## 2 Discrete-Time Distributed System and Its Mathematical Model  
In the multi-agent system with increasing communication distances, each agent updates its state and sends the state information to its neighbors according to the period \(T\). If an agent does not receive the state information sent by one of its neighbors in a period \(T\), then the agent updates its own state with the last received state information of the neighbor. Consider the digraph \(G=(V,E)\) with \(n\) agents, where \(V=\{1,2,\ldots,n\}\), \(E\) is the set of all directed links in digraph \(G\). Furthermore, if \((i,j)\in E\), agent \(j\) can receive the information sent by agent \(i\). In this paper, we have the following assumption on digraph \(G\). The digraph \(G\) contains a spanning tree. As we all know, the communication delay and the communication distance are positively correlated. In this paper, we take the approximately linear relationship between the time delay and the communication distance as an example, and we have \(m_{ji}(k)=\lfloor bC_{ji}(k)\rfloor\), where \(b>0\) is a constant, the integer \(m_{ji}(k)\in[0,k]\) is the delay in the directed link \((i,j)\), and \(C_{ji}(k)\) is the communication distance between agents \(i\) and \(j\). The consensus algorithm of the multi-agent system with increasing communication distances can be described as the discrete-time distributed system with non-distributed unbounded time-delays:  
\[x_{j}(k+1)=x_{j}(k)+c\sum_{i=1}^{n}a_{ji}\big{(}x_{i}(k-m_{ji}(k))-x_{j}(k) \big{)} \tag{1}\]  
for all \(j\in V\), where \(c\in(0,1)\) is a designable constant gain of system (1), and \(x_{j}\in R\) is the state of agent \(j\). In (1), \(\sum_{i=1}^{n}a_{ji}=1\), where \(a_{ji}>0\) when \((i,j)\in E\) and \(a_{ji}=0\) otherwise. For any \(j\in V\), the initial state \(x_{j}(0)\) is given. For the multi-agent system (1) with increasing communication distances, the following assumption means that the communication distance between agents mast increase slower than the speed of information transfer. In other words, for any \((i,j)\in E\), the following assumption guarantees that the agent \(j\) can receive the information sent by the agent \(i\). For any \((i,j)\in E\) and any integer \(k\geq 0\), \(m_{ji}(k)\in Z\bigcap[0,k]\) and \(k-m_{ji}(k)\to\infty\) as \(k\to\infty\). When the following definition is satisfied, system (1) is called achieving a consensus. **Definition 1**. : _System (1) achieves a consensus, if there exists a constant \(x^{*}\) such that_  
\[|x_{j}(k)-x^{*}|\to 0,k\to\infty \tag{2}\]  
_for all \(j\in V\)._  
Let \(M(k)\) be the maximum delay, i.e., \(M(k)=\max_{\{s\in Z\bigcap[0,k]\text{ and }i,j\in V\}}\{m_{j\bar{i}}(s)\}\). Assumption 2 shows that the maximum delay \(M(k)\) could be unbounded. In other words, when \(k\to\infty\), \(M(k)\to\infty\). Then, for all integer \(k\in[0,\infty)\), the system (1) cannot be described by one finite-dimensional time-delay system. To solve this problem in the modelling and consensus analysis, for any finite integer \(K>0\), system states are studied by building a finite-dimensional augmented model in the interval \([0,K]\). Let \(x(k)=[x_{1}(k),\dots,x_{n}(k)]^{T}\) for all integer \(k\in[0,\infty)\). Without loss of generality, let \(x(k)\equiv x(0)\) for all \(k\in\{0,-1,\dots\}\). Furthermore, for any integer \(K\in[0,\infty)\) and integer \(k\in[0,K]\), let \(y_{x,K}(k)=[x(k)^{T},x(k-1)^{T},\dots,x(k-M(K))^{T}]^{T}\). Then, for any integer \(K\in[1,\infty)\) and all integers \(k\in[0,K-1]\), system (1) is equivalent to  
\[y_{x,K}(k+1)=A_{K}(k)y_{x,K}(k), \tag{3}\]  
where  
\[A_{K}(k)=\left[ \begin{array}{cccc}\widehat{A_{K0}}(k)&\widehat{A_{K1}}(k)& \cdots&\widehat{A_{KM(K)-1}}(k)&\widehat{A_{KM(K)}}(k)\\ l&0_{nx}&\cdots&0_{nx}&0_{nx}\\ 0_{nx}&\cdots&0_{nx}&0_{nx}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0_{nx}&0_{nx}&\cdots&\hat{l}&0_{nx}\\ \end{array} \right].