# Switching Control of Closed-Loop Supply Chain Systems with Markov Jump Parameters
## 2 Closed-Loop Supply Chain Switching System Modeling
In the following, we consider the uncertainty of the system due to the recycling remanufacturing rate and give sufficient conditions for the closed-loop supply chain system (15) to be exponentially stable in mean square and have a disturbance rejection level of \(\gamma\) based on the idea of handling uncertainty in robust control thinking. **Theorem 3. **_The closed-loop supply chain system (15) with Markov jump parameter \(\{\sigma(k),k\geq 0\}\) and state transition probability matrix satisfying condition (10) is exponentially stable with mean square and interference suppression level \(\gamma\). If for a given positive constant \(\lambda\), there is a set of symmetric positive definite matrix \(X_{i}\), \(Y_{i}\), and parameter \(\varepsilon_{ai}\), \(\varepsilon_{c}\), which makes the followingoptimization problem solvable, and if the following problem is feasible, the state feedback controller \(K_{i}=Y_{i}X_{i}{}^{-1}\)._  
\[\left[ \begin{array}{ccccc}-e^{-\lambda}X_{i}&0&\left(A_{i}X_{i}+B_{1}Y_{i} \right)^{T}W_{i}&X_{i}C^{T}&X_{i}E_{ai}^{T}&X_{i}E_{c}^{T}\\ *&-e^{-\lambda}\gamma^{2}I&B_{2}^{T}W_{i}&0&0\\ *&*&\Xi_{i3}&0&0&0\\ *&*&*&\Xi_{i4}&0&0\\ *&*&*&*&-\varepsilon_{ai}&0\\ *&*&*&*&-\varepsilon_{c}\end{array} \right]<0 \tag{22}\]  
_where_  
\[\Xi_{i3}=-\chi+\varepsilon_{ai}W_{i}^{T}H_{ai}H_{ai}^{T}W_{i},\quad\Xi_{i4}=-e ^{-\lambda}I+\varepsilon_{c}W_{i}^{T}H_{c}H_{c}^{T}W_{i}.\]  
Proof. : It follows from Theorem 2 that a sufficient condition for a closed-loop supply chain system (15) to be mean square exponentially stable and have a disturbance suppression level \(\gamma\) is that the following inequality holds:  
\[\left[ \begin{array}{ccccc}-e^{-\lambda}X_{i}&0&\left(\left(A_{i}+\Delta A_{ i}\right)X_{i}+B_{1}Y_{i}\right)^{T}W_{i}&X_{i}(C+\Delta C)^{T}\\ *&-e^{-\lambda}\gamma^{2}I&B_{2}^{T}W_{i}&0\\ *&*&-\chi&0\\ *&*&*&-e^{-\lambda}\end{array} \right]<0 \tag{23}\]  
Since \(\Delta A_{i}=H_{ai}F_{ai}E_{ai}\), Equation (23) can be rewritten as  
\[\left[ \begin{array}{ccccc}-e^{-\lambda}X_{i}&0&\left(A_{i}X_{i}+B_{1}Y_{i} \right)^{T}W_{i}&X_{i}(C+\Delta C)^{T}\\ *&-e^{-\lambda}\gamma^{2}I&B_{2}^{T}W_{i}&0\\ *&*&-\chi&0\\ *&*&*&-e^{-\lambda}\end{array} \right] \tag{24}\] \[+\left[ \begin{array}{ccccc}0\\ 0\\ W_{i}^{T}H_{ai}\\ 0\end{array} \right]F_{ai}\big{[} \begin{array}{ccccc}E_{ai}X_{i}&0&0&0&0\\ \end{array} \big{]}+\left[ \begin{array}{ccccc}X_{i}E_{ai}^{T}\\ 0\\ 0\\ 0\end{array} \right]F_{ai}^{T}\big{[} \begin{array}{ccccc}0&0&H_{ai}^{T}W_{i}&0 \end{array} \big{]}<0\]  
By Lemma 1, if Equation (24) holds, then the following equation holds:  
\[\left[ \begin{array}{ccccc}-e^{-\lambda}X_{i}&0&\left(A_{i}X_{i}+B_{1}Y_{i} \right)^{T}W_{i}&X_{i}(C+\Delta C)^{T}\\ *&-e^{-\lambda}\gamma^{2}I&B_{2}^{T}W_{i}&0\\ *&*&-\chi&0\\ *&*&*&-e^{-\lambda}\end{array} \right] \tag{25}\] \[+\varepsilon_{ai}\left[ \begin{array}{ccccc}0\\ 0\\ W_{i}^{T}H_{ai}\\ 0\end{array} \right]\big{[} \begin{array}{ccccc}0&0&H_{ai}^{T}W_{i}&0\end{array}  \big{]}+\varepsilon_{ai}^{-1}\left[ \begin{array}{ccccc}X_{i}E_{ai}^{T}\\ 0\\ 0\\ 0\end{array} \right]\big{[} \begin{array}{ccccc}E_{ai}X_{i}&0&0&0\end{array}  \big{]}<0\]  
From [PERSON]'s complementary lemma, it follows that  
\[\left[ \begin{array}{ccccc}-e^{-\lambda}X_{i}&0&\left(A_{i}X_{i}+B_{1}Y_{i} \right)^{T}W_{i}&X_{i}(C+\Delta C)^{T}&X_{i}E_{ai}^{T}\\ *&-e^{-\lambda}\gamma^{2}I&B_{2}^{T}W_{i}&0&0\\ *&*&-\chi+\varepsilon_{ai}W_{i}^{T}H_{ai}H_{ai}^{T}W_{i}&0&0\\ *&*&*&-e^{-\lambda}I&0\\ *&*&*&*&-\varepsilon_{ai}\end{array} \right]<0 \tag{26}\]  
Since the uncertainty matrix still exists in the above equation, we continue to apply Lemma 1 to deal with the uncertainty matrix present in the inequality. Since \(\Delta C=H_{c}F_{c}E_{c}\), Equation (26) is rewritten as \[\left[ \begin{array}{ccccc}-e^{-\lambda}X_{i}&0&(A_{i}X_{i}+B_{1}Y_{i})^{T}W_{i}& X_{i}C^{T}&X_{i}E_{ai}^{T}\\ *&-e^{-\lambda}\gamma^{2}I&B_{2}^{T}W_{i}&0&0\\ *&*&-\chi+\varepsilon_{ai}W_{i}^{T}H_{ai}H_{ai}^{T}W_{i}&0&0\\ *&*&*&-e^{-\lambda}&0\\ *&*&*&*&-\varepsilon_{ai}\end{array} \right] \tag{27}\] \[+\left[