Necessary and Sufficient Conditions for Circle Formations of Mobile Agents with Coupling Delay via Sampled-Data Control

A circle forming problem for a group of mobile agents governed by first-order system is investigated, where each agent can only sense the relative angular positions of its neighboring two agents with time delay andmove on the one-dimensional space of a given circle. To solve this problem, a novel decentralized sampled-data control law is proposed. By combining algebraic graph theory with control theory, some necessary and sufficient conditions are established to guarantee that all themobile agents form a pregiven circle formation asymptotically. Moreover, the ranges of the sampling period and the coupling delay are determined, respectively. Finally, the theoretical results are demonstrated by numerical simulations.


Introduction
In recent years, decentralized control in networked multiagent systems has attracted considerable attention from various scientific communities [1][2][3][4][5] due to its broad applications in physics, biology, and engineering [6][7][8].Meanwhile, it should be noted that decentralized control has many advantages in achieving cooperative group performance, especially with low operational costs, high robustness, and flexible scalability.
As a popular research topic in decentralized control, formation control [9][10][11][12] refers to coordinating a group of agents such that they can form a predesigned geometrical configuration through local interactions so that some tasks can be finished by the collaboration of the agents.Forming circle formations becomes a benchmark problem, since on one hand circle formations are one of the simplest classes of formations with geometric shapes and on the other they are natural choices of the geometric shapes for a robotic team to exploit an area of interest [13][14][15].Research efforts have been made in the systems and control community.In [16], a novel pursuitbased approach has been presented to investigate collective motions and formations of a large number of agents with single-integrator kinematics and double-integrator dynamics on directed acyclic graphs, respectively.Furthermore, the problem of pattern formation based on complex Laplacians has been studied in [17].More recently, Lin et al. [18] have studied the leader-follower formation problem based on complex-valued Laplacians for graphs whose edges are attributed with complex weights and designed a novel linear control law to achieve the shape of a planar formation.In that work, the linear control law can only solve the formation problem asymptotically.Lou and Hong [19] have considered the distributed surrounding of a convex target set by a group of agents with switching communication graphs and proposed a distributed controller to make the agents surround a given set with equal distance and the desired projection angles specified by a complex-value adjacency matrix.
However, in some practical situations, it is more desirable for the multiagent systems to reach the formation in a finite time, such as when high precision performance and stringent convergence time are required.In [20], Xiao et al. have developed a novel finite-time formation control framework for multiagent systems.In their framework, the problems of time-invariant formation, time-varying formation, and trajectory tracking have been discussed, respectively, and some sufficient conditions for finite-time formation have been presented.
In addition, the coupling delay [21,22] between neighboring agents, which may deteriorate the system's performance or even destabilize it, is always unavoidable in real circumstance with practical reasons, such as the finite switching and spreading speed of the hardware and circuit implementation.Due to this observation, Qin et al. [23] have studied the consensus problem for second-order dynamic agents under directed arbitrarily switching topologies with communication delay.They have proven that consensus can be reached if the delay is small enough.Very recently, Chen et al. [24] have considered the consensus problem of nonlinear multiagent systems with state time delay and obtained some consensus results by designing an adaptive neural network control strategy.In their work, it should be noted that the approximation property of radial basis function neural networks is used to neutralize the uncertain nonlinear dynamics of agents.
In this paper, we investigate a circle formation problem of mobile agents with the coupling delay, where each agent is described by a kinematic point.Specifically, in the circle formation problem [25], all the agents move counterclockwise on the one-dimensional space of a given circle.We assume that each agent can only sense the relative angular positions of its neighboring two agents that are immediately in front of or behind it.The objective is to design appropriate decentralized control law such that all the agents can form a pregiven circle formation.Considering the limitations inherited in practical systems, such as the finite computing resource, we employ sampled-data control [26][27][28][29][30] when studying the circle formation problem of mobile agents with the coupling delay.Under the decentralized sampled-data control framework, the whole system is modeled in a hybrid fashion, and the continuous-time system is equivalently transformed into a discrete-time system.Furthermore, based on the discretetime system, some necessary and sufficient conditions are established to guarantee that all the mobile agents form a pregiven circle formation asymptotically.We emphasize that the formulation of circle formation problem in our paper mainly follows the work in [25].However, [25] has focused on the situation with the locomotion constraint that the mobile agents can only move forward but not backward which is motivated by several types of mobile robots, while this paper focuses on the case with time delay.Thus the way we deal with the circle formation problem with time delay here is quite different from that in [25].
The rest of the paper is organized as follows.In Section 2, some basic definitions in graph theory and the system model are provided.In Section 3, a novel decentralized sampleddata control law is proposed, based on which the main analytical results are obtained.In Section 4, numerical simulations are implemented to demonstrate the analytic results.Finally, the paper is concluded in Section 5.

Preliminaries
In this section, some basic definitions in graph theory and system model are firstly introduced for the subsequent use.Consider multiagent systems consist of  agents, which are initially located on a given circle and can only move on the circle.The agent indexes belong to a finite index set  = {1, 2, . . ., }, and we label the agents counterclockwise as shown in Figure 1.Each agent has the dynamics as follows: where   () is the position of agent  at time  measured by angles, and   () is the decentralized control of agent .Here, without loss of generality, it is assumed that the initial values of the agents satisfy which means that all the agents do not coincide in the beginning.
In the multiagent systems, each agent can communicate with several other agents which are defined as its neighbors, and the neighbor set of agent  is denoted by   .Specially, in the circle formation problem, each agent can only sense the relative angular positions of its neighboring two agents that are immediately in front of or behind it, and it follows that   = { + ,  − }, where Obviously, the interconnection topology between the agents in the circle formation problem is ring [31].Suppose   is the desired angular distance between agents  and  + .Then the pregiven circle formation can be described by a vector  = [ 1 ,  2 , . . .,   ] ∈   , where   > 0, and ∑  =1   = 2.Denote the auxiliary variable   as follows: where   is the angular distance between agents  and  + .The main objective of this paper is to design an appropriate decentralized control   () such that a group of agents form a pregiven circle formation asymptotically; that is, lim →∞   () =   , ∀ ∈ .In [24,25], Wang et al. adopted the following decentralized control to solve the circle formation problem: In this paper, we assume that each agent can only receive the neighbor information with the coupling delay.Specifically, by using periodic sampling technology and zero-order hold circuit, a decentralized sampled-data control induced from ( 5) is given as where ℎ > 0 is the sampling period,  is the coupling delay between neighboring agents, and it is assumed that  < ℎ.In the next section, we investigate the circle formation problem of the closed-loop system (1) and ( 6).

Main Results
In this section, the convergence analysis of circle formations is presented, and some necessary and sufficient conditions are established.
Lemma 2 (see [32]).Given the system where  ∈  × , then system (15) solves the consensus problem if and only if 1 is an algebraically simple eigenvalue of  and is the unique eigenvalue of maximum modulus.
Then the main result of the paper is given by the following theorem.
In order to solve the circle formation problem, one should guarantee that all the eigenvalues of   are located in the unit circle for  = 2, 3, . . ., .Actually, all the matrixes   have the same form; hence one can analyze them together.Consider the auxiliary matrix [ 1−(ℎ−) − 1 0 ], where  ∈ (0, 2], and its eigenvalues can be obtained by solving the characteristic equation Let the roots of (19) be  1 ,  2 , and it should be noted that there are two cases of the roots of (19).
According to Lemma 1, when the number of the agents is even, 2 is the maximum eigenvalue of the matrix ().Therefore, based on Theorem 3 and its proof, we have the following corollary.
Corollary 4. Consider a network with  agents governed by the form (1), and  is even; then the decentralized sampleddata control (6) solves the circle formation problem if and only if  < 1/2 and  < ℎ < 2 + 1.

Numerical Simulation
In this section, an example is provided to illustrate the effectiveness of the proposed theoretical results.
Furthermore, we choose  = 0.4, ℎ = 1.85, and then ℎ > 2 + 1, which do not meet for the condition in Corollary 4, and hence the multiagent system cannot achieve the pregiven circle formation, and the evolution of   () and   () is shown in Figure 3.

Conclusion
This paper has discussed the circle forming problem for a group of mobile agents which are governed by first-order dynamics.In this study, each agent can only sense the relative delayed angular positions of its neighboring two agents and move on the one-dimensional space of a given circle.With the help of periodic sampling technology and zero-order hold circuit, a novel decentralized sampled-data control has been proposed.By combining algebraic graph theory with control theory, some necessary and sufficient conditions have been established to guarantee that all the mobile agents form a pregiven circle formation asymptotically.Finally, the simulations have confirmed our theoretical results.In the future, we will focus on the situations where the agent has the intrinsic dynamic, and the decentralized sampled-data control needs to be designed.

Figure 2 :Figure 3 :
Figure 2: The evolution of   () and   () with the decentralized sampled-data control, when  = 0.4 and ℎ = 0.8 which meets the condition in Corollary 4.