Second-Order Multiagent Systems with Event-Driven Consensus Control

and Applied Analysis 3 defined as f(e(t), x(t), V(t)) = 0, where x(t), V(t) ∈ Rn are, respectively, the concatenation vector of the state variables in system (3). All agents update their consensus controls u i (t) at a series of event-times τ(s) (s = 0, 1, . . .) which are implicitly defined by f(e(τ(s)), x(τ(s)), V(τ(s))) = 0. Between control updates at two consecutive event-times, the eventdriven consensus control is held constant until the next event is triggered, that is, u i (t) = u i (τ (s)) , ∀t ∈ [τ (s) , τ (s + 1)) . (5) We say that the event-driven consensus problem of leaderless multi-agent system (3) is solved if an event-driven consensus control can be found to ensure that lim t→∞ 󵄨󵄨󵄨󵄨 x i (t) − x j (t) 󵄨󵄨󵄨󵄨 = 0, lim t→∞ 󵄨󵄨󵄨󵄨 V i (t) − V j (t) 󵄨󵄨󵄨󵄨 = 0, (6) for i, j ∈ V. Similarly, the event-driven consensus problem of leader-follower multi-agent system (3)-(4) is solved if lim t→∞ 󵄨󵄨󵄨󵄨xi (t) − x0 (t) 󵄨󵄨󵄨󵄨 = 0, lim t→∞ 󵄨󵄨󵄨󵄨Vi (t) − V0 (t) 󵄨󵄨󵄨󵄨 = 0, (7)


Introduction
Recently, synthesis and analysis of multi-agent systems have drawn great attention in many disciplines, such as mathematics, physics, computer science, systems biology, engineering, and social science.Roughly speaking, multi-agent systems are a class of networked dynamic systems consisting of a group of autonomous agents, which interact with each other locally and achieve an emergence behavior over a communication network.The controlled multi-agent systems have a broad range of applications including flocking and swarming in animal groups, vehicle formation, satellite reconfiguration, and unmanned aerial vehicles for rescue and surveillance.
Consensus problems have a long history originated from management science and statistics in 1960s [1].In the context of multi-agent systems, consensus generally means to reach an agreement regarding a certain quantity of interest that depends on the state of all agents [2].In the literature of consensus control of multi-agent systems, many works have been focused on an important issue, that is, to investigate the coordination behavior of agents governed by different order dynamics.Especially, first-order multi-agent systems are extensively considered as a representative multi-agent consensus model in, for example, [3][4][5][6][7][8] and references therein.More recently, the consensus problems of multi-agent systems with second-order dynamics (e.g., [9][10][11][12][13][14]) and high order dynamics (e.g., [15][16][17][18][19]) have been paid much attention, which is mainly because in many real applications mass-point models are invalid for agents and more complex dynamics should be considered.Generally, the dynamics of a secondorder multi-agent system is described by a second-order differential equation or difference equation, which contains both the position and the velocity information.
One potential application of multi-agent control is to equip each autonomous agent with a small embedded microprocessor to collect information from neighboring agents for actuating the controller updates.However, micro-processors are generally resource-and energy-limited [20], which requires a time-or an event-triggered scheduling strategy to update the control.A time-triggered update scheduling involves sampling at predefined time instances while an event-triggered one executes the control task whenever a certain error becomes large compared with the state norm.Time-triggered consensus problems were studied in [14,21,22] via data-sampled method.However, event-triggered strategies seem more favorable in applications.A distributed event-triggered control was considered for a first-order multi-agent system in [23][24][25].Up to date, there are few contributions devoted to designing an event-triggered consensus control for multi-agent systems with second-order dynamics.
In this paper, we consider an event-driven consensus problem of a second-order leaderless and leader-follower multi-agent system with a fixed directed communication network.Firstly, the event-driven consensus problem is formulated.Secondly, an event-driven consensus control is designed for each agent to achieve consensus.Then the closed-loop multi-agent system is proven to be input-to-state stable with respect to the measurement error and, simultaneously, a positive lower bound is found for the event-time between two consecutive actuation updates.

Preliminaries and Problem Formulation
2.1.Some Preliminaries.Let G = (V, E, ) be a weighted directed graph with a set of vertices V = {1, . . ., }, a set of arcs E ⊆ V×V, and a weighted adjacency matrix  = [  ] ∈ R × .In the mapping of graph G to the interconnection topology of a multi-agent system, vertex  ∈ V represents agent , and arc (, ), which starts from vertex  and ends on vertex , is in E if and only if agent  can receive information from agent .In this case, agent  is called a neighbor of agent , and, accordingly, N  = {|(, ) ∈ E} denotes the neighboring set of agent .The element   in the adjacency matrix  is associated with the arc (, ), that is,   > 0 if and only if (, ) ∈ E.Moreover, we assume that   = 0 for all  ∈ V.When a single leader-agent is involved in the multi-agent systems, a vertex 0 is added to represent the leader-agent and the interconnection topology is denoted by Definition 1 (see [26]).A path from vertex  to vertex  is a sequence of arcs ( 0 ,  1 ), ( 1 ,  2 ), . . ., ( −1 ,   ) in the directed graph G with distinct vertexes   ,  = 0, 1, . . .,  and  0 = ,   = .A directed graph G is strongly connected if there exists a path from vertex  to vertex , for every ,  ∈ V.
Definition 2 (see [26]).Vertex  is said to be reachable from vertex  if there exists a path from vertex  to vertex  in the directed graph G. Vertex  is globally reachable if there exists a path from every other vertex to vertex  in G.
According to Definitions 1 and 2, a directed graph G is strongly connected if and only if each vertex in G is globally reachable, which shows that the global reachability of a directed graph is much weaker than the strong connectedness.
A diagonal matrix  = diag( 1 , . . .,   ) ∈ R × is a degree matrix of G, whose diagonal elements   = ∑ ∈N    for  = 1, . . ., .Then the Laplacian matrix of a weighted directed graph is defined as  =  − . ( The next lemma shows an important property of Laplacian matrix  associated with directed graph G. Lemma 3 (see [4]).Laplacian matrix  has least one zero eigenvalue with 1  as its eigenvector, and all the non-zero eigenvalues of  have positive real parts. has a simple zero eigenvalue if and only if G has a globally reachable vertex.
In the leader-follower consensus literature, it is always assumed that the leader-agent is self-active, that is, the leader does not need information feedback from other agents and thus, the adjacency coefficients  0 = 0 for every followeragent  = 1, . . ., .For followers, we define a diagonal matrix  = diag{ 10 , . . .,  0 } to represent the leader-follower adjacency relationship.Let  =  + .
Lemma 4 (see [11]).If vertex 0 is globally reachable in ⃗ G, then all eigenvalues of  have positive real-parts.
A Schur-complement lemma will be used in the stability analysis of the close-loop multi-agent systems and is given to end this subsection.

Problem Formulation.
In a leaderless consensus problem, a group of  identical agents are moving with a continuoustime dynamics described by a second-order differential equation as follows: where   () ∈ R  , V  () ∈ R  , and   () ∈ R  are, respectively, the position, velocity, and control input of agent .
In a leader-follower consensus problem, the dynamics of follower-agents are given as (3) while the kinematics of the self-active leader is described by the following second-order differential equation: where  0 () ∈ R  , V 0 () ∈ R  , and  0 () ∈ R  are, respectively, the position, velocity, and acceleration.Here for notation simplicity, let  = 1.
When agents are equipped with resource-limited microprocessors, it is preferable to design an event-driven consensus controls for all agents such that the consensus controls need no update in continuous-time.For agent  ∈ V, we define a state measurement error   () and let () = col( 1 (), . . .,   ()) ∈ R  .Then an event-trigger condition is defined as ((), (), V()) = 0, where (), V() ∈ R  are, respectively, the concatenation vector of the state variables in system (3).All agents update their consensus controls   () at a series of event-times () ( = 0, 1, . ..) which are implicitly defined by ((()), (()), V(())) = 0. Between control updates at two consecutive event-times, the eventdriven consensus control is held constant until the next event is triggered, that is, We say that the event-driven consensus problem of leaderless multi-agent system (3) is solved if an event-driven consensus control can be found to ensure that lim for ,  ∈ V. Similarly, the event-driven consensus problem of leader-follower multi-agent system (3)-( 4) is solved if for  ∈ V.
From Lemma 6, the submatrix  of Λ is a full-rank matrix if and only if G has a global reachable vertex.Moreover, the eigenvalues of  have positive real-parts, or equivalently, − is Hurwitz stable.Therefore, there exists a positive definite matrix  ∈ R (−1)×(−1) such that Take a coordinate transformation and then system (10) becomes Essentially, system (17) can be regarded as a series interconnection of two subsystems: where ) . ( where Now a main result is obtained for system (10).
Theorem 7. Assume that the interconnection topology G associated with multi-agent system (3) has a globally reachable vertex.If the control gain  satisfies with  given in (15), then the event-driven consensus problem of the leaderless multi-agent system (3) is solved with the control (8), that is, Proof.For system (19), or equivalently (21), take a Lyapunov function where where  satisfies (15).Thus,  is positive definite with  given in (23).
On the other hand, for system (18), let  1 (0), V 1 (0) be the initial values of  1 (), V 1 () and take a variable change Then the solution can be described by the following integral equation: ) . (32) Since the system ( 19) is exponentially stable, based on the event-triggered condition (30), solution (32) has an exponential decay term with respect to time .Consequently, the solution is convergent to 0 as  → ∞, and thus, Abstract and Applied Analysis 5 as  → ∞.Furthermore, from the variable change ( 16), one has Therefore, when the event-driven consensus control ( 8) is applied to each agent, one has   () −   () → 0, V  () − V  () → 0 as  → ∞.The proof is complete.
Remark 8.The gain  in ( 23) can be taken without exact knowledge of the interconnection topology associated with the multi-agent system in real application of the control (8).
In fact, the spectral norm of  in ( 23) can be obtained by estimating the bound of the solution of Lyapunov equation (15) (see [28]), which is closely related with the Laplacian spectrum.Fortunately, there have been many results on the bounds of the eigenvalues of a Laplacian matrix [29].

Leader-Follower Consensus Control.
In the leaderfollowing problem, we assume that the state information, that is,  0 (), V 0 (), and  0 () of the leader can be measured in continuous-time by the followers.Thus, we propose the following event-driven consensus control of the follower : for  ∈ [(), ( + 1)).
Remark 10.For the leader-following problem (3)-( 4) under investigation, the leader is assumed to be self-active, which means that the leader is moving according to its own (predesigned) policy and needs no feedback information from any other agent.In some sense, the leader plays the role of an external commander of all the followers.Thus, no eventdriven strategy is applied to update the state of the leader.Though the followers are using the event-driven relative information from the neighboring followers, but they can obtain the real-time relative position and velocity measurements from the leader only if the followers are connected to the leader.Therefore, in the control (41), it would be preferable to assume that  0 () −   () and V 0 () − V  () are measured in continuous time.
Similar to Theorem 9, a conclusion about the lower bound of the event-times can also be found true, which is omitted here.

Simulations
Example 12. Consider four agents whose dynamics is described by (3).The interconnection topology is shown in Figure 1.Obviously, G has a globally vertex 4; however, it is not strongly connected.Assume that the weighted adjacency matrix  reduces to a 0 − 1 matrix.Then the Laplacian matrix  of G is which is an asymmetric matrix.A non-singular matrix  in Lemma 6 can be easily found as Then the submatrices  = ( The initial conditions of system (9) are (0) = col(2, −4, −9, 9), V(0) = col(2, 6, 3, −2) and (0) = 0.In the event triggered condition (30), we take  = 0.8.It can be seen that, from Figures 2 and 3, the four agents reach consensus on the position and velocity states with the proposed eventdriven consensus control (8).Additionally, since the vertex 4 has no link starting from it, the agent 4 is moved according to its own dynamics and the initial conditions, as shown in Figures 2, 3, and 5.The evolution of the measurement error vector () is depicted in Figure 4, which shows that the error () is bounded by the specified threshold ( min ()‖‖/2 2 √  2 + 1 max ()‖‖).In Figure 5, the consensus controllers are illustrated for the four agents, whose event-driven update frequencies are decreasing as time evolves.
Example 13.Consider four followers and one leader whose dynamics are, respectively, described by (3) and (4).The leader-follower interconnection topology ⃗ G is shown in Figure 6, which has a globally reachable vertex 0.
The acceleration of the active leader is assumed to be  0 () =  − sin().The initial values of system (3) is same as those in Example 12 and the initial values of the leader is given as  0 (0) = 0, V 0 (0) = 1 and  0 (0) = 0. Figures 7 and 8 show that the followers and the self-active leader reach consensus on the position and velocity under the event-driven control (41).The evolution of ‖()‖ is also presented in Figure 9.

Conclusions
An event-driven consensus problem of second-order multiagent systems with/without a self-active leader was considered in this paper.The consensus controllers have been proposed for all autonomous mobile agents based on an eventdriven control update strategy.The input-to-state stability of the closed-loop multi-agent system has been analyzed by employing an ISS Lyapunov function.Some numerical examples have been presented to validate the proposed eventdriven controls.However, it is noted that the event-driven condition depends on the states of the whole multi-agent group and all agents have identical event-times.The result is somewhat preliminary due to the centralized information gathering, so further work will be devoted to designing a decentralized event-driven consensus control for a secondorder multi-agent system in the future.