Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 901282, 8 pages

http://dx.doi.org/10.1155/2015/901282

## Distance Constrained Based Adaptive Flocking Control for Multiagent Networks with Time Delay

^{1}College of Science, Civil Aviation University of China, No. 2898, Jinbei Road, Tianjin 300300, China^{2}Department of Automation, Nankai University, Tianjin 300071, China

Received 6 June 2014; Revised 23 September 2014; Accepted 23 September 2014

Academic Editor: He Huang

Copyright © 2015 Qing Zhang 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

The flocking control of multiagent system is a new type of decentralized control method, which has aroused great attention. The paper includes a detailed research in terms of distance constrained based adaptive flocking control for multiagent system with time delay. Firstly, the program on the adaptive flocking with time delay of multiagent is proposed. Secondly, a kind of adaptive controllers and updating laws are presented. According to the Lyapunov stability theory, it is proved that the distance between agents can be larger than a constant during the motion evolution. What is more, velocities of each agent come to the same asymptotically. Finally, the analytical results can be verified by a numerical example.

#### 1. Introduction

In recent years, the research on the flocking behavior of multiagent system has attracted great attention. For a series of agents which can apply some simple rules and limited information of neighbors to organize into a coordinated state is called flocking phenomenon. There exist many forms of flocking behavior in nature, for example, flocking of birds, swarming of bacteria, and so on [1, 2]. With the development of technology and the importance of real application, the study of the flocking behaviors of multiagent system has caused attention from a lot of different backgrounds, for example, biology, computer science, physics, and so on [3–8].

There are a plenty of existing works contributing to the flocking problems. Three heuristic rules leading to emergence of the first computer animation of flocking were first reported by Reynolds in 1987 [5]. The essential flocking rules depict how a personal agent maneuvers based on the local flock mates’ positions as well as velocities. Vicsek et al. designed a simple flocking model of multiagents which can all move with the same speed but with different directions in the plane. The Vicsek model is a special version of a pattern introduced previously by Reynolds [6]. Jadbabaie et al. put forward the rigorous proof of the convergence for Vicsek model [7]. There are many generalized species of this model, such as a leader follower strategy which means one agent acts as a group leader and the other agents would just follow the leader and keep the cohesion/separation/alignment rules. Recently, various desired state flocking motions are deduced in most cases [9–13]. Zavlanos et al., especially, proposed connectivity preserving controllers, by designing novel interagent potentials, to realize the flocking of multiagent system under the initial connectivity assumption [12, 13].

The time delays of systems are a very common phenomenon in real life. Many factors, for example, finite signal transmission speeds and memory effects, can cause time delay in spreading and communication. Therefore, it should be considered to design the control scheme for multiagent system with time delay. The effect of exchange delays for consensus problems and formation problems has been discussed [14–19]. According to the matrix theory and the frequency analysis, Su et al. obtained the desired moving model with a delay-dependent formation control algorithm [20]. Yang et al. studied the virtual potential approach for stabilising a group of agents at a desired formation [21]. Adaptive control is a kind of very important method in the control of complex nonlinear systems [22–24]. A good adaptive control can adapt to the changes in a large range of parameters of controlled system. It can not only maintain stable operation of the system but also keep the optimal in degree. The literatures focus on flocking with collision avoidance. However, the distance between multiagents is required to be larger than a constant in reality. For example, bird to incite wings must have their own space. Robot teams and UAV (unmanned air vehicles) in the formation movement in order to avoid a collision must consider their size. So, in the flocking control, it is not enough only to require the distance between them to be greater than zero. Motivated by this fact and on the basis of the abovementioned works, distance constrained based adaptive flocking control for multiagent system with time delay is presented in this paper. The innovation of this paper is mainly in the following aspects: the adaptive controller being designed to achieve the adaptive flocking of multiagents with time delays and keeping the distance between multiagents to be larger than a constant . The stability of the adaptive flocking of the multiagent system with time delays is analyzed theoretically. A sufficient condition is given for the stability of the adaptive control system.

The rest of the paper was structured from the following aspects. In Section 2, the multiagent flocking problem and some preliminaries used throughout this paper are introduced. In Section 3, a controller is designed based on adaptive flocking control laws. In Section 4, the main theory results that velocities of each agent come to the same asymptotically and the distance between agents required to be larger than a constant are proved. In Section 5, a simulation case is also presented to verify the effectiveness of our theoretical results. In Section 6, the full text content is summarized and the further research in the aspects of adaptive flocking of multiagent networks is investigated.

#### 2. Preliminaries

A set of agents moving in an -dimensional Euclidean space are considered. The dynamics of each agent is characterized by the following dynamic system (see [25]):where and are the position vector and the velocity of agent , independently, is a nonlinear vector-valued continuous function which describes the intrinsic dynamics of agent , and is the control input acting on agent . Especially, the virtual leader for multiagent system (1) is a special agent described bywhere and are the position and velocity vector of the virtual leader, respectively. In this paper, an assumption has been made that all agents can get the information of the virtual leader. The information switching between multiagents with the leader exhibits time delays.

Define error vector

Based on the definition of , we have the following equations:

*Assumption 1. *Assume that there exists a nonnegative constant satisfying

*Definition 2. *Communication radius is defined as the biggest distance from which multiagents can get information of other agents. Hysteresis radius is the distance in which a new edge will not be added to the graph until the distance between any two agents which are not connected decreases to . Safe radius is the distance needed by the agent own activities.

*Definition 3 (dynamic graphs [10]). *We call a dynamic graph which consists of a set of vertices indexed by the set of agents and a time varying set of links , such that, for any ,(1)if , then ,(2)if , then .

Dynamic graphs meet the conditions that if and only if are called undirected, which constitute the key point of the paper. If any vertices and in an undirected graph are joined by a link , we can call them adjacent or neighbors at time and denote them by . Let be equivalently represented by a dynamic negative Laplacian matrix, where is a weighted adjacency matrix of the graph , which satisfies that if , , and , otherwise. A topological invariance of graphs, that is, graph connectivity, is of great interest for this paper. The switching process of dynamic graphs can be shown in Figure 1.