Consensus of Multiagent Networks with Intermittent Interaction and Directed Topology

Intermittent interaction control is introduced to solve the consensus problem for second-order multiagent networks due to the limited sensing abilities and environmental changes periodically. And, we get some sufficient conditions for the agents to reach consensus with linear protocol from the theoretical findings by using the Lyapunov control approach. Finally, the validity of the theoretical results is validated through the numerical example.


Introduction
The problem of coordinating the motion of multiagent networks has attracted increasing attention.Research on multiagent coordinated control problems not only helps in better understanding the general mechanisms and interconnection rules of natural collective phenomena but also carries out benefits in many practical applications of networked cyberphysical systems, such as tracking [1], flocking [2,3], and formation [4].Consensus, along with stability [5] and bifurcation [6], is a fundamental phenomenon in nature [7].Roughly speaking, consensus means that all agents in network will converge to some common state by negotiating with their neighbors.A consensus algorithm is an interaction rule on how agents update their states.
To realize consensus, many effective approaches were proposed [8][9][10].Since the network can be regarded as a graph, the issues can be depicted by the graph theory.The recent approaches concentrate on matrix analysis [11], convex analysis [12,13], and graph theory [14].The concept of spanning tree especially is widely used to describe the communicability between agents in networks that can guarantee the consensus [15].For more consensus problems, the reader may refer to [16][17][18][19][20][21] and the references therein.
As we know, sometimes only the intermittent states of its neighbors can be obtained by the agents to the transmission capacity, communication cost, sensing abilities, and the environmental changes.To decrease the control cost, only the intermittent states of its neighbors are obtained [22].This is mainly because such networks are constrained by the following operational characteristics: (i) they may not have a centralized entity for facilitating computation, communication, and timesynchronization, (ii) the network topology may not be completely known to the nodes of the network, and (iii) in the case of sensor networks, the computational power and energy resources may be very limited.Inspired by the above consideration, the goal in this setting is to design algorithms by exploiting partial state sampling at each node; it is possible to reduce the amount of data which needs to be transmitted in networks, thereby saving bandwidth and energy, extending the network lifetime, and reducing latency.Also, the linear local interaction protocol can guarantee the linear nature of distributed multiagent networks in real world and linear algorithm is simple and easy to implement so as to be widely used in practical engineering especially in the limited transmission environment.Using the Lyapunov control approach, some sententious conditions are obtained in this paper for reaching consensus in multiagent networks.
The rest of this paper is organized as follows.In Section 2, some preliminaries on the graph theory and the model formulation are given.The main results are established in Section 3. In Section 4, a numerical example is simulated to verify the theoretical analysis.Concise conclusions are finally drawn in Section 5.

Preliminaries and Model
2.1.Graph Theory.In this subsection, some basic concepts and result of algebraic grapy theory are introduced.Suppose that information exchange among agents in multiagent networks can be modeled by an interaction digraph.
Let  = (, , ) denote a directed graph with the set of nodes  = {1, 2, . . ., }, where  ⊆  ×  represents the edge set and  = (  ) × is the adjacency matrix with nonnegative elements   .A directed edge   in the network  is denoted by the ordered pair of nodes (, ), where  is the receiver and  is the sender, which means that node  can receive information from node .We always assume that there is no self-loop in network .An adjacency matrix  of a directed graph can be defined such that   is a nonnegative element if   ∈ , while   = 0 if   ∉ .The set of neighbors of node  is denoted by   = { ∈  : (, ) ∈ }.A sequence of edges of the form (,  1 ), ( 1 ,  2 ), . . ., (  , ) ∈  composes a directed path beginning with  and ending with  in the directed graph  with distinct nodes   ,  = 1, 2, . . ., , which means the node  is reachable from node .A directed graph is strongly connected if for any distinct nodes  and , there exists a directed path from node  to node .A directed graph has a directed spanning tree if there exists at least one node called root which has a directed path to all the other nodes [16].Let (generally nonsymmetrical) Laplacian matrix  = (  ) × associated with directed network  be defined by which ensure the diffusion property ∑  =1   = 0. Suppose  is irreducible.Then, 1  = 0  and there is a positive vector  = ( 1 ,  2 , . . .,   )  satisfying    = 0  and   1  = 1.In addition, there exists a positive definite diagonal matrix Ξ = diag( 1 ,  2 , . . .,   ) such that  = (Ξ +   Ξ)/2 is symmetric and ∑  =1   = ∑  =1   = 0 for all  = 1, 2, . . .,  [18].For simplicity, some mathematical notations are used throughout this paper.  (  ) denotes the identity (zero) matrix with  dimensions.Let 1  (0  ) be the vector with all  elements being 1(0).  is the -dimensional real vector space.The notation ⊗ denotes the Kronecker product.

Model Description.
The discretization process of a continuous-time system cannot entirely preserve the dynamics of the continuous-time part even small sampling period is adopted.So, we consider the following second-order multiagent networks of  agents in [19] with intermittent measurements.The th agent in the directed network  is governed by double-integrator dynamics where   () ∈   and V  () ∈   are the position and velocity states of the th agent, respectively. denotes the coupling strengths.() denotes the intermittent control as follows: where  > 0 is the control period and  > 0 is called the control width.Equivalently, model ( 2) can be rewritten as follows: In this paper, our goal is to design suitable ,  such that the network reaches consensus.In the following we present the following lemma and definitions.
Because the control gain () works intermittently by the control period and the control width, the consensus is discussed in the two different intervals, respectively.
Figure 1: The directed interaction topology of multiagent networks.

Conclusions
In this paper, we have considered the linear consensus of multiagent networks with periodic intermittent interaction and directed topology.We choose to show the consensus with linear local interaction protocols, partly for simplifying the problem.On the other hand, it is simple and easy to implement so as to be widely used in practical engineering.The tools from algebraic graph theory, matrix theory, and Lyapunov control approach have been adopted.It is shown that the consensus is determined commonly by the general algebraic connectivity, control period, and control width.And the states of agents converge exponentially.Project of CQCSTC (no.cstc2014jcyjA40041), the National Natural Science Foundation of China (no.60973114), and the Foundation of Chongqing University of Education (no.KY201318B).

2 Mathematical
Problems in Engineering

Figure 2 :
Figure 2: Position and velocity states of four agents in the network.