## 3 Stability Analysis
Prove that the tracking error is convergent. Next, we continue to discuss the convergence of \(e_{i}(\ell+1)=v_{d}(\ell+1)-v_{i}(\ell+1)\). We begin by defining the following vectors:  
\[\mathbf{e}(\ell) =\Big{[} \begin{array}{cccc}e_{1}(\ell)&e_{2}(\ell)&\cdots&e_{N} (\ell)\end{array} \Big{]}^{T} \tag{31}\] \[\mathbf{\xi}(\ell) =\Big{[} \begin{array}{cccc}\xi_{1}(\ell)&\xi_{2}(\ell)&\cdots& \xi_{N}(\ell)\end{array} \Big{]}^{T}\]  
The distributed error expressed in Equation (10) can be reformulated as follows:  
\[ \begin{split}\xi_{i}(\ell)&=\sum_{j\in N_{i}}a_{ij}\big{(}v_{d}( \ell)-v_{i}(\ell)-\big{(}v_{d}(\ell)-\tilde{v}_{j}(\ell)\big{)}\big{)}+d_{i}( \tilde{v}_{d}(\ell)-v_{i}(\ell))\\ &=\sum_{j\in N_{i}}a_{ij}\big{(}e_{i}(\ell)-e_{j}(\ell)-\beta_{j}(\ell) \chi_{j}(\ell)\big{)}+d_{i}(e_{i}(\ell)+\beta_{d}(l)\chi_{d}(l))\end{split} \]  
Taking the mathematical expectation yields  
\[ \begin{split}\mathbb{E}\{\xi_{i}(\ell)\}&=\sum_{j\in N_{i}}a_{ ij}\big{(}\mathbb{E}\{e_{i}(\ell)\}-\mathbb{E}\{e_{j}(\ell)\}-\mathbb{E}\{\beta_{j}( \ell)\}\chi_{j}(\ell)\big{)}\\ &+d_{i}(\mathbb{E}\{e_{i}(\ell)\}+\mathbb{E}\{\beta_{d}(l)\}\chi_{ d}(l))\\ &=\sum_{j\in N_{i}}a_{ij}\mathbb{E}\{e_{i}(\ell)\}-\sum_{j\in N_{i}} a_{ij}\mathbb{E}\{e_{j}(\ell)\}-\sum_{j\in N_{i}}a_{ij}\mathbb{E}\{\beta_{j}( \ell)\}\chi_{j}(\ell)\\ &+d_{i}\mathbb{E}\{e_{i}(\ell)\}+d_{i}\mathbb{E}\{\beta_{d}(\ell)\} \chi_{d}(\ell)\end{split}  \tag{32}\]Extending the aforementioned equation over the range \(i=1\) to \(i=N\), we obtain  
\[\mathbb{E}\{\xi(\ell)\}=(\mathcal{L}+\mathrm{D})\mathbb{E}\{\mathbf{e}(\ell)\}-A \boldsymbol{\beta}_{1}\boldsymbol{\chi}_{1}(\ell)+D\boldsymbol{\beta}_{2} \boldsymbol{\chi}_{d}(\ell) \tag{33}\]  
where \(\boldsymbol{\chi}_{1}(\ell)=\big{[}\boldsymbol{\chi}_{1}(\ell)\cdots\boldsymbol {\chi}_{j}(\ell)\big{]}^{T}\), \(\boldsymbol{\chi}_{d}(\ell)=[\boldsymbol{\chi}_{d}(\ell)\cdots\boldsymbol{ \chi}_{d}(\ell)]^{T}\), \(\boldsymbol{\beta}_{1}=\text{diag}\{\tilde{\beta}_{1},\tilde{\beta}_{2}\cdots \tilde{\beta}_{j}\}\), \(\boldsymbol{\beta}_{2}=\text{diag}\{\tilde{\beta}_{d},\tilde{\beta}_{d}\cdots \tilde{\beta}_{d}\}\), \(D=\text{diag}\{d_{1},d_{2},\cdots,d_{i}\}\)  
Rewrite \(e_{i}(\ell)\) as follows:  
\[ \begin{split} e_{i}(\ell+1)&=v_{d}(\ell+1)-v_{i}( \ell+1)\\ &=e_{i}(\ell)+\phi_{i}(\ell)\Delta F_{i}(\ell)+\Delta v_{d}(\ell+ 1)\end{split}  \tag{34}\]  
Substituting Control Algorithm (24) for (34) yields  
\[e_{i}(\ell+1)=e_{i}(\ell)+\phi_{i}(\ell)\mathbf{g}^{\mathrm{T}}\Big{[}\hat{ \mathbf{A}}_{ui}^{\mathrm{T}}(\ell)\hat{\mathbf{A}}_{ui}(\ell)+\lambda \mathbf{I}\Big{]}^{-1}\hat{\mathbf{A}}_{ui}^{\mathrm{T}}(\ell)\mathbf{E}( \ell)\xi_{i}(\ell)+\Delta v_{d}(\ell+1) \tag{35}\]  
Calculating the mathematical expectation, we obtain  
\[\mathbb{E}\{e_{i}(\ell+1)\}=\mathbb{E}\{e_{i}(\ell)\}+\phi_{i}(\ell)\Theta_{ i}(\ell)\mathbb{E}\{\xi_{i}(\ell)\}+\mathbb{E}\{\Delta v_{d}(\ell+1)\} \tag{36}\]  
where \(\Theta_{i}(\ell)=\mathbf{g}^{\mathrm{T}}\big{[}\hat{\mathbf{A}}_{ui}^{ \mathrm{T}}(\ell)\hat{\mathbf{A}}_{ui}(\ell)+\lambda\mathbf{I}\big{]}^{-1} \hat{\mathbf{A}}_{ui}^{\mathrm{T}}(\ell)\mathbf{E}(\ell)\). Along the same trajectory, (36) is extended to a vector form  
\[\mathbb{E}\{\mathbf{e}(\ell+1)\}=\mathbb{E}\{\mathbf{e}(\ell)\}+\boldsymbol{ \Theta}(\ell)\mathbb{E}\{\xi(\ell)\}+\mathbb{E}\{\Delta v_{d}(\ell+1)\} \tag{37}\]  
where \(\boldsymbol{\Theta}(\ell)=\text{diag}\{\phi_{1}(\ell)\Theta_{1}(\ell),\phi_{2 }(\ell)\Theta_{2}(\ell),\cdots\phi_{i}(\ell)\Theta_{i}(\ell)\}\). By replacing Control Algorithm (24) with (25), we obtain  
\[