## 4 Simulation Example  
In this section, a practical simulation example is given to illustrate the validity of the adaptive neural network finite-time control law.  
Consider a practical uncertain multi-agent system with unknown control gains as discussed in [43; 44], where each follower agent stands for a single-link robot. The communication topology graph of four single-link robots is given in Figure 1, where the leader agent is marked as \"0\". The leader agent is used to generate a tracking signal. The dynamics of the \(i\)th single-link robot are described as  
\[J_{i}\ddot{a}_{i}+B_{i}\dot{a}_{i}+M_{i}\sin(a_{i})=q_{i}u_{i}(t)+d_{i}(t),\ i=1,2,3,4 \tag{87}\]  
where \(a_{i}\) and \(\dot{a}_{i}\) are the angle and angular velocity of the single-link robot, respectively, \(u_{i}(t)\) is the actuator voltage input, \(J_{i}\) represents the rotational inertia, \(B_{i}\) represents the damping coefficient, \(M_{i}\) denotes the single-link robot's mass, and \(d_{i}(t)\) and \(q_{i}\) represent the unknown torque disturbance and unknown correlation coefficient, respectively. Let \(x_{i,1}=a_{i}\) and \(x_{i,2}=\dot{a}_{i}\); then, the system (87) can be changed as  
\[\begin{array}{l}x_{i,1}=x_{i,2}\\ x_{i,2}=\frac{q_{i}}{l_{i}}u_{i}(t)-\frac{B_{i}  
Figure 4: Curves of tracking errors \(e_{i1}\).  
Figure 3: Curves of system states \(x_{i2}\).  
Figure 2: Curves of system states \(x_{i1}\) and reference trajectory \(y_{d}\).  
Figure 5: Curves of control law \(u_{i}\).  
Figure 6: Curves of adaptive laws \(\widehat{\Xi}_{i1}\).  
Figure 7: Curves of adaptive laws \(\widehat{\Xi}_{i2}\).  
Figure 8: Curves of adaptive laws \(\hat{\Phi}_{i1}\).  
Figure 10: Curves of adaptive laws \(\hat{\Pi}_{i1}\).  
The asymptotic tracking performance curves for the four follower agents are shown in Figure 2. As can be seen from the figure, using the designed control law, the outputs \(x_{i1}\) of follower agents can track the trajectory of the leader agent in a finite time. The states \(x_{i2}\) of the four follower agents are given in Figure 3, and the curves for tracking errors \(e_{i1}\) are shown in Figure 4. From Figure 4, it is not difficult to see that the tracking errors of the four follower agents can converge to a small neighborhood of zero in a finite time. That is to say, the asymptotic tracking control of uncertain multi-agent systems can be achieved. Based on the results of Figures 2-4 it is shown that the finite time control law designed in this paper is effective. In other words, although the nonlinear multi-agent systems given in this work are affected by an unknown control direction and uncertain dynamics, the system can still achieve the desired control goal under the designed finite-time control law.  
Further, the curves for control laws \(u_{i}\) are given in Figure 5. As shown in Figure 5, during the initial period, the amplitude of the control law is relatively large. As described in Remark 3, we can only choose between tracking performance and control law amplitude. Obviously, our focus here is on tracking performance.  
In addition, the curves for adaptive laws \(\hat{\Xi}_{i1}\), \(\hat{\Xi}_{i2}\), \(\hat{\Phi}_{i1}\), \(\hat{\Phi}_{i2}\), and \(\hat{\Pi}_{i1}\) are shown in Figures 6-10 respectively. From these figures, we can find that all signals of the closed-loop system are bounded after a finite time. Further, in view of the results in Figures 2-10, we can deduce that the finite-time control law and adaptive laws designed in this paper can make the nonlinear multi-agent systems SGPFS. This also illustrates the correctness of the theoretical analysis results from another perspective.