# 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  
The parameters are defined as follows:  
\[\mathbf{e}(k,i) =\left[\varepsilon_{1}^{T}(k,i),e_{2}^{T}(k,i),\ldots,\varepsilon_{ N}^{T}(k,i)\right]^{T}\in\mathbb{R}^{mN\times mN}\] \[\mathbf{y}(k,i) =\left[y_{1}^{T}(k,i),y_{2}^{T}(k,i),\ldots,y_{N}^{T}(k,i)\right]^ {T}\in\mathbb{R}^{mN\times mN}\] \[\mathbf{u}(k,i) =:\left[u_{1}^{T}(k,i),u_{2}^{T}(k,i),\ldots,u_{N}^{T}(k,i)\right]^ {T}\in\mathbb{R}^{mN\times mN}\] \[\mathbf{\xi}(k,i) =\left[\xi_{1}^{T}(k,i),\xi_{2}^{T}(k,i),\ldots,\xi_{N}^{T}(k,i) \right]^{T}\in\mathbb{R}^{mN\times mN}\]  
From Lemma 1, the consensus error can be rewritten in the following form:  
\[\xi_{p}(k,i) =\sum_{j\in\mathbb{N}_{p}}a_{gj}(\overline{y}_{j}(k,i)-sgn(a_{gj} )\overline{y}_{p}(k,i))+d_{p}(y_{d}(k)-sgn(d_{p})\overline{y}_{p}(k,i) \tag{26}\] \[=\sum_{j\in\mathbb{N}_{p}}a_{pj}\delta_{j}(\delta_{j}\overline{y} _{j}(k,i)-\delta_{p}\overline{y}_{p}(k,i))+d_{p}(y_{d}(k)-\delta_{p}\overline {y}_{p}(k,i))\] \[=\sum_{j\in\mathbb{N}_{p}}a_{pj}\delta_{j}(e_{p}(k,i)-e_{j}(k,i)) +d_{p}e_{p}(k,i)\]  
Multiplying both sides of Equation (26) by \(\delta_{p}\), we obtain the following:  
\[\delta_{p}\xi_{p}(k,i)=\sum_{j\in\mathbb{N}_{p}}|a_{pj}|(e_{p}(k,i)-e_{j}(k,i) )+\big{|}d_{p}\big{|}e_{p}(k,i) \tag{27}\]  
Rewriting (27) as below:  
\[\Big{(}\Xi\otimes I_{m}\Big{)}\mathbf{\xi}\Big{(}k,i\Big{)}=\Big{(}(L+D)\otimes I _{m}\Big{)}\mathbf{e}\Big{(}k,i\Big{)} \tag{28}\]  
where \(\Xi=diag\{\delta_{1},\delta_{1},\ldots,\delta_{N}\}\), since \(\delta_{p}\in\{-1,1\}\), and hence \(\Xi\) is an invertible matrix. From Equation (28), we have  
\[\xi(k,i) =(\Xi\otimes I_{m})((L+D)\otimes I_{m})\mathbf{e}(k,i) \tag{29}\] \[\mathbf{y}(k,i) =\left[y_{1}^{T}(k,i),y_{2}^{T}(k,i),\ldots,y_{N}^{T}(k,i)\right]^ {T}\in\mathbb{R}^{mN\times mN}\]0  
Considering the two cases of event-triggered and non-event-triggered for BFMASs that is, when \(i=i_{s}\) and \(i_{s}<i<i_{s+1}\) hold, the control input is unchanged from the last triggered iteration. The following is obtained:  
\[\mathbf{y}(k,i) =\left[y_{1}^{T}(k,i),y_{2}^{T}(k,i),\ldots,y_{N}^{T}(k,i)\right]^ {T}\in\mathbb{R}^{mN\times mN}\]1 \[\mathbf{y}(k,i) =\left[y_{1}^{T}(k,i),y_{2}^{T}(k,i),\ldots,y_{N}^{T}(k,i)\right]^ {T}\in\mathbb{R}^{mN\times mN}\]2 \[\mathbf{y}(k,i) =\left[y_{1}^{T}(k,i),y_{2}^{T}(k,i),\ldots,y_{N}^{T}(k,i)\right]^ {T}\in\mathbb{R}^{mN\times mN}\]3  
where \(\Gamma(k,i)\triangleq\text{diag}\bigg{(}\frac{\Phi_{1}^{T}(k,i)}{\lambda_{1 }+\big{\|}\Phi_{1}(k,i)\big{\|}},\,\ldots,\,\frac{\Phi_{N}^{T}(k,i)}{\lambda_ {N}+\big{\|}\Phi_{N}(k,i)\big{\|}}\bigg{)}\), \(\rho\triangleq diag(\rho_{1},\rho_{2},\ldots,\rho_{N})\). Since \(e_{p}(k,i)=y_{d}-\delta_{p}\overline{y}_{p}(k,i)\), where \(\overline{y}_{p}(k,i)=y_{p}(k,i)-\overline{y}_{p}\), and the desired distance \(\widehat{y}_{p}\) among each agent and the virtual leader is a time-invariant constant, it follows that:  
\[\mathbf{y}(k,i) =\left[y_{1}^{T}(k,i),y_{2}^{T}(k,i),\ldots,y_{N}^{T}(k,i)\right]^ {T}\in\mathbb{R}^{mN\times mN}\]4 \[\mathbf{y}(k,i) =\left[y_{1}^{T}(k,i),y_{2}^{T}(k,i),\ldots,y_{N}^{T}(k,i)\right]^ {T}\in\mathbb{R}^{mN\times mN}\]5 \[\mathbf{y}(k,i) =\left[y_{1}^{T}(k,i),y_{2}^{T}(k,i),\ldots,y_{N}^{T}(k,i)\right]^ {T}\in\mathbb{R}^{mN\times mN}\]6When combined with (30), we obtain the following:  
\[\mathbf{y}(k,i) =\left[y_{1}^{T}(k,i),y_{2}^{T}(k,i),\ldots,y_{N}^{T}(k,i)\right]^ {T}\in\mathbb{R}^{mN\times mN}\]7  
where \(\Phi(k,i)\triangleq diag(\Phi_{1}(k,i),\,\Phi_{2}(k,i),\,\dots,\,\Phi_{N}(k,i))\). Taking the norm of both sides of Equation (32), we have  
\[ \begin{split}\|e(k+1,i)\|&=\|(I_{mN}-\rho H(k,i)(( L+D)\otimes I_{m}))e(k+1,i-1)\|\\ &\leq\|I_{mN}-\rho H(k,i)((L+D)\otimes I_{m})\|\|e(k+1,i-1)\|\end{split}  \tag{33}\]  
where \(\mathbf{H}(k,i)\triangleq\Phi(k,i)\Gamma(k,i)=diag\{H_{1}(k,i),H_{2}(k,i),\dots,H_ {N}(k,i)\}\),  
\(H_{p}(k,i)=\Phi_{p}(k,i)\hat{\Phi}_{p}^{T}(k,i)/\left(\lambda_{p}+\left\|\hat {\[PERSON]}_{p}(k,i)\right\|^{2}\right)\).