## 4 Main Result and Stability Analysis  
In this section, we will consider stabilization problem for the system \((\ref{eq:16})\). For this purpose, according to (6), (9), (10) and (13), we first propose the following equation:  
Figure 2: Subsystem quantization method. \[ \begin{array}{rcl}\bar{x}_{1i}&=&kq_{\chi}(\frac{-\sum_{j=0}^{N}a_{ij}(x_{1i}-x_{1 j})-\sum_{j=0}^{N}a_{ij}(x_{2i}-x_{2j})}{k})\\ &=&-kq_{\chi}(\frac{\left(\frac{\sum_{j}x_{1i}}{k}+\frac{\left(\frac{\sum_{j}x_{ 1i}}{k}\right)}{k}+\frac{\left(\frac{\sum_{j}x_{1i}}{k}\right)}{k})}{-Lx_{1i}- Lx_{2i}-e_{i}}\end{array}  \tag{19}\]  
According to (19), (9) and (10) is converted into following form:  
\[ \begin{array}{c}\left[\begin{matrix}\hat{x}_{1}\\ \hat{x}_{2}\end{matrix}\right]=\begin{bmatrix}0&I\\ -L&-L\end{bmatrix}\begin{bmatrix}x_{1i}\\ x_{2j}\end{bmatrix}-\begin{bmatrix}0\\ 1\end{bmatrix}(e_{i})\end{array}  \tag{20}\]  
where \(e_{i}=kq_{\chi}(\frac{\left(\frac{\sum_{j}x_{1i}}{k}+\frac{\sum_{j}x_{2i}} \right)}{k}-\left(Lx_{1i}+Lx_{2i}\right)\), \(\hat{L}= \begin{bmatrix}0&I\\ -L&-L\end{bmatrix} \), \(x_{1i}=x_{1i}(t)\), \(x_{2i}=x_{2i}(t)\), \(i=1,...,N\). We translate (9), (10) into the following form:  
\[\hat{x}_{1i}=x_{2i} \tag{21}\]  
\[\hat{x}_{2i}=-Lx_{1i}-Lx_{2i}-e_{i} \tag{22}\]  
then define the following \(T\), \(T^{+}\) (see the solution process in Appendix A) and \(\hat{T}^{+}\) matrix:  
\[T^{+}=\left[ \begin{array}{ccccc}-1&1&0&\cdots&0\\ -1&0&1&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots\\ -1&0&0&\cdots&1\end{array} \right]\]  
\[T^{+}=\left[ \begin{array}{ccccc}-\frac{1}{N}&-\frac{1}{N}&-\frac{1}{N}& \cdots&-\frac{1}{N}\\ \frac{N-1}{N}&-\frac{1}{N}&-\frac{1}{N}&\cdots&-\frac{1}{N}\\ -\frac{1}{N}&-\frac{N-1}{N}&-\frac{1}{N}&\cdots&-\frac{1}{N}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ -\frac{1}{N}&-\frac{1}{N}&-\frac{1}{N}&\cdots&\frac{N-1}{N}\end{array} \right]\]  
\[\hat{T}^{+}=\left[ \begin{array}{cc}T^{+}&0\\ 0&T^{+}\end{array} \right]\]  
where \(T\in R^{N\times(N+1)}\), \(T^{+}\in R^{(N+1)\times N}\) and \(\hat{T}^{+}\in R^{2(N+1)\times 2N}\). According to \(T\), we translate (14) and (15) into the following equations:  
\[\bar{x}_{1i}=Tx_{1i} \tag{23}\]  
\[\bar{x}_{2i}=Tx_{2i} \tag{24}\]  
where \(i\in 1,...,N\), \(\iota\in 0,...,N\). For convenience of writing, we translate (23), (24) into the following equation:  
\[\bar{z}=\hat{T}z \tag{25}\]  
where \(z=z(t)= \begin{bmatrix}x_{1i}\\ x_{2i}\end{bmatrix} \), \(z=z(t)= \begin{bmatrix}\bar{x}_{1i}\\ \bar{x}_{2i}\end{bmatrix} \), \(\hat{T}= \begin{bmatrix}T&0\\ 0&T\end{bmatrix} \), \(i=1,...,N\). According to (14) and (15), (19) is converted into the following form:\[ \begin{array}{rcl}\ddot{x}_{1i}&=&T\dot{x}_{1i}\\ &=&-TLx_{1i}-TLx_{2i}-T\dot{e}_{t}\\ &=&-TLT^{+}\dot{x}_{1i}-TLT^{+}\dot{x}_{2i}-T\dot{e}_{t}\\ &=&-L\dot{x}_{1i}-L\dot{x}_{2i}-T\dot{e}_{t}\end{array}  \tag{26}\]  
where \(\bar{L}=TLT^{+}\), \(e=e_{t}\), \(i=1,...,N\), \(\iota=0,...,N\). According to (14) and (15), (20) is converted into the following form:  
\[ \begin{bmatrix}\dot{x}_{1}\\ \dot{x}_{2}\end{bmatrix} = \begin{bmatrix}0&I\\ -\bar{L}&-\bar{L}\end{bmatrix}\begin{bmatrix}\dot{\pi}_{1}\\ \dot{\pi}_{2}\end{bmatrix} - \begin{bmatrix}0\\ T\end{bmatrix} (e) \tag{27}\]  
where  
\[\bar{L}=\left[ \begin{array}{cccc}l_{11}-l_{01}&l_{12}-l_{02}&\cdots&l_{1n}- l_{0n}\\ l_{21}-l_{01}&l_{22}-l_{02}&\cdots&l_{2n}-l_{0n}\\ \vdots&\vdots&\ddots&\vdots\\ l_{n1}-l_{01}&l_{n2}-l_{02}&\cdots&l_{nn}-l_{0n}\end{array} \right]\]  
We get following equation:  
\[\dot{z}=\bar{L}\bar{z}-\bar{E}e \tag{28}\]  
where \(x_{1}=x_{1i}\), \(x_{2}=x_{2i}\), \(\bar{L}= \begin{bmatrix}0&I\\ -\bar{L}&-\bar{L}\end{bmatrix} \), \(\bar{E}= \begin{bmatrix}0\\ T\end{bmatrix} \), \(\bar{L}\in R^{2N\times 2N}\), \(\bar{E}\in R^{2N\times(N+1)}\), \(\bar{z}= \begin{bmatrix}\dot{x}_{1}\\ \dot{\pi}_{2}\end{bmatrix} \), \(A=\bar{L},B=\bar{E}\). According to (7), The quantization error \(|e|\) of the novel adaptive dynamic quantizer is as follows:  
\[|e|\leq\left\{