## 3 Stability Analysis
\begin{split}\mathbb{E}\{\mathbf{e}(\ell+1)\}&= \mathbb{E}\{\mathbf{e}(\ell)\}+\boldsymbol{\Theta}(\ell)((\mathcal{L}+ \mathrm{D})\mathbb{E}\{\mathbf{e}(\ell)\}-A\boldsymbol{\beta}_{1}\boldsymbol{ \chi}_{1}(\ell)+D\boldsymbol{\beta}_{2}\boldsymbol{\chi}_{d}(\ell))+\mathbb{E }\{\Delta v_{d}(\ell+1)\}\\ &=(\mathbf{I}-\boldsymbol{\Theta}(\ell)(\mathcal{L}+\mathrm{D})) \mathbb{E}\{\mathbf{e}(\ell)\}-\boldsymbol{\Theta}(\ell)A\boldsymbol{\beta}_ {1}\boldsymbol{\chi}_{1}(\ell)\\ &+\boldsymbol{\Theta}(\ell)D\boldsymbol{\beta}_{2}\boldsymbol{ \chi}_{d}(\ell)+\mathbb{E}\{\Delta v_{d}(\ell+1)\}\end{split}  \tag{38}\]  
Applying the norm operation to both sides of Equation (38), we derive  
\[ \begin{split}\|\mathbb{E}\{\mathbf{e}(\ell+1)\}&\| \leq\|\mathbf{I}-\boldsymbol{\Theta}(\ell)(\mathcal{L}+\mathrm{D})\|\| \mathbb{E}\{\mathbf{e}(\ell)\}\|+\|\boldsymbol{\Theta}(\ell)\|\|A\boldsymbol{ \beta}_{1}\boldsymbol{\chi}_{1}(\ell)\|\\ &+\|\boldsymbol{\Theta}(\ell)\|\|\boldsymbol{\beta}_{2} \boldsymbol{\chi}_{d}(\ell)\|+\|\mathbb{E}\{\Delta v_{d}(\ell+1)\}\|\\ &\leq\|\mathbf{I}-\boldsymbol{\Theta}(\ell)(\mathcal{L}+ \mathrm{D})\|\|\mathbb{E}\{\mathbf{e}(\ell)\}\|+\Theta_{2}(\ell)\end{split}  \tag{39}\]  
where \(\Theta_{2}(\ell)=\|\boldsymbol{\Theta}(\ell)\|\|A\boldsymbol{\beta}_{1} \boldsymbol{\chi}_{1}(\ell)\|+\|\boldsymbol{\Theta}(\ell)\|\|D\boldsymbol{ \beta}_{2}\boldsymbol{\chi}_{d}(\ell)\|+\|\mathbb{E}\{\Delta v_{d}(\ell+1)\}\|\). To proceed, it is necessary to establish that the matrix \(\mathbf{I}-\boldsymbol{\Theta}(\ell)(\mathcal{L}+\mathrm{D})\) qualifies as an irreducible substochastic matrix. As a preliminary step, we examine the properties of the matrix \(M(k)\). Define the matrix \(\mathbf{C}\) as \(\mathbf{C}=\hat{\mathbf{A}}_{ui}^{T}(\ell)\hat{\mathbf{A}}_{ui}(\ell)+\lambda I\), then  
\[\Theta_{i}(\ell)=\mathbf{g}^{\mathrm{T}}\mathbf{C}^{-1}\hat{\mathbf{A}}_{ui}^{ \mathrm{T}}(\ell)\mathbf{E}(\ell)\]  
Since \(\hat{\mathbf{A}}_{ui}^{\mathrm{T}}(\ell)\hat{\mathbf{A}}_{ui}(\ell)\) is positive semidefinite and \(\lambda>0\), it follows that \(\mathbf{C}\) is positive definite. Consequently, its inverse \(\mathbf{C}^{-1}\) also possesses positive definiteness. Based on this, the following results can be derived:\[\mathbf{C}^{-1}=\frac{\mathbf{C}^{*}}{\det(\mathbf{C})}\] \[\mathbf{C}^{*}=\left[ \begin{array}{ccc}C_{11}&\cdots&C_{(N_{u})1} \\ \vdots&\ddots&\vdots\\ C_{1(N_{u})}&\cdots&C_{(N_{u})(N_{u})}\end{array} \right]\]  
Let \(\mathbf{C}\) be a matrix and \(\mathbf{C}^{*}\) be the adjoint matrix of \(\mathbf{C}\). The element \(C_{ij}\) is the algebraic cofactor of \(\mathbf{C}\). Hence, the following relationship holds:  
\[\Theta_{i}(\ell)=\mathbf{g}^{\mathbf{T}}\mathbf{C}^{-1}\mathbf{A}_{ui}^{ \mathrm{T}}(\ell)\mathbf{E}(\ell)=\text{diag}\left[\sum_{a=1}^{N_{u}}\frac{(N- a+1)C_{ai}\hat{\phi}_{i}(N-a+1)}{\det(\mathbf{C})}\right] \tag{40}\]  
and then  
\[\phi_{i}(\ell)\Theta_{i}(\ell)=\text{diag}\left[\sum_{a=1}^{N_{u}}\frac{(N-a+ 1)C_{ai}\hat{\phi}_{i}(k+a-1)\theta_{i}(k+a-1)}{\det(\mathbf{C})}\right] \tag{41}\]  
Given that both \(\phi_{i}(\ell)\) and \(\hat{\phi}_{i}(\ell)\) remain bounded for all time instants \(\ell\), it follows that the diagonal entries of the matrix defined in Equation (41) are likewise bounded. Moreover, the corresponding upper bounds do not depend on the specific value of \(\ell\), indicating time-invariant boundedness of these elements. As we know, \(\mathbf{C}\) is positive definite matrix; thus, \(\det(\mathbf{C})\) is a monic polynomial in \(\lambda\) of degree \(N_{u}\). Further, \(C_{11}>0\) and \(C_{a1}(a=2,3,\ldots,N_{u})\) are monic polynomials in \(\lambda\) of degree \((nN_{u}-1)\) and \((nN_{u}-2)\), respectively. Therefore there exists a constant \(\lambda_{\min}>0\) such that the sign of every entry in matrix \(\Theta_{i}(\ell)\) denoted by Equation (40) is the same as that of \(\frac{C_{11}}{\det(\mathbf{C})}\) for any \(\lambda\geq\lambda_{\min}\).