## 4 Simulation Results
Event-triggered networked predictive control (ENPC): This scheme only compensates for random communication constraints without active fault-tolerant control. As shown in Figure 8, once an actuator fault occurs, the system output fails to maintain a stable state near the reference value. This results in a large tracking error, with the deviation exceeding 1 for an extended period, as illustrated in Figure 9.
3. Event-triggered fault-tolerant control (EFTC): This scheme does not compensate for the random communication constraints of the system. As shown in Figure 10, the system output is quite bad compared to the previous two methods, while the tracking error has a deviation value exceeding 1  
Figure 5: Event trigger moments of agents. (**a**) Event trigger moment of agent 1. (**b**) Event trigger moment of agent 2. (**c**) Event trigger moment of agent 3. (**d**) Event trigger moment of agent 4. Figure 6: Output result of ENPFTC when the fault amplitude is 1.6. Figure 7: Tracking errors of ENPFTC when the fault amplitude is 1.6. Figure 11: Tracking errors of EFTTC. Figure 8: Output result of ENPC. Figure 10: Output result of EFTTC. Figure 9: Tracking errors of ENPC. In addition, the second numerical simulation is carried out using a networked multi-agent system in [9] to further verify the effectiveness of the proposed control method, and the communication topology \(\mathcal{G}\) of the control scheme is shown in Figure 12.  
The specific experimental parameters are set as follows:  
\[a_{21}=a_{32}=1,\]  
Figure 5:0  
Figure 5:1  
Figure 5:2  
Figure 5:3  
Figure 5:4  
Figure 5:5  
The system sampling frequency is 0.05 s, and the desired relative positions are set as follows:  
\[ \begin{cases}\delta_{\text{p}1}(t)=\begin{bmatrix}0&4+\sin(t)&-8\end{bmatrix} ^{\text{T}}\\ \delta_{\text{p}2}(t)=-\begin{bmatrix}\sqrt{3}(4+\sin(t))/2&(4+\sin(t))/2&8 \end{bmatrix}^{\text{T}}\\ \delta_{\text{p}3}(t)=-\begin{bmatrix}-\sqrt{3}(4+\sin(t))/2&(4+\sin(t))/2&8 \end{bmatrix}^{\text{T}}.\end{cases} \end{cases}\]  
At 20 s, the actuator fault applied to the agent 1 is as follows:  
\[f_{1}(t)= \begin{cases}0,&t<400\\ 4,&t\geq 400.\end{cases} \]  
1. ENPFTC: As shown in Figure 13, the system tends to stabilize after running for a period of time. At 20 s, agent 1 experiences an actuator fault, and the output shows fluctuations. Due to the compensation action of the proposed controller, the tracking error converges to the neighborhood of zero, while the maximum tracking error is less than 0.2, indicating that the designed fault-tolerant control method can actively compensate for actuator faults. In Figure 14, the number of event triggers are 516/700, 548/700 and 634/700, respectively, which show that the event trigger can reduce the data transmission volume by about 19%. 2. ENPC: As no fault handling measures are implemented, it can be observed from Figure 15 that the tracking error exceeded 40 after the occurrence of the fault. 3. EFTC: As can be observed in Figure 16, due to the influence of random communication constraints, the tracking error continues to oscillate persistently, failing to meet the required control performance. 4. Discussion: ENPC does not consider the impact of actuator faults on the system, while the EFTC method does not take network delay into account. Although both methods  
Figure 12: Communication topology \(\mathcal{G}\).