Consensus Control for a Kind of Dynamical Agents in Network

This paper discusses consensus control for a kind of dynamical agents in network. It is assumed that the agents distributed on a plane and their location coordinates are measured by remote sensor and transmitted to its neighbors. By designing the linear distributed control protocol, it is shown that the group of agents will achieves consensus. The simulations are given to show the effectiveness of our theoretical result.


Introduction
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 [1]- [11].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 [3] [4] 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 x u =  , the convergence analysis is provided in different types of the network topologies.Following the work of [3] [4], Guangming Xie [10] study the case where the agent is a point-mass distributed in a line, and its dynamics is described by the Newton's law ma F = .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.

Preliminaries
Consider a network of dynamical agents defined by a graph ( ) , , , represents the velocity of the i-th agent and i m is its mass and is a dynamical feedback matrix of the agent.F is an observation matrix of the agent by some remote sensor.
In what follows we simply assume that 1 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 1 The parameter δ 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 ; 0 t ≥ .We denote the initial locations and the initial velocities of the system as ( ) ( ) ( ) ( ) In this work, we discuss the collective behavior of the dynamical agents under a decentralized control law in the form that , , , where indexes { } 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 * 2 x R ∈ , which depends on the initial condition, such that for i M ∈ ( ) ( ) In our work, let (2) be ( ) where i N is the set of neighbors of agent i p . with the control protocol (4) can be decoupled into two identical linear systems of the form ) i dim x = , and it was discussed in [12].

Collective Behaviors of Dynamical Agents
Consider a group of dynamical agents in network associated with a graph ( ) The dynamical i p for each i M ∈ is described by linear dynamical equation (1) satisfying Assumption 2.1.Under control protocol (4) the dynamical equation of agent i p is written by ( ) is written in ( ) where , , , M τ τ τ τ ξ ξ ξ ξ =  , then the dynamic network is of the following form. where and L is the aforementioned Laplacian associated with the graph G .
The collective behavior problem of dynamical agents can be described in χ -consensus asymptotical con- sensus stability ( [3]).Let , 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 ( ) Ω converges to a constant matrix, thus we will explore some properties of the matrix Ω .
Lemma 3.1 The matrix Ω has two eigenvectors associated with zero eigenvalue.Let , r l v v be the right and left eigenvectors (denoted by matrices) of matrix Ω associated with zero eigenvalue, respectively.Then ( ) is connected if and only if its Laplacian satisfies that ( ) Then, there is only one zero eigenvalue of L, all the other ones are positive and real.By the definition of (8) one has ( ) Similarly, it is easy to check [ ] 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 δ in the C of (4) sa- where The dynamical behavior of the network ( 7) is characterized by the eigenvalues of Therefore, Ω has only two zero eigenvalues.Consider the characteristic polynomial of ; Construct the Routh array of criterion, for stability it is necessary that 1 Therefore, the dynamical network is stable if the following inequalities hold 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 0 a = .If 0 a > , one gets ( ) are defined in (12).Thus, one can consider the following inequalities J denote the Jordan form of two order associated with the eigenvalues 1 γ , and 2 γ .i J denote the Jordan form of four order associated with the eigenvalues 4 : , , , , , where ; 4 and it is obvious that k k k k < .Thus, by carefully examining (12) one finds that 0 c > and it further implies that 1 0 δ < and 2 0 δ > in (11).Thus we have the following.

Simulations
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 G of 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.

Remark 1 :
If then the two-dimension agent systems(1) of matrix L, then it is hold that Λ be the Jordan form associated with L, there exists an orthogonal matrix W such that the following formulae.

1 2
The last inequality obviously holds.Therefore, the solution of (18) leads set of the inequalities (15) for any a.

Theorem 3 . 1
Under conditions of Lemma 3.2 the control protocol (4) globally and asymptotically achieves 20)This implies the protocol(5) globally asymptotically achieve aggregation. Corollary 3.1 If the control gain k satisfies 12 21 control protocol (4) globally and asymptotically achieves the collective behavior of the dynamic agents.Under Assumption 2.1 one has 12 21 11 22

Figure 1 and its eigenvalues are 1 .Figure 1 .
Figure 1.A undirected graph G with M = 6 nodes.