# Edge-Event-Triggered Synchronization for Multi-Agent Systems with Nonlinear Controller Outputs
## 5 Simulations  
This section provides two numerical simulations to verify the effectiveness of the proposed policies. For a connected nonlinear multi-agent system with 4 agents and 4 edges, its connected topology is shown in Figure 1.  
Let the nonlinear dynamics of each agent be expressed as follows,  
\[\dot{x}_{i}(t)=\big{(}x+2\cos(x)\big{)}\sin(t)+u_{i}(t)\]  
Figure 1: Connected topology.  
where \(x_{i}(t)=[x_{i,1}\ x_{i,2}]^{T}\). For the controller, taking into account its inherent nonlinearity and the nonlinear effects caused by data communication quantization, the inherent nonlinear function and quantization function are provided respectively. Let the controllers' inherent nonlinearity function \(\mathcal{F}(\cdot)\) be  
\[\left\{\begin{aligned} \mathcal{F}_{1}(c\dot{t}_{i,1})=& c\dot{t}_{i,1} \Big{(}1-0.3\cos\big{(}0.5c\dot{t}_{i,1}+0.5\sin(c\dot{t}_{i,1})\\ &+0.2c\dot{t}_{i,1}\cos(0.5c\dot{t}_{i,1})+0.5\sin(3c\dot{t}_{i,1} )\big{)}\Big{)},\\ \mathcal{F}_{2}(c\dot{t}_{i,2})=& c\dot{t}_{i,2} \Big{(}1-0.3\cos\big{(}0.5c\dot{t}_{i,2}+\sin(c\dot{t}_{i,2})\big{)}\Big{)}. \end{aligned}\right.\]  
Let the logarithmic quantizer function \(g_{q}\big{(}\mathcal{F}_{q}(\cdot)\big{)}\), \(q=1,2\), satisfy  
\[g\big{(}\mathcal{F}_{q}(\cdot)\big{)}=\left\{\begin{array}{ll}0,&\text{if } \ \mathcal{F}_{q}(\cdot)=0,\\ \text{sign}(\mathcal{F}_{q}(\cdot))e^{\Delta\Big{[}\frac{\ln|\mathcal{F}_{q} (\cdot)|}{\Delta}+\frac{1}{2}\Big{]}},&\text{otherwise},\end{array}\right.\]  
with the quantization parameter \(\Delta=0.3\). Obviously, the nonlinear functions \(g_{q}\big{(}\mathcal{F}_{q}(\cdot)\big{)}\), \(q=1,2\), satisfy the properties P1-P3 with \(k_{1}=0.7\) and \(k_{2}=1.3\). The aforementioned vector functions \(\mathcal{F}(\cdot)\) and \(g(\cdot)\) correspond to Figure 2. Let the weight matrix be \(W=\text{diag}(0.85,0.97,1.12,0.78)\) with \(w_{\text{min}}=0.78\) and \(w_{\text{max}}=1.12\). Let the triggering thresholds be \(\sigma_{i}=0.1+0.01i\), \(i=1,2,3,4\). Let \(\beta_{1}=0.02\) and \(\beta_{2}=\beta_{3}=1\). The control gain is given by \(c=6.78\). It can be verified that all parameter settings meet the conditions of Theorem 1. The initial values of all agents are given by \([x_{1}(0)^{T}\ x_{2}(0)^{T}\ x_{3}(0)^{T}\ x_{4}(0)^{T}]^{T}=[-2\ 6\ 4\ -4\ -6\ 8\ 8\ -2]^{T}\). The results of edge-event- and edge-self-triggered policies are illustrated in Figures 3 and 4 and Tables 1 and 2, respectively. Here, we show the evolutionary state trajectories, the update times of all controllers, and the update values of all controllers. In subgraphs (a) of Figures 3 and 4, Xset-1 and Xset-2 represent the values of \([x_{1,1}\ x_{2,1}\ x_{3,1}\ x_{4,1}]^{T}\), and \([x_{1,2}\ x_{2,2}\ x_{3,2}\ x_{4,2}]^{T}\), respectively. It can be observed that the proposed control policies can prompt all agents to achieve state synchronization when the controllers are updated intermittently and the considered system includes controller output nonlinearities. Furthermore, Table 2 also shows the effectiveness of the trade-off between different system resources in the scenarios with nonlinear controller outputs, that is, the event-triggered policy has fewer sampling times, but requires continuous employment of sensors; the self-triggered policy has more sampling times and requires communication networks, but can save sensor resources.