\\ \eta(k{+}1)=\overset{\wedge}{C}_{k+1}\chi(k{+}1)\end{array} \right. \tag{13}\]  
Substituting Equation (12) into Equation (13), the state functions can be expressed as:  
\[\left\{ \begin{array}{c}\chi(k{+}2)=\overset{\wedge}{A}_{k+1}\overset{\wedge }{A}_{k}\chi(k{+}1)+\overset{\wedge}{A}_{k+1}\overset{\wedge}{B}_{k}\Delta u( k)+\overset{\wedge}{B}_{k+1}\Delta u(k{+}1)\\ \eta(k{+}1)=\overset{\wedge}{C}_{k+1}\overset{\wedge}{A}_{k}\chi(k)+\overset{ \wedge}{C}_{k+1}\overset{\wedge}{B}_{k}\Delta u(k)\end{array} \right. \tag{14}\]  
By recursion in the above way, Equation (15) can be solved. \[\left\{ \begin{array}{c}\eta(k{+}1)=\overset{\wedge}{C}_{k+1}\overset{\wedge }{A}_{k}\chi(k)+\overset{\wedge}{C}_{k+1}\overset{\wedge}{B}_{k}\Delta u(k) \\ \eta(k{+}2)=\overset{\wedge}{C}_{k+2}\overset{\wedge}{A}_{k+1}\overset{ \wedge}{A}_{k}\chi(k)+\overset{\wedge}{C}_{k+2}\overset{\wedge}{A}_{k+1} \overset{\wedge}{B}_{k}\Delta u(k)+\overset{\wedge}{C}_{k+2}\overset{ \wedge}{B}_{k+1}\Delta u(k)\\ \eta(k{+}3)=\overset{\wedge}{C}_{k+3}\overset{\wedge}{A}_{k+2}\overset{ \wedge}{A}_{k+1}\overset{\wedge}{A}_{k}\chi(k)+\overset{\wedge}{C}_{k+3} \overset{\wedge}{A}_{k+2}\overset{\wedge}{A}_{k+1}\overset{\wedge}{B}_{k} \Delta u(k)+\overset{\wedge}{C}_{k+3}\overset{\wedge}{A}_{k+2}\overset{ \wedge}{B}_{k+1}\Delta u(k)+\overset{\wedge}{C}_{k+3}\overset{\wedge}{A}_{k+2} \overset{\wedge}{B}_{k+1}\Delta u(k)\\ \vdots\\ \eta(k+N_{k})=\overset{\wedge}{C}_{k+N_{k}}\overset{N_{k}-1}{\underset{i=0} {\overset{\wedge}{A}}_{k+i}\chi(k)+\overset{\wedge}{C}_{k}\overset{\wedge}{B} _{k}\overset{N_{k}-1}{\underset{i=0}{\overset{\wedge}{A}}_{k+i}\Delta u(k)+ \cdots+\overset{\wedge}{C}_{k+2}\overset{\wedge}{B}_{k+N_{k}-1}\Delta u(k)} \\ \vdots\\ \eta(k+N_{p})=\overset{\wedge}{C}_{k+N_{p}}\overset{N_{k}-1}{\underset{i=0} {\overset{\wedge}{A}}_{k+i}\chi(k)+\overset{\wedge}{C}_{k}\overset{\wedge}{B} _{k}\overset{N_{k}-1}{\underset{i=1}{\overset{\wedge}{A}}_{k+i}\Delta u(k)+ \cdots+\overset{\wedge}{C}_{k}\overset{\wedge}{B}_{k+N_{k}-1}\overset{N_{k}- 1}{\underset{i=N_{k}}{\overset{\wedge}{A}}_{k+i}\Delta u(k)}}\end{array} \right. \tag{15}\]  
Equation (16) can be abbreviated as:  
\[Y(k)=\psi_{k}\chi(k)+\Theta_{k}\Delta U_{k} \tag{16}\]\[\text{where}\;Y(k)=\left[ \begin{array}{c}\eta(k+1)\\ \eta(k+2)\\ \cdots\\ \eta(k+N_{c})\\ \cdots\\ \eta(k+N_{p})\end{array} \right]_{N_{p}\times 1},\hat{\bar{C}}_{k}=\hat{\bar{C}}_{k+1 }=\cdots=\hat{\bar{C}}_{k+N_{p}},\psi_{k}=\left[ \begin{array}{c}\hat{\bar{C}}_{k} \hat{\bar{A}}_{k}\\ \hat{\bar{C}}_{k}\hat{\bar{A}}_{k+1}\hat{\bar{A}}_{k}\\ \cdots\\ \hat{\bar{C}}_{k}\hat{\bar{\prod}}_{i=0}^{N_{p}-1}\hat{\bar{A}}_{k+i}\\ \cdots\\ \hat{\bar{C}}_{k}\hat{\prod}_{i=0}^{N_{p}-1}\hat{\bar{A}}_{k+i}\\ \hat{\bar{C}}_{k}\hat{\prod}_{i=0}^{N_{p}-1}\hat{\bar{A}}_{k+i}\end{array} \right]_ {N_{p}\times 1},\]  
\[\Theta_{k}=\left[