## 4 Model Stability Analysis  
This section provides the solvability conditions for the \(H_{\infty}\) problem in a multi-agent-based supply chain system. The controller parameters that meet the requirements are obtained by solving linear matrix inequality (LMI) equations. For the system to achieve \(H_{\infty}\) performance \(\gamma\), the following conditions must be satisfied:  
The error system (18) is exponentially stable at \(\omega(k)=0\) (i.e., there exist scalars a and b such that \(\parallel\delta(k)\parallel\leq ae^{-b(k-b_{0})}\parallel\delta(k_{0})\parallel\)). The \(H_{\infty}\) norm of the transfer function from the disturbance input \(\omega(k)\) to the system output \(z(k)\) is less than \(\gamma\), that is, under the condition \(\delta(k)=0,k\in[0,t]\), \(\sum_{k=0}^{\infty}z^{\mathrm{T}}(k)z(k)\leq\gamma^{2}\sum_{k=0}^{\infty} \omega^{\mathrm{T}}(k)\omega(k)\),\(\forall\omega(t)\in l_{2}[0,\infty)\), where \(\gamma\) is the decay level of the bullwhip effect. To complete the proof, it is also necessary to introduce the following lemma. Lemma 1 ([32]): _([PERSON]'s inequality) For any vectors \(a\), \(b\) of appropriate dimensions and matrices \(N,X,Y,Z\) If \( \begin{bmatrix}X&Y\\ Y^{\mathrm{T}}&Z\end{bmatrix} \geqslant 0\), the following inequality holds_  
\[-2a^{\mathrm{T}}Nb\leqslant \begin{bmatrix}a\\ b\end{bmatrix} ^{\mathrm{T}} \begin{bmatrix}X&Y-N\\ Y^{\mathrm{T}}-N^{\mathrm{T}}&Z\end{bmatrix}\begin{bmatrix}a\\ b\end{bmatrix} \]  
For the system (10), choose the Lyapunov function as  
\[V(k)=V_{1}(k)+V_{2}(k)+V_{3}(k) \tag{19}\]where \(V_{1}(k)=\delta^{T}(k)P\delta^{T}(k)\)  
\[V_{2} =\sum_{\theta=k-\overline{\tau}_{1}}^{k-1}e^{\alpha(s-k+1)}\delta^{T} (\theta)\hat{Q}\delta(\theta)+\sum_{\theta=-\overline{\tau}_{2}}^{k-1}\sum_{s= k+\theta}^{k-1}e^{\alpha(s-k+1)}y^{T}(s)\overline{Z}y(s)\] \[+\sum_{\theta=-\overline{\tau}_{1}}^{k-1}\sum_{s=k+\theta}^{k-1}e ^{\alpha(s-k+1)}y^{T}(s)Zy(s)\] \[V_{3} =\sum_{\theta=k-\overline{\tau}_{2}}^{k-1}e^{\alpha(s-k+1)}\delta ^{T}(\theta)\overline{Q}\delta(\theta)+\sum_{\theta=-\overline{\tau}_{2}}^{-1} \sum_{s=k+\theta}^{k-1}e^{\alpha(s-k+1)}y^{T}(s)\overline{Z}y(s)\]  
where \(y(s)=\delta(k+1)-\delta(k)\) and \(P_{1},\hat{Q},\hat{Z},\hat{Q},\hat{Z}\) are symmetric positive definite matrices of appropriate dimensions. Based on the above conditions, we can derive the following lemma:  
**Theorem 1**. : _If Assumptions 1 and 2 hold, for a given scalar \(0\leq\tau_{1}\leq\tau_{1},0\leq\tau_{2}\leq\tau_{2},\gamma>0\), the system (18) has \(H_{\infty}\) performance \(\gamma\). If there exist symmetric positive definite matrices \(P_{1},\hat{Q},\hat{Z},\hat{Q},\bar{Z}\), symmetric semipositive definite matrices \(\hat{\mathbb{X}},\hat{\mathbb{Y}},\overline{\mathbb{X}},\overline{\mathbb{Y}}\), then_  
\[\left[ \begin{array}{cccc}\Omega&\zeta^{T}(k)P&\bar{\tau}_{1}\zeta^{T}(k)\hat {Z}&\bar{\tau}_{2}\zeta^{T}(k)\bar{Z}\\ *&-P&0&0\\ *&*&-\tau_{1}\hat{Z}&0\\ *&*&*&-\tau_{2}\bar{Z}\end{array} \right]<0 \tag{20}\]  
_and_  
\[\left[ \begin{array}{cccc}\hat{\mathbb{X}}&\hat{\mathbb{Y}}\\ *&\hat{Z}\end{array} \right]\geq 0,\left[ \begin{array}{cccc}\overline{ \mathbb{X}}&\overline{\mathbb{Y}}\\ *&\bar{Z}\end{array} \right]\geq 0 \tag{21}\]  
_where_  
\[\zeta^{T}(k)=\left[(\widehat{A}-I)&\widehat{B}&\overline{B}&B_{\omega}&-\widehat {B}\right]\]  
\[\Omega=\left[ \begin{array}{cccc}\Phi&\widehat{\mathcal{Y}}-P\widehat{B}& \overline{\mathcal{Y}}-P\overline{B}_{\overline{\tau}}&PB_{\omega}&-P\widehat {B}\\ -e^{-\alpha\tau_{1}}\widehat{Q}&0&0&0\\ *&-e^{-\alpha\overline{\tau}_{2}}\overline{Q}&0&0\\ *&*&-\gamma^{2}&0\\ *&*&*&I\end{array} \right]\]  
\[\Phi=[P(\widehat{A}_{r}-I)+(\widehat{A}_{r}-I)^{T}P+\overline{\tau}_{1}\widehat {X}+\widehat{Y}+\widehat{Y}^{T}+\overline{\tau_{2}}\overline{X}+\overline{ \mathbb{Y}}+\overline{\mathbb{Y}}^{T}+\widehat{Q}+\overline{Q}+I+(1-e^{-\alpha })P]\]  
**Proof of Theorem 1.