Complexity

Complexity / 2018 / Article
Special Issue

Advances in Modelling, Monitoring, and Control for Complex Industrial Systems

View this Special Issue

Research Article | Open Access

Volume 2018 |Article ID 7386729 | 11 pages | https://doi.org/10.1155/2018/7386729

Couple-Group Consensus: A Class of Delayed Heterogeneous Multiagent Systems in Competitive Networks

Academic Editor: Sing Kiong Nguang
Received19 Jun 2018
Accepted06 Aug 2018
Published03 Oct 2018

Abstract

This paper discusses the couple-group consensus issues of a class of heterogeneous multiagent systems containing first-order and second-order dynamic agents under the influence of both input and communication delays. In distinction to the existing works, a novel distributed coordination control protocol is proposed which is not only on the foundation of the competitive interaction between the agents but also has no virtual velocity estimation in the first-order dynamics. Furthermore, without the restrictive assumptions existing commonly in the related works, several sufficient algebraic criteria are established for the heterogeneous systems to realize couple-group consensus asymptotically. The obtained conclusions show that the achievement of the systems’ couple-group consensus intimately relates to the coupling weights between the agents, the systems control parameters, and the input time delays of the agents, while communication time delays between the agents are irrelevant to it. Finally, several simulations are illustrated to verify the effectiveness of the obtained theoretical results.

1. Introduction

As a fundamental issue of the coordinated control of multiagent systems (MASs), consensus problem of MASs has attracted wide attention in recent years due to its wide applications in various fields, including the mobile robot systems, distributed target tracking, and group decision-making. So far, lots of research works on various consensus problems have been constantly reported, such as in [15] and the refs. therein.

Group consensus, as an extension of consensus issue, implies that distinctive subconsensus states can be achieved in complex systems while separate subgroups cannot reach agreement. In the last few years, lots of research works on group consensus have widely emerged, such as in [614].

1.1. Related Works

Note that all the works mentioned above mainly focus on the MASs which constructed by multiple agents with homogeneous dynamics. Namely, all agents of the entire complex systems featured the identical dynamics. In fact, this case is very restricted because in the real world, the difference existed in the dynamics among agents is unavoidable. Meanwhile, we usually have such applications using multiple agents with different dynamics to achieve the desired goals for the case of reducing control cost. Therefore, more and more concerns are devoted by the researchers to heterogeneous MASs’ consensus problems, such as the [15], in which the necessary and sufficient criteria were proposed for the synchronization of the heterogeneous networks. Meanwhile, Wang et al. [16] also investigated the heterogeneous systems and addressed certain sufficient conditions to ensure the achievement of consensus. With regard of the time delays, Cui et al. discussed the consensus problem for the heterogeneous chaotic systems in [17]. Based on undirected topology, Goldin and Raisch addressed the consensus of the heterogeneous networks in [18]. In [19], Wang et al. studied the globally limited consensus of the heterogeneous systems and put forward certain criteria to guarantee the consensus. The consensus issues for the heterogeneous systems with time delays were also studied in [20, 21]. As for the discrete-time heterogeneous systems, Kim et al. [22] and Li et al. [23], respectively, discussed the consensus problems for the systems with accidental connection failures. The second-order consensus for the systems with Euler-Lagrange networks was investigated in [24], where the multiple consensus states were achieved. In [25], the effective protocols for achieving consensus of the heterogeneous systems were designed and certain conditions based on the state transformation method were obtained. Based on certain hypothesis, Liu et al. also investigated the heterogeneous MASs and its group consensus problems in [26]. Considering the influence of input delays or not, Wen et al. studied the group consensus of the systems with heterogeneous dynamics [27, 28], respectively. By utilizing the Lyapunov method, in [29], Qin et al. studied both linear and nonlinear heterogeneous systems as well as their group consensus issues.

1.2. Primary Motivations

Note that most of the research works mentioned above have been paid attention to consensus problems of the heterogeneous MASs. As we know, group consensus owns an important practical significance in the coordinated control of large scale and complex tasks. Hence, many further works for group consensus of heterogeneous MASs need to be concerned with. Meanwhile, it is known that communication and input time delays existing in the complex systems can usually affect and even destroy the stability of the system. Therefore, one motivation of this paper is to discuss the group consensus problems of a class of heterogeneous MASs with time delays.

The main contributions of this paper are summarized in the following three aspects. First, to our knowledge, there are rare works on the group consensus issue of heterogeneous MASs which considered both input and communication delays. Second, from a different point of view, we propose an original couple-group consensus protocols which established on the agents’ competitive interaction. It is distinguished with the aforementioned works which mainly modeled by the agents’ cooperative relationship [1528]. It is known that competitive interaction is also an important relationship in complex systems, such as in ecology, the problems of predator-prey on food chain. Meanwhile, in our control protocol, the first-order agents’ dynamics possess no virtual velocity, which been included in many related works (e.g., in [20, 2628]) for simplifying the process of analysis. Third, our results relax the following two conservative prerequisites existed in [20, 2529]: in-degree balance and the geometric multiplicity of the zero eigenvalue of the systems’ Laplacian matrix have no less than 2. Both of them limit the communication of the agents and the topology of the system. Based on matrix theory and frequency domain analysis, some criteria for ensuring the achievement of couple-group consensus are effectively proposed. The combination of the above innovative points makes the application scope of our results more general.

Notation. Throughout this paper, and indicate the real and complex numbers sets, respectively. , then its real part and modulus are defined as and . stands for the identity matrix with -dimension. and mean the determinant and the th eigenvalue of the matrix , individually.

2. Preliminaries and Problem Statements

2.1. Graph Theory

Using the graph theory, the agents and their information exchange in a MAS with agents can be represented by a digraph , where , , and denote the node set, the edge set, and the adjacency matrix, respectively. Noting that an undirected graph can be seen as a special digraph, we assume if throughout this paper. Namely, if and only if the node (agent) can receive the information from the node (agent) ; otherwise, . Meanwhile, the neighbour sets and the in-degree of node are defined as and , individually. Design in-degree matrix as , so is defined to be the Laplacian matrix.

2.2. Problem Statement

In this paper, we suppose a heterogeneous multiagent system consisting of agents which contained first-order and second-order dynamics. For convenience, assume the first and the remaining agents own second-order and first-order dynamics, respectively, then the system can be described as follows: where , , and . , , and represent position, velocity, and input control of agent , individually.

Regard a heterogeneous MAS, to each agent, its neighbor may has second-order and first-order dynamics, which may be distinguishingly denoted as and . Thus, the neighbors set of agent can be distinguished as . With regard that the agents’ dynamics in the systems are heterogeneous, then adjacency matrix can be denoted as where and represent the adjacency matrix consisting of all second-order or first-order agents, composes of coupling weights from first-order agents to second-order ones, and is the opposite of . The systems’ Laplacian matrix can be modified as in the Laplacian matrix and , the interactions of only second-order agents or first-order agents are included; and are in-degree matrix of agent , which includes information received from the neighbors of different orders.

Firstly, some fundamental definitions and lemma are introduced as below.

Definition 1. To a heterogeneous MASs, such as (1), which can be said to achieve couple-group consensus asymptotically when and only when the following two conditions are satisfied:

Definition 2. For a bipartite graph , where and denote its edge and vertex sets, it has the following two properties: (1) and and (2) , where and .

Lemma 1 [10]. Regarding a directed bipartite graph which contains a directed spanning tree, it then possesses the following two properties: and when . Meanwhile, for a undirected bipartite graph, . Where is the number of agents of systems, matrix .

3. Main Results

In [18], the authors investigated the group consensus of the heterogeneous systems with identical input time delay. The systems are described as follows:

For the sake of analysis, the velocity estimation is added to the first-order agents in (6). Meanwhile, the systems (5) and (6) are modeled by the cooperative relationship among the agents.

To realize group consensus and be distinguished to (5) and (6), we design an original distributed control protocol which utilizes the agents’ competitive interactions. It is showed as follows: where indicates communication delay between agent and agent and represents the identical input delay of the agents. Meanwhile, we suppose the control parameters , , and .

Theorem 1. Consider the MASs (7) and suppose the systems’ topology is an undirected bipartite graph, the systems’ couple-group consensus can be realized asymptotically if these conditions hold: and , where and .

Proof 1. Do Laplace transforms to (7) firstly, we get where and represent the Laplace transform of and , respectively.
From the (8), we have After a simple manipulation, it yields that Define , , and then from the (9) and (11), we have Define , then (13) can be rewritten as where Define . Based on the Lyapunov stability criterion, the systems can reach group consensus if the roots of are at or . Subsequently, we will discuss these two cases, respectively, according to the general Nyquist criterion.
When , . According to Lemma 1, one can know that zero is the simple eigenvalue of the matrix . Thus, the roots of are at the point .
When , set and Set , according to the general Nyquist criterion, when and only when the point is not enclosed by the Nyquist curve of ; ’s roots are located in the open left-half plane of the complex field. In other words, the group consensus can be reached in this situation. By the Gerschgorin disk theorem, one can obtain When , it follows Let . Since the point , cannot be encircled in , , then the following inequality is obtained: Based on the Euler formula and from (19), we have Thus, we can get It is easy to know that is monotonically increasing for . Thus, we obtain Here, as the control parameter is positive, the following inequation can be derived: Obviously, (23) can hold when these two inequations are satisfied: Based on (24), we can get because . According to (25), it follows that . Since , can be established.
Similarly, when , we can get the following inequation: Similarly, the point , cannot be encircled in , , thus it follows that Define , then from (27), which leads to After some manipulations, we have From (29), we know that will gradually increase as increases. Thus, is obtained. As , can be established.

In summary, that finishes the Proof 1 of Theorem 1.

Corollary 1. To the systems (7), if its topology is a bipartite digraph and contains a directed spanning tree, the systems’ couple-group consensus is said to be realized asymptotically if this two conditions hold: and , where and .

By Lemma 1 and combining the Proof 1 of Theorem 1, we can get the results in Corollary 1.

Remark 1. From the results in Theorem 1, they reveal that the coupling weights and the control parameters play key roles in systems’ couple-group consensus. However, communication delays have no effect on the realization of group consensus. Furthermore, they also imply that the smaller either the coupling weights between the agents or the control parameters, the greater the input time delays of the system can be tolerated.

Remark 2. The proposed protocol (7) is constructed by employing the competitive interaction among the agents. As nearly all of the relevant works rely on the collaborative relationship, such as in [1528], this paper investigates the group consensus of the heterogeneous complex systems from a new perspective. At the same time, the protocol (7) has no virtual velocity in first-order agents’ dynamics, which is included in some existing articles (e.g., in [20, 2628]) for the convenience of analysis, while it will cause extra calculations and degrade the flexibility of the systems. As known, relaxing some conditions may lead to a more challenging job.

Remark 3. Different from the works in [20, 2529], we further relax the following two conservative prerequisites: in-degree balance and the geometric multiplicity of the zero eigenvalue of the systems’ Laplacian matrix have no less than 2. As known, in-degree balance means that the interaction among the agents in different clusters is offset. That is to say, there is no actual communication between the subsystems [8]. It is a very restrictive condition. Meanwhile, the second assumption also makes some limitations on the topology of the systems. Furthermore, the existing researches either took no time delay into consideration, such as in [2427, 29], or merely consider the influence of input delay [28].

Next, the case of different input and communication delays will be discussed.

On the basis of (7), the systems models can be extended as follows: where is the input delay of agent and is the communication delay between the agents and .

Theorem 2. To the MASs (30), if systems’ topology is an undirected bipartite graph, couple-group consensus can be realized if the following two conditions hold: and if , ; otherwise, , , where and .

Corollary 2. To the MASs (30) with a directed bipartite topology and which contains a spanning tree, systems’ couple-group consensus can be realized asymptotically when these two conditions hold: and if , ; otherwise, , , where and .

The proof is omitted here due to the limitation of space.

Remark 4. The results in Theorem 2 show that the upper bound of the input time delay will vary with the different dynamics of the agents and decided by the control parameter and coupling weights between the agents with same dynamic. Similarly, communication delays between agents have no effect on the achievement of systems’ couple-group consensus.

Remark 5. The system topologies we considered in this paper are either an undirected bipartite graph or a directed bipartite graph containing a directed spanning tree. They seem to be a specific topology. But as a matter of fact, the systems’ group consensus can usually be reached with the help of some stronger conditions. For instance, in [20, 2529], the systems’ topology is also an undirected graph or a digraph containing a spanning tree. At the same time, for the purpose of realizing the group consensus, some extra assumptions are needed, which have been mentioned in Remarks 1 and 2. It will be a challenge work to investigate the group consensus problem of a class of heterogeneous MASs in a more general condition.

4. Simulation

In this section, the correctness of the obtained results will be illustrated by several simulations.

Regard the heterogeneous systems with bipartite topology as Figure 1. Hereinto, agents 1 and 2 and 3, 4, and 5 belong to the subgroups and , individually. For the generality, agents 1 and 5 are represented as second-order agents and the first-order agents include the remaining agents 2, 3, and 4. Therefore, each subgroup in Figure 1 is designed as heterogeneous.

Remark 6. Distinguish from the cases in [2426, 28], the dynamics of the agents in this paper are not restricted to be homogeneous within the same subgroup. Obviously, that is the special case of ours.

For simplicity, we set and select , , and . From the systems’ topology shown in Figure 1, it is easy to know that , , , , and .

Example 1. According to the group consensus criteria proposed in Theorem 1, the range of input time delay is calculated as . In this case, we select the identical input time delay as  s. Here, the conditions of Theorem 1 are now all satisfied. From the results in Theorem 1, we know that communication delay does not affect the systems’ couple-group consensus. Hence, for convenience, their values are set the same. Meanwhile, several different cases are considered to discover the impact of the delays including the communication and input delays on the systems’ convergence rate. Figures 2 and 3 show the trajectories of the agents in the system (7). They indicate that all agents of the heterogeneous system converge to two reverse subgroups, that is, the couple-group consensus is realized.

Remark 7. The results in Figures 2 and 3 reveal that the communication and input time delays are both capable to affect the trajectories of the agents. To illustrate, the shorter the delays, the faster the system converges. Hence, we can improve the convergence rate by reducing the time delays including communication time delays, the input ones, or both of them.

Example 2. From Theorem 2 and according to the given parameters, it has , , , , and . In the simulation, we choose  s,  s,  s,  s, and  s. Clearly, the conditions in Theorem 2 are all hold. Similar to the Example 1, we also set several sets of time delays, then the agents’ trajectories in the systems (30) under the influence of different input and communication time delays are illustrated as Figures 4 and 5. From them, we know that the couple-group consensus of each case is realized, respectively.

Example 3. Suppose the topology of the heterogeneous systems (7) is shown in Figure 6, which contains a directed spanning tree. We set the input time delays as  s that satisfies all the conditions proposed in Corollary 1. According to the trajectories of the agents shown in Figure 7, one can know that the couple-group consensus is achieved asymptotically. Similarly, regard Figure 6 as the topology of the systems (30). And set  s,  s,  s,  s, and  s that follow the conditions in Corollary 2. From the trajectories of the agents shown in Figure 8, the couple-group consensus with the influence of different input time delays is also achieved.

5. Conclusion

Considering the influence of both communication and input delays, an original distributed coordination control protocol based on agents’ competitive interaction is designed for realizing the consensus of heterogeneous systems. By utilizing matrix theory and general Nyquist criteria, some sufficient criteria as well as the input delays’ upper bound are theoretically presented. From the results, we find that the control parameters of systems, the coupling weights between the agents, and the input delay play key roles on the achievement of couple-group consensus of the heterogeneous systems, while communication delays are independent with it. Furthermore, the simulation results show that the systems’ convergence rate can be improved by reducing the time delays including communication, input, or both of them. In the future, this issues for the heterogeneous and/or hybrid MASs with switching interaction topology can be extended.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant nos. 61572091, 61673080, and 61876200, the Natural Science Foundation Project of Chongqing Science and Technology Commission under Grant no. cstc2018jcyjAX0112, the Key Theme Special Project of Chongqing Science and Technology Commission under Grant nos. cstc2017zdcy-zdyfx0091 and cstc2017rgzn-zdyfx0022, and the Doctoral Start-up Foundation of Chongqing University of Posts and Telecommunications under Grant A2014-20.

References

  1. R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1520–1533, 2004. View at: Publisher Site | Google Scholar
  2. F. Xiao, T. Chen, and H. Gao, “Synchronous hybrid event- and time-driven consensus in multiagent networks with time delays,” IEEE Transactions on Cybernetics, vol. 46, no. 5, pp. 1165–1174, 2016. View at: Publisher Site | Google Scholar
  3. H. Li, G. Chen, T. Huang, Z. Dong, W. Zhu, and L. Gao, “Event-triggered distributed average consensus over directed digital networks with limited communication bandwidth,” IEEE Transactions on Cybernetics, vol. 46, no. 12, pp. 3098–3110, 2016. View at: Publisher Site | Google Scholar
  4. Y. Chen, H. Dong, J. Lu, X. Sun, and K. Liu, “Robust consensus of nonlinear multiagent systems with switching topology and bounded noises,” IEEE Transactions on Cybernetics, vol. 46, no. 6, pp. 1276–1285, 2016. View at: Publisher Site | Google Scholar
  5. X. Wu, Y. Tang, J. Cao, and W. Zhang, “Distributed consensus of stochastic delayed multi-agent systems under asynchronous switching,” IEEE Transactions on Cybernetics, vol. 46, no. 8, pp. 1817–1827, 2016. View at: Publisher Site | Google Scholar
  6. J. Yu and L. Wang, “Group consensus in multi-agent systems with switching topologies and communication delays,” Systems & Control Letters, vol. 59, no. 6, pp. 340–348, 2010. View at: Publisher Site | Google Scholar
  7. Y. Han, W. Lu, and T. Chen, “Cluster consensus in discrete-time networks of multiagents with inter-cluster nonidentical inputs,” IEEE Transactions on Neural Networks and Learning Systems, vol. 24, no. 4, pp. 566–578, 2013. View at: Publisher Site | Google Scholar
  8. L. Ji, Q. Liu, and X. Liao, “On reaching group consensus for linearly coupled multi-agent networks,” Information Sciences, vol. 287, pp. 1–12, 2014. View at: Publisher Site | Google Scholar
  9. X. Liao and L. Ji, “On pinning group consensus for dynamical multi-agent networks with general connected topology,” Neurocomputing, vol. 135, pp. 262–267, 2014. View at: Publisher Site | Google Scholar
  10. Q. Wang and Y. Wang, “Cluster synchronization of a class of multi-agent systems with a bipartite graph topology,” Science China Information Sciences, vol. 57, no. 1, pp. 1–11, 2014. View at: Publisher Site | Google Scholar
  11. Y. Han, W. Lu, and T. Chen, “Achieving cluster consensus in continuous-time networks of multi-agents with inter-cluster non-identical inputs,” IEEE Transactions on Automatic Control, vol. 60, no. 3, pp. 793–798, 2015. View at: Publisher Site | Google Scholar
  12. Y. Shang, “Group consensus of multi-agent systems in directed networks with noises and time delays,” International Journal of Systems Science, vol. 46, no. 14, pp. 2481–2492, 2015. View at: Publisher Site | Google Scholar
  13. W. Xu and D. W. C. Ho, “Clustered event-triggered consensus analysis: an impulsive framework,” IEEE Transactions on Industrial Electronics, vol. 63, no. 11, pp. 7133–7143, 2016. View at: Publisher Site | Google Scholar
  14. H.-x. Hu, W. Yu, G. Wen, Q. Xuan, and J. Cao, “Reverse group consensus of multi-agent systems in the cooperation-competition network,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 63, no. 11, pp. 2036–2047, 2016. View at: Publisher Site | Google Scholar
  15. J. Lunze, “Synchronization of heterogeneous agents,” IEEE Transactions on Automatic Control, vol. 57, no. 11, pp. 2885–2890, 2012. View at: Publisher Site | Google Scholar
  16. L. Wang, M. Z. Q. Chen, and Q.-G. Wang, “Bounded synchronization of a heterogeneous complex switched network,” Automatica, vol. 56, pp. 19–24, 2015. View at: Publisher Site | Google Scholar
  17. B. Cui, T. Ma, Y. Song, C. Feng, F. L. Lewis, and C. Zhao, “Distributed adaptive consensus control of heterogeneous multi-agent chaotic systems with unknown time delays,” IET Control Theory & Applications, vol. 9, no. 16, pp. 2414–2422, 2015. View at: Publisher Site | Google Scholar
  18. D. Goldin and J. Raisch, “Consensus for agents with double integrator dynamics in heterogeneous networks,” Asian Journal of Control, vol. 16, no. 1, pp. 30–39, 2014. View at: Publisher Site | Google Scholar
  19. L. Wang, M. Z. Q. Chen, W.-j. Feng, and Q.-g. Wang, “Global bounded consensus in heterogeneous multi-agent systems with directed communication graph,” IET Control Theory & Applications, vol. 9, no. 1, pp. 147–152, 2015. View at: Publisher Site | Google Scholar
  20. C.-L. Liu and F. Liu, “Dynamical consensus seeking of heterogeneous multi-agent systems under input delays,” International Journal of Communication Systems, vol. 26, no. 10, pp. 1243–1258, 2013. View at: Publisher Site | Google Scholar
  21. Y.-J. Sun, G.-l. Zhang, and J. Zeng, “Consensus analysis for a class of heterogeneous multiagent systems with time delay based on frequency domain method,” Mathematical Problems in Engineering, vol. 2014, Article ID 248684, 7 pages, 2014. View at: Publisher Site | Google Scholar
  22. J. M. Kim, Y. H. Choi, and J. B. Park, “Leaderless and leader-following consensus for heterogeneous multi-agent systems with random link failures,” IET Control Theory & Applications, vol. 8, no. 1, pp. 51–60, 2014. View at: Publisher Site | Google Scholar
  23. S. Li, G. Feng, X. Luo, and X. Guan, “Output consensus of heterogeneous linear discrete-time multiagent systems with structural uncertainties,” IEEE Transactions on Cybernetics, vol. 45, no. 12, pp. 2868–2879, 2015. View at: Publisher Site | Google Scholar
  24. H. Hu, W. Yu, Q. Xuan, C. Zhang, and G. Xie, “Group consensus for heterogeneous multi-agent systems with parametric uncertainties,” Neurocomputing, vol. 142, pp. 383–392, 2014. View at: Publisher Site | Google Scholar
  25. Y. Zheng and L. Wang, “A novel group consensus protocol for heterogeneous multi-agent systems,” International Journal of Control, vol. 88, no. 11, pp. 2347–2353, 2015. View at: Publisher Site | Google Scholar
  26. C. Liu, Q. Zhou, and X. Hu, “Group consensus of heterogeneous multi-agent systems with fixed topologies,” International Journal of Intelligent Computing and Cybernetics, vol. 8, no. 4, pp. 294–311, 2015. View at: Publisher Site | Google Scholar
  27. G. Wen, J. Huang, C. Wang, Z. Chen, and Z. Peng, “Group consensus control for heterogeneous multi-agent systems with fixed and switching topologies,” International Journal of Control, vol. 89, no. 2, pp. 259–269, 2016. View at: Publisher Site | Google Scholar
  28. G. Wen, Y. Yu, Z. Peng, and H. Wang, “Dynamical group consensus of heterogenous multi-agent systems with input time delays,” Neurocomputing, vol. 175, pp. 278–286, 2016. View at: Publisher Site | Google Scholar
  29. J. Qin, Q. Ma, H. Gao, Y. Shi, and Y. Kang, “On group synchronization for interacting clusters of heterogeneous systems,” IEEE Transactions on Cybernetics, vol. 47, no. 12, pp. 4122–4133, 2017. View at: Publisher Site | Google Scholar

Copyright © 2018 Lianghao Ji 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.


More related articles

363 Views | 242 Downloads | 1 Citation
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.