Distributed Event-Triggered Control of Multiagent Systems with Time-Varying Topology

This paper studies the consensus of first-order discrete-time multiagent systems, where the interaction topology is time-varying. The event-triggered control is used to update the control input of each agent, and the event-triggering condition is designed based on the combination of the relative states of each agent to its neighbors. By applying the common Lyapunov function method, a sufficient condition for consensus, which is expressed as a group of linear matrix inequalities, is obtained and the feasibility of these linear matrix inequalities is further analyzed. Simulation examples are provided to explain the effectiveness of the theoretical results.


Introduction
The distributed control of multiagent systems has received much attention because of its wide applications in scheduling of automated highway systems, cooperation control of multiple vehicles/robots, design of sensor networks, and so forth, and consensus is one of the hot topics in distributed control of multiagent systems [1][2][3].
Consensus of multiagent systems has been studied by researchers from different viewpoints, such as the dynamics of agents [4][5][6][7], the interaction topology [8][9][10], the convergence rate [11], and the information transmission [12][13][14][15].In [12], consensus of first-order multiagent systems is considered in the cases with sampling delay and without sampling delay, respectively, and some sufficient and necessary conditions are provided in the case of fixed topology.In [13], consensus is analyzed for second-order multiagent systems with time-varying sampling intervals by applying matrix theory and Lyapunov stability theory, respectively.In [14], consensus of second-order multiagent systems is also discussed, where the sampling periods of agents are different and the topology at each time may not have spanning tree.In [15], consensus of second-order multiagent systems with uniform and nonuniform sampling periods is considered, where only position information is transmitted among agents.It is noted that the work in [12][13][14][15] all assumes that the information received by agents at each time will be transmitted to controllers and thus consensus in these cases can be viewed as time-based consensus.
Compared with the time-based consensus, event-based consensus is more realistic, where the information received by agents is transmitted to controllers only when some events occur.Event-triggered control has some advantages such as reducing the number of transmissions and thus it is more suitable for cooperative control over networks with limited bandwidth [16].There has been some work on event-based consensus of first-order continuous-time multiagent systems [17][18][19][20][21][22] and second-order continuous-time multiagent systems [20,[23][24][25].In [17], a centralized event-triggering condition is provided and a sufficient condition of consensus is established in the cases of fixed topology and switching topology, where the topology at each time is strongly connected and balanced.In [18][19][20], a distributed event-triggering condition is given and some sufficient conditions for consensus are obtained in the case of fixed topology.Moreover, to avoid continuously monitoring whether the event-triggering condition is satisfied, the self-trigged control, where the next update time of controller is predetermined, is also discussed.In [21,22], each agent can only obtain information at some discrete times, and if the information at these discrete times satisfies the distributed event-triggering condition, then they will be used to update control input.Furthermore, sufficient conditions for consensus in the cases of fixed and switching topology, where the topology at each time is connected, are provided.In [20,23,24], some centralized or distributed event-triggering conditions are designed for second-order multiagent systems with fixed topology, where the position and velocity information of each agent share a common event-triggering condition.In addition, the leader-follower consensus based on event-triggered control is considered for the case of fixed topology in [25].
However, there is little work [26][27][28] on the event-based consensus of discrete-time multiagent systems.In [26], a centralized event-triggering condition is provided for firstorder multiagent systems and some sufficient conditions are obtained.In [27,28], the event-based consensus of heterogeneous discrete-time multiagent systems is discussed.Moreover, the above three papers all consider the fixed topology case.Based on the above observation, we study the event-based consensus of first-order discrete-time multiagent systems in the case of time-varying topology.The main contribution of our work is to design a distributed eventtriggering condition and obtain a sufficient condition of consensus in the case of time-varying topology.
This paper is organized as follows.In Section 2, we present some concepts in graph theory and formulate the model to be studied.In Section 3, main results are stated.In Section 4, simulations are provided to illustrate the effectiveness of the theoretical results.Conclusion remarks are made in Section 5.

Preliminaries
2.1.Graph Theory.Graph plays a key role in modeling the interaction topology among agents.We first introduce some basic definitions in graph theory [29].
A directed graph G consists of a vertex set V(G) and an edge set E(G), where = }, and the associated index set is denoted by (G, ) = { : A directed graph G is strongly connected if there is a path from every vertex to every other vertex.A directed tree is a directed graph, where every vertex except one special vertex has exactly one parent vertex, and the special vertex, called root vertex, has no parent vertices and can be connected to any other vertices via paths.A subgraph , then G  is said to be an induced subgraph of G and G  is also said to be induced by V(G  ).A spanning tree of G is a directed tree which is a spanning subgraph of G. G is said to have a spanning tree if some edges form a spanning tree of G.
A matrix is called nonnegative if each of its elements is nonnegative.A weighted directed graph G() is a directed graph G plus a nonnegative matrix  = [  ] ∈ R × , where then G() is also called undirected, and G() is said to be connected if it, as a directed graph, is strongly connected.The Laplacian matrix  = [  ] ∈ R × of G() is defined as The Laplacian matrix of G() has the following properties.
Different from the previous work on consensus, not all the information obtained by agent  is used by controller   in our work.For agent , when it receives new information, it will validate whether some condition, which is called event-triggering condition, is satisfied.If the event-triggering condition is satisfied, then the received information will not be used by its controller, or else the received information will be transmitted to its controller.The following eventtriggering condition is considered: where   (   ) is dependent on the information received at time    ; namely, where  > 0.
Remark 3. If ( 3) is satisfied at all times, then the consensus under consideration becomes the usual consensus of discrete time multiagent systems.
Remark 4. In the following analysis, we use another topology graph Ĝ(), which is different from the actual topology graph G().Ĝ() is defined as follows: Ĝ() has the same vertex set as For facilitating the following analysis, we assume that { Ĝ(0), Ĝ(1), . ..} is a finite set.
Hence, we can obtain the following main result by analyzing the stability of system (9).Because the topology graph is time-varying, we apply the common Lyapunov function to analyze the stability of system (9).Theorem 6. Assume Ĝ() has a spanning tree at all times.Multiagent system (2) under event-triggering condition (3) and controller (4) solves a consensus problem asymptotically if there exists  > 0 such that the following linear matrix inequalities are satisfied: where  = max{ 1 , . . .,   }. where ) .
Remark 7. The feasibility of LMI ( 11) is explained as follows: where ( 2 ) denotes a matrix, each element of which is the same order infinitesimal of  2 .By Schur complement, Hence, if there exists  > 0 such that     +   > 0,  = 1, 2, . . ., , then (11) must be feasible in the case that  and  are small.

Simulations
Consider four agents and their interaction topology is timevarying.For convenience, assume that the topology is G 1 and G 2 at odd and even times, respectively.The Laplacian matrices of  1 and  2 are By the LMI Toolbox in Matlab, (11) is feasible for  = 0.1 and  = 0.1.Hence, by Theorem 6, consensus can be reached asymptotically for controller (4) with  = 0.1 and eventtriggering condition (3) with max{  } < 0.1.The state trajectories of four agents in the case of  = 0.1,  1 = 0.01,  2 = 0.02,  3 = 0.03, and  4 = 0.05 are shown in Figure 1, which validate the result of Theorem 6.

Conclusion
This paper has studied the consensus problem of firstorder discrete-time multiagent systems with time-varying topology.Based on the designed event-triggering condition, each agent updates its control input only at event times.By applying a state transformation, consensus is transformed into the asymptotical stability of a time-varying system.In virtue of the common Lyapunov function method, we obtain that if the event-triggering conditions and controllers satisfy some linear matrix inequalities, then consensus can be reached asymptotically, and we further analyze the feasibility of these linear matrix inequalities.Our future work will focus on time delays and another time-varying tropology, namely, the jointly connected case.