# Switching Control of Closed-Loop Supply Chain Systems with Markov Jump Parameters
## 2 Closed-Loop Supply Chain Switching System Modeling
: _For any disturbance initial condition level \(\omega(t)\in l_{2}\), the system (15) is called mean square exponentially stable and has a dry disturbance suppression level \(\gamma\) if the system mean square index is stable and satisfies the constant \(\left\|y(k)\right\|_{2}\leq\gamma\|\omega(k)\|_{2}\), where \(\left\|y(k)\right\|_{2}=\left[\sum\limits_{k=0}^{\infty}E\big{[}y^{T}(k)y(k) \big{]}\right]^{\frac{1}{2}}\)._  
The main results are given below. **Theorem 1**. : _The closed-loop supply chain system (17) with Markov jump parameters \(\{\sigma(k),k\geq 0\}\) 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 exists a set of symmetric positive definite matrices \(P_{i}i\in S\), such that the following optimization problem can be solved:_  
\[ \begin{array}{l}\min\limits_{\gamma}\\ S.t\end{array} \]  
\[\Sigma_{i1}=\left[ \begin{array}{cc}(A_{i}+B_{1}K_{i})^{T}G_{i}(A_{i}+B_{1}K _{i})+e^{-\lambda}C^{T}C-e^{-\lambda}P_{i}&(A_{i}+B_{1}K_{i})^{T}G_{i}B_{2}\\ *&B_{2}^{T}G_{i}B_{2}-e^{-\lambda}\gamma^{2}I\end{array} \right]<0. \tag{18}\]  
_where \(G_{i}=\sum\limits_{j=1}^{4}p_{ij}P_{j}\),_  
Proof. : We first consider the system at \(\omega(k)=0\). Additionally, by [PERSON]'s complementary lemma, if Equation (18) holds, then  
\[\Sigma_{i0}=(A_{i}+B_{1}K_{i})^{T}G_{i}(A_{i}+B_{1}K_{i})-e^{-\lambda}P_{i}<0.\]  
Construct the following Lyapunov functional:  
\[V(x_{k},k)=x^{T}(k)P_{\sigma(k)}x(k)\]  
**Note 5. ** The condition in Theorem 1 is not linear matrix inequality, because it contains the coupling term of the product of the controller gain matrix \(K_{i}\) and the Lyapunov functional matrix. Therefore, Theorem 1 is just the result of the mean square stability of the system in theory and has no practical operational significance. The gain matrix \(K_{i}\) of the controller can be obtained only by converting Equation (18) into linear matrix inequalities through appropriate matrix transformation. **Theorem 2. **_The closed-loop supply chain system (17) with Markov jump parameter \(\{\sigma(k),k\geq 0\}\) and state transition probability matrix satisfying condition (10) is mean square exponential stable and has disturbance suppression level \(\gamma\). If for a given positive constant \(\lambda\), there exists a set of symmetric positive definite matrices \(X_{i}\), \(Y_{i}\), \(i\in S\), such that the following optimization problem can be solved, and if the following problem is feasible, the state feedback controller \(K_{i}=Y_{i}X_{i}{}^{-1}\):_  
\[\Sigma_{i2}=\left[ \begin{array}{cccc}-e^{-\lambda}X_{i}&0&(A_{i}X_{i}+B_{1}Y _{i})^{T}W_{i}&X_{i}C^{T}\\ *&-e^{-\lambda}\gamma^{2}I&B_{2}^{T}W_{i}&0\\ *&*&-\chi&0\\ *&*&*&-e^{-\lambda}I\end{array} \right]<0. \tag{19}\]  
**Proof. ** Let  
\[W_{i}=\big{(} \begin{array}{cccc}\sqrt{p_{11}}I&\sqrt{p_{12}}I&\sqrt{p_{13}}I &\sqrt{p_{14}}I\end{array} \big{)},\chi=diag\big{\{} \begin{array}{cccc}X_{1} &X_{2}&X_{3}&X_{4}\end{array} \big{\}}\]  
then  
\[G_{i}=W_{i}\mathcal{P}W_{i}^{T},\mathcal{P}=diag\big{\{} \begin{array}{cccc}P_ {1}&P_{2}&P_{3}&P_{4}\end{array} \big{\}}\]  
It is clear that  
\[\Sigma_{i1} =\left[ \begin{array}{cc}e^{-\lambda}C^{T}C-e^{-\lambda}P_{i}&0 \\ 0&-e^{-\lambda}\gamma^{2}I\end{array} \right] \tag{20}\] \[+\left[ \begin{array}{cc}(A_{i}+B_{1}K_{i})^{T}W_{i}\\ B_{2}^{T}W_{i}\end{array} \right]\mathcal{P}\big{[} \begin{array}{cc}W_{i}^{T }(A_{i}+B_{1}K_{i})&B_{2}\end{array} \big{]}\]  
By [PERSON]'s complementary lemma, the above equation is equivalent to  
\[\left[ \begin{array}{cccc}-e^{-\lambda}P_{i}&0&(A_{i}+B_{1}K_{i})^{T}W_{i}&C ^{T}\\ *&-e^{-\lambda}\gamma^{2}I&B_{2}^{T}W_{i}&0\\ *&*&-\mathcal{P}-1&0\\ *&*&*&-e^{-\lambda}I\end{array} \right]<0 \tag{21}\]  
The above equation is multiplied by \(diag\big{\{} \begin{array}{cccc}X_{i}&I&I&I\end{array} \big{\}}\) and its transpose, \(K_{i}X_{i}=Y_{i}\), on the left and right sides, respectively, to obtain Equation (19).