Adjacent-Compensation Consensus Algorithm in Asynchronously Coupled Form for Second-Order Multiagent Network under Communication Delay

and Applied Analysis 3 where τ > 0 is the communication delay, while the synchronously coupled form is given by


Introduction
Coordination control of multiple autonomous dynamic agents has attracted much attention in recent years for its broad application in automated highway systems, air traffic control, congestion control in the Internet, and so on.Consensus problem, which requires the agents' outputs to reach a common value without central or global communication, is one of the most important and fundamental collective types of behavior in distributed coordination control of multiagent systems and has been extensively studied in many research societies in recent years, such as biology, robotics, and sensor networks.
In the past decade, consensus problem has been thoroughly studied for the multiagent systems with diverse agent's dynamics, such as single integrator, double integrator, and high-order integrators.Different consensus algorithms have been designed [1][2][3][4][5][6], and many conditions have been obtained for the agents converging to the consensus asymptotically.
Due to the information exchange between neighboring agents, communication delay should be investigated in multiagent network.Subjected to communication delays, consensus algorithms are usually divided into synchronously coupled and asynchronously coupled forms.In synchronously coupled consensus algorithm, self-delays introduced for each agent in the coordination part are equal to the corresponding communication delays.Besides, each agent uses its delayed state with the delay value different from the corresponding communication delay or uses its current state to compare it with its delayed neighboring agents' states in the asynchronously coupled consensus algorithm.
Moreover, asynchronously coupled consensus algorithm has been analyzed thoroughly for the first-order and secondorder multiagent systems with stationary consensus algorithm by using many different analysis methods including frequency-domain analysis [16,17], Lyapunov functions [18][19][20], and the concept of delayed and hierarchical graphs [21][22][23][24].It has been proved that the agents with the stationary consensus algorithms could achieve an asymptotic consensus with arbitrary communication delay by choosing proper control parameters.However, for second-order multiagent 2 Abstract and Applied Analysis systems with the dynamical consensus algorithm composed of the position and the velocity consensus coordination control parts, Yang et al. [25] obtained sufficient consensus conditions for the second-order dynamic agents with timevarying communication delays based on small- stability theorem, and C.-L. Liu and F. Liu [26] got sufficient and necessary consensus condition for two coupled second-order dynamic agents with identical time-invariant communication delay by using frequency-domain analysis.In addition, Münz et al. [27] investigated the multiagent systems with agents' dynamics described by strictly stable linear systems under diverse communication delays.By analyzing the convex sets of the frequency-domain feedback matrix, set-valued consensus conditions have been obtained for the system under synchronously coupled and asynchronously coupled consensus algorithms, respectively [27].
In this paper, a new asynchronously coupled consensus algorithm associated with adjacent compensations is proposed for the second-order multiagent systems with communication delay to track a dynamic leader.The consensus algorithm is composed of the position and the velocity consensus coordination control parts, and the compensations are composed of each agent's neighboring agents' delayed state compensation in the normal asynchronously coupled consensus algorithm.To prove the effectiveness of the proposed algorithm, we firstly investigate the secondorder multiagent systems with a spanning tree topology that has the leader as a root, and it is proved that the agents can achieve a dynamical consensus without any relationship to the communication delay.Under the general connected topology that is composed of the agents and the leader, delay-dependent sufficient condition is also obtained for the multiagent systems with communication delay.

Problem Formulation
2.1.Agent's Dynamics and Interconnection Topology.Secondorder agent's dynamic is given by where   ∈ , V  ∈ , and   ∈  are the position, velocity, and acceleration, respectively, of the agent .As we all know, the interconnection topology of multiagent systems (1) is usually described as a digraph.Agents can be considered as the nodes of the digraph, while the information flow between neighboring agents can be regarded as a directed edge between the neighboring nodes in the digraph.
A weighted digraph  = (, , ) of order  consists of a set of vertices  = {1, . . ., }, a set of edges  ⊆  × , and a weighted adjacency matrix  = [  ] ∈  × with   ≥ 0. The node indexes belong to a finite index set I = {1, . . ., }.A directed edge from  to  in  is denoted by   = (, ) ∈ , which means that the node  can obtain information from the node .Assume that   > 0 ⇔   ∈  and   = 0 for all  ∈ {1, . . ., }.The set of neighbors of node  is denoted by   = { ∈  : (, ) ∈ }.The Laplacian matrix of the digraph  is defined as  =  −  = [  ] ∈  × , where  = diag{∑  =1   ,  = 1, . . ., } is the degree matrix.In the digraph , a directed path from node  1 to node   is a sequence of ordered edges of the form ( 1 ,  2 ), . . ., ( −1 ,   ), where   ∈ .A digraph is said to have a spanning tree, if there exists a node such that there is a directed path from this node to every other node.

Adjacent-Compensation Consensus Algorithm.
In this paper, we consider the dynamical consensus problem of the second-order agents (1).Without loss of generality, we analyze the second-order agents (1) following a dynamic leader, and the leader's dynamic is determined by where  0 ∈  is the position and V 0 ∈  is a constant denoting the desired velocity for all agents (1).Under the leaderfollowing coordination control framework, the agents' states are required to converge to the leader's states asymptotically; that is, lim We adopt the following consensus algorithm, which consists of the position and the velocity coordination control parts [4]: where  > 0,  > 0,   denotes the neighbors of the agent ,   > 0 is the adjacency element of  in the digraph  = (, , ),   = ∑ ∈    , and   is defined as With communication delay, the asynchronously coupled form of the algorithm (4) is given by where  > 0 is the communication delay, while the synchronously coupled form is given by Remark 1.The second-order agents (1) under synchronously coupled consensus algorithm (7) also achieve a dynamical consensus if the communication delay and the control parameters satisfy some certain conditions [8].With asynchronously coupled consensus algorithm (6), the agents may not converge to an asymptotic consensus or the final consensus state of the dynamic agents (1) is stationary, and consensus conditions depend on the communication delay [25,26].
So far, consensus algorithms with compensations related to delayed states of neighboring agents, desired target, and dynamical leader have been added to consensus algorithm (6) in order to remain the original control objective.Similar to the compensation-based consensus algorithms in [28], we modify the asynchronously coupled consensus algorithm (6) by introducing the delayed state compensations as follows: where  > 0 and the compensations −(  (−2−)−  (−− )) and −(V  ( − 2 − ) − V  ( −  − )) are added to improve the system's performance.Compared with the algorithm in [28], the algorithm (8) in this paper is more general with an adjustable value .
Remark 2. Obviously, the adjustable delay value  in algorithm (8) can present different communication-delay robustness, and the best value  can be obtained based on numerical simulations.
Besides, we take the following assumption to make the algorithm (8) reasonable.
Assumption 3.Each second-order agent has at least one neighboring agent.

Main Results
First of all, we present consensus criterion for the multiagent systems (9) with a spanning tree topology that has the leader as a root.
Theorem 4. Assume that the interconnection topology composed of the second-order agents (9) and the dynamic leader is a spanning tree.Then, all the agents in system (9) asymptotically converge to the leader's state with arbitrary communication delay.
Proof.When the interconnection topology of  agents and a leader is a spanning tree, the leader must be the root and we assume that the direct edge from agent  to  satisfies  < .Hence, the characteristic equation (11) becomes where  = diag{  ,  = 1, . . ., },  = diag{  ,  = 1, . . ., }, and Hence, (12) equals The roots of ( 14) have negative real parts; that is, the roots of the characteristic equation (12) lie on the open left half complex plane.Therefore, the agents in system (9) converge to the leader's states asymptotically.
Remark 5. To our delight, the system (9) under a special topology can tolerate arbitrary communication delay.
In the following, we investigate the consensus problem of second-order dynamic agents under a general interconnection topology in the leader-following framework.Theorem 6. Assume that the agents in system (9) without communication delay converge to the leader's states asymptotically and Assumption 3 holds.Let where   () = ((1 + )/ holds for  ∈ .
Proof.Without communication delay, (11) becomes and the above equation is equivalent to Under the assumption that the agents without communication delay converge to a dynamical consensus asymptotically, the roots of (18) all lie on the open left half complex plane, and rank( + ) = ; that is, the interconnection topology of  agents and the leader has a spanning tree.With communication delay, ( 11) can be rewritten as Letting   () =  2 + (1 + )(1 −   (e − + e −(+) − e −(2+) )), we get (0) = (1 −   ) ̸ = 0. Thus, (20) can be rewritten as The above equation is equivalent to where () is defined in (15).Obviously,  From condition (16), holds for all  ∈ .Therefore, 1+()(1−e − )(1−e −(+) ) is nonsingular for Re() ≥ 0; that is, the roots of the characteristic equation (11) all lie on the open left half complex plane.Hence, the agents in the system (9) converge to the leader's states asymptotically.Theorem 6 is proved.

Simulation
Example 7. Consider a multiagent network with five secondorder agents and a leader given by ( 9), and the interconnection topology described in Figure 1 is a spanning tree.The weights of the directed edges are  1 = 0.4,  21 = 0.1,  43 = 0.8,  3 = 1.5, and  53 = 0.2.Choose the control parameters as  = 1.2 and  = 0.8.Then, the agents in the system (9) can reach a dynamical consensus asymptotically under arbitrary communication delay (see Figure 2).
It should be noted that the dynamical consensus convergence for the agents with synchronously coupled consensus algorithm under the same control parameters and interconnection topology must depend on communication delay strictly.
To illustrate the function of adjustable value , we choose diverse  and get an algorithm with the best delay robustness.Figure 6 shows the largest communication delay our proposed algorithm can bear under different .When  = 3(s), the largest communication delay the agents in system (9) can bear is  max = 1.47(s), which is much larger than that the synchronously coupled algorithm can achieve.

Conclusion
In this paper, we consider the dynamical consensus problem of the second-order multiagent systems under communication delay and adopt the leader-following consensus algorithm which is composed of the position and the velocity consensus coordination control parts.By introducing neighboring agents' delayed state compensation into the normal asynchronously coupled consensus algorithm, the secondorder agents can achieve a dynamical consensus as the synchronously coupled algorithm.Firstly, we consider secondorder dynamic agents under a spanning tree topology that has the leader as the root and use frequency-domain analysis to prove the delay-independent consensus convergence.Then, we investigate the consensus seeking under the general interconnection topology in the leader-following coordination control framework.Delay-dependent consensus condition is obtained for the second-order agents converging to the leader's states asymptotically.It should be noted that choosing proper adjustable delay introduced in our proposed algorithm can obtain different delay robustness.In a word, our proposed algorithm can tolerate much higher communication delay than synchronously coupled consensus algorithm.