## 3 Problem Statement  
Considering the GWCF, we can write the equation of motion in the vertical direction as  
\[ \begin{array}{c}F_{i}(t)-f_{i}-m_{i}g-m_{i}\dot{x}_{i}(t)=0\\ f_{i}=\mu_{i}N_{i}\quad i=1,...,N\end{array}  \tag{8}\]  
\(F_{i}(t)\) represents the pulling force of each hanging point, \(\mu_{i}\) is the coefficient of sliding friction of an object at each hanging point, \(N_{i}\) is the pressure on the orbit of the object at each hanging point, \(f_{i}\) is the sliding friction force in the vertical direction of an object at each hanging point, \(m_{i}\) is the mass of an object at each hanging point, \(g\) is the acceleration of gravity. To obtain the state space model of the GWCF synchronized rise. First, let us take the state variables as \(x_{1i}=x_{i}(t)\) and \(x_{2i}=\dot{x}_{i}(t)\). Then the state space equation of the input quantized is as follows by the newly proposed adaptive quantizer (6)  
\[\dot{x}_{1i}=x_{2i} \tag{9}\]  
\[\dot{x}_{2i}=kq_{\chi}(\frac{u_{i}}{k}) \tag{10}\]  
where \(u_{i}=\frac{F_{i}}{m_{i}}-\frac{f_{i}}{m_{i}}-g\), \(i=1,...,N\), \(x_{1i}=col(x_{11}(t),...,x_{1N}(t))\), \(x_{2i}=col(x_{21}(t),...,x_{2N}(t))\) is the state, \(u_{i}=u_{i}(t)\) is the input. \[\dot{x}_{10}=x_{20} \tag{11}\]  
\[\dot{x}_{20}=u_{0} \tag{12}\]  
where \(x_{10}\in R\) and \(x_{20}\in R\) are an exogenous signal, \(x_{10}=x_{10}(t)\), \(x_{20}=x_{20}(t)\), \(u_{0}=u_{0}(t)\). The plants (9), (10) and the exosystems (11), (12) together can be viewed as multi-agent systems of \(N+1\) agents with (11), (12) as the leader and the N subsystems of (9), (10)as N followers. Given the plant(9), (10), the exosystem (11), (12), we can define a digraph \(g=(\
u,\epsilon)\). To describe our control law, denote the adjacency matrix of digraph \(g\) by \(A=\left[a_{ij}\right]\in R^{(N+1)\times(N+1)}\) where \(a_{ii}=0,a_{ij}=1\Leftrightarrow(j,i)\in\epsilon\), and \(a_{ij}=0\Leftrightarrow(j,i)\
otin\epsilon\) for \(i,j=0,1,...,N\). According to the paper [25], we can get the following control law:  
\[u_{i}=-\sum_{j=0}^{N}a_{ij}(x_{1i}-x_{1j})-\sum_{j=0}^{N}a_{ij}(x_{2i}-x_{2j}) \tag{13}\]  
where \(i=0,...,N\), the control law of the form (13) is called distributed control law. It can only access to the information of its neighbors and itself. Our problem is described as follows:  
**Problem 1**. : _Given the multi-agent system (9)-(12) under the distributed control law (13), the state of followers eventually tends to leader's state by using the newly proposed adaptive dynamic quantization, i.e., \(\lim_{t\rightarrow\infty}|x_{1i}-x_{10}|=0\), \(\lim_{t\rightarrow\infty}|x_{2i}-x_{20}|=0\), \(i=1,...,N\)._  
The structure of the system quantization method is shown in Figure 2.  
Following the framework in [40], for a given plant to stabilization problem of a well-defined augmented system, we will first convert the leader-following consensus problem of multi-agent systems. \[\tilde{x}_{1i}=x_{1i}-x_{10}; \tag{14}\]  
\[\tilde{x}_{2i}=x_{2i}-x_{20}; \tag{15}\]  
where \(i\in 1,...,N\). Then, according to (9)-(15) we can get the following form:  
\[\hat{z}=A\tilde{z}+Be \tag{16}\]  
where matrix \(A\) and matrix \(B\) are Hurwitz, \(A\) is Laplacian matrix, according to Laplacian property we get \(x_{1i}\) and \(x_{2i}\) to receive signals from itself and its neighbor (that is, only nodes near the leader can accept the leader's information), \(e=e_{i}\) is quantization error, the specific values of matrix \(A\), \(B\) and \(e\) are introduced later. \(\tilde{z}=\tilde{z}_{i}= \begin{bmatrix}\tilde{x}_{1i}\\ \tilde{x}_{2i}\end{bmatrix} \), \(i=1,...,N\).