## 4 Simulation And Discussion  
In order to verify the efficiency of our proposed consensus algorithm, we consider a system consisting of the six heterogeneous following agents, given by  
\[\left\{ \begin{array}{l}\dot{x}_{1}(t)=\left[\begin{array}{cc}0&2\\ -2&-3\end{array}\right]x_{1}(t)+\left[\begin{array}{c}0\\ 1\end{array}\right]u_{1}(t),\\ y_{1}(t)=\left[\begin{array}{cc}0&0.1\end{array}\right]x_{1}(t),\end{array} \right.\]  
\[\left\{ \begin{array}{l}\dot{x}_{2}(t)=\left[\begin{array}{cc}0.4&-0.2\\ 2&-1\end{array}\right]x_{2}(t)+\left[\begin{array}{c}0\\ 0.4\end{array}\right]u_{2}(t),\\ y_{2}(t)=\left[\begin{array}{cc}0.2&1\end{array}\right]x_{2}(t),\end{array} \right.\]  
\[\left\{\begin{array}{l}\dot{x}_{i}(t)=\left[ \begin{array}{ccc}1&-0.5&0\\ 0.1&0&0.2\\ 1&-2&-3\end{array} \right]x_{i}(t)+\left[ \begin{array}{c}0.1\\ 0\\ 1\end{array} \right]u_{i}(t),\\ y_{i}(t)=\left[ \begin{array}{cc}0.1&0.2&0.4\end{array} \right]x_{i}(t), \hskip 14.226378 pti=\,3,4,\]  
\[\left\{\begin{array}{l}\dot{x}_{i}(t)=\left[ \begin{array}{cccc}1&0.2&0.3&0. 5\\ 0.2&0.3&0&0\\ 0.1&0&0.2&1\\ 1&-4&-1&-3\end{array} \right]x_{i}(t)+\left[ \begin{array}{c}0.5\\ 0\\ 0\\ 1\end{array} \right]u_{i}(t),\\ y_{i}(t)=\left[ \begin{array}{cccc}0&0&0&0.2\end{array} \right]x_{i}(t), \hskip 14.226378 pti=\,5,6.\]  
The leader's trajectory is chosen as \(y_{d}(t)=0.25t^{2}(5-t),t\in[0,5]\). Figure 1: Location of \(\beta\lambda_{i}\). The digraph formed by the six following agents and the leader is shown in Figure 2, where vertex 0 represents the leader and the vertices 1-6 represent the following agents. The Laplacian for six following agents is  
\[L=\left[ \begin{array}{cccccc}2&0&-1&-1&0&0\\ -1&1&0&0&0&0\\ 0&-1&1&0&0&0\\ 0&0&0&1&-1&0\\ 0&0&-1&0&2&-1\\ 0&0&0&-1&0&1\end{array} \right],\]  
and \(S=\operatorname{diag}\{1.5,0,0,2,0,0\}\), so the eigenvalues of \(L+S\) are \(2.6624\pm 0.5623j,1.8774\pm 0.7449j,0.3551,0.6753\). The parameters of fractional-power ILC algorithm (14) are chosen as \(\Gamma=\operatorname{diag}\{4,1,1,1,2,2\}\) and \(K=\operatorname{diag}\{6,1.2,1.5,1.5,3,3\}\), and it is easy to verify that \(\rho\left(I_{mn}-CBT((L+S)\otimes I_{m})\right)=0.86<1\), i.e., these parameters chosen above satisfy the condition (15) of Theorem 1. It is seen from Assumption 2 that the initial tracking errors are assumed to be zero, so all agents are reset to the same initial position after each iteration. The control input signals for the first iteration of all agents are set to 0. We choose \(\alpha\) as 0.7, and the tracking processes at the 10 th and 70 th iteration are represented by Figures 3 and 4, respectively. Figure 5 shows the control input signals when the six following agents fully track the leader's trajectory. In addition, Figure 6 illustrates the evolution of each agent's tracking error \(e_{i,k}(t)\). Set \(\max_{t\in[0,T]}\left|e_{i,k}\left(t\right)\right|<10^{-3}\) as the precision requirement, Figure 6 clearly shows that all the following agents track the leader's trajectory when the iteration reaches 70 times. Then, we choose \(\alpha=0.85\); other parameters remain unchanged. Figure 7 shows the tracking errors of the six following agents, and the novel ILC consensus algorithm needs to iterate 80 times so that all agents completely track the leader's trajectory with \(\alpha=0.85\). Obviously, selecting different control parameters yields distinct consensus convergence rate, and how to get the best control parameters for the algorithm could refer to the main idea of the agent-based simulator for tourist urban routes in the work by the authors of [26]. In order to compare the tracking performance of the novel ILC consensus algorithm based on fractional-power error signals with that of the traditional PD-type ILC algorithm, we choose \(\alpha=1\), so the algorithm (14) is converted into the traditional PD-type ILC algorithm.