## 5 Example  
In this section, a simulation example is provided for illustration. We consider the form of the second-order integrator multi-agent system with followers state quantized by the newly proposed adaptive dynamic quantizer is as follows. The exosystem is in the form of (11), (12). \[ \begin{array}{c}\dot{x}_{1i}=x_{2i}\\ \dot{x}_{2i}=q_{\chi}(\frac{F_{i}}{m_{i}}-\frac{f_{i}}{m_{i}}-g)\end{array}  \tag{50}\]  
where \(u_{0}=0\). \(q_{\chi}(\cdot)\) is the state quantization by quantizer (1), where \(d=0.02,\delta=0.2\), \(x_{th}=14\), \(\alpha=0.84\). The initial quantization parameter is \(k_{0}\), so that the initial condition after transformation is in ellipse \(R_{1}\) (46). Reduction ratio \(\pi\) and update interval \(\psi\) are calculated by (47), (49). A network of three followers is modeled by (50), which is shown in the Figure 3. where node 0 represents the exogenous signal (leader) and other nodes represent followers. \[\hat{L}=\left[ \begin{array}{cccccccc}0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1\\ 0&0&0&0&0&0&0&0\\ 1&-2&1&0&1&-2&1&0\\ 0&1&-2&1&0&1&-2&1\\ 1&0&1&-2&1&0&1&-2\end{array} \right]\]  
where \(L\) is the Laplacian matrix after conversion of adjacent matrix. Figure 3: The communication graph. where \(-\tilde{L}\) is Hurwitz, the matrix \(\eta\) and \(\theta\) are calculated by linear matrix inequality \(\theta=-\tilde{L}\eta-\eta\tilde{L}^{T}\), obtain as follows:  
\[\eta=\left[ \begin{array}{cccccc}3.9375&2.25&1.3125&-1.5&0&0\\ 2.25&5.25&2.25&0&-1.5&0\\ 1.3125&2.25&3.9375&0&0&-1.5\\ -1.5&0&0&2.625&0.75&0.375\\ 0&-1.5&0&0.75&3&0.75\\ 0&0&-1.5&0.375&0.75&2.625\end{array} \right]\]  
\[\theta=\left[ \begin{array}{cccccc}3&0&0&0&0&0\\ 0&3&0&0&0&0\\ 0&0&3&0&0&0\\ 0&0&0&3&0&0\\ 0&0&0&0&3&0\\ 0&0&0&0&0&3\end{array} \right]\]  
where \(\lambda_{max}(\eta)=8.8,\lambda_{min}(\eta)=0.5\), \(\lambda_{min}(\theta)=3\), \(M=1600\), \(\|\eta\tilde{E}\|=8.58\), according to \((40)\), \(\phi_{max}=96k\), \(x_{10}(0)=0\), \(x_{1}(0)=[-1,2,1]\), \(x_{20}(0)=5\), \(x_{2}(0)=[1.2,1,1.2]\), \(\psi=2.46\), \(\pi=0.757\), \(\epsilon=2\), \(k_{0}=1\). Figure 4 shows the change of dynamic parameter \(k\) over time. The error of the novel adaptive dynamic quantizer is less than that of the dynamic uniform quantizer proposed in [25]. To show more clearly, we simplify the leader-following system as follows:  
\[ \begin{array}{c}x_{1}=x_{2}\\ x_{2}=q(u)\end{array}  \tag{51}\]  
where \(q(u)\) is input quantized by quantizer, \(x_{1}(0)=-1\), \(x_{2}(0)=1.2\), the parameters of the leader \((11)\) and \((12)\), the novel adaptive dynamic quantizer \((6)\) and the dynamic uniform quantizer \((4)\) are the same as those mentioned above. We get the error comparison between the two dynamic quantizers, as shown in Figure 5.  
Figure 4: Zooming variable \(k\). Let \(e1=q_{\chi}(u)-u\), \(e2=q_{\mu}(u)-u\), \(e1\) and \(e2\) are the quantization errors under the action of the new adaptive dynamic quantizer and uniform quantizer [25], respectively. Figure 5 show the comparison between the quantization error of the newly proposed dynamic quantizer and the quantization error of the dynamic uniform quantizer in the system (51). These simulation results confirm that \(e1\leq 0.12\) and \(e2\leq 0.42\). Moreover, we can get the error of the novel dynamic quantizer at any time is less than dynamic uniform quantizer error. The following is applied to the multi-agent system shown in Figure 3, where the follower's input is quantized by the dynamic quantizer (6) proposed in this paper. Figure 6 shows the speed of three followers hanging points. The final velocity tends to be 1. Figure 7 shows the errors between the speed of three followers' hanging points and the speed of leaders' hanging points. The final error tends to be 0. Figure 5: Dynamic quantizer quantization error. Figure 6: The Speed of Three Followers Hanging Points. Figure 7: Errors between the Speed of Three Followers’ Hanging Points and the Speed of Leaders’ Hanging Points.