^{1}

^{1}

^{2}

^{3}

This paper investigates the flocking problem in multi-agent system with time-varying delay and a virtual leader. Each agent here is subject to nonlinear dynamics. For the system, the corresponding algorithm with time-varying delay is proposed. Under the assumption that the initial network is connected, it is proved that the distance between agents is in the desired distance. The theoretical deduction shows that the stable flocking motion is achieved.

Flocking is a collective behavior of large number of interacting agents with a common group objective. Examples of these agents include birds, fish, penguins, ants, bees, and crowds. Many scientists from rather diverse disciplines, including physics, mathematics, control engineering and biology, have been interested in flocking problem [

A flocking problem concerning multiple leaders in which followers use the position of flocking center to keep their connections is studied in [

In practice, time delay is inevitable and would damage the stability of system. Jing et al. [

The rest of this paper is organized as follows. Some basic preliminaries and flocking algorithms are presented in Section 2. Section 3 gives the nonlinear leader-following multi-agent models. Algorithms and main results are presented in Section 4. Section 5 concludes the paper and offers suggestions for future work.

In this section, some related preliminary knowledge are introduced. For any vector

Graphs with self-loops will not be considered in this paper. The weight adjacency matrix

1) The eigenvalues of L satisfy

2) The Laplacian matrix L is a positive semi-definite matrix that satisfies the following sum-of-squares property:

Lemma 1. [

Lemma 2. For any vectors

Consider the multi-agent system described by

where ^{th} agent, respectively.

For the systems with virtual leader available, the dynamics of virtual leader is described as

where

Assumption (A): There exists a positive constant

Supposed that all agents have the same sensing radius

Definition 1: Given a constant

1) Initial links are generated by

2) If

3) If

The neighboring set of agent i is divided into four regions, named collision region, separation region, alignment region and attraction region, in which

Definition 2.

1)

2)

3)

Definition 3. The pairwise bounded potential function

which satisfies

1)

2)

According to conditions,

where the parameters

For system (1) with virtual leader (2), the flocking algorithm can be described by

where

The control input (6) can be equivalently rewritten as

Denote

Definition 4. Flocking motion with a virtual leader is said to be achieved asymptotically for systems (1) and (2), if for any initial state, there is

To demonstrate the validity of control protocol (7), the following positive semi-definite function is constructed

where

Theorem 1. Consider a multi-agent system modeled by dynamics (1) and (2), driven by control protocol (5). Suppose that the network is initially connected and

1)

2) No collision occurs among agents for all

3) Flocking motion with a virtual leader is achieved asymptotically.

Proof: Denote the topology switching time sequence as

From Lemma 2, there is

For a positive constant k, one has

Assume that

which implies that

By definition (2), one has

Hence there is no edge lost. In addition, from the definite of potential function, one has

Similar to the above analysis, taking the time derivative of

one has

Similarly, from

is positively invariant, where

and

Since

has

principle that if the initial condition lies in Ω, then the corresponding trajectories will converge to the largest invariant set inside the region

From (8),

Since

Thus, unless the inital configuration of the agents is close enough to the global minimum, almost every final configuration locally minimizes each agent’s global potential. which implies

Then the flocking is achieved. This completes the proof of part (3), thus Theorem 1 hold.

Remark 1. If

This paper mainly discusses the flocking problem of multi-agent system with a virtual leader and time-varying delay. Unlike most existing flocking algorithms, each agent here is subject to nonlinear dynamics. The corresponding algorithms with the time-varying delay are proposed. Under the assumption that the initial network is connected, the theoretical deduction is made. The related topic over the directed network or the jointly connected network will be studied in future.

We thank the Editor and the referee for their comments. This work was supported by the National Nature Sci- ence Foundation of China under Grants 61503053, 61472374 and 61304197, the Natural Science Foundation Pro- ject of CQ CSTC, China (Grant No. cstc2013jcyjA40018), the Youth Science Research Project of CQUPT, China (Grant Nos. A2012-78 and A2012-82), the Doctor Start-up Foundation of CQUPT, China (Grant Nos. A2012-23 and A2012-26), the Natural Science Fundation of CQJW, China (Grant Nos. KJ130506 and KJ1400435), and Training Programme Foundation for the Talents of Higher Education Commission. This support is greatly appreciated.

FenglanSun,RuiWang,YongfuLi,FengLiu, (2016) Flocking for Leader-Following Multi-Agent Systems with Time-Varying Delay. Intelligent Control and Automation,07,9-15. doi: 10.4236/ica.2016.71002