## 1 Introduction  
The multi-agent system has become a very popular topic in the control community in recent years. This is mainly due to its broad applications in practical systems, such as formation control (see [1, 2]), time synchronization (see [3, 4, 5]), etc. In the study of the multi-agent system, a fundamental problem is to design a distributed controller such that the states or the outputs of all agents reach an agreement whereas every agent has only access to the information of its neighboring agents. Such a problem, which is called the consensus problem, has been studied widely (see [6, 7]). It is a well-known fact that delay is unavoidable in many practical systems, which can lead to oscillatory instability, chaos and bifurcation (see [8, 9, 10]). In the multi-agent system, many factors such as the packet loss, the channel congestion, and the communication distance can cause delays. When the number of agents is large and the distance between these agents is small, the delay caused by the communication distance is very small and is often ignored. However, in order to perform precise tasks, the state between agents needs to be exactly synchronized. For example, in the field of the military reconnaissance, if a distributed multi-agent system needs to cooperatively observe targets, the agents need to achieve exact time synchronization. In this type of system, the time synchronization error between agents needs to reach the microsecond or nanosecond level (see [11]), and the smaller the time synchronization error is, the better the cooperative task of agents is completed. In this case, although the delay caused by the communication distance is very small, which may only reach the millisecond or microsecond level, it still cannot be ignored. Furthermore, when the number of agents in the system is small and the distance between agents is far, the density of the network is small. In this kind of network, the delay caused by the packet loss and the channel congestion decreases (see [12]). On the contrary, the delay caused by the communication distance increases and has a greater impact on the communication time-delay. Therefore, this paper mainly studies the influence of the time-delay caused by the communication distance on the consensus of the distributed multi-agent system. Generally, time delays studied in multi-agent systems can be divided into constant delays (see [13; 14]), time-varying delays (see [15; 16; 17]) and random delays (see [18]). In most of the researches, the above-mentioned delays were assumed to be bounded. To solve the consensus problem of multi-agent systems with bounded time-delays, researchers use the following common methods. In [15; 18], by expanding the dimension of the system by a finite multiple, the multi-agent system with bounded time-delays is transformed into the multi-agent system without time-delay. Furthermore, the Lyapunov function is used to solve the consensus problem. In [14; 16], the consensus problem of the system is transformed into a stability problem, and then the Lyapunov-Krasovskii functional with double integral is used to get conditions to ensure that the derivative of the Lyapunov-Krasovskii functional is negative definite. These conditions are related to time-delays and are sufficient conditions to ensure the stability of the system. In the Lyapunov-Krasovskii functional, since integration limits of the double integral are related to time-delays, the derivative of the Lyapunov-Krasovskii functional with double integral contains a positive quadratic term whose coefficient is the time-delay. The boundedness of the time-delay guarantees the existence of conditions to ensure that the derivative of the Lyapunov-Krasovskii functional is negative definite. In [13; 17], under the discrete-time multi-agent system, after the consensus problem of the system is transformed into a stability problem, the discrete Lyapunov-Krasovskii functional with double summation is used to get stability conditions related to time-delays.