## 4 Main Result and Stability Analysis
\begin{array}{ll}(\delta|\frac{U_{1}}{\bar{L}}|+(1-\delta)d)k &|U|<kx_{th}\\ \frac{1}{2}k\alpha&kx_{th}\leq|U|\leq kM\end{array} \right. \tag{29}\]  
where \(U=U_{i}=Lx_{1i}+Lx_{2i}\), \(i=0,...,1\), \(k\in N^{+}\), \(kM\) is the maximum quantization range, \(kx_{th}\) is the switching value of dynamic logarithmic quantizer and dynamic uniform quantizer in a novel adaptive dynamic quantizer. The parameters \(d(d>0)\) and \(\rho(0<\rho<1)\), \(\delta(\delta=\frac{1-\rho}{1+\rho})\) determines the size of the deadzone for \(q_{t}(u)\). Where \(\lfloor a\rfloor\) denotes the greatest integer that is less than or equal to \(a\), \(\alpha(\alpha>0)\) is the parameter of uniform quantizer and determines the quantization density. We defined \(\frac{1}{2}k\alpha=kq_{t}(x_{th})-kx_{th}\). where \(q_{t}(\cdot)\) is the logarithmic quantizer. **Theorem 1**: _The leader-following consensus problem of system (9)-(12) is transformed into the problem of augment system (16) origin stabilization. We can solve Problem 1, i.e., get the following equation:_  
\[\lim_{t\rightarrow\infty}z=0 \tag{30}\]  
Proof: We can define the nonsingular special \(P\) matrix. \[P=\left[ \begin{array}{cc}\rho&0\\ 0&\rho\end{array} \right] \tag{31}\]  
where \(\rho=\left[ \begin{array}{cc}T\\ \frac{1}{\sqrt{N}}1^{T}\end{array} \right]\), \(1=[1,1,...,1]^{T}\). The inverse matrix of the \(P\) matrix is shown below:  
\[P^{-1}=\left[ \begin{array}{cc}\rho^{-1}&0\\ 0&\rho^{-1}\end{array} \right] \tag{32}\]  
Clearly the eigenvalues of \(P\bar{L}P^{+}\) are same as those of \(\bar{L}\), we have \[ \begin{array}{rcl}\dot{P}\dot{L}P^{-1}&=&\begin{bmatrix}0&0&I&0\\ 0&0&\cdots&0\\ -TLT^{+}&\vdots&-TLT^{+}&\vdots\\ 0&0&\cdot&0\end{bmatrix}\\ &=&\begin{bmatrix}0&0&I&0\\ 0&0&\cdots&0\\ -L&\vdots&-L&\vdots\\ 0&0&\cdot&0\end{bmatrix}\end{array}  \tag{33}\]  
The eigenvalues of (33) are the solutions of the characteristic polynomial. \[det(\lambda I-P\hat{L}P^{-1})=\lambda^{2}det(\lambda I-\hat{L})=0 \tag{34}\]  
Therefore \(\hat{L}\) has the same eigenvalues of \(\hat{L}\) but the two zero eigenvalue. The existence of negative definite matrix \(\tau\) makes positive definite matrix symmetric matrix \(\eta\) and \(\theta\) to satisfy the following equation:  
\[\tau^{T}\eta+\eta\tau=-\theta \tag{35}\]  
We can choose following Lyapunov function candidate. \[v=\bar{z}^{T}\eta\bar{z} \tag{36}\]  
where \(v=v(t)\), \(\lambda_{min}(\eta)\) and \(\lambda_{max}(\eta)\) are some know class \(K_{\infty}\) functions, and they are the smallest and largest eigenvalues of matrix \(\eta\), respectively. \[\lambda_{min}(\eta)|\bar{z}|^{2}\leq v\leq\lambda_{max}(\eta)|\bar{z}|^{2} \tag{37}\]  
According to (28), By taking the derivate of (36), we get the following inequality:  
\[ \begin{array}{rcl}\dot{v}&\leq&-\bar{z}^{T}\theta\bar{z}-2\bar{z}^{T}\eta \bar{z}e\\ &\leq&-\lambda_{min}(\theta)|\bar{z}|^{2}+2|\bar{z}^{T}||\eta\bar{\bar{z}}||e |\\ &=&-\lambda_{min}(\theta)|\bar{z}|(|\bar{z}|-\frac{2\|\eta\bar{\bar{z}}\| |e|}{\lambda_{min}(\theta)})\end{array}  \tag{38}\]  
According to Equation (29), \(e\) is a piecewise function, so we get the following inequality:  
\[\begin{array}{rcl}\dot{v}\leq min(-\lambda_{min}(\theta)|\bar{z}|(|\bar{z}| -\frac{\|\eta\bar{\bar{z}}\||k\omega|\sqrt{M}}{\lambda_{min}(\theta)}),\\ -\lambda_{min}(\theta)|\bar{z}|(|\bar{z}|-\frac{2\|\eta\bar{\bar{z}}\|(| \delta|\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{\bar{ \bar{\bar{\bar{\barbar{\barbar{\barbarbarbar{\Under the condition of (41), we obtain the inequality \(\dot{v}<0\), and transform the inequality of (38) into the following form:  
\[\dot{v}\leq-\lambda_{min}(\theta)(\frac{\epsilon}{1+\epsilon})\frac{v}{\lambda_ {max}(\eta)} \tag{42}\]  
We replace the following variable for facilitate writing:  
\[a=\frac{\lambda_{min}(\theta)}{\lambda_{max}(\eta)}\frac{\epsilon}{1+\epsilon} \tag{43}\]  
Then by applying the Comparison Lemma [39],we can provide the following estimates of the convergence rate as in [40].