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We are concerned with the consensus problems for networks of second-order agents, where each agent can only access the relative position information from its neighbours. We aim to find the largest tolerable input delay such that the system consensus can be reached. We introduce a protocol with time-delay and fixed topology. A sufficient and necessary condition is given to guarantee the consensus. By using eigenvector-eigenvalue method and frequency domain method, it is proved that the largest tolerable time-delay is only related to the eigenvalues of the graph Laplacian. And simulation results are also provided to demonstrate the effectiveness of our theoretical results.

In the last few years, consensus problems have attracted a great deal of attention owing to their enormous potential applications including formation control, attitude alignment of clusters of satellites, and flocking. And a large number of results have been obtained for consensus problems of multiagent systems [

In reality, due to packet loss and asynchronous clocks of the agents, there unavoidably exist communication delays in the exchange of the agent states [

In this paper, we investigate the consensus problems for networks of second-order agents with time-delay and fixed topology using the method of [

At first, we introduce some preliminary knowledge of graph theory for the following analysis (referring to [

If the undirected graph

Zero is one eigenvalue of

The remaining

Suppose that the multiagent system under consideration consists of

We consider the system with fixed topology and time-delay. In this paper, we are expected to calculate the largest tolerable time-delay. Using the following consensus protocol and considering the communication time-delays are the same, we can get

Here, the proposed protocol only uses the relative position information. The variable

Let

Using protocol (

Let

Since

Assume the communication topology

Denote the nonzero eigenvalues of

Consider a network of second-order agents with a fixed topology

For

For

Consider the equation

For (

Assume the communication topology

From Lemma

Now we prove that

Set

In this section, by presenting some numerical simulations, we will test and verify the theoretical results obtained in the previous section. These simulations are performed with four agents, whose initial conditions are set randomly. We use the fixed topology which is showed in Figure

Topology.

(a, b) Velocity-time.

(a, b) Position-time.

In this paper, we investigate the consensus problem for networks of second-order agents with fixed topology and time-delay and without relative velocity measurement. For undirected networks with fixed topology and time-delay, a sufficient and necessary condition is also given by a frequency domain analysis that established a direct relation between the largest tolerable time-delay and the eigenvalues of the graph Laplacian. Simulation results are provided to demonstrate the effectiveness of our theoretical results.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (61203080, 61573082), the Foundation of State Key Laboratory of Networking and Switching Technology Foundation (SKLNST2011105, SKLNST2013109), the National Program 863 of China (2014AA4032), the National Program 973 of China (613237201506), and State Key Laboratory of Intelligent Control and Decision of Complex Systems.

_{∞}consensus control in directed networks of agents with time-delay