# Switching Control of Closed-Loop Supply Chain Systems with Markov Jump Parameters
## 2 Closed-Loop Supply Chain Switching System Modeling
** The bullwhip effect is the most significant performance indicator of a closed-loop supply chain analysis, which generates production control and recovery remanufacturing control through inventory levels \(u(k)\), suppressing the perturbations of uncertain system demand \(\omega(k)\), thereby minimizing the system operating costs \(y(k)\), and this level of suppression can be described by the following equation:  
\[\frac{\|y\|_{2}}{\|\omega\|_{2}}\leq\gamma \tag{13}\]  
The smaller the \(\gamma\), the better the performance of the system, and it is easy to see that Equation (13) is precisely the condition for the level of disturbance suppression in the \(H_{\infty}\) control to satisfy the gain of \(l_{2}\). In essence, the study of the suppression of the bullwhip effect in supply chain systems can be incorporated into the framework of the study of \(H_{\infty}\) control. **Note 4. ** This paper uses the idea of robust control to study the control problem of \(H_{\infty}\) for switching systems with Markov jump parameters. Assumption (12) is a common assumption condition in robust control. In a closed-loop supply chain system, the following state feedback control law is designed. \[u(k)=\hat{b}+K_{\sigma(k)}x(k) \tag{14}\]  
The closed-loop supply chain switching system can be described as follows:  
\[\left\{ \begin{array}{l}x(k+1)=(A_{\sigma(k)}+\Delta A_{\sigma(k)}+B_{1}K_{ \sigma(k)})x(k)+B_{2}\omega(k)\\ y(k)=(C+\Delta C)x(k)\end{array} \right. \tag{15}\]  
The initial conditions of system (15) are given as follows:  
\[x(0)=x_{0} \tag{16}\]  
The objective of this paper is described as follows: to establish a switching system for a closed-loop supply chain with Markov jump parameters, on this basis, design a robust \(H_{\infty}\) controller to make the closed-loop supply chain system with remanufacturing and abandonment stable under the condition of satisfying \(H_{\infty}\) performance. **3. Stability Analysis and Controller Design**  
In this section, we give sufficient conditions for the mean square exponential stability of closed-loop supply chain systems with Markov jump parameters and give the design method of the state feedback control law. The following lemma is used in this paper. **Lemma 1** ([26]). : _Given matrices of appropriate dimensions \(G=G^{T}\), \(H\), \(E\), for all matrices \(F\) satisfying \(F^{T}F\leq I\), such that, the following inequality holds:_  
\[G+HFE+E^{T}F^{T}H^{T}<0\]  
_Then, there exists the scalar \(\varepsilon>0\), such that the following inequality holds:_  
\[G+\varepsilon HH^{T}+\varepsilon^{-1}E^{T}E<0\]  
**Lemma 2** ([31]). : _(Sulhur's Complementary Lemma) For a given symmetric matrix \(R=\left[ \begin{array}{cc}R_{11}&R_{12}\\ R_{12}^{\top}&R_{22}\end{array} \right]\), where \(R_{11}\) is \(r\times r\) dimensional, the following three conditions are equivalent_  
\( \begin{array}{l}\mbox{(1) $R<0$.}\\ \mbox{(2) $R_{11}<0$, $R_{22}-R_{12}^{T}R_{11}^{-1}R_{12}<0$.}\\ \mbox{(3) $R_{22}<0$, $R_{11}-R_{12}R_{22}^{-1}R_{12}^{T}<0$.}\end{array} \)__  
In the following, we consider the following nominal system, i.e., the stability and robust \(H_{\infty}\) control of the system in the case of \(\Delta A_{\sigma(k)}\equiv 0\), \(\Delta\mbox{C}_{\sigma(k)}\equiv 0\), with the nominal system modeled as  
\[\left\{ \begin{array}{l}x(k+1)=(A_{\sigma(k)}+B_{1}K_{\sigma(k)})x(k)+B_{2} \omega(k)\\ y(k)=Cx(k)\end{array} \right. \tag{17}\]  
We begin by introducing the following two definitions:  
**Definition 1** ([32]). : _For any initial condition \((x_{0},0)\), if there exist constants \(\alpha\) and \(\lambda\) such that_  
\[E\Big{\{}\|x(k)\|^{2}\Big{\}}\leq\alpha e^{-\lambda k}\|x_{0}\|^{2},\quad\ k\to\infty\]  
_holds, the system (15) is said to be mean square exponentially stable._  
**Definition 2** ([32]).