## 5 Simulation  
Firstly, consider the multi-agent system in Section 2.2.1. and let Assumptions 1-3 hold. Next, the learning law (10) is applied and the learning gains satisfy the condition (23). The results of numerical simulations are discussed in this section. In order to illustrate the effectiveness of the proposed dropout compensation ILC method, we considered a heterogeneous multi-agent system consisting of four agents in the undirected graph as shown in Figure 3. It can be seen that the graph is a connected graph and satisfies Assumption 3. Only the second agent and the third agent can obtain the desired reference trajectory. The Laplacian matrix \(\mathcal{L}\) of graph \(\mathcal{G}\) and the reference-accessibility matrix \(\Phi\) were  
\[\mathcal{L}=\left[ \begin{array}{cccc}4&-1.7&-2.3&0\\ -1.7&1.7&0&0\\ -2.3&0&3.8&-1.5\\ 0&0&-1.5&1.5\\ \end{array} \right],\] \[\Phi=\left[ \begin{array}{cccc}0&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&0\\ \end{array} \right].\]  
Then it was obtained that the information interaction weight between agents can be obtained, as shown in Table 1.  
For every agent in Figure 3, the system parameters were defined in Table 2.  
Let \(\Gamma_{1}=-0.813\), \(\Gamma_{2}=1.31\), \(\Gamma_{3}=0.57\), and \(\Gamma_{4}=0.92\). Then, we can obtain the parameters in Table 3 for Theorem 1. Therefore, the learning gains satisfied the second condition in Theorem 1. To perform our simulation tests, we considered the desired reference trajectory as \begin{table}
\begin{tabular}{c c c c} \hline \(K_{1}=4\) & \(K_{2}=1.7\) & \(K_{3}=3.8\) & \(K_{4}=1.5\) \\ \(\varphi_{1}=0\) & \(\varphi_{2}=1\) & \(\varphi_{3}=1\) & \(\varphi_{4}=0\) \\ \hline \end{tabular}
\end{table} 
Table 1: Interaction weight of agent. \begin{table}
\begin{tabular}{c c c c} \hline \(A_{1}=-0.16\) & \(A_{2}=-0.18\) & \(A_{3}=-0.12\) & \(A_{4}=-0.32\) \\ \(B_{1}=0.416\) & \(B_{2}=-0.15\) & \(B_{3}=0.6\) & \(B_{4}=0.05\) \\ \(C_{1}=-0.38\) & \(C_{2}=-0.7\) & \(C_{3}=0.53\) & \(C_{4}=0.59\) \\ \hline \end{tabular}
\end{table} 
Table 2: System parameters. Figure 3: Undirected graph of multi-agent system. \begin{table}
\begin{tabular}{c c c c} \hline \(Q_{1}=0.514\) & \(Q_{2}=0.371\) & \(Q_{3}=0.870\) & \(Q_{4}=0.041\) \\ \(S_{1}=0\) & \(S_{2}=0.083\) & \(S_{3}=0.283\) & \(S_{4}=0\) \\ \(P_{1,1}=1.486\) & \(P_{2,1}=1.629\) & \(P_{3,1}=1.130\) & \(P_{4,1}=1.959\) \\ \(P_{1,2}=-14.563\) & \(P_{2,2}=-17.005\) & \(P_{3,2}=-9.514\) & \(P_{4,2}=-27.168\) \\ \(P_{1,3}=2.095\) & \(P_{2,3}=2.347\) & \(P_{3,3}=1.534\) & \(P_{4,3}=3.431\) \\ \(P_{1,4}=-2.184\) & \(P_{2,4}=-3.083\) & \(P_{3,4}=-0.542\) & \(P_{4,4}=-6.617\) \\ \(P_{1,5}=2.997\) & \(P_{2,5}=3.900\) & \(P_{3,5}=1.402\) & \(P_{4,5}=6.755\) \\ \hline \end{tabular}
\end{table} 
Table 3: Convergence condition parameter. \[r(t)=1.5+\sin\frac{\pi t}{25} \tag{33}\]  
and the desired output deviation of each agent as  
\[ \begin{array}{l}d_{1}(t)=0,d_{2}(t)=-1.5-\sin\frac{\pi t}{25},\\ d_{3}(t)=1.5+\sin\frac{\pi t}{25},d_{4}(t)=0.03t.\end{array}  \tag{34}\]  
Set the initial state of four agents as: \(x_{1}(0)=-4.277\), \(x_{2}(0)=0\), \(x_{3}(0)=6.1333\), and \(x_{4}(0)=2.806\), which satisfies Assumption 2. When the iteration step is \(500\) and \(t=1,2,\cdots,150\), the trajectory of the multi-agent system without packet loss is shown as Figure 4. It can be seen that the desired formation is well achieved to track the desired reference trajectory by the desired output deviation. In order to measure the formation accuracy quantitatively, the disagreement among all agents on their output errors was defined as: \(error(k)=\frac{1}{n^{T}}\sum\limits_{i=1}^{n}\sum\limits_{t=1}^{T}\left|e_{i, k}(t)\right|\). When \(\lim\limits_{k\rightarrow\infty}error(k)=0\), multi-agent system achieved formation tacking. Figure 5 shows that formation performance of the system at the first \(500\) iterations. Obviously, when the iteration is around \(350\) the system error can converge to zero and achieve formation tracking. Figure 4: Multi-agent system trajectory at iteration \(500\).