Mathematical Problems in Engineering

Volume 2016, Article ID 1056594, 8 pages

http://dx.doi.org/10.1155/2016/1056594

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

^{1}School of Mechanical Electronic & Information Engineering, China University of Mining & Technology, Beijing, Beijing 100083, China^{2}Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, China^{3}Intelligent Control Laboratory, College of Engineering, Peking University, Beijing 100871, China

Received 1 February 2016; Accepted 2 June 2016

Academic Editor: Weizhong Dai

Copyright © 2016 Jianwei Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 and move 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 the mobile 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.

#### 1. Introduction

In recent years, decentralized control in networked multiagent systems has attracted considerable attention from various scientific communities [1–5] due to its broad applications in physics, biology, and engineering [6–8]. 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 [9–12] 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 [13–15]. Research efforts have been made in the systems and control community. In [16], a novel pursuit-based 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 [26–30] 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 discrete-time 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 sampled-data 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.

*Notations.* Throughout this paper, denotes the empty set, and denote transpose and inverse, respectively. For , is the eigenvalue of the matrix . Moreover, denotes a block diagonal matrix with the matrices , on the main diagonal. If the range of the indices is clear from the context, this notation is abbreviated by .

#### 2. 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 , 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 satisfywhich means that all the agents do not coincide in the beginning.