On the General Consensus Protocol in Multiagent Networks with Double-Integrator Dynamics and Coupling Time Delay

Abstract

This paper considers the problem of the convergence of the consensus algorithm for multiple agents in a directed network where each agent is governed by double-integrator dynamics and coupling time delay. The advantage of this protocol is that almost all the existing linear local interaction consensus protocols can be considered as special cases of the present paper. By combining algebraic graph theory and matrix theory and studying the distribution of the eigenvalues of the associated characteristic equation, some necessary and sufficient conditions are derived for reaching the second-order consensus. Finally, an illustrative example is also given to support the theoretical results.

1. Introduction

Recently, coordinated control of multiple agents has attracted a great deal of attention in many fields such as biology, physics, robotics, and control engineering [1, 2]. Research on multiagent coordinated control problems not only helps in better understanding the general mechanisms and interconnection rules of natural collective phenomena, but also benefits many practical applications of networked cyberphysical systems, such as the coordination and control of distributed sensor networks [3], formation control in multirobots [4], unmanned autonomous vehicles (UAVs) formations [5, 6], flocking [7], complex networks [8, 9], and so on [10, 11]. A critical problem in coordinated control of multiple agents is to find control laws such that all agents can reach an agreement regarding a certain quantity of interest that depends on the states of all agents. This problem is usually called the consensus problem [12]. In recent years, numerous studies have been conducted on this problem, some special second-order consensus protocols were presented, and some consensus conditions were obtained [13–18].

In the references on the consensus problem, multiagent systems with double-integrator dynamics have been paid great attentions because of their importance in practice. In [13], the authors studied some necessary and sufficient conditions for second-order consensus multiagent system, and revealed that both the real and imaginary parts of the eigenvalues of the Laplacian matrix of the corresponding directed network play key roles in reaching consensus. In [14], Zhu et al. discussed the consensus problem of coupled second-order linear harmonic oscillators with local interaction, which means that the states of all agents converge to the same periodic function. In [15], Zhu discussed a more general linear form of consensus protocols for multiagent systems with double-integrator dynamics. In [16], Li et al. investigated the final consensus convergence state of multiagent dynamical systems by using a kind of generalized linear local interaction protocols without time delay. However, most of the protocols discussed in the existing literature have neither addressed the consensus of positive exponential dynamics, namely, each agent’s state converges to a positive exponential function, nor considered the time delay in network. So, the observation provides us the motivation of this paper to investigate the consensus of second-order agents in directed networks double-integrators and coupling time delay.

In this paper, we consider the convergence of the second-order consensus of multiagent systems composed of coupled double-integrators dynamics and coupling time delay. For this protocol, all the agents in the fixed directed network topology are governed by double-integrator dynamics. The advantage of this protocol is that almost all the existing linear local interaction consensus protocols can be considered as special cases of the present paper [13–16]. By combining algebraic graph theory and matrix theory and studying the distribution of the eigenvalues of the associated characteristic equation, we give explicitly the effect of delays for second-order consensus. The contribution of this paper is to obtain a necessary and sufficient condition that the second-order consensus can be achieved in a delayed multiagent system with a directed spanning tree if and only if the time delay is less than a certain critical value.

The rest of this paper is outlined 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, several numerical examples are simulated to verify the theoretical analysis. Conclusions are finally drawn in Section 5.

2. Preliminaries

In this subsection, some basic concepts and results about algebraic grapy theory are introduced. For more details about algebraic graph theory, please refer to [19]. Suppose that information exchange among agents in multiagent systems can be modeled by an interaction digraph. Let 𝑔=(𝑉,𝜀,𝐴) denote a directed graph, where 𝑉={1,2,…,𝑁} denotes the node set, 𝜀⊆𝑉×𝑉 represents the edge set, and 𝐴=(𝑎_𝑖𝑗)_𝑁×𝑁 is the adjacency matrix. A directed edge 𝜀_𝑖𝑗 in the network 𝑔 is denoted by the ordered pair of nodes (𝑖,𝑗), where 𝑖 is the head and 𝑗 is the tail, which means that node 𝑖 can receive information from node 𝑗 [20]. The elements of the adjacency matrix 𝐴 are defined such that 𝑎_𝑖𝑗=1⇔𝜀_𝑖𝑗∈𝜀, while 𝑎_𝑖𝑗=0⇔𝜀_𝑖𝑗∉𝜀. We always assume that there is no self-loop in network 𝑔; that is, 𝑎_𝑖𝑖=0 for all 𝑖∈𝑉. Weighted adjacency matrix 𝐴 of a weighted directed graph can be defined such that 𝑎_𝑖𝑗 is a positive weight if 𝜀_𝑖𝑗∈𝜀, while 𝑎_𝑖𝑗=0 if 𝜀_𝑖𝑗∉𝜀. The set of neighbors of node 𝑖 is denoted by 𝑁_𝑖={𝑗∈𝑉∶(𝑖,𝑗)∈𝜀}.

If there is a sequence of edges of the form (𝑖,𝑗₁),(𝑗₁,𝑗₂),…,(𝑗_𝑚,𝑗)∈𝜀 composing a directed path beginning with 𝑖 and ending with 𝑗 in the directed graph 𝑔 with distinct nodes 𝑗_𝑘,𝑘=1,2,…,𝑚, then the node 𝑗 is said to be reachable from node 𝑖. A directed graph is strongly connected if for any distinct nodes 𝑖 and 𝑗, there exists a directed path from node 𝑖 to node 𝑗. 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 [21]. A directed graph is balanced if ∑^𝑛_𝑗=1𝑎_𝑖𝑗=∑^𝑛_𝑗=1𝑎_𝑗𝑖 for all 𝑖. Let (generally nonsymmetrical) Laplacian matrix 𝐿=(𝑙_𝑖𝑗)_𝑁×𝑁 associated with directed network 𝑔 be defined as 𝑙_𝑖𝑖=∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗 and 𝑙_𝑖𝑗=−𝑎_𝑖𝑗, where 𝑖≠𝑗. Especially, for an undirected graph, 𝐿 is symmetric positive semidefinite. However, 𝐿 for a directed graph does not have this property.

2.1. Notations

Some mathematical notations are used for simplicity throughout this paper. 𝐼_𝑛(𝑂_𝑛) denotes the identity (zero) matrix with 𝑛 dimensions. Let 𝟏_𝑛(𝟎_𝑛) be a column vector with 𝑛 elements being 1 (0). Re(⋅) and Im(⋅) represent the real part and imaginary part of a complex, respectively. 𝐶^𝑛 is the 𝑛-dimensional complex vector space.

2.2. Consensus Protocols

Suppose that the 𝑖th agent in the network 𝑔 is modeled by double-integrator dynamics and coupling time delay 𝑟̇_𝑖(𝑡)=𝑣_𝑖(𝑡),𝑣̇_𝑖(𝑡)=𝑢_𝑖(𝑡),𝑖=1,2,…,𝑁, (1) where 𝑟_𝑖(𝑡)∈𝑅 is the position state, 𝑣_𝑖(𝑡)∈𝑅 is the velocity state, and 𝑢_𝑖(𝑡)∈𝑅 is the control input, which is designed based on local information exchange. In this paper, the consensus protocol in a multiagents system is as follows: 𝑢_𝑖(𝑡)=−𝛼𝑟_𝑖(𝑡)−𝛽𝑣_𝑖(𝑡)+𝛾_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑟_𝑗(𝑡−𝜏)−𝑟_𝑖(𝑡−𝜏)+𝜉_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑣_𝑗(𝑡−𝜏)−𝑣_𝑖(𝑡−𝜏),𝑖=1,2,…,𝑁, (2) where  𝛼>0(𝛽>0)  denotes the position (velocity) damping gain, 𝛾>0(𝜉>0) represents the coupling strength of positions (velocities) between neighboring agents, and 𝜏>0 is the communication time-delay.

Equivalently, system (2) can be rewritten as follows: 𝑢_𝑖(𝑡)=−𝛼𝑟_𝑖(𝑡)−𝛽𝑣_𝑖(𝑡)−𝛾_{𝑗∈𝑁𝑖}𝐿_𝑖𝑗𝑟_𝑗(𝑡−𝜏)−𝜉_{𝑗∈𝑁𝑖}𝐿_𝑖𝑗𝑣_𝑗(𝑡−𝜏),𝑖=1,2,…,𝑁. (3)

Let 𝐱(𝑡)=[𝑟₁(𝑡),𝑟₂(𝑡),…,𝑟_𝑁(𝑡)]^𝑇 and 𝐯(𝑡)=[𝑣₁(𝑡),𝑣₂(𝑡),…,𝑣_𝑁(𝑡)]^𝑇. Then, network (1) can be rewritten in a compact matrix form as 𝐱̇(𝑡)𝐯̇(𝑡)𝐱(𝑡)𝐯(𝑡)=𝑄, (4) where 𝑄=^{𝑂𝑁𝐼𝑁}_{−𝛼𝐼𝑁−𝛾𝐿𝑒−𝜆𝜏−𝛽𝐼𝑁−𝜉𝐿𝑒−𝜆𝜏}, and 𝐿∈𝑅^𝑁×𝑁 is the (nonsymmetric) Laplacian matrix associated with directed graph 𝑔.

Definition 1. Second-order consensus in the multiagent system (1) under control input (2) is said to be achieved if for any initial conditions 𝑟_𝑖(0),  𝑣_𝑖(0),  lim_𝑡→∞|𝑟_𝑖(𝑡)−𝑟_𝑗(𝑡)|=0  and lim_𝑡→∞|𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)|=0,  ∀𝑖,𝑗=1,2,…,𝑁.

3. Main Results

In delayed systems, the time delays can be regarded as the bifurcation parameters. In [17, 18], it was found that Hopf bifurcation occurs when time delays pass through some critical values where the conditions for local asymptotical stability of the equilibrium are not satisfied. Similarly, in this section, we aim to find the maximum time delay with which the consensus can be achieved in the multiagent system (1). Before reaching our main results, we first give the following lemmas.

Lemma 2 (see [5]). Let 𝐿 be the (nonsymmetric) Laplacian matrix associated with directed network 𝑔. Then, 𝐿 has a simple zero eigenvalue and its other eigenvalues have positive real parts if and only if 𝑔 has a directed spanning tree.
The characteristic equation for system (4) is det𝜆𝐼_𝑁−𝐼_𝑁𝛼𝐼_𝑁+𝛾𝐿𝑒^−𝜆𝜏𝜆𝐼_𝑁+𝛽𝐼_𝑁+𝜉𝐿𝑒^−𝜆𝜏=det𝜆𝐼_𝑁𝜆𝐼_𝑁+𝛽𝐼_𝑁+𝜉𝐿𝑒^−𝜆𝜏+𝛼𝐼_𝑁+𝛾𝐿𝑒^−𝜆𝜏=det𝜆²𝐼_𝑁+𝛽𝜆𝐼_𝑁+𝛼𝐼_𝑁+𝑒^−𝜆𝜏𝜆𝐼_𝑁=𝜉𝐿+𝛾𝐿_𝑁𝑖=1𝜆²+𝛽𝜆+𝛼−𝜆𝜉𝜇_𝑖+𝛾𝜇_𝑖𝑒^−𝜆𝜏=0. (5) Let 𝑔_𝑖(𝜆)=𝜆²+𝛽𝜆+𝛼−(𝜆𝜉𝜇_𝑖+𝛾𝜇_𝑖)𝑒^−𝜆𝜏 and 𝑔(𝜆)=∏^𝑁_𝑖=1𝑔_𝑖(𝜆). From (5), it is easy to see that 𝐿 has a zero eigenvalue of algebraic multiplicity 𝑚 if and only if 𝑔(𝜆)=0 has a zero eigenvalue of algebraic multiplicity 2𝑚.
In the following, one considers the case that 𝑔_𝑖(𝜆) has a pair of purely imaginary roots ±𝑖𝜔_𝑖. If  𝑖𝜔_𝑖(𝜔_𝑖>0) is a root of 𝑔_𝑖(𝜆)=0, substituting it into 𝑔_𝑖(𝜆), one can obtain −𝜔²_𝑖+𝛽𝑖𝜔_𝑖+𝛼−𝑖𝜔_𝑖𝜉𝜇_𝑖+𝛾𝜇_𝑖𝑒^{−𝑖𝜔𝜏}=0. (6) Separating the real and imaginary parts of (6) leads to 𝑅𝜇_𝑖𝛾−𝑙𝜇_𝑖𝜔_𝑖𝜉cos𝜔_𝑖𝜏+𝑅𝜇_𝑖𝜔_𝑖𝜉+𝑙𝜇_𝑖𝛾sin𝜔_𝑖𝜏=−𝜔²_𝑖𝑅𝜇+𝛼,_𝑖𝜔_𝑖𝜉+𝑙𝜇_𝑖𝛾cos𝜔_𝑖𝜏−𝑅𝜇_𝑖𝛾−𝑙𝜇_𝑖𝜔_𝑖𝜉sin𝜔_𝑖𝜏=𝛽𝜔_𝑖. (7) By squaring and adding the previous equations, it follows that 𝜔⁴_𝑖−𝜇_𝑖²𝜉²+2𝛼−𝛽²𝜔²_𝑖−𝜇_𝑖²𝛾²+𝛼²=0. (8) By solving (8), one can obtain 𝜔²_𝑖=𝜇_𝑖²𝜉²+2𝛼−𝛽²±𝜇_𝑖²𝜉²+2𝛼−𝛽²²+4𝜇_𝑖²𝑟²−𝛼²2. (9)

Lemma 3. (1)  If ‖𝜇_𝑖‖²𝜉²+2𝛼<𝛽² and ‖𝜇_𝑖‖²𝛾²<𝛼², then (8) has no positive root.
(2)  If ‖𝜇_𝑖‖²𝜉²+2𝛼>𝛽² and ‖𝜇_𝑖‖²𝛾²>𝛼², then (8) has only one positive root.
(3)  If ‖𝜇_𝑖‖²𝜉²+2𝛼>𝛽² and ‖𝜇_𝑖‖²𝛾²<𝛼², then (8) has two positive roots.
Suppose that (8) has positive roots; without loss of generality, one assumes that it has two positive roots defined by 𝜔_𝑖𝑘, (𝑘=1,2). By (7), one has cos𝜔_𝑖𝑘𝜏=−𝜔²_𝑖𝑘+𝛼𝑅𝜇_𝑖𝛾−𝑙𝜇_𝑖𝜔_𝑖𝑘𝜉+𝛽𝜔_𝑖𝑘𝑅𝜇_𝑖𝜔_𝑖𝑘𝜉+𝑙𝜇_𝑖𝛾𝜔²_𝑖𝑘−𝛼²+𝛽²𝜔²_𝑖𝑘 (10) Thus, denoting 𝜏^𝑗_𝑖𝑘=1𝜔_𝑖𝑘arccos−𝜔²_𝑖𝑘+𝛼𝑅𝜇_𝑖𝛾−𝑙𝜇_𝑖𝜔_𝑖𝑘𝜉+𝛽𝜔_𝑖𝑘𝑅𝜇_𝑖𝜔_𝑖𝑘𝜉+𝑙𝜇_𝑖𝛾𝜔²_𝑖𝑘−𝛼²+𝛽²𝜔²_𝑖𝑘+2𝑗𝜋𝜔_𝑖𝑘, (11) where 𝑗=0,1,…, then ±𝑖𝜔_𝑖 is a pair of purely imaginary roots of 𝑔_𝑖(𝜆)=0 with 𝜏^𝑗_𝑖𝑘.
Define 𝜏⁰_𝑖=min_𝑘=1,2{𝜏_𝑖𝑘} and 𝜏⁰=min_{𝑖=2,…,𝑁}{𝜏⁰_𝑖}.

Lemma 4 (see [15]). Consider the exponential polynomial 𝑃𝜆,𝑒^−𝜆𝜏,…,𝑒^{−𝜆𝜏𝑚}=𝜆^𝑛+𝑝₁⁽⁰⁾𝜆^𝑛−1+⋯+𝑝⁽⁰⁾_𝑛−1𝜆+𝑝_𝑛⁽⁰⁾+𝑝₁⁽¹⁾𝜆^𝑛−1+⋯+𝑝⁽¹⁾_𝑛−1𝜆+𝑝_𝑛⁽¹⁾𝑒^−𝜆𝜏+⋯+𝑝₁^(𝑚)𝜆^𝑛−1+⋯+𝑝^(𝑚)_𝑛−1𝜆+𝑝_𝑛^(𝑚)𝑒^{−𝜆𝜏𝑚}, (12) where 𝜏_𝑖≥0 (𝑖=1,2,…,𝑚) and 𝑝_𝑗^(𝑖)(𝑗=1,2,…,𝑚) are constants. As (𝜏₁,𝜏₂,…,𝜏_𝑚) vary, the sum of the order of the zeros of 𝑃(𝜆,𝑒^−𝜆𝜏,…,𝑒^{−𝜆𝜏𝑚}) on the open right half plane can change only if a zero appears on or crosses the imaginary axis.
Let 𝛿=max_{2≤𝑖≤𝑁}−2𝛼𝛽²+4Re𝜇_𝑖𝛼𝛽𝜉−2Re²𝜇_𝑖𝛼𝜉²−2Re𝜇_𝑖𝛽³𝜉+2Re𝜇_𝑖𝛽²𝛾+Re²𝜇_𝑖𝛽²𝜉²−2Re³𝜇_𝑖𝛽𝜉³−22Re²𝜇_𝑖+Im²𝜇_𝑖𝜇𝛽𝛾𝜉+2Re_𝑖Re²𝜇_𝑖+Im²𝜇_𝑖𝜉²𝛾+2Im²𝜇_𝑖𝛾². (13)

Lemma 5. Suppose that the network contains a directed spanning tree, and all the roots of (5) have negative real parts if and only if the following conditions are satisfied: 𝛿<0 and 𝜏∈[0,𝜏₀).

Proof. For 𝜏=0, (5) can be rewritten in the following form: ^𝑁_𝑖=1𝜆²+𝛽−𝜉𝜇_𝑖𝜆+𝛼−𝛾𝜇_𝑖=0. (14) Therefore, the roots of (14) satisfy 𝜆²+𝛽−𝜉𝜇_𝑖𝜆+𝛼−𝛾𝜇_𝑖=0. (15) We know that each eigenvalue of −𝐿, 𝜇_𝑖, corresponds to two eigenvalues of 𝑄, denoted by 𝜆_𝑖±. Their relationships can be described by 𝜆_𝑖±=−𝛽+𝜉𝜇_𝑖±𝛽−𝜉𝜇_𝑖²−4𝛼−𝛾𝜇_𝑖2,𝑖=1,2,…,𝑁. (16) From Lemma 2, one knows that −𝐿 has a simple zero eigenvalue, and all the other eigenvalues have negative parts if and only if the directed network 𝑔 has a directed spanning tree. That is to say, Re(𝜆_𝑖±)<0, 𝑖=2,…,𝑁. For convenience, let (𝛽−𝜉𝜇_𝑖)²−4(𝛼−𝛾𝜇_𝑖)=𝑐+𝐢𝑑, where 𝑐,𝑑 are real numbers, and 𝐢=√−1. It follows from (16) that Re(𝜆_𝑖±)<0 implies (−𝛽+𝜉Re(𝜇_𝑖)±𝑐)/2<0, 𝑖=2,…,𝑁. This is equivalent to the following two inequalities 𝜉Re(𝜇_𝑖)−𝛽<𝑐<−𝜉Re(𝜇_𝑖)+𝛽 and Re(𝜇_𝑖)<0  which hold simultaneously. Therefore, 𝑐²<𝜉²Re²𝜇_𝑖+𝛽²−2𝜉𝛽Re𝜇_𝑖,𝑖=2,3,…,𝑁. (17)
Also note that (𝛽−𝜉𝜇_𝑖)²−4(𝛼−𝛾𝜇_𝑖)=𝑐²−𝑑²+𝐢2𝑐𝑑. By some calculations and separating the real and imaginary parts, one obtains 𝛽²+𝜉²Re²𝜇_𝑖−𝜉²Im²𝜇_𝑖−2𝛽𝜉Re𝜇_𝑖−4𝛼+4𝛾Re𝜇_𝑖=𝑐²−𝑑²,𝜉²Re𝜇_𝑖Im𝜇_𝑖−𝛽𝜉Im𝜇_𝑖+2𝛾Im𝜇_𝑖=𝑐𝑑. (18) It follows from (18) that 𝑐⁴−𝐵𝑐²−𝐴=0, (19) where 𝐵=𝛽²+𝜉²Re²𝜇_𝑖−𝜉²Im²𝜇_𝑖−2𝛽𝜉Re𝜇_𝑖−4𝛼+4𝛾Re𝜇_𝑖,𝐴=𝜉²Re𝜇_𝑖Im𝜇_𝑖−𝛽𝜉Im𝜇_𝑖+2𝛾Im𝜇_𝑖². (20) Solving (19) and combining (17) yield 𝑐²=𝐵±𝐵²+4𝐴2<𝜉²Re²𝜇_𝑖+𝛽²−2𝜉𝛽Re𝜇_𝑖,𝑖=2,3,…,𝑁. (21) After some simplifications, we can derive that the inequality 𝛿<0 in Lemma 5 holds. Using Lemma 4 and similar to the approach in [22, 23], we complete the proof.

Summarizing the previous discussions, we have the following theorem.

Theorem 6. Suppose that the network contains a directed spanning tree; we have the following results.(1)If max{‖𝜇_𝑖‖²𝜉²+2𝛼}<𝛽²,max{‖𝜇_𝑖‖²𝑟²}<𝛼², and 𝛿<0, the second-order consensus in system (1) is achieved.(2)If min{‖𝜇_𝑖‖²𝜉²+2𝛼}>𝛽², 𝛿<0 and 𝜏∈[0,𝜏₀), the second-order consensus in system (1) is achieved.

Corollary 7. For 𝛼=𝛽=0, 𝛾,𝜉>0, and 𝜏>0, second-order consensus in multiagent system (1) can be achieved if and only if the network contains a directed spanning tree and 𝜉²𝛾>max_{2≤𝑖≤𝑁}Im²𝜇_𝑖−Re𝜇_𝑖Re²𝜇_𝑖+Im²𝜇_𝑖,𝜏<𝜏₀=min_{2≤𝑖≤𝑁}𝜃_𝑖1𝜔_𝑖1, (22) where 𝜇_𝑖 are nonzero eigenvalues of matrix −𝐿, 𝑖=2,3,…,𝑁. In [13], Yu et al. studied the second-order consensus under the linear protocol: 𝑢_𝑖(𝑡)=𝛾_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑟_𝑗(𝑡)−𝑟_𝑖(𝑡)+𝜉_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑣_𝑗(𝑡)−𝑣_𝑖(𝑡),𝑖=1,2,…,𝑁, (23) and obtained the similar conclusions as Corollary 7. Therefore, Theorem 6 can be viewed as the extension of the results in [13].

Corollary 8. For 𝜏=0, second-order consensus in multiagent system (1) can be achieved if and only if the network contains a directed spanning tree and 𝛿<0. In [15], Zhu investigated the second-order consensus under the linear protocol: 𝑢_𝑖(𝑡)=−𝛼𝑟_𝑖(𝑡)−𝛽𝑣_𝑖(𝑡)+𝛾_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑟_𝑗(𝑡)−𝑟_𝑖(𝑡)+𝜉_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑣_𝑗(𝑡)−𝑣_𝑖(𝑡),𝑖=1,2,…,𝑁, (24) and obtained the similar conclusions as Corollary 8. Therefore, Theorem 6 can be viewed as the extension of the results in [15].

4. Simulation Examples

In this section, some numerical results of simulating system (1) are given to verify the theorems obtained earlier. Consider the network (1) with the topology shown in Figure 1. In Figure 1, the adjacency matrix 𝐴 and the Laplacian matrix 𝐿 of directed network 𝑔 are given in (25). Moreover, it is easily to see that there exists a directed spanning tree Figure 1. With simple calculations, we obtain the eigenvalues of −𝐿, 𝜇₁=0,  𝜇₂=−1+𝐢,  𝜇₃=−1−𝐢,  𝜇₄=−2,  and  𝜇₅=−2. Consider ,.𝐴=0110000100000100000101000𝐿=2−1−10001−100001−100001−10−1001 (25) Let us set the parameters of the system (1) as follows: 𝛼=0.1,  𝛽=0.2,  𝛾=0.5, and  𝜉=1.  According to (11) and Lemma 5, we obtain 𝜏₀=0.3939 and𝛿=−0.3962<0.

Figure 1 

The directed interaction topology of five agents.

Consider 𝜏=0.39<𝜏₀; from Theorem 6, we know that the second-order consensus can be reached. The position and velocity states of all the agents are shown in Figure 2. It is easy to see that the consensus of all agents is achieved.

(a)

(b)

Figure 2 

Velocity and position states of five agents in a network under linear consensus protocols where 𝜏=0.39<𝜏0.

Consider 𝜏=0.394>𝜏₀; from Theorem 6, we know that the second-order consensus cannot be reached. The position and velocity states of all the agents are shown in Figure 3. It is easy to see that the consensus of all agents is not achieved.

(a)

(b)

Figure 3 

Velocity and position states of five agents in a network under periodic consensus protocols where 𝜏=0.394>𝜏0.

5. Conclusions

In this paper, the convergence of the second-order consensus of multiagent systems composed of coupled double-integrators dynamics and coupling time delay has been studied. The advantage of the considered protocol is that it can be treated as the extensions of most linear local interaction protocols in the existing literatures. By combining algebraic graph theory and matrix theory and studying the distribution of the eigenvalues of the associated characteristic equation, some necessary and sufficient conditions are derived for reaching the second-order consensus. Several simulation results further validate the effectiveness of theoretical analysis.

Acknowledgments

This work was supported in part by Project no. CDJXS12180002 supported by the Fundamental Research Funds for the Central Universities, in part by the National Natural Science Foundation of China under Grant 60973114 and Grant 61170249, in part by the Research Fund of Preferential Development Domain for the Doctoral Program of Ministry of Education of China under Grant 201101911130005, in part by the State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, under Grant 2007DA10512711206, and in part by the Program for Changjiang Scholars.