Sage Journals: Discover world-class research

Abstract

In this study, a new formation behavior problem for second-order multi-agent systems with time delay is investigated in the presence of antagonistic interactions. We first proposed a formation behavior protocol with time delay for each agent in the antagonistic network. Then, by a frequency-domain analysis, a sufficient and necessary condition is derived to guarantee the consensus stability of the multi-agent system. It is shown that the agents in the same group form their own desired formation while keeping a desired relative position with other groups. Finally, numerical simulations are provided to illustrate the obtained results.

Keywords

Formation behavior time delay antagonistic interactions multi-agent systems second-order

Introduction

In the past few years, formation behavior problems of multi-agent systems have attracted considerable attention from various fields, such as robotics, biology, physics, engineering, as well as computer science. Formation behaviors widely exist in the nature world, which makes individual agents to perform their own tasks by collecting neighbor information and maximize the coordination functions. For example, the formation of a flock of geese can be adjusted automatically so as to evade enemies or overcome obstacles, when they flying from one place to another place. In addition, fish herd or wild animals act like organisms when foraging for food and escaping predators, and they will reach a desired formation instead of dispersing randomly.

Inspired by these natural phenomena, numerous researchers have devoted to the study of formation behavior.^1–17 Animal formation behaviors to forage food and avoid predators, such as flocking or schooling, require the individuals to stay in the group and yet simultaneously keep a separation distance from other members of the group.^11,12 Chu et al.¹² developed an anisotropic swarm model and presented the results concerning the average behavior, aggregation and cohesiveness, and complete stability of the swarm. So far, there are many improvements have been made to solve formation behavior problem in a practical application. Balch and Arkin¹⁵ theoretically analyzed and evaluated the formation behavior problem that enable a robotic team to reach navigational goals, avoid hazards, and simultaneously remain in formation. Three popular methods are mentioned in the literature, namely, behavior-based method, virtual structure approach, and leader-follower strategy.

In real circumstance, communication delays exist ubiquitously. It is well known that communication delays between agents may frequently occur due to the congestion of communication channels, the asymmetry of interaction, and the finite transmission speed. So, a lot of works about the consensus problems of multi-agent systems have taken into account the effects of the communication time delays.^7,8,10,13,18 And the linear constrained consensus problems on second-order multi-agent systems in unbalanced networks with communication delays are studied.¹³ Moreover, Liu et al.⁷ investigate consensus of second-order dynamics with time-varying communication delays. Sufficient conditions are obtained in terms of linear matrix inequality by Lyapunov theory.

For the existing results, a great number of efforts on the formation behavior problems are focused on cooperative systems to exchange information in some studies.^9–13 While in several real-world scenarios, some agents may collaborate while others compete.^14–25 As far as we know, less attention has been paid to these antagonistic networks in multi-agent systems. Antagonistic interactions are represented as signed graphs, that is, graphs which can assume also as negative weights. A positive/negative weight can be associated to a allied/adversary relationship between the agents linked by the edge. Hu et al.²⁴ analyzed the formation behavior problems on antagonistic networks in directed and undirected systems; all agents be divided into two groups, and within each group, agents converge to their desired formations in an opposite direction.

Motivated by the results mentioned above, this article focuses on the antagonistic interaction behavior of networks with time delay for second-order multi-agent systems, and the agents collaborate in the same group while competing in different groups. Then, a new algorithm is proposed for each agent to achieve an antagonistic formation, where agents in the allied groups converge to their own desired formation with an opposed relationships to other group agents. By the frequency-domain analysis, a equivalent system is considered to analyze the stability of the close-loop multi-agent system, and the maximum delay is related to the maximum eigenvalue of the graph Laplacian.

In this article, some notation will be used. $ℝ^{m}$ denotes the set of all m-dimensional real column vectors and $C^{m}$ represents the complex column vectors with m-dimension. 1 represents a column vector and its all elements are a parameter one, $I_{n}$ represents a $n \times n$ identity matrix, 0 denotes a zero value or zero matrix with an appropriate dimension, and $\otimes$ represents the Kronecker product.

Graph theory

Let $G (V, E, A)$ be a signed graph of order n, where $V = {s_{1}, s_{2}, \dots, s_{n}}$ is the set of nodes, $E \subseteq V \times V$ is the set of edges, and $A = [a_{ij}] \in ℝ^{n \times n}$ is an adjacency matrix of the weight graph. The node indexes belong to a finite index set $I = {1, 2, \dots, n}$ . We assume that $a_{ii} = 0$ for all $i \in I$ . An edge of G is denoted by $e_{ij} = (s_{i}, s_{j})$ . The sets of all neighbor nodes $s_{i}$ are denoted by $N_{i} = {s_{j} | (s_{i}, s_{j}) \in E}$ . Moreover, the element $a_{ij}$ of $A$ is nonzero if and only if the edge $(s_{i}, s_{j}) \in E$ , whose head is i and tail is j in the directed graphs while the order is irrelevant for undirected graphs. For the signed graph G, there exists a sign mapping $γ$ : $E \to {+, -}$ such that the edge set $E = E^{+} \cup E^{-}$ . $E^{+} = {(i, j) | a_{ij} > 0}$ and $E^{-} = {(i, j) | a_{ij} < 0}$ are the sets of positive and negative edges, respectively.

The undirected graph is connected if there is a path from every node to every other node. A path $P$ of G is a sequence of ordered edges: $P = {(s_{i_{1}}, s_{i_{2}}), (s_{i_{2}}, s_{i_{3}}), \dots, (s_{i_{q - 1}}, s_{i_{q}})} \subset E$ , in which all nodes $s_{i_{1}}, \dots, s_{i_{q}}$ are distinct. For an undirected graph, consider a given signed adjacency matrix $A$ . The definition used in this article for a Laplacian L in the case of signed $A$ is $L = C - A$ , where in the connectivity matrix $C$ , the weights are in absolute value. The elements of L are therefore

l_{ij} = {\begin{matrix} \sum_{j = 1}^{n} a_{ij} & i = j \\ - a_{ij} & i \neq j \end{matrix}

Obviously, the Laplacian of any undirected graph is symmetric. We consider the antagonistic networks in the signed graphs, which leads to the fact that Laplacian matrix is special. With the transformation D, $L_{D}$ is the new Laplacian of the transformed system. Section “Antagonistic formation behavior of agents” will give the detailed analysis.

Lemma 1.²⁶

If the undirected graph G is connected, then the Laplacian L of G has the following properties:

0 is a simple eigenvalue of L associated with the eigenvector $1$ .

The other $n - 1$ eigenvalues are all positive and real.

Antagonistic formation behavior of agents

For the simplicity of presentation, we take four-group antagonistic networks of agents into consideration. At first, consider a multi-agent system that consists of n agents, each agent is regarded as a node in an undirected graph G. Modeling the dynamics of ith agent as follows

\begin{matrix} {\overset{\cdot}{x}}_{i} (t) = v_{i} (t) \\ {\overset{\cdot}{v}}_{i} (t) = u_{i} (t) \end{matrix}

(1)

where $x_{i}$ and $v_{i}$ are, respectively, the position and velocity of agent i, respectively, and $u_{i}$ denotes the control input.

The n agents belong to the four hostile groups $V_{1}$ , $V_{2}$ , $V_{3}$ , and $V_{4}$ , where $V = V_{1} \cup V_{2} \cup V_{3} \cup V_{4}$ . Agents collaborate in the same group while they are opposed in different groups. Therefore, the antagonistic interaction network of the multi-agent system can be described by a signed graph $G (V, E, A)$ . In our considered problem, agents in the allied groups achieve their own desired formations while keeping an antagonistic interaction with other groups. For example, in Figure 1, the solid in blue and dash in red edges denote the cooperation and confrontation relationships, respectively, between agents. Obviously, it is very easy to use the positive and negative weights to describe such relationships from signed graph theory. In this model, the agents are required to move in four contrary directions surrounded by a desired location to form the four-group antagonistic formation.

Figure 1.

The antagonistic interaction network G.

In the antagonistic formation behavior problems, the desired formation vectors $d_{i}$ and $d_{j}$ are assigned to each agent $i \in V$ ( $j \in V$ ) from starting. All agents have to move to four desired positions; simultaneously, the agents in same group have to achieve their desired formation by collecting neighbor information. We say that the antagonistic interactions of networks are achieved if and only if the states of agents satisfy the following condition

lim_{t \to \infty} {(x_{i} (t) - d_{i}) - (x_{j} (t) - d_{j})} = 0

(2)

for any $i, j \in V_{k}$ , $k = 1, 2, 3, 4$ . And there is only antagonistic relationship between different groups.

In this article, our aim is to propose a formation law using local and neighbor information to enable the multi-agent systems to form four antagonistic groups, where each group converges to their desired formation. To solve such problem, the control law is presented as follows

\begin{matrix} u_{i} (t) = - 2 b_{i} v_{i} (t) + \sum_{s_{j} \in N_{i}} K_{q} sgn (a_{ij}) a_{ij} \\ [(x_{j} (t - τ) - d_{j}) - K_{q} (x_{i} (t - τ) - d_{i})] \end{matrix}

(3)

where $sgn (\cdot)$ is a sign function, $τ$ denotes the time delay from any agent j to agent i, and $k_{i}$ denotes the feedback gain of ith agent.

$K_{q}$ has four different forms, and each form shows a collaborate/compete relationship. Between group 1 and group 2 of agent, we have $K_{q} = K_{2} = diag {- 1, 1}$ . In a similar way, between group 1 and group 3 of agent, $K_{q} = K_{3} = diag {- 1, - 1}$ . And between group 1 and group 4 of agent, $K_{q} = K_{4} = diag {1, - 1}$ . Obviously, if the agent be part of a certain group, there exists a cooperative relationship with each other, and we have $K_{q} = K_{1} = diag {1, 1}$ .

If $x'_{j} (t - τ) = x_{j} (t - τ) - d_{j}$ and $x'_{i} (t - τ) = x_{i} (t - τ) - d_{i}$ , the control protocol (3) can be written as follows

\begin{array}{l} u_{i} (t) = - 2 b_{i} v_{i} (t) + \sum_{s_{j} \in N_{i}} K_{q} s g n (a_{i j}) a_{i j} x_{j}^{'} (t - τ) - \sum_{s_{j} \in N_{i}} s g n (a_{i j}) a_{i j} x_{i}^{'} (t - τ) \end{array}

(4)

Then, we consider the transformation D and denote $D = diag {D_{1}, D_{2}, D_{3}, D_{4}, \dots, D_{i}, \dots, D_{n}}$ , where $D_{1} = diag {1, 1}$ , $D_{2} = diag {- 1, 1}$ , $D_{3} = diag {- 1, - 1}$ , and $D_{4} = diag {1, - 1}$ ; they represent four groups of antagonistic networks, respectively. If there exists a cooperative relationship between two agents, then $D_{i}$ is similar to that group. For example, if agent i and agent $1$ are in the same group, we have $D_{i} = D_{1} = diag {1, 1}$

\begin{matrix} x_{di} = D_{i} x_{i} v_{di} = D_{i} v_{i} u_{di} = D_{i} u_{i} \end{matrix}

Since $D_{i}^{- 1} = D_{i}$ and $D_{i} K_{q} D_{i}^{- 1} = I$ , thus after transformation for each agent, system (4) can be written as follows

\begin{matrix} u_{di} (t) = - 2 b_{i} v_{di} + \sum_{s_{j} \in N_{i}} sgn (a_{ij}) a_{ij} x'_{dj} (t - τ) \\ - \sum_{s_{j} \in N_{i}} sgn (a_{ij}) a_{ij} x'_{di} (t - τ) \end{matrix}

(5)

Let

\begin{matrix} ε_{D} (t) = [{x'_{d 1}}^{T} (t), {\bar{v}}_{d 1}^{T} (t), {x'_{d 2}}^{T} (t), \\ {\bar{v}}_{d 2}^{T} (t), \dots, {x'_{dn}}^{T} (t), {\bar{v}}_{dn}^{T} (t {)]}^{T} \\ {\bar{v}}_{di} (t) = \frac{v_{di} (t)}{b_{i}} + x'_{di} (t), i = 1, 2, \dots, n \end{matrix}

(6)

By simple calculation, we have

\begin{matrix} \overset{\cdot}{x}'_{di} (t) = v_{di} (t) = b_{i} ({\bar{v}}_{di} (t) - x'_{di} (t)) \\ {\overset{\cdot}{\bar{v}}}_{di} (t) = \frac{1}{b_{i}} {\overset{\cdot}{v}}_{di} (t) + \overset{\cdot}{x}'_{di} (t) \\ = \frac{1}{b_{i}} {- 2 b_{i} v_{di} (t) + \sum_{s_{j} \in N_{i}} sgn (a_{ij}) a_{ij} [x'_{dj} (t - τ) - x'_{di} (t - τ)]} + b_{i} ({\bar{v}}_{di} (t) - x'_{di} (t)) \\ = b_{i} x'_{di} (t) - b_{i} {\bar{v}}_{di} (t) + \frac{1}{b_{i}} \sum_{s_{j} \in N_{i}} sgn (a_{ij}) a_{ij} [x'_{dj} (t - τ) - x'_{di} (t - τ)] \end{matrix}

Then, the network dynamics can be denoted as

{\overset{\cdot}{ε}}_{D} (t) = (I_{n} \otimes A) ε_{D} (t) - (L_{D} \otimes B) ε_{D} (t - τ)

(7)

where

A = [\begin{matrix} - b_{i} & 0 & b_{i} & 0 \\ 0 & - b_{i} & 0 & b_{i} \\ b_{i} & 0 & - b_{i} & 0 \\ 0 & b_{i} & 0 & - b_{i} \end{matrix}] B = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{1}{b_{i}} & 0 & 0 & 0 \\ \frac{1}{b_{i}} & 0 & 0 & 0 \end{matrix}]

and where $L_{D}$ is the new Laplacian of the transformed system, that is to say, $L \to DLD$ is the similarity transformation, and it can be obtained that L and $L_{D}$ have the same eigenvalue. However, $L_{D}$ satisfies the condition of the Laplacian of a weighted undirected graph. From the above transformation, the antagonistic networks have been converted to consensus problems.

Remark 1

In this section, we consider the four-group antagonistic networks of agents. All the agents will be spontaneously divided into four groups, and the agents in the same group form a desired formation while keeping a desired relative position with other groups, that is to say, agents cooperate with each other in the same group while competing in different groups.

Stability analysis of interaction network

Lemma 2

Let

H (x) = \frac{arc \tan (x) (1 + x^{2})}{x}

Then, for any $x \in (0, 1)$ , $H (x)$ is a decreasing function of x.

Theorem 1

Suppose that the communication topology G is undirected and connected. Then, using protocol (4), four-group antagonistic behavior is achieved for the multi-agent systems (2) if and only if $τ < τ^{*}$ , where

τ^{*} = \frac{arc \cos \frac{\sqrt{4 b_{\max}^{4} + λ_{\max}^{2}} - 2 b_{\max}^{2}}{λ_{\max}}}{\sqrt{\sqrt{4 b_{\max}^{4} + λ_{\max}^{2}} - 2 b_{\max}^{2}}}

(8)

where $λ_{\max}$ is the largest eigenvalue of L, $b_{\max}$ is the largest feedback gains, and $ζ$ is the positive root of the following equation

b_{\max}^{2} λ_{\max} ζ^{2} + ζ - b_{\max}^{2} λ_{\max} = 0

(9)

Proof

Consider system (7). Since $1_{4 n}^{T} (I_{n} \otimes A) = 0$ and $1_{4 n}^{T} (L_{D} \otimes B) = 0$ , by Lemma 1. Thus

1_{4 n}^{T} {\overset{\cdot}{ε}}_{D} (t) = 1_{4 n}^{T} [(I_{n} \otimes A) ε_{D} (t) - (L_{D} \otimes B) ε_{D} (t - τ)] = 0_{4 n} .

It follows that $\sum_{i = 1}^{n} [\overset{\cdot}{x}'_{di} (t) + {\overset{\cdot}{\bar{v}}}_{di} (t)] = 0$ . Thus, $\sum_{i = 1}^{n} [_{di} x' (t) + {\bar{v}}_{di} (t)]$ is a invariant. Let

\begin{matrix} α = \frac{1}{4 n} \sum_{i = 1}^{n} [x'_{di} (t) + {\bar{v}}_{di} (t)] = \frac{1}{4 n} \sum_{i = 1}^{n} [x'_{di} (0) + {\bar{v}}_{di} (0)] \\ ε_{D} (t) = α 1_{4 n} + \bar{θ} (t) \end{matrix}

Then, system (7) is equivalent to

\overset{\cdot}{\bar{θ}} (t) = (I_{n} \otimes A) \bar{θ} (t) - (L_{D} \otimes B) \bar{θ} (t - τ)

(10)

where the disagreement vector $\bar{θ} (t)$ satisfies $lim_{t \to + \infty} \bar{θ} (t - τ) = c 1_{4 n}$ (c is a constant).

Moreover, there exists an orthogonal matrix Q satisfying

\bar{Λ} = Q^{T} L_{D} Q = diag {0, λ_{2}, λ_{3}, \dots, λ_{n}}

where $λ_{2}, \dots, λ_{n}$ is the nonzero eigenvalues of L; since L and $L_{D}$ are similarity matrixes, so the nonzero eigenvalues of $L_{D}$ is also $λ_{2}, λ_{3}, \dots, λ_{n}$ . Denote $Λ = diag {λ_{2}, λ_{3}, \dots, λ_{n}}$ , and then, system (7) is also equivalent to

\overset{\cdot}{θ} (t) = (I_{n - 1} \otimes A) θ (t) - (Λ \otimes B) θ (t - τ)

(11)

where

(Q^{T} \otimes I_{4}) \bar{θ} (t) = [0 0 0 0 θ^{T} (t)]^{T}

Then, analyze the equivalent system (11) instead. Note that $X (s) = G_{τ} (s) x (0)$ where

G_{τ} (s) = (s I_{4 n - 4} - I_{n - 1} \otimes A + Λ \otimes B e^{- τ s})^{- 1}

(12)

We define $φ_{τ} (s) = (s I_{4 n - 4} - I_{n - 1} \otimes A + Λ \otimes B e^{- τ s})$ and $φ_{τ} (s)$ can be denoted as

φ_{τ} (s) = [\begin{matrix} sI - A + λ_{2} e^{- τ s} \\ ⋱ \\ sI - A + λ_{n} e^{- τ s} \end{matrix}]

Then, calculating the determinant of $φ_{τ} (s)$ , we have

\det (φ_{τ} (s)) = Π_{i = 2}^{n} [(s^{2} + 2 b_{i} s + λ_{i} e^{- τ s})^{2}]

(13)

In order to guarantee the stability of system (11), all the roots of $φ_{τ} (s)$ must be on the open left half-plane, which holds if and only if all the roots of the following equation are all located on the open left half-plane

s^{2} + 2 b_{i} s + λ_{i} e^{- τ s} = 0, i = 2, \dots, n

(14)

Then, consider the critical condition by the frequency-domain analysis to guarantee that $φ_{τ} (s)$ has at least one root on the imaginary axis. Set $s = j ω$ in equation (14), and we have

λ_{i} \cos (τ ω) - ω^{2} = 0

(15)

2 b_{i} ω - λ_{i} \sin (τ ω) = 0

(16)

From equation (16), we have

ω = \frac{λ_{i} \sin (τ ω)}{2 b_{i}}

(17)

Then, by inserting equation (17) into equation (15), we have

λ_{i} \cos (τ ω) - \frac{λ_{i}^{2} \overset{2}{\sin} (τ ω)}{4 b_{i}^{2}} = 0

(18)

Let $h_{i} = b_{i}^{2} / λ_{i}$ and $y = \overset{2}{\tan} (τ ω / 2) \geq 0$ . Since

\begin{matrix} \cos (τ ω) = \frac{1 - \overset{2}{\tan} (\frac{τ ω}{2})}{1 + \overset{2}{\tan} (\frac{τ ω}{2})} \\ \sin (τ ω) = \frac{2 \tan (\frac{τ ω}{2})}{1 + \overset{2}{\tan} (\frac{τ ω}{2})} \end{matrix}

(19)

It follows that

F (y, h_{i}) = h_{i} y^{2} + y - h_{i} = 0

(20)

Suppose that $b_{i} > 0$ and $λ_{i} > 0$ , hence $h_{i} > 0$ . Then, by Vieta’s theorem, we have

\begin{matrix} y_{1} + y_{2} = \frac{1}{h_{i}} > 0 \\ y_{1} y_{2} = - 1 < 0 \end{matrix}

(21)

Hence, equation (20) has only one positive real root assumed as $ζ_{i}$ and another root located on the open left half-plane. Moreover, since

F (0, h_{i}) = - h_{i} < 0, F (1, h_{i}) = 1 > 0

then $ζ_{i}$ must be in the interval (0,1). Since

ζ_{i} = \overset{2}{\tan} (\frac{T ω}{2}), τ ω = \pm 2 arc \tan \sqrt{ζ_{i}} + 2 π l

By inserting equation (19) into equation (17), we have

ω = \frac{\pm \sqrt{ζ_{i}}}{b_{i} (1 + ζ_{i})}, i = 2, \dots, n

(22)

It follows that

τ_{i} = \frac{2 b_{i} (1 + ζ_{i}) arc \tan \sqrt{ζ_{i}}}{\sqrt{ζ_{i}}}

Next, we calculate the smallest $τ^{*}$ . From equation (20), take the partial of $ζ_{i}$ with respect to $h_{i}$ , we have

\frac{d ζ_{i}}{d h_{i}} = - \frac{dF / d h_{i}}{dF / d ζ_{i}} = - \frac{ζ_{i}^{2} - 1}{2 h_{i} ζ_{i} + 1} > 0

It is obvious that $ζ_{i}$ is an increasing function of $h_{i}$ and $h_{i} \in (0, + \infty)$ . In addition, $τ_{i}$ is the decreasing function of $ζ_{i}$ . Thus, $τ_{i}$ is a decreasing function of $λ_{i}$ and $λ_{i} \in (0, + \infty)$ . Then, we consider the function relationship between $τ_{i}$ and $b_{i}$ . Since $τ_{i}$ is a decreasing function of $ζ_{i}$ from Lemma 2 and $ζ_{i}$ is an increasing function of $h_{i}$ . Moreover, $h_{i}$ is an increasing function of $b_{i}$ . Thus, $τ_{i}$ is a decreasing function of $b_{i}$ and $b_{i} \in (0, + \infty)$ . Therefore

τ^{*} = \frac{2 b_{\max} (1 + ζ) arc \tan \sqrt{ζ}}{\sqrt{ζ}}

where $ζ$ is the positive root of equation (9). Now, we prove equation (8). From the analysis above, $τ$ gets its minimum when $b_{i} = b_{\max}$ and $λ_{i} = λ_{\max}$ . Since $\overset{2}{\cos} (τ ω) = 1 - \overset{2}{\sin} (τ ω)$ , set $b_{i} = b_{\max}$ and $λ_{i} = λ_{\max}$ in equation (18)

λ_{\max} \overset{2}{\cos} (τ^{*} ω) + 4 b_{\max}^{2} \cos (τ^{*} ω) - λ_{\max} = 0

(23)

It follows that

τ^{*} ω = ar ccos \frac{\sqrt{4 b_{\max}^{4} + λ_{\max}^{2}} - 2 b_{\max}^{2}}{λ_{\max}}

(24)

And equation (17) can be also rewritten as

ω = \sqrt{\sqrt{4 b_{\max}^{4} + λ_{\max}^{2}} - 2 b_{\max}^{2}}

(25)

Therefore

τ^{*} = \frac{arc \cos \frac{\sqrt{4 b_{\max}^{4} + λ_{\max}^{2}} - 2 b_{\max}^{2}}{λ_{\max}}}{\sqrt{\sqrt{4 b_{\max}^{4} + λ_{\max}^{2}} - 2 b_{\max}^{2}}}

Moreover, we consider the condition when $τ = 0$ in equation (14)

s^{2} + 2 b_{i} s + λ_{i} = 0, i = 1, 2, \dots, n

By the Routh stability criterion, it can be easily obtained that all the roots of equation (14) are all located on the open left half-plane.

Simulation

In this section, a numerical simulation is provided to demonstrate the four-group antagonistic networks under the proposed formation behavior algorithm (3). The following communication topologies associated with second-order multi-agent systems are given in Figure 2.

Figure 2.

The communication topology of the multi-agent system.

Example 1

Consider a multi-agent system with eight agents (agents 1 and 2; 3 and 4; 5 and 6; and 7 and 8 represent four antagonistic groups), and the initial value of the agents is set randomly. Moreover, the solid and dashed lines show the cooperative and confrontation relationship between agents, respectively, and we can assume that the weight $a_{ij} = 1$ or $- 1$ if there is an interaction $(i, j)$ in the network G. In Figure 3, We can see the trajectories of all second-order agents under the antagonistic interactions. It is obvious that eight agents eventually form four antagonistic groups and the agents in same groups converge to same position after sequences of behavior.

Figure 3.

Formation behaviors of four antagonistic groups. Boxes (a) and (b) are the initial position and Box (c) is the final position.

Figure 4(a) and (b), respectively, shows the state trajectories from X and Y directions. Obviously, all of agents reach a consensus of four-group antagonistic interaction behaviors. Since the multi-agents are second-order system, it naturally considers the velocity trajectories as shown in Figure 5(a) and (b).

Figure 4.

State trajectories of all agents in different directions and different lines denote eight agents: (a) state trajectories in X direction and (b) state trajectories in Y direction.

Figure 5.

Velocity trajectories of all agents in different directions. The velocity trajectories with different lines denote eight agents: (a) velocity trajectories in X direction and (b) velocity trajectories in Y direction.

By equation (8), we easily obtained $τ^{*} = 0.1808$ . From Figure 6, one can observe that consensus can be achieved asymptotically when $τ < τ^{*}$ , whereas the network system tends to oscillate when $τ > τ^{*}$ as shown in Figure 7.

Figure 6.

The behavior of agents when $τ < τ^{*}$ .

Figure 7.

The behavior of agents when $τ > τ^{*}$ .

Conclusion

In this article, antagonistic interaction behaviors of networks are investigated for second-order multi-agent systems with time delay. To solve this problem, we propose a distributed formation control algorithm for each agents to realize antagonistic interaction behaviors. Under the designed protocol, all agents belong to four groups, where agents in the allied groups achieve their own desired formations while keeping an antagonistic interaction with other groups. By the frequency-domain analysis, a sufficient and necessary condition is derived to ensure all agents reach consensus, and it is proved that the largest tolerable delay is related to the largest eigenvalue of the graph Laplacian. Finally, numerical simulations are showed to prove the effectiveness of our theoretical results.

Footnotes

Academic Editor: Zebo Zhou

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Zhang

Liu

Zeng

et al . Consensus analysis of continuous-time second-order multi-agent systems with nonuniform time-delays and switching topologies. Asian J Control 2013; 15(5): 1516–1523.

Lin

Jia

Multi-agent consensus with diverse time-delays and jointly connected topologies. Automatica 2011; 47(4): 848–856.

Zhang

Liu

Zeng

et al . Consensus of second-order multi-agent systems with nonuniform time delays. Chinese Phys B 2013; 22(5): 252–255.

Lin

Ren

Song

Distributed multi-agent optimization subject to nonidentical constraints and communication delays. Automatica 2016; 65(3): 120–131.

Lin

Jia

Distributed robust H_∞ consensus control in directed networks of agents with time-delay. Syst Control Lett 2008; 57(8): 643–653.

Jia

YM.

H-infinity consensus of a class of high-order multi-agent systems with time-delay. IET Control Theory A 2011; 5(1): 247–253.

Liu

Xie

Wang

Consensus for multi-agent systems under double integrator dynamic with time-varying communication delays. Int J Robust Nonlin 2012; 22(17): 1881–1898.

Lin

Ren

Farrell

JA.

Distributed optimization with the consideration of switching directed graphs, nonuniform step-sizes, finite-time convergence, and convex set constraint. IEEE T Automat Contr. Available online, 2017.

Beard

Lawton

Hadaegh

. A feedback architecture for formation control. In: Proceedings of the 2000 American control conference, Chicago, IL, 28–30 June 2000, pp.4087–4091. New York: IEEE.

10.

Lin

Ren

Gao

Distributed velocity-constrained consensus of discrete-time multi-agent systems with nonconvex constraints, switching topologies, and delays. IEEE T Automat Contr 2016; PP(99): 1.

11.

Couzin

Krause

Franks

NR.

Effective leadership and decision-making in animal groups on the move. Nature 2005; 433(7025): 513–516.

12.

Chu

Wang

Chen

TW.

Self-organized motion in anisotropic swarms. J Contr Theor Appl 2003; 1(1): 77–81.

13.

Lin

Ren

Constrained consensus in unbalanced networks with communication delays. IEEE T Automat Contr 2014; 59(3): 775–781.

14.

Lin

Dai

Song

Consensus stability of a class of second-order multi-agent systems with nonuniform time-delays. J Frankl Inst 2014; 351(3): 1571–1576.

15.

Balch

Arkin

RC.

Behavior-based formation control for multi-robot teams. IEEE T Robotics Autom 1998; 14(6): 926–939.

16.

Couzin

Krause

Self-organization and collective behavior in vertebrates. Adv Stud Behav 2003; 32: 1–75.

17.

Lin

Ren

Distributed robust H_∞ constrained control. Syst Control Lett 2017; 104: 45–48.

18.

Gharesifard

Jorge

Distributed continuous-time convex optimization on weight-balanced digraphs. IEEE T Automat Contr 2014; 59(3): 781–786.

19.

Nedic

Ozdaglar

Parrilo

PA.

Constrained consensus and optimization in multi-agent networks. IEEE T Automat Contr 2010; 55(4): 922–938.

20.

Easley

Kleinberg

Networks, crowds, and markets. Cambridge: Cambridge University Press, 2010.

21.

Wang

Nicola

. A control perspective for centralized and distributed convex optimization. In: 2011 50th IEEE conference on decision and control and European control conference, Orlando, FL, 12–15 December 2011, pp.3800–3805. New York: IEEE.

22.

Altifini

Consensus problems on networks with antagonistic interactions. IEEE T Automat Contr 2013; 58(4): 935–946.

23.

Smith

HL.

Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, vol. 41. Providence, RI: American Mathematical Society, 2008.

24.

Xiao

Zhou

ZY.

Formation control over antagonistic networks. In: 2013 32nd Chinese control conference, Xi’an, China, 26–28 July 2013, pp.6879–6884. New York: IEEE.

25.

Zhou

Liu

et al . Equality constrained robust measurement fusion for adaptive Kalman filter based heterogeneous multi-sensor navigation. IEEE T Aero Elec Sys 2013; 49(4): 2146–2157.

26.

Godsil

Royle

Algebraic graph theory. New York: Springer-Verlag, 2001.

Formation behaviors of networks with antagonistic interactions of agents

Abstract

Keywords

Introduction

Graph theory

Lemma 1. 26

Antagonistic formation behavior of agents

Remark 1

Stability analysis of interaction network

Lemma 2

Theorem 1

Proof

Simulation

Example 1

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References

Lemma 1.²⁶