Received 27 November 2015; accepted 24 January 2016; published 27 January 2016
Distributed coordination of network of dynamic agents has attracted a great attention in recent years. Modeling and exploring these coordinated dynamic agents have become an important issue in physics, biophysics, systems biology, applied mathematics, mechanics, computer science and control theory  - . How and when coordinated dynamic agents achieve aggregation is one of the interesting topics in the research area. Such problem may also be described as a consensus control problem.
To describe the collective behavior of agents in a large scale network, the agent in the network usually is modeled by a very simple mathematical model, which is an approximation of real objects. Saber and Murray   proposed a systematical framework of consensus problems in networks of dynamic agents. In their work the dynamics of the agent is modelled by a simple scalar continuous-time integrator, the convergence analysis is provided in different types of the network topologies. Following the work of   , Guangming Xie  study the case where the agent is a point-mass distributed in a line, and its dynamics is described by the Newton’s law. In their work the dynamic agents connected by a network, which is characterized by a graph and each agent is Lyapunov stable. They show that by means of a simple linear control protocol based on the structure of the graph, the dynamical agents will eventually achieve aggregation, i.e. all agents will gradually move into a fixed position, meanwhile their velocities converge to zero.
In our work a similar problem is studied under the condition that the agents move in a plane. The agents may represent the vehicles or mobile robots spread over a wild area and they communicate by means of some remote sensors with certain error. When the agents are moving in a plane, the collective behavior conditions will depend on the communicated error and the algebraic characterization of the communicated network topology, as well as the dynamical behavior of agents.
This paper is organized as follows. In Section 2, we recall some properties on graph theory and give the problem formulation. In Section 3 the main results of this paper are given and some simulation results are presented in Section 4. Final section is a conclusion.
Consider a network of dynamical agents defined by a graph. The node set consists of dynamical agents. The dynamics of for is described as follows.
Let be the coordinate of dynamical agent in, then the dynamical equation of agent is
where indicates the location of agent in the plane, represents the velocity of the i-th agent and is its mass and is a dynamical feedback matrix of the agent. F is an ob-
servation matrix of the agent by some remote sensor.
In what follows we simply assume that for all and. Let which means that the location information of the i-th agent is only measured by some remote sensor and is transmitted to its
neighbors through the network. The matrix C is assumed to be an orthogonal matrix in the form.
The parameter will indicates that the network transmitted error or the coordinates used for sensor could be different from that of the agents.
For the dynamic agent (1) in network we have following assumption.
Assumption 2.1 The dynamics (1) is Lyapunov stable when it disconnected with its neighbors, meaning that the dynamical agent as an autonomous will gradually stop by moving a finite distance for any non-zero initial velocity.
The collective behavior of dynamical agents in network can be described by
;. We denote the initial locations and the initial velocities of the system as, respectively.
In this work, we discuss the collective behavior of the dynamical agents under a decentralized control law in the form that
We claim that a group of dynamical agents associated with asymptotically achieve the collective behavior under control protocol (2). That is to say, for any initial conditions of the agents,;, there will exist a fixed position, which depends on the initial condition, such that for
In our work, let (2) be
where is the set of neighbors of agent.
Remark 1: If we choose and, then the two-dimension agent systems (1) with the control protocol (4) can be decoupled into two identical linear systems of the form
for. i.e., and it was discussed in  .
3. Collective Behaviors of Dynamical Agents
Consider a group of dynamical agents in network associated with a graph. The node set consists of dynamic. The dynamical for each is described by linear dynamical equation (1) satisfying Assumption 2.1. Under control protocol (4) the dynamical equation of agent is written by
Denote, then (5) is written in
Let, then the dynamic network is of the following form.
and L is the aforementioned Laplacian associated with the graph.
The collective behavior problem of dynamical agents can be described in -consensus asymptotical consensus stability ( ). Let be a map, for,. The group of dynamical agents is called -consensus asymptotically stable under control protocol (4) if let for a given, then for each agent in network its state variables meets the properties of (3).
As dynamics (7) is a standard linear time-invariant dynamical system, its trajectory can be described by
The consensus asymptotical stability implies that the matrix converges to a constant matrix, thus we will explore some properties of the matrix.
Lemma 3.1 The matrix has two eigenvectors associated with zero eigenvalue. Let be the right and left eigenvectors (denoted by matrices) of matrix associated with zero eigenvalue, respectively. Then
Proof: It is well known that the graph is connected if and only if its Laplacian satisfies that. Moreover, is an eigenvector of L associated with eigenvalue, i.e.,. Then, there is only one zero eigenvalue of L, all the other ones are positive and real. By the definition of (8) one has
Thus, represented two right-eigenvectors of associated with zero-eigenvalue. Similarly, it is easy to check represents two left-eigenvectors of and ,
The following Lemma is key to our work.
Lemma 3.2 If the control gain k in dynamical agent (1) satisfies Assumption 2.1, and in the C of (4) satisfies
where, , , and denotes the biggest eigenvalue of matrix L, then it is hold that
Proof: Denote the eigenvalues of L by, and let be the Jordan form associated with L, there exists an orthogonal matrix W such that.
One can verify the following formulae.
The dynamical behavior of the network (7) is characterized by the eigenvalues of for.
First we discuss the block with. By Assumption 2.1, one has and, its four characteristic eigenvalues must satisfy, ,.
For, one has. As,. Therefore, has only two zero eigenvalues.
Consider the characteristic polynomial of
Construct the Routh array of
with By the Routh-Hurwith
criterion, for stability it is necessary that. Therefore, the dynamical network is stable if the following inequalities hold
By (14) one has
The inequalities (15) can be rewritten as the following form by using the conditions of Lemma 3.2 and the Equations (16)-(17).
We can further show that the second inequality in above implies the first one. Obviously, it is true when. If, one gets
where are defined in (12).
Thus, one can consider the following inequalities
The last inequality obviously holds. Therefore, the solution of (18) leads.
If, one can obtain and. So we can get that with a similar
computing process. It shows that is the solution set of the inequalities (15) for any a.
Therefore, are Hurwitz.
By one denotes right-eigenvectors of associated with eigenvalues, respectively. Thus,
where denote the Jordan form of two order associated with the eigenvalues, and. denote the Jordan form of four order associated with the eigenvalues, , and for all.
Let, where are row left-eigenvectors of,
As, one has
Let and, one has
Due to the fact that, and satisfy the property. ,
Theorem 3.1 Under conditions of Lemma 3.2 the control protocol (4) globally and asymptotically achieves the collective behavior of the dynamic agents.
Proof: As and, it follows that
and it is obvious that
This implies the protocol (5) globally asymptotically achieve aggregation.
Corollary 3.1 If the control gain k satisfies and, then the control protocol (4) globally and asymptotically achieves the collective behavior of the dynamic agents.
Under Assumption 2.1 one has. Thus, by carefully examining (12) one finds that and it further implies that and in (11). Thus we have the following.
Corollary 3.2 The dynamical agents achieve collective behavior if in control protocol (4). Again, the -map is defined by (19) and (20).
We study some examples to show that our results are effective. The network of dynamic agents is described in Figure 1.
We can obtain the Laplacian matrix L of the graph of Figure 1 and its eigenvalues are,, ,.
We consider that the dynamic agent (1) in the network has and observation matrix. Thus, it is Lyapunov stable and satisfies Assumption 2.1. One can get, ,
Figure 1. A undirected graph with M = 6 nodes.
, , and the belongs to the range of parameters i.e..
When a control protocol (4) is applied into the agents in network, the collective behavior of dynamic agents takes place according to our result.
Figure 2 gives simulation results of the collective behavior of the agents with initial conditions, , , , , , , , , , and the initial velocities, , , , , , , , , , ,.
It is found that when the agents approach to, the speeds of agents tend to zero.
We discuss the consensus control of dynamical agents in network which associated with a graph. When the
Figure 2. State and velocity trajectories of the agents in.
agents are moving in a plane, the aggregation of the dynamical agents are depended on not only the communicated error, but also the algebraic characterization of the communicated network graph and the dynamical properties of agents.
This work was supported by the Natural Science Foundation of Jiangsu Higher Education Institutions of China (Grant no. 13KJB110015).
 Saber, R.O. and Murray, R.M. (2004) Consensus Problems in Networks of Agents with Switching Topology and Time-Delays. IEEE Transactions on Automatic Control, 49, 1520-1533.
 Liu, Y. and Passino, K.M. (2003) Stability Analysis of M-Dimensional Asynchronous Swarms with a Fixed Communication Topology. IEEE Transactions on Automatic Control, 48, 76-95.
 Savkin, A.V. (2004) Coordinate Collective Motion of Groups of Autonomous Mobile Robots: Analysis of Vicsek’s Model. IEEE Transactions on Automatic Control, 49, 981-983.