# Switching Control of Closed-Loop Supply Chain Systems with Markov Jump Parameters
## 2 Closed-Loop Supply Chain Switching System Modeling
In order to establish a closed-loop supply chain production-inventory switching system model, let \(x(k)=\left[ \begin{array}{cc}x_{1}^{T}(k)&x_{2}^{T}(k)\end{array} \right]^{T}\); considering Equations (1) and (3), the following closed-loop supply chain model can be obtained:  
\[\left\{ \begin{array}{l}x(k+1)=(A_{1}+\Delta A_{1})x(k)+B_{1}u(k)+b+B_{2} \omega(k)\\ y(k)=(C+\Delta C)x(k)\end{array} \right. \tag{6}\]  
where  
\[A_{1}=\left[ \begin{array}{cc}1-\rho_{1}&0\\ 0&1-\rho_{2}\end{array} \right]\), \(\Delta A_{1}=\left[ \begin{array}{cc}0&\alpha\\ 0&-\alpha-\beta\end{array} \right]\), \(b=\left[ \begin{array}{c}-d\\ 0\end{array} \right]\)  
\[B_{1}=\left[ \begin{array}{cc}1&0\\ 0&1\end{array} \right]\), \(B_{2}=\left[ \begin{array}{cc}-1&0\\ 0&0\end{array} \right]\), \(u(k)=\left[ \begin{array}{c}u_{1}(k)\\ u_{2}(k)\end{array} \right]\)  
Considering Equations (1) and (4), the supply chain model of the closed-loop system is as follows:  
\[\left\{ \begin{array}{l}x(k+1)=(A_{2}+\Delta A_{2})x(k)+B_{1}u(k)+b+B_{2} \omega(k)\\ y(k)=(C+\Delta C)x(k)\end{array} \right. \tag{7}\]where  
\[A_{2}=\left[ \begin{array}{cc}1-\rho_{1}&0\\ 0&1\end{array} \right],\Delta A_{2}=\left[ \begin{array}{cc}0&\alpha\\ 0&0\end{array} \right].\]  
Similarly, combining Equation (2) with Equation (3), the following closed-loop supply chain system can be obtained:  
\[\left\{ \begin{array}{l}x(k+1)=(A_{3}+\Delta A_{3})x(k)+B_{1}u(k)+b+B_{2} \omega(k)\\ y(k)=(C+\Delta C)x(k)\end{array} \right. \tag{8}\]  
where  
\[A_{3}=\left[ \begin{array}{cc}1&0\\ 0&1-\rho_{2}\end{array} \right],\Delta A_{3}=\left[ \begin{array}{cc}0&\alpha \\ 0&-\alpha-\beta\end{array} \right]\]  
Finally, considering Equations (2) and (4), the following closed-loop supply chain system is obtained:  
\[\left\{ \begin{array}{l}x(k+1)=(A_{4}+\Delta A_{4})x(k)+B_{1}u(k)+b+B_{2} \omega(k)\\ y(k)=(C+\Delta C)x(k)\end{array} \right. \tag{9}\]  
where  
\[A_{4}=\left[ \begin{array}{cc}1&0\\ 0&1\end{array} \right],\Delta A_{4}=\left[ \begin{array}{cc}0&\alpha\\ 0&0\end{array} \right]\]  
Suppose that the switch between the 4 subsystems is determined by the Markov process \(\{\sigma(k),k\geq 0\}\) and that the transfer probabilities of \(P=\left[P_{ij}\right]_{(ij\in S)^{\prime}}\), \(\{\sigma(k),k\geq 0\}\), satisfy  
\[P[\sigma(k+1)=j|\sigma(k)=i]=P_{ij}\;,\;\forall i,j\in S \tag{10}\]  
and meet \(P_{ij}\geq 0\), \(\sum\limits_{j=1}^{4}P_{ij}=1\). The closed-loop supply chain system can be rewritten as a switching control system as follows:  
\[\left\{ \begin{array}{l}x(k+1)=(A_{\sigma(k)}+\Delta A_{\sigma(k)})x(k)+B_{1 }u(k)+b+B_{2}\omega(k)\\ y(k)=(C+\Delta C)x(k)\end{array} \right. \tag{11}\]  
where \(\Delta A_{i}\), \(\Delta C\) are uncertain matrices. Suppose it satisfies \(\Delta A_{i}=H_{ai}F_{ai}E_{ai}\), where \(F_{ai}\) is unknown matrices, and satisfies  
\[F_{ai}^{\rm T}F_{ai}\leq I,\quad F_{ci}^{\rm T}F_{ci}\leq I. \tag{12}\]  
\(H_{ai}\), \(E_{ai}\), \(H_{ci}\), and \(E_{ci}\) are known matrices. In the following, \(\Delta A_{i}\), \(\Delta C\) is treated in that form, when \(i=1,\;3\;,\Delta A_{i}=H_{ai}F_{ai}E_{ai}\), where  
\[H_{ai}=\left[ \begin{array}{cc}1&0\\ 0&1\end{array} \right],E_{ai}=\left[ \begin{array}{cc}0&1\\ 0&-1\end{array} \right],F_{ai}=\left[ \begin{array}{cc}\alpha&0\\ 0&\alpha+\beta\end{array} \right]\]  
When \(i=2,4\), \(\Delta A_{i}=H_{ai}F_{ai}E_{ai}\), where  
\[H_{ai}=\left[ \begin{array}{cc}1&0\\ 0&0\end{array} \right],\quad E_{ai}=\left[ \begin{array}{cc}1&0\\ 0&1\end{array} \right]\]\(\Delta C=H_{c}F_{c}E_{c}\), where  
\(H_{c}=\left[ \begin{array}{cc}c_{r}&0\\ 0&c_{0}\end{array} \right]\), \(F_{c}=\left[ \begin{array}{cc}\alpha&0\\ 0&\beta\end{array} \right]\), \(E_{c}=\left[ \begin{array}{cc}1&0\\ 0&1\end{array} \right]\)  
**Note 3.