Multiple Dynamic Targets Encirclement Control of Multiagent Systems

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 467060, 6 pages http://dx.doi.org/10.1155/2015/467060 Research Article Multiple Dynamic Targets Encirclement Control of Multiagent Systems Wenguang Zhang, Jizhen Liu, and Deliang Zeng State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China Correspondence should be addressed to Wenguang Zhang; Received 28 September 2015; Accepted 9 November 2015 Academic Editor: Peng Lin Copyright © 2015 Wenguang 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. This paper develops the distributed encirclement control problem of multiagent systems, in which each agent tracks multiple targets, each target can be tracked by one agent, and the numbers of the agents and the targets are the same or not. Firstly, an encirclement control protocol is proposed for multiagent systems, and this protocol contains some estimators. Secondly, some conditions are derived, under which multiagent systems can achieve encirclement control by circular formation. Finally, numerical simulations are provided to illustrate the obtained results. 1. Introduction Distributed coordination control of multiagent systems has attracted a great number of researchers from different backgrounds, such as physics, biology, control theory, robotics, and computer [1–17]. Multiagent systems arise in wide areas, including movement of flocks of birds or schools of fish, molecular conformation problems, cooperative control of unmanned aerial vehicles, formation control of mobile robots, and power systems. For instance, Olfati-Saber and Murray [2] presented two consensus protocols to solve agreement problems in a network of continuous-time and discrete-time integrator agents and investigated a systematical framework of consensus problem in networks of agents with a simple scalar continuous-time integrator in three cases. Lin and Jia [3–5] studied consensus problems for first-order or second-order multiagent systems with time varying communication delays and switching topology. In [6, 7], 𝐻∞ consensus problems were, respectively, investigated for the first-order and high-order multiagent systems, and they gave the conditions of satisfying 𝐻∞ based on linear matrix inequality. In [8], the constrained consensus problem of multiagent systems in dynamically changing unbalanced networks with communication delays has been studied. It has been shown that the error auxiliary vanishes as time evolves and the linear main body has an exponential convergence rate to a vector as a separate system. In some situations, encirclement control for multiple targets can be studied in a distributed manner. However, the work on this problem is rare currently. In [9, 10], they only considered the fixed targets. In [9], a group of unmanned aerial vehicles surrounding one target by using decentralized nonlinear model predictive control was studied. In [10], Chen et al. used the leader-follower framework to make the followers surround the stationary leaders with a fixed communication graph. A multiagent cooperative control problem in which agents move collectively to surround multiple targets was studied in [11], and the proposed control law works not only for stationary targets but also for dynamic ones. But in that paper, it is assumed that the numbers of the agents and the targets are the same. This paper will focus on the study of the distributed encirclement control and tracking problems of multiple dynamic targets by graph theory. We suppose that each agent tracks multiple targets and each target only can be tracked by one agent. Firstly, we design a control protocol including some estimators. Secondly, the required conditions to realize 2 Mathematical Problems in Engineering encirclement are proposed by Lyapunov theory. Finally, we prove this theory to be effective by the simulation. The rest of this paper is organized as follows. In Section 2, we introduce some basic notations and some concepts in graph theory. In Section 3, the model to be researched is formulated and a distributed encirclement control protocol is proposed. In Section 4, the main results are stated and derived. In Section 5, numerical simulations are provided to demonstrate the effectiveness of the obtained theoretical results. In Section 6, we conclude this paper. 2. Notations and Preliminaries Let 𝐺(V, E, A) be an undirected graph, where V = {𝑠1 , 𝑠2 , . . . , 𝑠𝑛 } is the set of nodes and E ∈ V × V is the set of edges. The node indexes belong to a finite index set I = {1, 2, . . . , 𝑛} and 𝑁𝑖 = {𝑠𝑗 ∈ V(𝑠𝑖 , 𝑠𝑗 ) ∈ E} is defined as the neighbourhood set of 𝑠𝑖 . A = [𝑎𝑖𝑗 ] ∈ R𝑛×𝑛 is a symmetric weighted adjacency matrix, where the element 𝑎𝑖𝑗 represents the weight from node 𝑠𝑖 to node 𝑠𝑗 . When 𝑠𝑗 ∈ 𝑁𝑖 , then 𝑎𝑖𝑗 > 0, or else 𝑎𝑖𝑗 = 0. In the undirected graph, any (𝑠𝑖 , 𝑠𝑗 ) ∈ E ⇔ (𝑠𝑗 , 𝑠𝑖 ) ∈ E. The graph Laplacian with the diagraph is defined as 𝐿 = [𝑙𝑖𝑗 ], where 𝑙𝑖𝑖 = ∑𝑛𝑗=1 𝑎𝑖𝑗 and 𝑙𝑖𝑗 = −𝑎𝑖𝑗 , 𝑖 ≠ 𝑗. If there is a path from every node to every other node, the graph is said to be connected and undirected. Lemma 1 (see [18]). If the undirected graph 𝐺 is connected, then its Laplacian 𝐿 satisfies the following: (1) Zero is a simple eigenvalue of 𝐿, and 1𝑛 is the corresponding eigenvector, and 𝐿1𝑛 = 0. (2) The remaining 𝑛 − 1 eigenvalues of 𝐿 all have positive real parts. And 𝐿 is a symmetric matrix and the eigenvalues 0 = 𝜆 min = 𝜆 1 ≤ 𝜆 2 ⋅ ⋅ ⋅ ≤ 𝜆 𝑛 = 𝜆 max . Lemma 2 (see [19]). Suppose there is a positive definite Lyapunov function 𝑉(𝑥, 𝑡) defined on 𝑈 × 𝑅+ , where 𝑈 ∈ 𝑈0 is the neighbourhood of the origin. There are positive real ̇ 𝑡) + 𝑐𝑉𝛼 (𝑥, 𝑡) is constants 𝑐 > 0 and 0 < 𝛼 < 1, such that 𝑉(𝑥, negative semidefinite on 𝑈. Then, 𝑉(𝑥, 𝑡) is locally finite-time convergent with a settling time 𝑉1−𝛼 (𝑥0 (𝑡)) . 𝑇≤ 𝑐 (1 − 𝛼) (1) To simplify the analysis, we will consider the dynamics in polar coordinate system corresponding to system (2): 𝑙𝑖̇ (𝑡) = V𝑖 (𝑡) , 𝜃𝑖̇ (𝑡) = 𝜔𝑖 (𝑡) , (3) 𝑖 ∈ I, where 𝑙𝑖 (𝑡) ∈ 𝑅 and 𝜃𝑖 (𝑡) ∈ 𝑅, respectively, denote the radius and angle of the 𝑖th agent in the polar coordinate system which regards the geometric center 𝑃 = (1/𝑚) ∑𝑚 𝑖=1 𝑟𝑖 (𝑡) as the origin. 𝑟𝑖 (𝑡) represents the position of the 𝑖th target at time 𝑡. Obviously, 𝑦𝑖 (𝑡) = 𝑝𝑖 (𝑡) + [𝑙𝑖 (𝑡)cos(𝜃𝑖 (𝑡)), 𝑙𝑖 (𝑡)sin(𝜃𝑖 (𝑡))]𝑇 , where 𝑝𝑖 (𝑡) denotes the estimated value of the distance from the 𝑖th agent to the geometric center 𝑃. We say the control protocol 𝑢𝑖 (𝑡) can solve the distributed encirclement problems of system (2) if the states of agents satisfy 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 𝑚 lim {󵄩󵄩󵄩𝑦𝑖 (𝑡) − ∑ 𝑟𝑘 (𝑡)󵄩󵄩󵄩 𝑡→∞ 󵄩 󵄩󵄩 𝑚 𝑘=1 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 𝑚 󵄩 − 𝑘 max {󵄩󵄩𝑟𝑖 (𝑡) − ∑ 𝑟𝑘 (𝑡)󵄩󵄩󵄩}} = 0, 󵄩󵄩 󵄩󵄩 𝑖∈I 𝑚 (4) 𝑘=1 󵄩 󵄩 lim [𝜃𝑖 (𝑡) − 𝜃𝑗 (𝑡) − 𝑡→∞ 2𝜋 (𝑖 (...truncated)