Research Article  Open Access
Average Consensus in Networks of Neutral Dynamical Agents with Fixed and Switching Topologies
Abstract
This paper addresses the average consensus problem of neutral multiagent systems in undirected networks with fixed and switching topologies. For the case of fixed topology, necessary and sufficient conditions to average consensus are established by decoupling the neutral multiagent system in terms of the eigenvalues of the Laplacian matrix. For the case of switching topology, sufficient conditions to average consensus are given in terms of linear matrix inequalities to determine the allowable upper bound of the timevarying communication delay. Finally, two examples are worked out to explain the effectiveness of the theoretical results.
1. Introduction
In the last few years, more and more researchers have focused their attention on the study of multiagent systems due to many applications in unmanned aerial vehicles, satellite cluster, automated highways, and mobile robots. In all cases, the aim is to control a group of agents connected through a wireless network. Many profound theoretical results have been established for cooperative control of multiagent systems. In cooperative control of multiagent systems, an important issue is to design appropriate control strategies or protocols based on local information such that the group of dynamic agents can reach an agreement on certain quantities of interest. Such a problem is usually called a consensus problem, which has attracted many people in recent years.
Consensus problem has a long history in the field of computer science. In the field of control theory, a discretetime model of agents all moving in the plane with the same speed was proposed by Vicsek et al. [1]. A theoretical explanation of the consensus behavior in the Vecsek model was put forward by Jadbabaie et al. [2], where the alignment of a network of agent with switching topologies that are periodically connected was analyzed. OlfatiSaber and Murray [3] discussed the consensus problem for networks of dynamic agents with switching topologies and time delays in a continuoustime model by defining a disagreement function. Some more relaxable consensus conditions were presented by Ren and Beard [4] under dynamically changing interaction topologies.
In recent years, more and more consensus results with single integrator models were established. Xiao and Wang studied the consensus problem of discretetime multiagent systems with time delay [5, 6]. Sun et al. [7, 8] studied the average consensus problem in undirected networks of multiagent systems with fixed and switching topologies as well as multiple timevarying communication delays. Yu et al. [9] investigated the average consensus of multiagent systems with the problem of packet losses when both the firstorder neighbors’ information and the secondorder neighbors’ information are used. The consensus problem in directed networks via nonlinear protocols was premeditated by Liu et al. [10], and consensus of nonlinear multiagent systems with observerbased protocols was studied by Hu et al. [11].
Unlike the single integrator case, a spanning tree is a necessary rather than a sufficient condition for consensus seeking with double integrator dynamics. However, the extension of consensus algorithms from the firstorder case to the secondorder case is nontrivial [12]. The general consensus protocol of multiagent systems with secondorder dynamics was investigated by Zhu et al. [13], where necessary and sufficient conditions to consensus were given. Tian and Liu [14] considered the robust consensus of secondorder multiagent systems with different input delays and asymmetric interconnection perturbations based on the frequencydomain analysis method. Yu et al. [15] investigated the secondorder consensus for nonlinear multiagent systems with directed topologies. Song et al. [16] studied the secondorder leaderfollowing consensus of nonlinear multiagent systems by pinning control. Consensus algorithms designed for double integrator dynamics were provided by Abdessameud and Tayebi [17]. Zheng et al. [18] investigated the consensus problem of heterogeneous multiagent systems. Some sufficient conditions to consensus were established for both cases of undirected graphs and leaderfollowing networks.
It is well known that neutral systems play an important role in engineering area. If the agent’s dynamical equation takes the neutral form, what about the consensus of multiple agents in networks? To the best of our knowledge, consensus problem of neutral multiagent systems is not resolved by now. Therefore, it is necessary to study the consensus of neutral multiagent systems, which is the main purpose of this paper. We make the following contributions. Firstly, average consensus for neutral multiagent systems in undirected networks is proposed. Secondly, the convergence of the given proposed protocols is analyzed, and sufficient and necessary or sufficient conditions to average consensus of neutral multiagent systems have been established.
The remainder of this paper is organized as follows. In Section 2, graph theory and some important definitions and lemmas are given. The neutral multiagent system model is also established. The average consensus analysis of the neutral multiagent system with fixed and switching topologies is given in Section 3. Two examples are provided in Section 4. Finally, conclusions are drawn in Section 5.
2. Preliminaries and Problem Statement
As usual, the communication network is modeled through an undirected weighted graph defined by a set of nodes , a set of edges , and a weighted adjacency matrix with nonnegative adjacency elements . The node indexes belong to a finite index set , and represents the th agent. An edge of is denoted by . The adjacency elements associated with the edges of the graph are positive, that is, if and only if . Moreover, we assume and , for . The set of neighbors of node is denoted by . An undirected graph is called connected if any two distinct nodes of the graph can be connected via a path that follows the edges of the graph. The Laplacian matrix of the graph is defined by
It is easy to see that is symmetric and , where and are appropriate dimensional vectors with entries and . We now define an operator as Let represent the space collecting all vectors in with zero average. The eigenstructure of the operator is completely characterized by the next lemma proved in Bensoussan and Menaldi [19].
Lemma 1. (a) The operator is symmetric. It has strictly negative eigenvalues and the corresponding eigenvectors form an orthogonal basis for ; (b) for , if and only if .
Lemma 1 shows that the operator or the Laplacian matrix is invertible on the subspace . Now, consider the following neutral multiagent system model:where and , , , denote the state and the control input of the th agent at time , respectively, and and are constants. Consider the delayed Laplacian protocol as follows:where the timevarying delay is a piecewise continuous function satisfying , for . Then, the neutral multiagent system takes the formRewrite (5) in the following operator form:where .
As a result of the supposed piecewise continuity of the delay, the Cauchy problem associated with (6) is defined because it admits unique solution, depending on the initial condition. Set . The initial condition may be given by a continuously differentiable function defined on . In this paper, assume without loss of generality that , for .
The average consensus for the neutral multiagent system (6) is defined as follows.
Definition 2. The neutral multiagent system (6) is said to achieve consensus ifwhere is a constant. If , the neutral multiagent system (6) is said to achieve average consensus.
Remark 3. For the neutral multiagent system (6), once it achieves consensus, it must achieve average consensus since the involved graph is undirected. In fact, if (7) holds, noting that , we have which implies that is an invariant quantity. Therefore, Since , for , we have that .
By Lemma 1 and the following decomposition , where , , we can easily get the following lemma.
Lemma 4. The function is a solution of (6) if and only if and are solutions of the following:
Remark 5. By Lemma 4, we know that the spaces and are positively invariant for (6). Furthermore, by Remark 3, the average state is a constant in time and equals . Thus, the problem of average consensus for (6) is simplified to prove that as ; that is, the zero solution of (10) is asymptotically stable.
3. Main Results
In this section, we will consider the average consensus of neutral multiagent systems with fixed and switching topologies, respectively.
3.1. Networks with Fixed Topology
Theorem 6. Let be an undirected connected weighted graph. The neutral multiagent system (6) achieves average consensus if and only if all the following scalar differential equations are asymptotically stable, where are eigenvalues of the operator .
Proof. Let be a set of eigenfunctions of corresponding to the eigenvalues that forms a basis for . Then, there exist suitable functions such that By multiplying () on both sides of the first equation in (10), we have that the lefthand side and the righthand side of (10), respectively, take the following forms:By (14) and (15), one can obtainTherefore, the neutral multiagent system (6) can achieve average consensus; that is, (10) is asymptotically stable, if and only if all equations (16) are asymptotically stable for . The proof of Theorem 6 is completed.
For the particular case , the following lemma is required.
Lemma 7 (see [20]). The following scalar neutral differential equationis asymptotically stable if and only if
Based on Lemma 7, we have the following explicit result.
Corollary 8. Let be an undirected connected weighted graph. The neutral multiagent systemcan achieve average consensus if and only if and , where
Proof. By Theorem 6, system (19) can achieve average consensus if and only if all the following scalar differential equationsare asymptotically stable. By Lemma 7, all equations (21) are asymptotically stable if and only if Noting that , for , all equations (21) are asymptotically stable if and only if The proof of Corollary 8 is completed.
3.2. Networks with Switching Topologies
Note that system (5) can be rewritten in the following matrix form:where and are defined as above and the timevarying delay satisfies the assumptions () and with and .
We are here interested in discussing such a problem of whether a network with switching topology can still solve average consensus. In this case, the following switched system is considered:where is the Laplacian matrix of the undirected graph that belongs to set and is a finite collection of graphs of order with an index set . The map is a switching signal that determines the network topology.
Definition 9. The neutral multiagent system (25) is said to achieve average consensus under arbitrary switching if each solution of system (25) satisfies with .
Let us first take the following reducedorder transformation: The system (25) can be transformed into the following reducedorder disagreement system:where and with Define the operator as The operator is stable when and
Before giving the main result, we need the following lemma.
Lemma 10 (see [7]). For any real differentiable vector function and any constant matrix , the following inequality holds: where satisfies .
Theorem 11. Assume that () holds and . System (25) can achieve average consensus under arbitrary switching, if there exist symmetric positive definite matrices , , , , and of appropriate dimensions such that the following linear matrix inequality holds:where .
Proof. In spite of the existence of switching topology, similar to the discussion given in Remark 3, system (25) achieves average consensus once it solves consensus. Therefore, based on the above analysis, it is sufficient to prove that system (27) is asymptotically stable. Let be the common Lyapunov function for system (27), where Assume that the th subsystem is activated at time ; that is, . In the sequel, by considering the righthand side derivative of , that is, , along the trajectory of the solution of system (27), we have Noting that and following Lemma 10, we have In addition, Therefore, Then, a new bound of is given as follows: where , and is defined by (32). Noting that the matrix is negative definite, there exists such that . Let . Then, for any switching signal, the zero solution of (27) is asymptotically stable since the operator is stable. Thus, for any timevarying delay satisfying (), system (25) achieves average consensus under any switching. The proof of Theorem 11 is completed.
When or nothing has been known about the derivative of , construct the common Lyapunov function , where , , , and are defined as above. Similar to the proof of Theorem 11, we can easily obtain the following corollary.
Corollary 12. Assume that for some and . System (25) achieves average consensus under arbitrary switching, if there exist symmetric positive definite matrices , , , and of appropriate dimensions such that the following linear matrix inequality holds:
Remark 13. In Theorem 11 and Corollary 12, average consensus of the neutral multiagent system is determined by a linear matrix inequality rather than the connectivity of each switching topology. Similar to the analysis given in [7], it is not surprising that the connectivity of each switching topology is necessary to guarantee the feasibility of linear matrix inequalities (32) and (39).
Remark 14. Theorem 11 can also be extended to the case of directed graphs. Following the proof of Theorem 11, if each switching topology is a directed graph, the given linear matrix inequalities (32) guarantee consensus of the neutral multiagent system (25).
4. Examples
We now work out two examples to illustrate the theoretical results in this paper.
Example 1. Consider the neutral multiagent system (19) with the topology given by Figure 1.
The Laplacian matrix takes the form We can get , , and . When , we have by Corollary 8.
Example 2. Consider an undirected network with the following switching topologies shown in Figure 2.
Solving (32) gives the following estimate on for different and , which guarantees system (25) associated with the switching topology given in Figure 2 achieves average consensus under arbitrary switching. When is fixed to 0.1, the estimate on for different is shown in Table 1.
When is fixed to 0.1, the estimate on for different is shown in Table 2.


5. Conclusion
The average consensus problem of neutral multiagent systems in undirected networks with fixed and switching topologies is studied in this paper. Firstly, for the case of fixed topology, we establish sufficient and necessary conditions to average consensus by decoupling the neutral multiagent system in terms of the eigenvalues of the Laplacian matrix. Secondly, by using a linear matrix inequality method, average consensus criteria in terms of linear matrix inequality are given to determine the allowable upper bound of the timevarying communication delay for the case of switching topology. Finally, two examples are worked out to illustrate the theoretical results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Natural Science Foundation of Shandong Province under Grant no. JQ201119 and the National Natural Science Foundation of China under Grant nos. 61174217, 61374074, and 61473133.
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Copyright © 2015 Yingying Wu and Yuangong Sun. 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.