## 1 Introduction
However, to make the system achieving a consensus, assumptions of topologies in these results are quite strict. In this paper, the multi-agent system with increasing communication distances is described as the discrete-time distributed system with non-distributed unbounded time-delays. The first goal of this paper is to use distributed algorithms to make the discrete-time system achieving a consensus under a weaker topology assumption. This paper uses a time-delay system to model the discrete-time system, and the maximum delay in the time-delay system tends to infinity as time goes on. Then, the consensus problem of the time-delay system is studied through a new method. It can be noticed that, for any finite integer \(k>0\), the maximum delay is bounded in the interval \([0,k]\). Hence, for any finite integer \(k>0\), the finite-dimensional augmented model of the time-delay system can be built in the interval \([0,k]\). Furthermore, these finite dimensional system models are used to study the system state. By using this method, under the fixed directed topology containing a spanning tree, which is weaker than topology assumptions in existing results (see [25; 26]), the system is proved to achieve a consensus, if a mild assumption satisfied by the rate of the maximum delay tending to infinity. Since the maximum delay and the maximum communication distance are positively correlated, the assumption of the growth rate of the maximum delay also is the constraint on the growth rate of the maximum communication distance. The second goal of this paper is to study the convergent rate of the system. This paper reveals a new feature that the rate of the system achieving a consensus and the growth rate of the maximum delay are negatively correlated. In other words, rate of the system achieving a consensus and the growth rate of the communication distances between agents are negatively correlated. These theoretical results are verified by simulation results. Furthermore, these results are not only applicable to the multi-agent system with increasing communication distances, but also to any multi-agent system that can be described as the multi-agent system with non-distributed unbounded time-varying delays. The rest of this paper is organized as follows. In the second section, the time-delay system model of the multi-agent system with increasing communication distances is introduced. Furthermore, for any integer \(k\geq 0\), the finite-dimensional augmented model of the time-delay system in interval \([0,k]\) is built. In the third section, conditions and the rate of the system achieving a consensus is studied. The fourth section uses numerical simulations to verify theoretical results given in the paper. Basic symbols and concepts: \(R\) means the set of real number, and \(Z\) means the set of integers. \(R^{m\times n}\) means \(m\times n\)-dimensional real space. \(0_{m\times n}\) means the \(m\times n\)-dimensional 0 matrix. The unit matrix of an appropriate dimension is represented by \(I\). For any matrix \(A\), the \(i-j\) entry of the matrix is represented by \(A_{ij}\). Let \(\Lambda(A)=\{A^{\prime}|A^{\prime}_{ij}=A_{ij}\text{ or }0\}\). For any vector \(x\in R^{n}\), the \(i\)th entry of \(x\) is represented by \(x_{i}\). For functions \(f(x),g(x),x\in D\), where \(D\) is the definition domain, \(f(x)=\text{O}(g(x))\) means that there exists a positive real number \(c_{1}\) such that \(f(x)\leq c_{1}g(x)\) for all \(x\in D\). For an \(x_{0}\in D\), \(f(x)=\text{o}(g(x))\) as \(x\to x_{0}\) means that \(\frac{f(x)}{g(x)}\to 0\) as \(x\to x_{0}\), where if the definition domain \(D\) is unbounded, \(x_{0}\) could be infinity. Furthermore, \(f^{-1}(x)\) means the inverse function of \(f(x)\). A function \(f(x)\) is called as a \(\mathcal{K}_{\infty}\) class function, if \(f:[0,+\infty)\mapsto[0,+\infty)\), \(f(0)=0\) and \(f(x)\) is strictly increasing. \(\max_{j}\{\cdot\}\) and \(\min_{j}\{\cdot\}\) represent maximum and minimum functions, respectively. For \(a\in R\) and \(b\in Z\), \(\lfloor a\rfloor=\max\{b|b\leq a\}\).