## 4 Model Stability Analysis
\begin{cases}\delta(k+1)=\hat{A}_{r}\delta(k)+\hat{B}(\delta(k-\tau_{1r})-\delta (0))+\overline{B}_{r\prime(h)}\delta(k-\tau_{2r})+B_{\omega}\omega(k)\\ z(k)=\delta(k)\end{cases}  \tag{32}\]  
For the system (32), choose the Lyapunov function as  
\[V_{\sigma(h)}(k)=V_{1\sigma(h)}(k)+V_{2\sigma(h)}(k)+V_{3\sigma(h)}(k) \tag{33}\]  
where  
\[ \begin{split}& V_{1\sigma(h)}(k)=\delta^{\mathsf{T}}(k)P_{ \sigma(h)}\delta(k)\\ & V_{2\sigma(h)}=\sum_{\theta=k-\overline{\tau_{1}}}^{k-1}e^{ \mathsf{\tilde{f}}(k-s-1)}\delta^{\mathsf{T}}(\theta)\widehat{Q}_{\sigma(h)} \delta(\theta)+\sum_{\theta=-\overline{\tau_{1}}}^{-1}\sum_{s=k+\theta}^{k-1 }e^{\beta(k-s-1)}y^{\mathsf{T}}(s)\hat{Z}_{\sigma(h)}y(s)\\ &+\sum_{\theta=0}^{k-1}e^{\mathsf{\tilde{f}}(s-k+1)}\delta^{ \mathsf{T}}(0)\delta(0)\\ & V_{3\sigma(h)}=\sum_{\theta=k-\overline{\tau_{2}}}^{k-1}e^{ \mathsf{\tilde{f}}(k-s-1)}\delta^{\mathsf{T}}(\theta)\overline{Q}_{\sigma(h)} \delta(\theta)+\sum_{\theta=-\overline{\tau_{2}}}^{-1}\sum_{s=k+\theta}^{k-1 }e^{\beta(k-s-1)}y^{\mathsf{T}}(s)\overline{Z}_{\sigma(h)}y(s)\\ & y(s)=\delta(k+1)-\delta(k)\text{, and }P_{\sigma(h)},\widehat{Q}_{\sigma(h)},\overline{Z}_{\sigma(h)},\overline{Q}_{\sigma(h)},\overline{Z}_{\sigma(h)} \text{ is a symmetric positive definite matrix of appropriate dimension.}\end{split} \]  
Based on the above conditions, we can derive the following lemma  
Theorem 3.2: _If Assumptions 1 and 2 hold, for a given scalar \(0\leq\tau_{1}\leq\bar{\tau}_{1},0\leq\tau_{2}\leq\bar{\tau}_{2},\gamma>0\), the system (18) has \(H_{\infty}\) performance \(\gamma\). If there exist symmetric positive definite matrices \(P_{\sigma(h)},\widehat{Q}_{\sigma(h)},\hat{Z}_{\sigma(h)},\overline{Q}_{ \sigma(h)},\overline{Z}_{\sigma(h)}\),-symmetric semipositive definite matrices \(\widehat{\mathsf{X}}_{\sigma(h)},\hat{\mathsf{Y}}_{\sigma(h)},\overline{ \mathsf{X}}_{\sigma(h)},\overline{\mathsf{Y}}_{\sigma(h)}\), then_  
\[ \begin{bmatrix}\Omega&\zeta^{\mathsf{T}}(k)P_{\sigma(h)}&\tau_{1}\zeta^{ \mathsf{T}}(k)\hat{Z}_{\sigma(h)}&\tau_{2}\zeta^{\mathsf{T}}(k)\overline{Z}_{ \sigma(h)}\\ &-P_{\sigma(h)}&0&0\\ &*&-\tau_{1}\tilde{Z}_{\sigma(h)}&0\\ &*&*&-\overline{\tau}_{2}\overline{Z}_{\sigma(h)}\end{bmatrix} <0 \tag{34}\]  
_and_  
\[ \begin{bmatrix}\widehat{\mathsf{X}}_{\sigma(h)}&\widehat{\mathsf{Y}}_{\sigma( h)}\\ &\widetilde{Z}_{\sigma(h)}\end{bmatrix} \geq 0, \begin{bmatrix}\overline{\mathsf{X}}_{ \sigma(h)}&\overline{\mathsf{Y}}_{\sigma(h)}\\ &\overline{Z}_{\sigma(h)}\end{bmatrix} \geq 0 \tag{35}\]  
_where_  
\[ \begin{split}&\zeta^{\mathsf{T}}(k)=\begin{bmatrix}(\widehat{A}_{r }-I)&\widehat{B}&\overline{B}_{r\sigma(h)}&B_{\omega}&-\widehat{B}\end{bmatrix} \\ &\Omega=\begin{bmatrix}\Phi&\widehat{\mathcal{Y}}-P\widehat{B}&\overline{ \mathcal{Y}}-P\overline{B}_{r\sigma(h)}&PB_{\omega}&-P\widehat{B}\\ &-e^{\theta\bar{\tau}_{1}}\widehat{Q}&0&0\\ &*&e^{\theta\bar{\tau}_{2}}\overline{Q}&0&0\\ &*&-\gamma^{2}&0\\ &*&*&I\end{bmatrix}\\ &\Phi=[P_{\sigma(h)}(\widehat{A}_{r}-I)+(\widehat{A}_{r}-I)^{T}P_{\sigma(h)}+ \tau_{1}\widehat{X}_{\sigma(h)}+\widehat{Y}_{\sigma(h)}+\widehat{Y}_{\sigma(h)}^ {T}\\ &+\overline{\tau}_{2}\overline{X}_{\sigma(h)}+\overline{Y}_{\sigma(h)}+ \overline{Y}_{\sigma(h)}^{T}+\widehat{Q}_{\sigma(h)}+\overline{Q}_{\sigma(h)}+ I+(1-e^{\beta})P_{\sigma(h)}]\end{split} \]  
The proof of Theorem 3.2 is similar to that of Theorem 3.1 and is omitted here for space reasons. Theorems 3.1 and 3.2 provide the \(H_{\infty}\) consensus conditions for supply chain systems under uncertain demand. The gains of the controller can be determined by employing methods similar to those presented in [33; 34]. When a node in the supply chain fails, the system's stability is compromised, rendering it non-stabilizable. Additionally, the disruption and restoration of nodes result in changes to the network's topology, which subsequently alters the corresponding Laplacian matrix. To address both unforeseen events and the impacts of structural upgrades in the supply chain, we demonstrate that the system can remain stabilizable under certain conditions.