Mathematical Problems in Engineering

Volume 2011, Article ID 791052, 11 pages

http://dx.doi.org/10.1155/2011/791052

## Collective Coordination of High-Order Dynamic Multiagent Systems Guided by Multiple Leaders

School of Mathematical Sciences/Institute of Computational Science, University of Electronic Science and Technology of China, Sichuan, Chengdu 611731, China

Received 6 December 2010; Revised 20 March 2011; Accepted 17 April 2011

Academic Editor: John Burns

Copyright © 2011 Xin-Lei Feng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For second-order and high-order dynamic multiagent systems with multiple leaders, the coordination schemes that all the follower agents flock to the polytope region formed by multiple leaders are considered. Necessary and sufficient conditions which the follower agents can enter the polytope region by the leaders are obtained. Finally, numerical examples are given to illustrate our theoretical results.

#### 1. Introduction

Recently, collective coordinations of multiagent systems have received significant attention due to their potential impact in numerous civilian, homeland security, and military applications, and so forth. For example, Wei et al. [1] described a multiagent recommender system in which the agents form a marketplace and compete to provide the best recommendation for a given user. De Meo et al. [2] presented an XML-based multiagent system for supporting e-recruitment services, in which the various agents collaborate to extract data and rank them according to user queries and needs.

Consensus plays an important role in achieving distributed coordination. The basic idea of consensus is that a team of vehicles reaches an agreement on a common value by negotiating with their neighbors. Consensus algorithms are studied for both first-order dynamics [3–5] and high-order dynamics [6–10].

Formal study of consensus problems in groups of experts originated in management science and statistics in 1960s. Distributed computation over networks has a tradition in systems and control theory starting with the pioneering work of Borkar and Varaiya [11] and Tsitsiklis [12] in 1980s. In 1995, Vicsek et al. [13] provided a formal analysis of emergence of alignment in the simplified model of flocking. Due to providing Vicsek Model, [13] has an important influence on the development of the multiagent systems consensus theory. On the study of consensus of continuous-time system, the classical model of consensus is provided by Olfati-Saber and Murray [14] in 2004.

In multiagent coordination, leader-follower is an important architecture. Hu and Yuan [15] presented a first-order dynamic collective coordination algorithm of multiagent systems guided by multiple leaders, which make all the follower agents flock to the polytope region formed by the leaders. In this paper, we consider the second-order and high-order dynamic collective coordination algorithms of multiagent systems guided by multiple leaders.

#### 2. Preliminaries

A directed graph (digraph) of order n consists of a set of nodes and a set of edges . is an edge of if and only if . Accordingly, node is a neighbor of node . The set of neighbors of node is denoted by . Suppose that there are nodes in the graph. The weighted adjacency matrix is defined as , and if and only if . A graph with the property that implies is said to be undirected. The Laplacian matrix is defined as , for . Moreover, matrix is symmetric if an undirected graph has symmetric weights, that is, .

In this paper, we consider a system consisting of follower-agents and leaders, and the interconnection topology among them can be described by an undirected graph . Where each follower-agent (or leader) is regarded as a node in a graph , and each available information channel between the follower-agent (or leader) and the follower-agent (or leader) corresponds to a couple of edges . is said that and is connected. Moreover, the interconnection topology among follower-agents can be described by an undirected graph . The undirected graph is connected; we mean that at least one node in each component of is connected to the nodes who are leaders. A diagonal matrix to be a leader adjacency matrix associate with with diagonal elements such that each is some positive number if agent is connected to the leader node and 0 otherwise.

Let , is said to be convex if whenever and . A vector sum is called a convex combination of , if the coefficients are all nonnegative and . The intersection of all convex sets containing S is the convex hull of S. The convex hull of a finite set of points is a polytope [16].

The Kronecker product of and is denoted by and is defined to be the block matrix [17]

Lemma 2.1 (see [4]). * (i) All the eigenvalues of Laplacian matrix have nonnegative real parts; (ii) Zero is an eigenvalue of with (where is the column vector of all ones) as the corresponding right eigenvector. Furthermore, zero is a simple eigenvalue of if and only if graph has a directed spanning tree.*

#### 3. Coordination Algorithms That All the Follower-Agents Flock to the Polytope Region Formed by the Leaders

A continuous-time second-order dynamics of follower-agents is described as follows: where are the position and velocity of follower-agent . We consider the following dynamical protocol: where are the position and velocity of the leader , nonnegative constant if and only if follower-agent is connected to leader . The objective of this paper is to lead all the follower-agents to enter the polytope region formed by the leaders, namely, , will be contained in a convex hull of , as .

Furthermore, (3.1) and (3.2) can be rewritten as vector form: where , is a diagonal matrix with diagonal entry , and .

Theorem 3.1. *For the multiagent systems given by (3.1) and (3.2), the follower-agents can enter the polytope region formed by the leaders if and only if is connected. *

*Proof. **Sufficiency*: Let
Then, (3.3) can be rewritten as
Further, (3.5) can be rewritten as
Let be an eigenvalue of , and be the corresponding eigenvector of . Then we get
which implies
Let be the eigenvalues of . Thus,
That is, each eigenvalue of , ([11, Lemma 1]), corresponds to two eigenvalues of , denoted by
From (3.10), we can obtain that all the eigenvalues of have negative real parts. By (3.6), we can get
which implies that and when . Therefore,
We also know that is a row-stochastic matrix which is a nonnegative matrix and the sum of the entries in every row equals 1 [15]. So the follower-agents can enter the polytope region formed by the leaders.*Necessity*: If is not connected, by the definition of connectivity, then some follower-agents, without loss of generality, are denoted by , will not get information from the leaders, and the other follower-agents . Then these follower-agents will not get any position information about leaders. Therefore, follower-agents can not enter the polytope region formed by the leaders. So must be connected.

*Remark 3.2. *(1) For switching network topologies of multiagents systems, if topologies are finite, and the shift is made in turn, then the system given by (3.1) and (3.2) is still asymptotically convergence.

(2) The system given by (3.1) can solve formation control of network. This can be made by the following protocol.

where can be decomposed into for any , and are some specified constants. Let Then Protocol (3.13) has the same form as (3.2), and the system will asymptotically converge. By choosing suitable and , we can obtain on appropriate formation of network.

In the following, a continuous-time high-order dynamics of follower-agents system are considered.

Consider multiagent systems with l-th order dynamics given by where , are the states of follower-agents, is the control input, and denotes the -th derivation of , with , . We consider the following dynamical protocol: where , , are the states of the leaders, denotes the -th derivation of .

Furthermore, the above equations can be rewritten as vector form: where , and .

Denote and as the stacked vector of the follower-agents' states and the leaders' states, respectively, and . Then the above equations can be rewritten as where Let . Then (3.20) can be written as

For high-order dynamic systems (3.23), we have the following theorem.

Theorem 3.3. *For the multiagent system (3.23), the follower-agents can enter the polytope region formed by the leaders, if and only if is connected and is Hurwitz stable, where , are eigenvalues of .*

*Proof. **Sufficiency*. Let be an eigenvalue of , and be the corresponding eigenvector of . Then we get
Furthermore, we have
Let , be the eigenvalues of . Then we get characteristic equation of
By (3.23), we get
If is Hurwitz stable, then all the eigenvalues of have negative real parts. Therefore, , when . So
Thus,
is a row stochastic matrix which is a nonnegative matrix and the sum of the entries in every row equals 1, so the follower-agents can enter the region formed by the leaders.*Necessity*: Similar to the Proof of Theorem 3.1.

#### 4. Simulation

In this section, simulation examples are presented to illustrate the proposed algorithms introduced in Section 3.

*Example 4.1. *We consider a system of five follower-agents guarded by three leaders with the topology in Figure 1. The corresponding weights of edges of are shown in Figure 1. Moreover, the initial positions and velocities of follower-agents and leaders are given as follows:

Figure 2 is position trajectories of the agents. In Figure 2, the red lines are the trajectories of the three leaders, and the others are the trajectories of the five follower-agents. From Figure 2, we can obtain that the five follower-agents can enter the polytope region formed by three leaders as the time gradually increasing.

The velocities of leaders have no effect on follower-agents' flocking to the polytope region formed by leaders. We consider the following simulation for the same multiagent system as the above example. The network topology of multiagents is still in Figure 1. The initial positions and velocities of follower-agents, the initial positions of leaders and the corresponding weights of edges of are the same as the above example, and only the velocities of leaders are changed as Figure 3 is trajectories of the agents. The red lines are the trajectories of the three leaders, and the others are the trajectories of the five follower-agents. From Figure 3, though the velocities of leaders are different, we can still obtain that the five follower-agents can enter the polytope region formed by three leaders as the time gradually increasing. The final states of the follower-agents are consistent with Theorem 3.1.

*Example 4.2. *We consider a system of five follower-agents guarded by four leaders with the topology in Figure 4. The corresponding weights of edges of are shown in Figure 4. Moreover, the initial positions and velocities of leaders are given as follows:
Likewise, the initial positions and velocities of follower-agents are the same as those in Example 4.1. The topology of follower-agents is not connected, according to the definition in Preliminaries, but is connected.

In Figure 5, the red lines are the trajectories of the four leaders, and the others are the trajectories of the five follower-agents. From Figure 5, we can obtain that the five follower-agents can enter the polytope region formed by four leaders as the time gradually increasing.

#### 5. Conclusion

In this paper, we consider the second order and high-order dynamic collective coordination algorithms of multiagent systems guided by multiple leaders. We give the necessary and sufficient conditions which follower-agents can enter the polytope region formed by leaders. Numerical examples are given to illustrate our theoretical results.

#### Acknowledgments

The authors would like to express their great gratitude to the referees and the Editor Professor John Burns for their constructive comments and suggestions that lead to the enhancement of this paper. This research was supported by 973 Program (2007CB311002), Sichuan Province Sci. & Tech. Research Project (2009GZ0004, 2009HH0025).

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