## 4 Consensus of Leader-Follower Multi-Agent Systems with Unknown Parameters
### Numerical Simulations  
Therefore, two examples of leader-follower multi-agent systems are given to show the proposed control method, and they are given as follows:  
**Example 3**.: _The leader-follower network among five agents is given in Figure 6a, where\"0\" denotes the leader, \"1,2,3,4\" denote the followers, the \(1\)th and \(2\)th followers are connected with leader, and the \(3\)th and \(4\)th followers are only connected with \(2\)th follower._  
The initial values \((x_{i}(0),y_{i}(0),z_{i}(0))\) and the parameters \((\hat{a}_{i}(0),\hat{b}_{i}(0),\hat{c}_{i}(0))\) of leader and followers are given by  
\[\begin{cases}(x_{0}(0),y_{0}(0),z_{0}(0))=(0.1,0.2,0.1),(x_{1}(0),y_{1}(0),z_{ 1}(0))=(0.2,0.1,0.3),\\ (x_{2}(0),y_{2}(0),z_{2}(0))=(0.1,1.1,1.3),(x_{3}(0),y_{3}(0),z_{3}(0))=(1.2,0.5,0.1),\\ (x_{4}(0),y_{4}(0),z_{4}(0))=(0.6,0.7,0.2),k_{ij}^{(x)}=4.1,k_{ij}^{(y)}=2.2,\\ (a_{0}(0),b_{0}(0),c_{0}(0))=(0.9,0.1,1),(\hat{a}_{1}(0),\hat{b}_{1}(0),\hat{c }_{1}(0))=(0.8,0.2,0.7),\\ (\hat{a}_{2}(0),\hat{b}_{2}(0),\hat{c}_{2}(0))=(0.7,0.4,0.6),(\hat{a}_{3}(0), \hat{b}_{3}(0),\hat{c}_{3}(0))=(0.71,0.35,0.64),\\ (\hat{a}_{4}(0),\hat{b}_{4}(0),\hat{c}_{4}(0))=(0.68,0.5,0.6).\end{cases} \tag{19}\]  
The numerical simulation results based on Matlab R2020a are shown in Figure 7. All agents have reached consensus, and followers have also achieved accurate identification of the parameters of the leader system.  
Figure 6: The network structure of leader–follower multi-agent. (**a**) Five agents in Example 3. (**b**) Six agents in Example 4.  
Figure 7: The numerical simulation results of leader-follower systems in Example 3. (**a**) The errors of multi-agent systems. (**b**) The parameters estimation of multi-agent systems.  
**Example 4**.: _The leader-follower network among six agents is given in Figure 6b, where \"0\" denotes the leader and \"1,2,3,4,5\" denote the followers. Two followers are connected with leader, the \(3\)th follower is connected with \(1\)th and \(2\)th follows, and \(5\)th follower is connected with \(4\)th, \(3\)th, and \(2\)th followers._  
Similarly, the initial values \((x_{i}(0),y_{i}(0),z_{i}(0))\) of leader and followers are given in Equation (19), and the initials of the \(5\)th follower are given by \((x_{5}(0),y_{5}(0),z_{5}(0))=(0.5,0.4,0.3)\); the initial estimated parameters are given by \((\hat{a}_{5}(0),\hat{b}_{5}(0),\hat{c}_{5}(0))=(0.58,0.4,0.7)\).  
The numerical simulation results based on Matlab R2020a are shown in Figure 8. All agents have reached consensus, and the followers have also achieved accurate identification of the parameters \(a,b,\) and \(c\) of the leader system.  
**Remark 3**.: _In the network of the agents in Figure 6b, the \(3\)th agent communicates with \(1\)th and \(2\)th agents. Therefore, the parameters identification can choose the parameters of \(1\)th or \(2\)th agents to track and identify. Similarly, the \(5\)th agent communicates with the \(2\)th, \(3\)th, and \(4\)th agents, and then one of these agents can be selected for the parameters tracking and identification._  
**Remark 4**.: _Although there are only two simple leader-follower multi-agent systems are provided to verify the feasibility of linear feedback and adaptive method, the proposed method can be also extended to other complex leader-follower multi-agent systems, and the multi-agent systems can complete consensus as \(t\to+\infty\)._