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Mathematical Problems in Engineering
Volume 2011, Article ID 791052, 11 pages
http://dx.doi.org/10.1155/2011/791052
Research Article

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 [35] and high-order dynamics [610].

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) 𝐺=(𝑉,𝐸) of order n consists of a set of nodes 𝑉={1,,𝑛} and a set of edges 𝐸=𝑉×𝑉. (𝑗,𝑖) is an edge of 𝐺 if and only if (𝑗,𝑖)𝐺. Accordingly, node 𝑗 is a neighbor of node 𝑖. The set of neighbors of node 𝑖 is denoted by 𝒩𝑖(𝑡). Suppose that there are 𝑛 nodes in the graph. The weighted adjacency matrix 𝐴𝑛×𝑛 is defined as 𝑎𝑖𝑖=0,𝑎𝑖𝑗0, and 𝑎𝑖𝑗>0 if and only if (𝑗,𝑖)𝐸. A graph with the property that (𝑖,𝑗)𝐸 implies (𝑗,𝑖)𝐸 is said to be undirected. The Laplacian matrix 𝐿𝑛×𝑛 is defined as 𝑙𝑖𝑖=𝑗𝑖𝑎𝑖𝑗,𝑙𝑖𝑗=𝑎𝑖𝑗, for 𝑖𝑗. Moreover, matrix 𝐿 is symmetric if an undirected graph has symmetric weights, that is, 𝑎𝑖𝑗=𝑎𝑗𝑖.

In this paper, we consider a system consisting of 𝑛 follower-agents and 𝑘 leaders, and the interconnection topology among them can be described by an undirected graph 𝐺. Where each follower-agent (or leader) is regarded as a node in a graph 𝐺=(𝑉,𝐸), and each available information channel between the follower-agent (or leader) 𝑖 and the follower-agent (or leader) 𝑗 corresponds to a couple of edges (𝑖,𝑗),(𝑗,𝑖)𝐸. (𝑖,𝑗)𝐸 is said that 𝑖 and 𝑗 is connected. Moreover, the interconnection topology among follower-agents can be described by an undirected graph 𝐺. The undirected graph 𝐺 is connected; we mean that at least one node in each component of 𝐺 is connected to the nodes who are leaders. A diagonal matrix 𝐵 to be a leader adjacency matrix associate with 𝐺 with diagonal elements 𝑏𝑖(𝑖{1,,𝑛}) such that each 𝑏𝑖 is some positive number if agent 𝑖 is connected to the leader node and 0 otherwise.

Let 𝑆𝑚, 𝑆 is said to be convex if (1𝛾)𝑥+𝛾𝑦𝑆 whenever 𝑥𝑆,𝑦𝑆 and 0<𝛾<1. A vector sum 𝛾1𝑥1+𝛾2𝑥2++𝛾𝑛𝑥𝑛 is called a convex combination of 𝑥1,,𝑥𝑛, if the coefficients 𝛾𝑖 are all nonnegative and 𝛾1++𝛾𝑛=1. The intersection of all convex sets containing S is the convex hull of S. The convex hull of a finite set of points 𝑥1,,𝑥𝑛𝑚 is a polytope [16].

The Kronecker product of 𝐴=[𝑎𝑖𝑗]𝑀𝑚,𝑛(𝐹) and 𝐵=[𝑏𝑖𝑗]𝑀𝑝,𝑞(𝐹) is denoted by 𝐴𝐵 and is defined to be the block matrix [17]

Lemma 2.1 (see [4]). (i) All the eigenvalues of Laplacian matrix 𝐿 have nonnegative real parts; (ii) Zero is an eigenvalue of 𝐿 with 1𝑛 (where 1𝑛 is the 𝑛×1 column vector of all ones) as the corresponding right eigenvector. Furthermore, zero is a simple eigenvalue of 𝐿 if and only if graph 𝐺 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 𝑛 follower-agents is described as follows:̇𝑥𝑖=𝑣𝑖,̇𝑣𝑖=𝑢𝑖,(3.1) where 𝑥𝑖,𝑣𝑖𝑚 are the position and velocity of follower-agent 𝑖. We consider the following dynamical protocol:𝑢𝑖=𝑗𝒩𝑖𝑎𝑖𝑗𝑥𝑗𝑥𝑖+𝑘𝑞=1𝑏𝑞𝑖𝑥𝑞0𝑥𝑖+𝑗𝒩𝑖𝑎𝑖𝑗𝑣𝑗𝑣𝑖+𝑘𝑞=1𝑏𝑞𝑖𝑣𝑞0𝑣𝑖,(3.2) where 𝑥𝑗0,𝑣𝑗0(𝑗=1,,𝑘) are the position and velocity of the leader 𝑗, nonnegative constant 𝑏𝑞𝑖>0 if and only if follower-agent 𝑖 is connected to leader 𝑞(𝑞=1,,𝑘). The objective of this paper is to lead all the follower-agents to enter the polytope region formed by the leaders, namely, 𝑥𝑖,𝑖=1,,𝑛, will be contained in a convex hull of 𝑥𝑞0,𝑞=1,,𝑘, as 𝑡.

Furthermore, (3.1) and (3.2) can be rewritten as vector form:̇̇𝑥=𝑣,𝑣=𝐻𝐼𝑚𝐵𝐼𝑥+𝑘1𝑛𝐼𝑚𝑥0𝐻𝐼𝑚𝐵𝐼𝑣+𝑘1𝑛𝐼𝑚𝑣0,(3.3) where 𝐻=𝐿+𝐵(1𝑘𝐼𝑛), 𝐵𝑞𝑛×𝑛 is a diagonal matrix with diagonal entry 𝑏𝑞𝑖, and 𝐵=[𝐵1𝐵𝑘]𝑛×𝑛𝑘.

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 𝐺 is connected.

Proof. Sufficiency: Let 𝐻𝑥=𝑥1𝐵𝐼𝑘1𝑛𝐼𝑚𝑥0,𝐻𝑣=𝑣1𝐵𝐼𝑘1𝑛𝐼𝑚𝑣0.(3.4) Then, (3.3) can be rewritten as ̇𝑥=̇𝑣,𝑣=𝐻𝐼𝑚𝑥𝐻𝐼𝑚𝑣.(3.5) Further, (3.5) can be rewritten as ̇𝑥𝑣=0𝑚𝑛×𝑚𝑛𝐼𝑚𝑛𝐻𝐼𝑚𝐻𝐼𝑚𝑥𝑣.(3.6) Let 𝜆 be an eigenvalue of Γ=0𝑚𝑛×𝑚𝑛𝐼𝑚𝑛𝐻𝐼𝑚𝐻𝐼𝑚, and 𝑓𝑔 be the corresponding eigenvector of Γ. Then we get 0𝑚𝑛×𝑚𝑛𝐼𝑚𝑛𝐻𝐼𝑚𝐻𝐼𝑚𝑥=𝜆𝑥,(3.7) which implies 𝑔=𝜆𝑓,𝐻𝐼𝑚𝑓𝐻𝐼𝑚𝑔=𝜆𝑔.(3.8) Let 𝜇𝑖,𝑖=1,,𝑛 be the eigenvalues of 𝐻𝐼𝑚. Thus, 𝐻𝐼𝑚𝑓𝐻𝐼𝑚𝜆𝑓=𝜆2𝑓.(3.9) That is, each eigenvalue of 𝐻𝐼𝑚, 𝜇𝑖>0 ([11, Lemma 1]), corresponds to two eigenvalues of Γ, denoted by 𝜆2𝑖1,2𝑖=𝜇𝑖±𝜇2𝑖4𝜇𝑖2.(3.10) From (3.10), we can obtain that all the eigenvalues of Γ have negative real parts. By (3.6), we can get 𝑥𝑣=𝑒Γ𝑡𝑥(0)𝑣(0),(3.11) which implies that 𝑥0 and 𝑣0 when 𝑡. Therefore, 𝐻𝑥=1𝐵𝐼𝑘1𝑛𝐼𝑚𝑥0𝐻,𝑣=1𝐵𝐼𝑘1𝑛𝐼𝑚𝑣0.(3.12) We also know that [𝐻1𝐵(𝐼𝑘1𝑛)]𝐼𝑚 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 𝐺 is not connected, by the definition of connectivity, then some follower-agents, without loss of generality, are denoted by 𝑥𝑖,𝑖=1,,𝑙(0<𝑙<𝑛), will not get information from the leaders, and the other follower-agents 𝑥𝑗,𝑗=𝑙+1,,𝑛. Then these follower-agents 𝑥𝑖,𝑖=1,,𝑙 will not get any position information about leaders. Therefore, follower-agents 𝑥𝑖,𝑖=1,,𝑙 can not enter the polytope region formed by the leaders. So 𝐺 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.

𝑢𝑖=𝑗𝒩𝑖𝑎𝑖𝑗𝑥𝑗𝑥𝑖𝑓𝑖𝑗+𝑘𝑞=1𝑏𝑞𝑖𝑥𝑞0𝑥𝑖+𝑗𝒩𝑖𝑎𝑖𝑗𝑣𝑗𝑣𝑖+𝑘𝑞=1𝑏𝑞𝑖𝑣𝑞0𝑣𝑖,(3.13) where 𝑓𝑖𝑗 can be decomposed into 𝑓𝑖𝑗=𝑓𝑖𝑓𝑗 for any 𝑖,𝑗=1,,𝑛,  𝑓𝑖 and 𝑓𝑗 are some specified constants. Let̃𝑥=𝑥𝑖𝑓𝑖,𝑣𝑖0=𝑣𝑗0,𝑖,𝑗=1,,𝑚.(3.14) Then Protocol (3.13) has the same form as (3.2), and the system will asymptotically converge. By choosing suitable 𝑓𝑖 and 𝑓𝑗, we can obtain on appropriate formation of network.

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

Consider multiagent systems with l-th (𝑙3) order dynamics given bẏ𝑥𝑖(0)=𝑥𝑖(1),̇𝑥𝑖(𝑙2)=𝑥𝑖(𝑙1),̇𝑥𝑖(𝑙1)=𝑢𝑖,(3.15) where 𝑥𝑖(𝑑)𝑚,  𝑑=0,,𝑙1 are the states of follower-agents, 𝑢𝑖𝑚 is the control input, and 𝑥𝑖(𝑑) denotes the 𝑘-th derivation of 𝑥𝑖, with 𝑥𝑖(0)=𝑥𝑖, 𝑖=1,,𝑛. We consider the following dynamical protocol:𝑢𝑖=𝑗𝒩𝑖𝑎𝑖𝑗𝑙1𝑑=𝑜𝑥𝑗(𝑑)𝑥𝑖(𝑑)+𝑘𝑞=1𝑏𝑞𝑖𝑙1𝑑=0𝑥(𝑑)𝑞0𝑥𝑖(𝑑),(3.16) where 𝑥(𝑑)𝑞0,  𝑞=1,,𝑘, are the states of the leaders, 𝑥(𝑑)𝑞0 denotes the 𝑑-th derivation of 𝑥𝑞0.

Furthermore, the above equations can be rewritten as vector form:̇𝑥(0)=𝑥(1),̇𝑥(𝑙2)=𝑥(𝑙1),(3.17)̇𝑥(𝑙1)=𝑙1𝑑=𝑜𝐻𝐼𝑚𝑥(𝑑)+𝑙1𝑑=𝑜𝐵𝐼𝑘𝐼𝑛𝐼𝑚𝑥0(𝑑),(3.18) where 𝐻=𝐿+𝐵(1𝑘𝐼𝑛),  𝑥=[𝑥𝑇1𝑥𝑇𝑛]𝑇 and 𝑥0=[𝑥𝑇10𝑥𝑇𝑘0]𝑇.

Denote 𝑥=[𝑥𝑇1𝑥𝑇𝑙]𝑇 and 𝑥0=[𝑥𝑇01𝑥𝑇0𝑙]𝑇 as the stacked vector of the follower-agents' states and the leaders' states, respectively,𝑥𝑠=𝑥1(𝑠1)𝑇𝑥𝑛(𝑠1)𝑇𝑇,𝑥0𝑠=𝑥(𝑠1)10𝑇𝑥(𝑠1)𝑘0𝑇𝑇,𝑠=1,,𝑙,(3.19) and 𝑥0=𝑥01. Then the above equations can be rewritten as𝑥=Γ𝑥+Υ𝑥0,(3.20) where 0Γ=𝑚𝑛×𝑚𝑛𝐼𝑚𝑛0𝑚𝑛×𝑚𝑛0𝑚𝑛×𝑚𝑛0𝑚𝑛×𝑚𝑛0𝑚𝑛×𝑚𝑛𝐼𝑚𝑛0𝑚𝑛×𝑚𝑛0𝑚𝑛×𝑚𝑛0𝑚𝑛×𝑚𝑛0𝑚𝑛×𝑚𝑛0𝑚𝑛×𝑚𝑛𝐻𝐼𝑚𝐻𝐼𝑚𝐻𝐼𝑚𝐻𝐼𝑚,(3.21)0Υ=𝑛𝑚×𝑘𝑚0𝑛𝑚×𝑘𝑚0𝑛𝑚×𝑘𝑚0𝑛𝑚×𝑘𝑚0𝑛𝑚×𝑘𝑚0𝑛𝑚×𝑘𝑚0𝑛𝑚×𝑘𝑚0𝑛𝑚×𝑘𝑚0𝑛𝑚×𝑘𝑚0𝑛𝑚×𝑘𝑚0𝑛𝑚×𝑘𝑚0𝑛𝑚×𝑘𝑚𝐵𝐼ΛΛΛΛ,Λ=𝑘1𝑛𝐼𝑚.(3.22) Let ̃𝑥=𝑥𝐼𝑙{[𝐻1𝐵(𝐼𝑘1𝑛)]𝐼𝑚}𝑥0. Then (3.20) can be written aṡ̃𝑥=Γ̃𝑥.(3.23)

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 𝐺 is connected and 𝜆𝑙+𝜇𝑖𝜆𝑙1++𝜇𝑖 is Hurwitz stable, where 𝜇𝑖,   𝑖=1,,𝑚𝑛 are eigenvalues of 𝐻𝐼𝑚.

Proof. Sufficiency. Let 𝜆 be an eigenvalue of Γ, and ̃𝑥=[̃𝑥𝑇1̃𝑥𝑇𝑙]𝑇 be the corresponding eigenvector of Γ. Then we get ̃𝑥2=𝜆̃𝑥1,̃𝑥3=𝜆̃𝑥2,̃𝑥𝑙=𝜆̃𝑥𝑖1𝐻𝐼𝑚̃𝑥1𝐻𝐼𝑚̃𝑥2𝐻𝐼𝑚̃𝑥𝑙=𝜆̃𝑥𝑙.(3.24) Furthermore, we have 𝐻𝐼𝑚̃𝑥1𝜆𝐻𝐼𝑚̃𝑥1𝜆𝑙1𝐻𝐼𝑚̃𝑥𝑙=𝜆𝑙̃𝑥𝑙.(3.25) Let 𝜇𝑖,𝑖=1,,𝑛 be the eigenvalues of 𝐻𝐼𝑚. Then we get characteristic equation of Γ𝜆𝑙+𝜆𝑙1𝜇𝑖++𝜇𝑖=0.(3.26) By (3.23), we get ̃𝑥=𝑒Γ𝑡̃𝑥(0).(3.27) If 𝜆𝑙+𝜆𝑙1𝜇𝑖++𝜇𝑖 is Hurwitz stable, then all the eigenvalues of Γ have negative real parts. Therefore, ̃𝑥0, when 𝑡. So 𝑥=𝐼𝑙𝐻1𝐵𝐼𝑘1𝑛𝐼𝑚𝑥0.(3.28) Thus, 𝐻𝑥=1𝐵𝐼𝑘1𝑛𝐼𝑚𝑥0,,𝑥(𝑙)=𝐻1𝐵𝐼𝑘1𝑛𝐼𝑚𝑥0𝑙.(3.29)[𝐻1𝐵(𝐼𝑘1𝑛)]𝐼𝑚 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 𝐺1 in Figure 1. The corresponding weights of edges of 𝐺1 are shown in Figure 1. Moreover, the initial positions and velocities of follower-agents and leaders are given as follows: 𝑥122(0)=,𝑥233(0)=,𝑥311(0)=,𝑥411(0)=,𝑥522,𝑣(0)=122(0)=,𝑣233(0)=,𝑣311(0)=,𝑣411(0)=,𝑣533,𝑥(0)=1023(0)=,𝑥2032(0)=,𝑥3043,𝑣(0)=10=0.10.4,𝑣20=0.10.4,𝑣30=,.0.10.4𝐵=100000000000000000000000001000000000000000000000000000000000000000000100000(4.1)

791052.fig.001
Figure 1: Network topology 𝐺1 of a multiagent system.

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 𝑡 gradually increasing.

791052.fig.002
Figure 2: Trajectories of the agents in the multiagent system with topology 𝐺1 under the condition that the leaders have the same velocity.

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 𝐺1 in Figure 1. The initial positions and velocities of follower-agents, the initial positions of leaders and the corresponding weights of edges of 𝐺1 are the same as the above example, and only the velocities of leaders are changed as𝑣10=12,𝑣20=23,𝑣30=12,(4.2) 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 𝑡 gradually increasing. The final states of the follower-agents are consistent with Theorem 3.1.

791052.fig.003
Figure 3: Trajectories of the agents in the multiagent system with topology 𝐺1 under the condition that the leaders have different velocities.

Example 4.2. We consider a system of five follower-agents guarded by four leaders with the topology 𝐺2 in Figure 4. The corresponding weights of edges of 𝐺2 are shown in Figure 4. Moreover, the initial positions and velocities of leaders are given as follows: 𝑥1023(0)=,𝑥2032(0)=,𝑥3043(0)=,𝑥4063,𝑣(0)=10(0)=0.10.4,𝑣20(0)=0.10.4,𝑣30(0)=0.10.4,𝑣4031,(0)=(4.3)𝐵=1000000000000000000000000000000100000000000000000000000000000000000000000000001000000000010000000000.(4.4) Likewise, the initial positions and velocities of follower-agents are the same as those in Example 4.1. The topology 𝐺2 of follower-agents is not connected, according to the definition in Preliminaries, but 𝐺2 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 𝑡 gradually increasing.

791052.fig.004
Figure 4: Network topology 𝐺2 of a multiagent system.
791052.fig.005
Figure 5: Trajectories of the agents in the multiagent system with topology 𝐺2.

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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