## 4 Convergence Analysis  
As we know, when we say that an iterative learning algorithm converges, it means that the system error trends to zero as the number of iterations approaches infinity. For the proposed dropout compensation ILC method, we will analyze the convergence of formation tracking for the multi-agent systems in this section. **Theorem 1. **_Consider the multi-agent system (1), let Assumptions 1-3 hold and the learning law (10) be applied. Given that any one of the following conditions is satisfied for all \(i=1,2,\cdots,n\),_  
\[ \begin{array}{l}1.\ (1>A_{i}>0)\wedge(P_{i,1}>0)\wedge(P_{i,2}>0)\wedge(P_{i,3}>0) \wedge\ (P_{i,4}>0)\wedge(P_{i,5}>0),\\ 2.\ (-1<A_{i}<0)\wedge(P_{i,1}>0)\wedge(P_{i,2}<0)\wedge(P_{i,3}>0)\wedge(P_{i,4} <0)\wedge(P_{i,5}>0),\end{array}  \tag{23}\]  
_where_  
\[ \begin{array}{c}P_{i,1}=2-Q_{i},\\ P_{i,2}=-16(1-A_{i})^{2}+4(1-A_{i})(3-A_{i})Q_{i}+(A_{i}-2){Q_{i}}^{2},\\ P_{i,3}=2(1-A_{i})^{2}-(1-A_{i})Q_{i},\\ P_{i,4}=-4(1-A_{i})^{2}+2(1-A_{i})(3-A_{i})Q_{i}+(A_{i}-2){Q_{i}}^{2},\\ P_{i,5}=4(1-A_{i})^{2}-4(1-A_{i})Q_{i}+S_{i},\\ Q_{i}=\Gamma_{i}B_{i}C_{i}(K_{i}+\varphi_{i}),\\ S_{i}=(\Gamma_{i}B_{i}C_{i})^{2}(2K_{i}\varphi_{i}+{\varphi_{i}}^{2}),\\ K_{i}=\sum\limits_{j\in N_{i}}a_{i,j}.\end{array}  \tag{24}\]  
_then the formation tracking objective (7) is achieved._  
**Proof. ** Let \(S(\mathbb{z},z)\)\(=\)\(\det\{(\mathbb{z}-1)I+zH(z)[\mathcal{L}^{\prime}(\mathbb{z})+\Phi]\}\). The characteristic equation is \(S(\mathbb{z},z)=0\). When \(\mathbb{z}=1\), \(\mathcal{L}^{\prime}(1)=\mathcal{L}\). Thus, \(S(1,z)\)\(=\)\(\det\{zH(z)(\mathcal{L}+\Phi)\}\). In Remark 3, \(\mathcal{L}+\Phi\) proves to be nonsingular. Therefore, matrix \(zH(z)(\mathcal{L}+\Phi)\) is nonsingular also and \(S(1,z)\
eq 0\). \(\Box\)  
When \(\mathbb{z}\
eq 1\), we need to prove that the modules of all roots of (25) are smaller than units. \[\det\left\{I+\frac{zH(z)[\mathcal{L}(\mathbb{z})+\Phi]}{\mathbb{z}-1}\right\}=0 \tag{25}\]  
Let \(F(\mathbb{z},z)=\frac{zH(z)[\mathcal{L}(\mathbb{z})+\Phi]}{\mathbb{z}-1}\), based on the general Nyquist stability criterion [23], the modules of all roots of (25) are smaller than units, if the eigenloci \(\lambda[F(\omega,\beta)]\) of  
\[F(\omega,\beta)=\frac{e^{\beta H}(e^{\beta})[\mathcal{L}(e^{i\omega})+\Phi]} {e^{i\omega}-1} \tag{26}\]  
does not enclose the point \((-1,j0)\) for \(\beta\in[-\pi,\pi)\), \(\omega\in[-\pi,\pi)\), and \(\omega\
eq 0\). Based on [PERSON]'s disk theorem, we have \(\lambda[F(\omega,\beta)]\in\)\(\cup_{i=1}^{n}F_{i}\) for all \(\beta\in[-\pi,\pi)\), \(\omega\in[-\pi,\pi)\), and \(\omega\
eq 0\), where  
\[ \begin{array}{c}F_{i}=\{\varsigma:\varsigma\in C,\mid&\varsigma-\frac{e^{i \beta H_{i}(e^{i\beta})(\sum_{j\in N_{i}}a_{i,j}+\varphi_{i})}}{e^{i\omega}-1} \bigg{\rvert}\\ &\leq\sum\limits_{j\in N_{i}}\left|\frac{a_{i,j}e^{i\beta H_{i}(e^{i\beta})[e^ {i\omega}\sum_{j\in N_{i}}(e^{i\omega})]}}{e^{i\omega}(e^{i\omega}-1)}\right| \right\}\end{array}  \tag{27}\]Since \(\xi_{ij}(e^{j\omega})\) can only be 0 or 1, \(e^{j\omega}\xi_{ij}(e^{j\omega})+1-\xi_{ij}(e^{j\omega})\) is 1 or \(e^{i\omega}\), respectively. Taking \(H_{i}(e^{j\beta})\) as \(\frac{\Gamma_{i}\beta C_{i}}{(e^{\beta}-A_{i})}\), we can rewrite (27) as  
\[F_{i}=\left\{\varsigma:\varsigma\in C_{i}\right|\varsigma-\frac{e^{j\beta} \Gamma_{i}B_{i}C_{i}(K_{i}+\varphi_{i})}{(e^{j\omega}-1)(e^{j\beta}-A_{i})} \Bigg{|}\leq\left|\frac{K_{i}\Gamma_{i}B_{i}C_{i}}{(e^{j\omega}-1)(e^{j\beta}- A_{i})}\right|\right\} \tag{28}\]  
So the eigenloci \(\lambda[F(\omega,\beta)]\) does not enclose the point \((-1,j0)\) for \(\beta\in[-\pi,\pi)\), \(\omega\in[-\pi,\pi)\), and \(\omega\
eq 0\) when the point \((-a,j0)\) with \(a\geq 1\) is not in the disc \(F_{i}\) for all \(i=1,2,\cdots,n,\beta\in[-\pi,\pi)\), \(\omega\in[-\pi,\pi)\), and \(\omega\
eq 0\).