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This paper addresses the consensus of second-order multiagent systems with general topology and time delay based on the nearest neighbor rule. By using the Laplace transform technique, it is proved that the second-order multi-agent system in the presence of time-delay can reach consensus if the network topology contains a globally reachable node and time delay is bounded. The bound of time-delay only depends on eigenvalues of the Laplacian matrix of the system. The main contribution of this paper is that the accurate state of the consensus center and the upper bound of the communication delay to make the agents reach consensus are given. Some numerical simulations are given to illustrate the theoretical results.

The wide applications of multiagent systems abounded in nature and engineering area have stimulated a great deal of interests in studying cooperative and coordinated control problems. The mechanisms operational principles of multi-agent systems can provide useful ideas for developing distributed cooperative control, formation control, unmanned air vehicles, and sensing networks, and so forth. Recently, the study of collective dynamics and coordination of multi-agent systems becomes a hot topic and has attracted many researchers from mathematics, physics, biology, sociology, control science, computer science, artificial intelligence, and so forth [

However, study of some fundamental issues concerning the coordinated control of multi-agent systems, such as consensus, stability, synchronization, and controllability analysis. As in its usual sense, consensus means multiple agents can reach an agreement on a common value by regulating their neighbors. The consensus problem was firstly studied by DeGroot [

Many papers on consensus problems of first-order multi-agent system were presented (see, e.g., [

In the view of [

The rest of this paper is organized as follows. In Section

Consider the second-order multi-agent system with time-delay and general topology is described by

Let

For the Laplacian matrix

Laplacian matrix

For Laplacian matrix

For a column vector denoted as

In this section, we will analyze the consensus of second-order multi-agent systems with general topology and time-delay based on the nearest neighbor rule. For this, we use the Laplace transform technique into (

For simplicity and without loss of generality, we will discuss the case

For second-order multi-agent system (

Denoting

Similarly, for

By the inverse Laplace transform, we can obtain

Since

Notice that if the network topology is undirected with a globally reachable node and the adjacency matrix is symmetric, then the eigenvalues of the Laplacian matrix

Compared with the eigenvalue analysis in the time and frequency domain, such as the [

For second-order multi-agent system (

Consider the second-order multi-agent system with time-delay and general topology is described by

By the same way, the similar results are easy to obtain by the Laplace transform.

For second-order multi-agent system (

For second-order multi-agent system (

In this section, we present some numerical simulations to illustrate the consensus of second-order multi-agent systems with time delay. These simulations are given with ten agents, whose initial positions are zero and velocities are given randomly. The network topologies contain at least one globally reachable node in these simulations. Figure

The consensus in system (

Figure

The consensus in system (

In Figure

The consensus in system (

Figure

The position states in system (

In this paper, we have introduced the Laplace transform to investigate the consensus of second-order multi-agent system with time-delay in general topology. We have proved that the system can achieve consensus when the network topology contains a globally reachable node as well as when time-delay is less than a critical value which only depends on eigenvalues of the Laplacian matrix of network topology.

The authors would like to thank the associate editor and the anonymous reviewer for their valuable comments and suggestions, which significantly contributed to improving the quality of the paper. This work was supported by the National Natural Science Foundation of China under Grants nos. 60774089, 10972003, and 61203150, the Beijing Natural Science Foundation Program (1102016, 4122019), Science and Technology Development Plan Project of Beijing Education Commission (no. KM201310009011); and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR201108055). This work was also supported by the Foundation Grant of Guangxi Key Laboratory of Automobile Components and Vehicle Technology (13-A-03-01), the Opening Project of Guangxi Key Laboratory of Automobile Components and Vehicle Technology (2012KFZD03).