# Switching Control of Closed-Loop Supply Chain Systems with Markov Jump Parameters
## 5 Simulation Examples  
Considering the scrap recycling data of a domestic steel mill [33], the following model parameters are set according to the actual situation and the historical data of the enterprise: the decay rate of the available commodity warehouse and the recycled commodity warehouse are \(\rho_{1}=0.06\), \(\rho_{2}=0.08\), and the cost parameter matrix \(C=\left[\begin{array}{cc}2.8&0\\ 0&4.1\end{array}\right]\), \(c_{r}=1.2\), \(c_{0}=0.3\), and the initial values are set to \(x_{1}(0)=5\), \(x_{2}(0)=10\), (unit \(10^{6}\) tons). \(d(k)\) satisfies Assumption 2. The remanufacturing rate \(\alpha\) and the obsolescence rate \(\beta\) are considered as uncertain parameters and satisfy Assumption 3.  
The system parameters are as follows:  
\[A_{1}=\left[\begin{array}{cc}1-\rho_{1}&0\\ 0&1-\rho_{2}\end{array}\right],\Delta A_{1}=\left[\begin{array}{cc}0& \alpha\\ 0&-\alpha-\beta\end{array}\right],A_{2}=\left[\begin{array}{cc}1-\rho_{1}&0 \\ 0&1\end{array}\right],\Delta A_{2}=\left[\begin{array}{cc}0&\alpha\\ 0&0\end{array}\right],\]  
\[A_{3}=\left[\begin{array}{cc}1&0\\ 0&1-\rho_{2}\end{array}\right],\Delta A_{3}=\left[\begin{array}{cc}0&\alpha \\ 0&-\alpha-\beta\end{array}\right],A_{4}=\left[\begin{array}{cc}1&0\\ 0&1\end{array}\right],\Delta A_{4}=\left[\begin{array}{cc}0&\alpha\\ 0&0\end{array}\right],\]State Transfer Matrix  
\[P=\left[\begin{array}{cccc}0.11&0.72&0.04&0.13\\ 0.15&0.65&0.05&0.15\\ 0.12&0.68&0.05&0.15\\ 0.12&0.68&0.05&0.15\end{array}\right].\]  
Take another  
\[E_{a1}=E_{a3}=\left[\begin{array}{cc}0&1\\ 0&-1\end{array}\right],E_{a2}=E_{a4}=E_{c}=\left[\begin{array}{cc}0&1\\ 0&1\end{array}\right],H_{ai}=\left[\begin{array}{cc}1&0\\ 0&1\end{array}\right],H_{c}=\left[\begin{array}{cc}1.2&0.3\end{array}\right],\]  
From the Matlab toolbox, the state feedback controller is obtained as follows:  
\[K_{1}=\left[\begin{array}{cc}1.3784&-0.0210\\ -0.0062&0.0573\end{array}\right]\times 10^{-3},K_{2}=\left[\begin{array}{cc}2. 2074&-0.001\\ -0.0001&0.0231\end{array}\right]\times 10^{-3},\]  
\[K_{3}=\left[\begin{array}{cc}0.0340&-0.010\\ -0.0010&0.0065\end{array}\right]\times 10^{-3},K_{4}=\left[\begin{array}{cc}0. 3491&-0.000\\ -0.0000&0.0301\end{array}\right]\times 10^{-3},\]  
In the following, we consider the robust \(H_{\infty}\) control problem for a system with both external demand uncertainty and remanufacturing process uncertainty. Assuming that the external demand uncertainty \(\omega(k)\) is a sinusoidal disturbance, Figure 1 represents the switching signal of a supply chain switching system with Markov jump parameters.  
Figure 2 shows the change in stock levels of available commodity warehouses in the presence of external disturbances to the system in the form of sinusoidal disturbances and in the presence of recovery and remanu  
Figure 1: The switching signal of CLSC.  
Figure 3 shows the change in stock levels in the recycled goods warehouse in the presence of external disturbances in the form of sinusoidal disturbances in the system and in the presence of uncertainty in recycling and remanufacturing.  
Figure 2: The state of the serviceable stock.  
From the simulation results, it can be seen that the robust \(H_{\infty}\) controller designed in this paper can effectively suppress the uncertain demand disturbance in the recycling and remanufacturing process for the closed-loop supply chain system with Markov jump parameters.