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The flocking control of multiagent system is a new type of decentralized control method, which has aroused great attention. The paper includes a detailed research in terms of distance constrained based adaptive flocking control for multiagent system with time delay. Firstly, the program on the adaptive flocking with time delay of multiagent is proposed. Secondly, a kind of adaptive controllers and updating laws are presented. According to the Lyapunov stability theory, it is proved that the distance between agents can be larger than a constant during the motion evolution. What is more, velocities of each agent come to the same asymptotically. Finally, the analytical results can be verified by a numerical example.

In recent years, the research on the flocking behavior of multiagent system has attracted great attention. For a series of agents which can apply some simple rules and limited information of neighbors to organize into a coordinated state is called flocking phenomenon. There exist many forms of flocking behavior in nature, for example, flocking of birds, swarming of bacteria, and so on [

There are a plenty of existing works contributing to the flocking problems. Three heuristic rules leading to emergence of the first computer animation of flocking were first reported by Reynolds in 1987 [

The time delays of systems are a very common phenomenon in real life. Many factors, for example, finite signal transmission speeds and memory effects, can cause time delay in spreading and communication. Therefore, it should be considered to design the control scheme for multiagent system with time delay. The effect of exchange delays for consensus problems and formation problems has been discussed [

The rest of the paper was structured from the following aspects. In Section

A set of

Define error vector

Based on the definition of

Assume that there exists a nonnegative constant

Communication radius

We call

if

if

Dynamic graphs

The switching process of dynamic graphs according to Definition

We say that a dynamic graph

Hence, the problem which is mentioned can be formally stated as follows.

Consider the set of connected graphs

In view of any dynamic graph

Based on the definition of

From equations of (

From (

We will give some main results for our proposed scheme in next chapter.

For the considered multiagent system, we can assume that

For the multiagent systems (

Consider the following semipositive definite function:

The generalized time derivative of

As

Because

Clearly, Theorem

Under control law (

The size of the set of links

Take a group of

The number of switching times of the closed loop system is finite by Corollary

Referring to the analysis method of [

From (

Because

We can define

The generalized time derivative of

If this is not true, we have

Then, there obviously exists a real number

Finally, we prove that the distance of each gent is bigger than

In this section, an example was presented to show the effectiveness of our proposed algorithm. The potential function

Initial configuration.

End configuration.

Velocity errors.

Changing curve of control laws.

Distance of multiagents.

In this paper, the adaptive flocking of multiagents with time delay is studied. A novel adaptive flocking control method for multiagents is proposed, and the control law is designed depending on functions of the state information and the external signal. By the control law, all agents can follow the virtual leader and can ensure freedom from collisions between neighboring agents. Some theoretical results are attained, and a numerical example is given to show the practicability of the proposed method. The distance constrained based adaptive flocking control for multiagent system with time delay is proposed in this paper, which has received the expected results. But there are still some problems to be resolved in future research. Firstly, the research of the distance constrained based adaptive flocking control for multiagent system with time delay is achieved on the basics of the strongly connected network. We can study the adaptive flocking control for multiagent system based on the networks which contains a directed spanning tree. Secondly, we can apply the algorithm proposed in this paper to the concrete platform of multiple mobile robots or wireless sensor network, which can combine theory with practice better.

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

This work is supported by National Natural Science Foundation of China (61174094, 61039001), National Natural Science Foundation of Tianjin (14JCYBJC18700), Technology and Innovation Fund Project of Civil Aviation University Of China (SY-1415), and Basic Research Projects of High Education (ZXH2010D011, ZXH2012 K002).