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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 848942, 14 pages
Leader-Following Consensus in Networks of Agents with Nonuniform Time-Varying Delays
1School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
2School of Science, Chongqing Jiaotong University, Chongqing 400074, China
3School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
Received 30 October 2011; Revised 26 March 2012; Accepted 27 March 2012
Academic Editor: Victoria Vampa
Copyright © 2012 Zhao-Jun Tang 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 is concerned with a leader-following consensus problem for networks of agents with fixed and switching topologies as well as nonuniform time-varying communication delays. By employing Lyapunov-Razumikhin function, a necessary and sufficient condition is derived in the case of fixed topology, and a sufficient condition is obtained in the case when the interconnection topology is switched and satisfies certain condition. Simulation results are provided to illustrate the theoretical results.
In recent years, consensus problems of multiagent systems have received compelling attention from various research communities. This is mainly due to their wide applications in many areas such as synchronization of coupled oscillators, flocking, rendezvous, distributed sensor fusion in sensor networks, and distributed formation control (see ). The study of consensus problems is of great importance to understand many complex phenomena related to animal behaviors, such as flocking of birds, schooling of fish, and swarming of bees.
Consensus problems have a long history in the field of computer science, particularly in automata theory and distributed computation . Vicsek et al.  proposed a simple model of a system of several autonomous agents, and demonstrated by simulation that all agents eventually reach an agreement. Jadbabaie et al.  provided a theoretical explanation for the observed behavior of the Vicsek model. Up to now, a variety of topics related to consensus problems have been addressed such as consensus with switching topologies, consensus with time-delays, finite-time consensus, consensus over random networks, consensus with measurement noises.
In multiagent systems, time-varying delays may arise naturally, for example, because of the moving of the agents, the congestion of the communication channels, the asymmetry of interactions, and the finite transmission speed due to the physical characteristic of the medium transmitting the information . It has been observed from numerical experiments that consensus protocols without considering time delays may lead to unexpected instability. Therefore, it is important and meaningful to consider the consensus problems when communication is affected by time-delays. Consensus problems with communication delays have been addressed by many researchers. Olfati-Saber and Murray  studied the average consensus of first-order multiagent systems with constant and uniform communication time delays under fixed topology. The upper bound on the admissible delays was derived by means of frequency domain approach, and it was shown that it is inversely proportional to the largest eigenvalue of the Laplacian matrix of the network topology. Bliman and Ferrari-Trecate  generalized the results of Reference  in considering uniform and nonuniform time-varying time-delays. Consensus in networks of agents with single-integrator dynamics and double-integrator dynamics as well as multiple time-varying delays was addressed in [5, 7–10]. Note that the maximum allowable upper bound of time-delays in these literatures is presented in terms of linear matrix inequalities (LMIs). By employing Lyapunov-Razuminkhin function, Hu and Hong  and Hu and Lin  investigated the consensus problems of second-order multiagent systems with fixed and switching topologies in the presence of uniform communication delays.
In this paper, we are interested in the leader-following consensus of multiagent systems with fixed and switching topologies as well as multiple time-varying delays. With the help of Lyapunov-Razumikhin function, we derive the maximal allowable upper bound of communication delays such that all the agents can follow the considered leader. In , the authors also studied the leader-following consensus problem with multiple time-varying delays, but the proposed protocol requires that the time-delays can be detectable. We assume that the time-delays are all unknown in the present paper. The objective of this paper is to generalize the results of  by considering nonuniform time-varying communication delays. Obviously, it is more practical to consider nonuniform time-delays than uniform time-delays. It is worthy to note that we derive an explicit formula for the bound of the allowable time-delays by means of Lyapunov-Razumikhin function whereas the bound is presented in terms of LMIs in [5, 7–10].
The following notations will be used throughout this paper. Let be an identity matrix with appropriate dimension. i is the imaginary unit. For a given matrix , denotes its transpose; denotes its spectral norm; denotes the set of all eigenvalues of ; and denote its maximum and minimum eigenvalues, respectively. denotes the Euclidean norm for a given vector. Given a complex number , and are its real part and imaginary part, respectively. A matrix is said to be positive stable if all its eigenvalues have positive real parts.
2. Problem Formulation
Consider a multiagent system consisting of agents, and a leader. We first describe the interconnection topology among the agents by a simple digraph , where is the set of nodes representing the agents and is the set of edges of the graph. An edge of is denoted by , representing that agent can directly receive information from agent . A path in a digraph is a sequence of ordered edges of the form , . We say that node is reachable from node if there is a path from node to node . The set of neighbors of node is denoted by .
The weighted adjacency matrix of the digraph is denoted by , where if and , otherwise. Moreover, we assume that for all . The indegree and outdegree of node are defined as and , respectively. A digraph is said to be balanced if . The Laplacian matrix associated with digraph is defined as
The definition of clearly implies that must have a zero eigenvalue corresponding to a eigenvector , where . Moreover, 0 is a simple eigenvalue of if and only if has a spanning tree .
In order to study a leader-following problem, we also concern another digraph , which consists of digraph , node 0, and edges from some nodes to node 0. We say that node 0 is globally reachable in if node 0 is reachable from any node in . The leader adjacency matrix associated with is defined as a diagonal matrix with diagonal elements , where if is an edge of and , otherwise.
In this paper, we consider the following double integrator system of agents: where , , denote the position, velocity, and control input of agent , respectively. The dynamics of the leader is expressed as follows:
where is the desired constant velocity.
Let denote the communication time-delay from agent to agent . Similarly to , we assume that communication delays between agents are symmetrical, that is, . Our control goal is to let the agents follow the considered leader in the sense of both position and velocity, namely, as . For this end, we study the following neighbor-based protocol: where is a control parameter.
The communication topology among the group of agents may change dynamically due to link failure or creation, for instance, because of the limited detection range of agents, existence of the obstacles. In order to describe the switching topologies, we define a piecewise constant switching signal ( in short):, where denotes the total number of all possible interaction topologies. The collection of all possible interaction topologies is a finite set. For convenience, let
be the collection of independent time-delays affecting the communication links, where are piecewise continuous functions. It is clear that because the delays are symmetrical. We assume that the nonuniform time-varying delays are uniformly bounded; namely, there exists a constant such that , . The associated edges, with the time-delay and switching signal , define a subgraph . The corresponding Laplacian matrix associated with and the leader adjacency matrix associated with are denoted by and , respectively. It is clear that To illustrate these relationships, an example is given as follows.
Example 2.1. Consider a multiagent system consisting of four agents and a leader with the interconnection topology shown in Figure 1. We assume that has weights, and , , . The subgraphs , , are shown in Figure 2, and one can obtain the following:
and , . Obviously, (2.6) is true.
Write , . With protocol (2.4), (2.2) can be written in the following matrix form:
Let , . We can obtain an error dynamics of system (2.8) as follows: where
Before ending this section, we introduce Lyapunov-Razumikhin Theorem, which plays a key role in the convergence analysis of system (2.9).
Consider the following system: where , and . Let be a Banach space of continuous functions defined on an interval , taking values in with the topology of uniform convergence, and with a norm .
Lemma 2.2 (Lyapunov-Razumikhin Theorem ). Let , , and be continuous, nonnegative, nondecreasing functions with , , for and . For system (2.11), suppose that the function takes bounded sets of in bounded sets of . If there is a continuous function such that In addition, there exists a continuous nondecreasing function with , such that the derivative of along the solution of (2.11) satisfies then the solution is uniformly asymptotically stable.
3. Main Results
3.1. Fixed Interconnection Topology
To derive a delay-dependent stability criteria, we make the following model transformation. With the observation that it follows from (3.1) that where . Substituting (3.3) into system (3.1) leads to Noting that for , we have where
The process of transforming a system represented by (3.1) to one represented by (3.5) is known as a model transformation. The stability of the system represented by (3.5) implies the stability of the original system .
To get the main result of this subsection, we need the following lemmas.
Lemma 3.1 (see ). Given a complex-coefficient polynomial, where , , , , is Hurwitz stable if and only if and .
Lemma 3.2. Let Then is Hurwitz stable if and only if is positive stable and
Proof. Note that the characteristic polynomial of is given by
where we have used Schur formula  to obtain the second equality. Let be the th eigenvalue of , and we have
Denote that . It follows from Lemma 3.1 that is Hurwitz stable if and only if and . Therefore, all eigenvalues of have negative real parts if and only if and for any , which implies the conclusion.
Lemma 3.3 (see ). The matrix is positive stable if and only if node 0 is globally reachable in .
Now we state one of our main results.
Theorem 3.4. Consider system (3.1) and take where , is the control parameter in protocol (2.4). Then, there exists a constant (which will be defined in the following (3.20)) such that when , namely, the agents can follow the leader (in the sense of both position and velocity), if and only if node 0 is globally reachable in .
Proof. (Sufficiency). Since node 0 is globally reachable in , it follows from Lemma 3.3 that is positive stable. Thus, it follows from (3.12) and Lemma 3.2 that is Hurwitz stable. Hence, by Lyapunov theorem , there exists a positive definite matrix such that
Take a Lyapunov-Razumikhin function
Along the solution of system (2.9), from (3.5), we have
Note that holds for any appropriate positive definite matrix . Then, we have
Take for some constant . In the case of
by recalling that . As a result, if
then for some constant . Therefore, the conclusion follows by Lemma 2.2.
(Necessity). System (3.1) is asymptotically stable for any time delays , . In particular, let , . By (3.1), the system is asymptotically stable, and hence all eigenvalues of have negative real parts. Therefore, it follows from Lemma 3.2 that is positive stable, and the conclusion follows by Lemma 3.3.
Remark 3.5. From the proof of Theorem 3.4, we can see that many zoom techniques have to be applied during the derivation of , and hence our estimate may be very conservative.
3.2. Time-Varying Topology
In this subsection, we consider the case of switching topologies. Similar to the case of fixed topology, we can obtain that Then, from (2.9), we have by noting that for , . Denote the following:
where . Then (3.22) can be rewritten as
To obtain the main result of this subsection, we introduce the following assumption.
Assumption 3.6. The weights of digraph satisfy the following conditions:
(1), ,(2), ,
where namely, denotes the index set of neighbors of vertex 0.
Remark 3.8. In the study of leader-following consensus for second-order multiagent systems with switching topologies [11, 21–24], it was assumed that is balanced and vertex 0 is globally reachable in so that a common Lyapunov function can be established. The theoretical base is the conclusion that is positive definite if is balanced, and vertex 0 is globally reachable in . Noticing that and denote the out-degree and in-degree of node in , respectively, (Assumption 3.6) contains that the out-degree of node is greater or equal to its in-degree for each as a special case. Then (Assumption 3.6) is much weaker than the balanced constraint on , and the corresponding results in the above literatures can be improved accordingly.
For convenience, denote that , , which are well defined by noting that the set is finite. The main result of this subsection is as follows.
Proof. Take a Lyapunov-Razumikhin function with positive definite matrix
Similar to the analysis in the proof of Theorem 3.4, we have
Take for some constant . In the case of we have where According to Schur complement , is positive definite for any if satisfies (3.25). Hence, if which is well defined by noting that the set is finite, then for some . Therefore, the conclusion follows by Lemma 2.2.
Remark 3.10. For the first-order multiagent systems, it was shown that the consensus can be achieved provided that the network topology jointly contains a spanning tree [14, 25]. However, if the group of agents is governed by second-order dynamics, the consensus depends not only on the topology condition but also on the coupling strength between neighboring agents, and it was shown that consensus may fail to be achieved even if the network topology contains a spanning tree . It should be pointed out that (Assumption 3.6) is not necessary to ensure the consensus, and it is of great interest to consider the more general condition on the network topology.
In this section, two examples are provided to illustrate the theoretical results. For simplicity, we assume that each interconnection topology has 0-1 weights in the following two examples.
Example 4.1. Consider a multiagent system consisting of a leader and four agents with fixed topology given in Figure 1. It is clear that node 0 is globally reachable in . By simple calculation, we have and . Let , , . The simulation results are obtained with . Figure 3 shows that the four agents can follow the considered leader.
Example 4.2. Consider a multiagent system consisting of a leader and four agents. The interconnection topology of the multiagent system switches every 1 in the sequence described as Figure 4. It is clear that the weights of and satisfy (Assumption 3.6) and node 0 is globally reachable in and . Let , , . The simulation results are obtained with . It can be seen from Figure 5 that the four agents can follow the considered leader.
In this paper, we study a leader-following consensus problem of second-order multiagent systems with fixed and switching topologies as well as nonuniform time-varying communication delays. With the help of Lyapunov-Razumikhin function, an explicit formula for the upper bound of admissible delays is obtained for both fixed and switching topologies. Future research issues will include the cases when the communication delays are asymmetric, and the velocity of the considered leader is time-varying.
The authors would like to thank the anonymous reviewers for their insightful comments and suggestions. This work was supported by NSFC (60973015, 61170311, 61104104), NSFC Tianyuan foundation (11126104), Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), Sichuan Province Sci.&Tech. Research Project (12ZC1802).
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