# Event-Triggered MFAILC Bipartite Formation Control for Multi-Agent Systems Under DoS Attacks
## 4 Proof of Convergence
### Convergence Analysis of the Tracking Error for Agents
For each block matrix in (34), performing the decomposition yields (35)  
\[ \begin{split} H_{p}(k,i)=\frac{1}{\lambda_{p}+\left\|\hat{\[PERSON]} _{p}(k,i)\right\|^{2}}\\ &\times\left\{\begin{bmatrix}\sum\limits_{l=1}^{m}\phi_{p,1l}(k,i)\hat{\phi}_{p,1l}(k,i)&0&0\\ 0&\ddots&0\\ 0&0&\sum\limits_{l=1}^{m}\phi_{p,ml}(k,i)\hat{\phi}_{p,ml}(k,i)\\ \end{bmatrix}\\ &+\begin{bmatrix}0&\dots&\sum\limits_{l=1}^{m}\phi_{p,1l}(k,i)\hat{\phi}_{ p,ml}(k,i)\\ \vdots&\ddots&\vdots\\ \sum\limits_{l=1}^{m}\phi_{p,ml}(k,i)\hat{\phi}_{p,1l}(k,i)&\dots&0\\ \end{bmatrix}\right\}\end{split}  \tag{35}\]  
\[ \begin{split}\Delta\overline{A}_{p}(k,i)+\overline{B}_{p}(k,i), \quad\overline{A}_{p}(k,i),\overline{B}_{p}(k,i)\in\mathbb{R}^{m\times m}\end{split} \]  
Rewriting \(H(k,i)\) as below  
\[ \begin{split}\mathbf{H}(k,i)&=diag\{\overline{A}_{1}(k,i), \overline{A}_{2}(k,i),\dots,\overline{A}_{N}(k,i)\}\\ &+diag\{\overline{B}_{1}(k,i),\overline{B}_{2}(k,i),\dots, \overline{B}_{N}(k,i)\}\\ &=\overline{A}(k,i)+\overline{B}(k,i)\end{split}  \tag{36}\]  
Combining Equations (36) and (34), we can obtain  
\[ \begin{split}\|\mathbf{e}(k+1,i)\|&\leq\|I_{mN}-\rho\mathbf{H}(k,i)((L+D) \otimes I_{m})\|\|\mathbf{e}(k+1,i-1)\|\\ &\leq(\|I_{mN}-\rho T(k,i)((L+D)\otimes I_{m})\|\\ &+\|\rho\mathbf{F}(k,i)((L+D)\otimes I_{m})\|)\|\mathbf{e}(k+1,i-1)\| \end{split}  \tag{37}\]  
Taking the \(v\) norm of (37)'s two sides, we obtain  
\[ \begin{split}\|\mathbf{e}(k+1,i)\|_{v}&\leq(\|I_{mN}-\rho T (k,i)((L+D)\otimes I_{m})\|_{v}\|\\ &+\|\rho\|\|F(k,i)\|_{v}\Big{\|}\Big{(}\Big{(}L+D\Big{)}\otimes I _{m}\Big{)}\Big{\|}_{v}\Big{)}\Big{\|}\mathbf{e}(k+1,i-1)\Big{\|}_{v}\end{split}  \tag{38}\]Defining \(\mathcal{Q}(k,i)=I_{mN}-\rho T(k,i)((L+D)\otimes I_{m})\) and verifying the boundedness of \(\mathcal{Q}(k,i)\) as the following part. Since \(\phi_{p,ml}(k,i)\) and \(\hat{\phi}_{p,ml}(k,i)\) have the same sign, \(\phi_{p,ml}(k,i)\hat{\phi}_{p,ml}(k,i)>0\).Therefore, when \(\lambda_{p}>\bar{\lambda}\), there is a \(\bar{\lambda}\) such that we have  
\[0 <\frac{\sum_{l=1}^{m}\phi_{p,dl}(k,i)\hat{\phi}_{p,dl}(k,l)}{ \lambda_{p}+\left\|\hat{\Phi}_{p}(k,i)\right\|}=\frac{\sum_{l=1}^{m}\left|\phi_ {p,dl}(k,i)\right|\left|\hat{\phi}_{p,dl}(k,l)\right|}{\lambda_{p}+\left\|\hat{ \Phi}_{p}(k,i)\right\|} \tag{39}\] \[\leq\frac{\sum_{l=1}^{m}\left|\phi_{p,dl}(k,i)\right|\left\|\hat{ \phi}_{p,dl}(k,l)\right|}{\bar{\lambda}+\left\|\hat{\Phi}_{p}(k,i)\right\|^{2 }}<1\]  
From the above, the block matrix \(\bar{A}(k,i)<1\). Theorem 2 states that \(\mathcal{G}\) is a graph with strong connections. Combining Equation (39), we select a \(\rho_{p}\) such that \(\rho_{p}<\tilde{\rho}=1/\left(\max_{p=1,2,\ldots,N}\sum_{p\in\mathcal{N}_{l}} \left|a_{p}\right|+\left|d_{p}\right|\right)\) is an irreducible sub-stochastic matrix that contains diagonal components that are positive, and at least one row of \(\mathcal{Q}(k,i)\) is strictly less than \(1\). Thus, the conclusion is that \(\mathcal{Q}(k,i)\) is bounded. Since the matrix \(\bar{B}(k,i)\) has zero diagonal elements, by the Gerschgorin Circle Theorem, we have:  
\[z_{p,q}=\left\{z\mid\left|z-0\right|\leq\sum_{h=1,h\
eq q}^{m}\left|\frac{ \sum_{l=1}^{m}\phi_{p,dl}(k,i)\hat{\phi}_{p,dl}(k,i)}{\lambda_{p}+\left\|\hat{ \Phi}(k,i)\right\|}\right|\right\} \tag{40}\]  
where \(p\in\mathcal{V}\), \(q=1,2  
where \(\lfloor\bullet\rfloor\) is the floor function. For \(\lim_{i\to\infty}\lVert e(k+1,i)\rVert_{v}=0\), it is required that \(\left(\beta+\gamma\right)<1\), i.e., \(\rho_{p}\) satisfies the following conditions:  
\[\lVert\rho\rVert\leq\overline{p}=\frac{1-\beta}{\varsigma\lVert(L+D)\otimes I _{m}\rVert_{v}} \tag{45}\]  
From the above, it can be concluded that when \(\rho_{p}<\min\{\tilde{\rho},\overline{p}\}\), we have \(\lim_{i\to\infty}\lVert e(k+1,i)\rVert_{v}=0\) and converging. From Equation (28), it follows that \(\lVert e(k+1,i)\rVert_{v}\) is bounded.