## 4 Consensus of Leader-Follower Multi-Agent Systems with Unknown Parameters
### Discussions of the Consensus for Multi-Agent Systems  
In order to demonstrate the advantages of the proposed control method, the analysis of control costs and parameters adaptive laws are discussed with the initial errors:  
\[ \begin{cases}|e_{01}^{(x)}(0)|=0.1,|e_{01}^{(y)}(0)|=0.1,|e_{01}^{(z)}(0)|=0.2 ;\\ |e_{02}^{(x)}(0)|=0.0,|e_{02}^{(y)}(0)|=0.9,|e_{02}^{(z)}(0)|=1.2;\\ |e_{23}^{(x)}(0)|=1.1,|e_{23}^{(x)}(0)|=0.6,|e_{23}^{(z)}(0)|=1.2;\\ |e_{24}^{(x)}(0)|=0.5,|e_{24}^{(y)}(0)|=0.4,|e_{24}^{(z)}(0)|=1.1,\end{cases}  \tag{20}\]  
and the initial parameters errors  
\[ \begin{cases}|e_{01}^{(a)}(0)|=0.1,|e_{01}^{(b)}(0)|=0.1,|e_{01}^{(c)}(0)|=0.3 ;\\ |e_{02}^{(a)}(0)|=0.2,|e_{02}^{(b)}(0)|=0.3,|e_{02}^{(c)}(0)|=0.4;\\ |e_{23}^{(a)}(0)|=0.19,|e_{23}^{(b)}(0)|=0.25,|e_{23}^{(c)}(0)|=0.36;\\ |e_{24}^{(a)}(0)|=0.22,|e_{24}^{(b)}(0)|=0.4,|e_{24}^{(c)}(0)|=0.4.\end{cases}  \tag{21}\]  
Figure 8: The numerical simulation results of leader–follower systems in Example 4. **(a)** The errors of multi-agent systems. **(b)** The parameters estimation of multi-agent systems. The costs of different leader-follower multi-agent systems are shown in Figure 9, where all the costs of controllers and the update rate of parameters \(|\hat{a}_{j}|,|\hat{b}_{j}|\) and \(|\hat{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{ \dot{\dot{\dot{ }}}}}}}}}}}}}}|\) also tend to zero as \(t\rightarrow\infty\). In addition, some comparison results of the controllers and adaptive laws of parameters are given in Tables 1 and 2, respectively. For the consensus control of multi-agent systems with unknown parameters, the control costs are limited, which are not only related to the controller but also to the initial errors of the parameters. The larger the initial errors among the leader-follower multi-agent systems, the higher the costs of the controllers, and the larger the parameters errors, the higher the costs of the adaptive controllers. The costs of the controllers and the parameters adaptive laws will also approach zero as \(t\rightarrow\infty\). Similarly, the cost consumption of the network model of Example 4 has been analyzed through the cost analysis of the controllers and parameters adaptive laws. In Example 4, \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Agent**} & \multicolumn{2}{c}{**Initial Errors**} & \multicolumn{2}{c}{**Control Time \(t\)**} & \multicolumn{1}{c}{**The Costs of \(\{|\hat{a}_{ij}|,\)** \\  & \(|e_{ij}^{(x)}(0)|,|e_{ij}^{(y)}(0)|,\) & \(|\hat{\dot{b}_{ij}}|\leqslant 0.01,\)** & \(|\hat{\dot{b}_{ij}}|,|\hat{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{ \dot{\dot{\dot{\dot{\dot{ }}}}}}}}}}}}}}|\) and Total Cost \\ \hline
1 & 0.1, 0.1, 0.2 & 14.19, 12.00 & \(|\)812.92, 311.68, 1124.60 \\ \hline
2 & 0.0, 0.9, 1.2 & 12.29, 8.49 & [1640.54, 1600.14], 3240.68 \\ \hline
3 & 1.1, 0.6, 1.2 & 21.98, 18.21 & [1814.09, 1145.51], 2959.60 \\ \hline
4 & 0.5, 0.4, 1.1 & 22.01, 20.24 & [1806.64, 1418.36], 3225.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The results of control costs for all agents in Example 3.  
the network topology of followers 1, 2, and 3 is the same as Example 3, and the corresponding initial conditions are the same as Example 3, so their cost results are the same, as shown in Tables 1 and 2. In Figure 10, the controllers and adaptive update laws tend to zero as the synchronization of the intelligent agent is completed, and parameters identification and system errors can both be completed as \(t\rightarrow\infty\). The followers 3, 4, and 5 have no communication with the leader 0, but follower 3 can establish communication with followers 1 and 2, while follower 5 can communicate with followers 2, 3, and 4. Follower 3 synchronizes with follower 1 when follower 1 and the leader are synchronized, so followers 3 and 5 can also synchronize and identify the leader's parameters \(a,b,c\), respectively. Therefore, the consensus control between the leader and followers has been achieved.