# Hierarchical Scheduling of Complex Dynamic Supply Chain
## 3 Robust Control of Complex Dynamic Supply Chain  
In this section, the optimal resource scheduling scheme is designed based on the principle of \(\mathcal{H}^{\infty}\) control scheme, which is called the hierarchical scheduling of complex dynamic supply chain. Firstly, we give a general result without considering the influence of market disturbance on dynamic supply chain. Secondly, we study the control problem of complex dynamic supply chain under the interference of market disturbance on dynamic supply chain. When there is no external disturbance, i.e., \(w(k)=0\), if there is any positive numbers \(\alpha,a_{i},\eta_{i},\epsilon_{i}\). Any positive definite symmetric matrix \(X\) and any matrix \(Y\) make inequality  
\[\left[ \begin{array}{cccccccc}-X&*&*&*&*&*&*&*\\ AX+BY&-\alpha^{-1}X&*&*&*&*&*\\ A_{1}X+B_{1}Y&0&-a_{1}^{-1}X&*&*&*&*&*\\ \vdots&\vdots&\vdots&\ddots&*&*&*\\ A_{l}X+B_{l}Y&0&0&\cdots&-a_{l}^{-1}X&*&*&*\\ A_{l+1}X&0&0&\cdots&0&-a_{l+1}^{-1}X&*&*\\ \vdots&\vdots&\vdots&&\vdots&\vdots&\ddots&*\\ A_{h}X&0&0&\cdots&0&0&\cdots&-a_{h}^{-1}X\end{array} \right]<0 \tag{9}\]  
hold, then the closed-loop supply chain (8) can still be gradually stable through the supply volume  
\[u(k)=YX^{-1}x(k), \tag{10}\]  
where  
\[\alpha= 1+\epsilon_{1}, \tag{11}\] \[a_{i}= \begin{cases}(1+\eta_{1})\prod_{j=1}^{i}\bigl{(}1+\epsilon_{j}^{- 1}\bigr{)}(1+\epsilon_{i+1}),&(i=1,2,\cdots,l),\\ \Bigl{(}1+\epsilon_{1}^{-1}\Bigr{)}\prod_{j=1}^{i-l}\Bigl{(}1+\eta_{j}^{-1} \Bigr{)}(1+\eta_{i-l+1}),&(i=l+1,l+2,\cdots,h).\end{cases} \]  
The Lyapunov function is constructed as follows  
\[V(k)=x^{\top}(k)Px(k)+\sum_{i=1}^{h}x^{\top}(k-i)Q_{i}x(k-i), \tag{12}\]  
where \(P>0,Q_{l}(i=1,2,\cdots,h)\) are determined later. Considering that the interference input is \(w(k)=0\), the closed-loop supply chain (8) is used to make difference to obtain:  
\[\Delta V(k)= V(x(k+1),k+1)-V(x(k),k) \tag{13}\] \[= x^{\mathrm{T}}(k+1)Px(k+1)+\sum_{i=1}^{h}x^{\mathrm{T}}(k+1-i)Q_{ i}x(k+1-i)\] \[-x^{\mathrm{T}}(k)Px(k)-\sum_{i=1}^{h}x^{\mathrm{T}}(k-i)Q_{i}x(k -i)\] \[= x^{\mathrm{T}}(k+1)Px(k+1)+x^{\mathrm{T}}(k)(Q_{i}-P)x(k)\] \[+\sum_{i=1}^{h-1}x^{\mathrm{T}}(k-i)(Q_{i+1}-Q_{i})x(k-i)-x^{ \mathrm{T}}(k-h)Q_{h}x(k-h).\]  
The following results can be obtained through simple calculation. \[x^{\rm T}(k+1)Px(k+1)= \Bigg{[}(A+BK)x(k)+\sum_{i=1}^{l}(A_{i}+B_{i}K)x(k-i) \tag{14}\] \[+\sum_{i=l+1}^{h}A_{i}x(k-i)\Bigg{]}^{\rm T}P\Bigg{[}(A+BK)x(k)\] \[+\sum_{i=1}^{l}(A_{i}+B_{i}K)x(k-i)+\sum_{i=l+1}^{h}A_{i}x(k-i) \Bigg{]}\] \[\leqslant (1+\epsilon_{1})x^{\rm T}(k)(A+BK)^{\rm T}P(A+BK)x(k)\] \[\Delta V(k)= V(x(k+1),k+1)-V(x(k),k) \tag{13}\]0 \[\Delta V(k)= V(x(k+1),k+1)-V(x(k),k) \tag{13}\]1 \[\Delta V(k)= V(x(k+1),k+1)-V(x(k),k) \tag{13}\]2 \[\Delta V(k)= V(x(k+1),k+1)-V(x(k),k) \tag{13}\]3 \[\Delta V(k)= V(x(k+1),k+1)-V(x(k),k) \tag{13}\]4 \[\Delta V(k)= V(x(k+1),k+1)-V(x(k),k) \tag{13}\]5  
Repeated use of Cauchy matrix inequality leads to  
\[\Delta V(k)= V(x(k+1),k+1)-V(x(k),k) \tag{13}\]6 \[\Delta V(k)= V(x(k+1),k+1)-V(x(k),k) \tag{13}\]7 \[\Delta V(k)= V(x(k+1),k+1)-V(x(k),k) \tag{13}\]8 \[\Delta V(k)= V(x(k+1),k+1)-V(x(k),k) \tag{13}\]9  
Combining the above two expressions, we can obtain  
\[= x^{\mathrm{T}}(k+1)Px(k+1)+\sum_{i=1}^{h}x^{\mathrm{T}}(k+1-i)Q_{ i}x(k+1-i)\]0 \[= x^{\mathrm{T}}(k+1)Px(k+1)+\sum_{i=1}^{h}x^{\mathrm{T}}(k+1-i)Q_{ i}x(k+1-i)\]1 \[= x^{\mathrm{T}}(k+1)Px(k+1)+\sum_{i=1}^{h}x^{\mathrm{T}}(k+1-i)Q_{ i}x(k+1-i)\]2 \[= x^{\mathrm{T}}(k+1)Px(k+1)+\sum_{i=1}^{h}x^{\mathrm{T}}(k+1-i)Q_{ i}x(k+1-i)\]3  
Let  
\[= x^{\mathrm{T}}(k+1)Px(k+1)+\sum_{i=1}^{h}x^{\mathrm{T}}(k+1-i)Q_{ i}x(k+1-i)\]4 \[Q_{i}= \begin{cases}Q_{i+1}+a_{i}(A_{i}+B_{i}K)^{\rm T}P(A_{i}+B_{i}K),&i =1,2,\cdots,l,\\ Q_{i+1}+a_{i}A_{i}^{\rm T}PA_{i},&i=l+1,l+2,\cdots,h-1.\end{cases}