Elsevier

Neurocomputing

Volume 422, 21 January 2021, Pages 277-286
Neurocomputing

The variant d-path Laplacian based consensus protocols for networked harmonic oscillators

https://doi.org/10.1016/j.neucom.2020.09.053Get rights and content

Abstract

In this paper, we address the consensus protocol design problem of the networked harmonic oscillators interacting through a directed graph with peer pressure from the perspective of the output information, in which the peer pressure is in the expression of the variant d-path Laplacian. For networked harmonic oscillators without time delay, sufficient conditions in terms of coupling strength are given, while for networked harmonic oscillators with time delay, sufficient conditions in the term of coupling strength and the upper bound on the time delay are proposed. Finally, simulation examples are provided to verify the theoretical results.

Introduction

Nowadays, the research on cooperation control of networked multi-agent systems (MASs) absorb considerable attention from different fields, such as traffic control, formation flying, robotic sensor networks, automated web navigation, and medical quality [1], [2], [3], [4], [5]. Consensus, as one of the typical cooperation behaviors, means that all agents in a MAS can achieve certain common interest by sharing information with the neighbors. To date, many techniques have been introduced to the investigations on the consensus of MASs [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. Ma et al. [6] investigated the influence of communication topology on the negotiation of a linear MAS. In [7], the authors studied the consensus of a MAS with nonlinear dynamics and directed topologies. Then, Han and Li extended the results in [7] to the second-order consensus problems of discrete-time nonlinear systems in [8]. In addition, communication time delay, which is induced by the limitation of communication ranges or physical devices, has also drawn special attention due to [9], [10], [11], [12]. Lin et al. declared that second-order systems with dynamically varying topologies can tolerate any boundary time delay in [13], while in [14], it showed that the consensus cannot be achieved if the time delay is not mutually prime with the cycle of the communication topology. In particular, distributed optimization in multi-agent networks is widely used in data regression and resource allocation. Shi et al. studied the augmented Lagrange algorithm for distributed optimization in [17]. Droge et al. explored the relationship between dual decomposition and the consensus-based distributed optimisation in [18].

As stated in [19], [20], the harmonic oscillators play a fundamental role in some physical systems. Hence, there are many literature about the coordination of networked harmonic oscillators or coupled harmonic oscillators, which means a series of harmonic oscillators communicating with each other through a network. In 2006, Olfati et al. investigated the swarm of sphere system with polynomial dynamical in [21], which exhibited synchronization behavior closely relating to the coupled oscillators. In 2008, Ren studied the synchronization of coupled second-order linear harmonic oscillators with fixed network topology or switched network topology in [22]. In 2009, in [23], Su et al. relaxed the connectivity premise in [22] and solved the leader-following synchronization of coupled harmonic oscillators without any assumptions on the connectivity of the followers. Then, in 2010, Papachristodoulou et al. discussed the effect of time delay on the synchronization of harmonic oscillators in [24]. In recent years, more and more new elements are added into the coordination of networked harmonic oscillators. In [25], sampled data with measurement noise were considered in the networked harmonic oscillators, while in [26], input saturation and external disturbances were taken into consideration in the coordination of networked harmonic oscillators. On the other hand, the networked harmonic oscillator subsystems can also provide application examples for studying multi-agent systems with repetitive motions such as patrol, surveillance, artificial intelligent mobile robots, and driverless cars [27], [28], [29], [30]. To the best knowledge of the authors, the existing coordination results on networked harmonic oscillators all focus on the case that each harmonic oscillator communicates only with its direct neighbors.

However, in some cases, the indirect relationship among a community can also have a non-negligible or even amazing effect on the collective behavior of it. For example, as found in [31], the weak ties in the social networks can have a strong effect on each other. This famous phenomenon is called “the strong effectiveness of weak ties”, demonstrates the important function of indirect peers. A tie of two points is referred as a local bridge of degree n, where n represents the shortest path between these two points. For a fixed point, its weak tie is a point having large shortest path between them. A similar concept of weak tie in social networks is the peer pressure, which means that the decision of an individual not only can be influenced by individuals directly linked to it but also those undirectly linked to it [32]. In [32], Estrada and Vargas-Estrada showed that the peer pressure level can determine how fast a social group reaches consensus. In other words, the weak ties and the peer pressure mentioned above are both used to describe the long-range influences of a network. In [33], Gambuzza et al. introduced a mathematical tool known as the transformed d-path Laplacian defined in [32], [34] to describe the long-range interactions among the multiple agents interacting on an undirected graph. With the help of the transformed d-path Laplacian, some interesting results got. The definition of the transformed d-path Laplacian was given based on the irreducible shortest paths. Motivated by these findings, we will switch our attentions to the consensus problem of the networked harmonic oscillators communicating through a directed network with long-range influences. It is obvious that the long-range interaction on the directed graph is much more difficult than that on the undirected graph. Hence, the irreducible shortest paths based d-path Laplacian would be much more hard to accomplish.

In this paper, we will introduce a simple and straightforward approach to describe the long-range interactions among agents on a directed graph. That is the power function of the adjacent matrix. In order to distinguish it to the transformed d-path Laplacian defined for the long-rang interactions on undirected graphs in [32], [34], we name the long-range interactions on directed graphs by using the power function of the adjacent matrix as the variant d-path Laplacian. Concisely, the main significance is as follows.

  • First, a variant d-path Laplacian is defined to describe the long-range interactions in directed graphs. The introduction of the variant d-path Laplacian for the directed graphs induces two advantages. On the one hand, it simplifies the acquirement of the d-path Laplacian by using the adjacency matrix to replace the irreducible shortest paths. On the other hand, in the variant d-path Laplacian matrix associated with an unweighted directed graph, for two agents with multiple different shortest paths and of course these shortest paths having the same length, the corresponding element in the variant d-path Laplacian matrix is the number of the shortest paths, but not one as in the d-path Laplacian matrix defined by the references [32], [33]. This way of the definition in the variant d-path Laplacian matrix is more suitable in the social networks, since one person can put pressure or influence to another indirectly via two or more different people.

  • Second, consensus of the networked harmonic oscillators with directed long-range interactions will be solved by using the output information.

  • Third, the observer based consensus of the networked harmonic oscillators with time delay communicating through a directed graph with variant d-path Laplacian matrix will be solved. Sufficient conditions in term of the coupling strength and the upper bound on the time delay will be proposed.

The rest of this paper is organized as follows. Section II introduces the notations needed for this paper and proposes problem formulation which will be addressed. Section III is the main results of this paper, giving the sufficient conditions for networked harmonic oscillators with or without time delay. Section IV provides some numerical simulations to verify the theoretical results and Section V summarizes the paper.

Section snippets

Notation

Throughout the paper, IN denotes the identity matrix of order N. 1N is an N-dimensional column vector with all entries being 1. For any matrix A,AT and A represent the transposition and conjugated transposition of A, respectively. j is the imaginary unit.

Graph theory

We use a directed graph G=(υ,ϑ) to describe the directed interactions among agents, where υ={1,2,,N} represents the set of nodes and ϑυ×υ represents the set of edges. If there exists a directed edge from node i to node j, we denote (i,j)ϑ. A

Main results

Taking the fact that output information is easily acquired than state information, in this section, the observer-based control protocol on the basis of the variant d-path Laplacian is designed for the consensus of networked harmonic oscillators.

Simulations

In this section, we will propose some numerical simulations to verify the obtained theoretical achievement. Here, we choose α=1, then the parameter matrices of harmonic oscillators are given:A=01-10,B=01,C=10.

By Lemma 3 and Lemma 4, it follows from the Riccati Eq. (7), (9) thatP=3112,S-1=2226.

The topology of a directed multi-agent system with five agents is described by Fig. 2(a) and actual synchronization region of A-σBK is given in Fig. 2(b).

Example 1

For the variant d-path Laplacian based consensus

Conclusion

In this paper, a networked harmonic oscillators with directed long-distance interactions is considered and the concept of the variant d-path Laplacian is introduced to describe this kind of directed long-distance interactions. Based on the variant d-path Laplacian, an observer-based consensus protocol without or with time delay is constructed. Then, sufficient conditions for consensus are provided.

CRediT authorship contribution statement

Rong Fang: Conceptualization, Methodology, Software, Investigation, Writing - review & editing. Xiaoling Wang: Conceptualization, Methodology, Investigation, Writing - original draft. Housheng Su: Investigation, Writing - original draft. Min Xiao: Investigation, Writing - original draft.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 61803209, 61991412, 61873318, and 61573194, in part by the Natural Science Foundation of Jiangsu Province under Grant Nos. BK20180752 and BK20181389, in part by the University Science Research Project of Jiangsu Province under Grant No. 18KJB120006, in part by the Scientific Foundation of Nanjing University of Posts and Telecommunications (NUPTSF) under Grant No. NY218121, and the Program for

Rong Fang is currently a Master with the College of Automation and the College of Artificial Intelligence, Nanjing University of Posts and Telecommunications, Nanjing. Her current research interests include the coordination control of multi-agent systems and uncertain networked systems.

References (46)

  • W. Ren

    Synchronization of coupled harmonic oscillators with local interaction

    Automatica

    (2008)
  • H. Su et al.

    Synchronization of coupled harmonic oscillators in a dynamic proximity network

    Automatica

    (2009)
  • L.V. Gambuzza et al.

    Second-order consensus protocols based on transformed d-path laplacians

    Applied Mathematics and Computation

    (2019)
  • E. Estrada

    Path laplacian matrices: introduction and application to the analysis of consensus in networks

    Linear Algebra and its Applications

    (2012)
  • Z. Li et al.

    Distributed consensus of linear multi-agent systems with adaptive dynamic protocols

    Automatica

    (2013)
  • S.E. Tuna

    Synchronizing linear systems via partial-state coupling

    Automatica

    (2008)
  • H. Su et al.

    Adaptive second-order consensus of networked mobile agents with nonlinear dynamics

    Automatica

    (2011)
  • C. Wang et al.

    A new solution for freeway congestion: Cooperative speed limit control using distributed reinforcement learning

    IEEE Access

    (2019)
  • X. Dong et al.

    Time-varying formation tracking for second-order multi-agent systems subjected to switching topologies with application to quadrotor formation flying

    IEEE Transactions on Industrial Electronics

    (2016)
  • D. Saldana et al.

    Predicting environmental boundary behaviors with a mobile robot

    IEEE Robotics and Automation Letters

    (2016)
  • J. Varghese et al.

    Automated web navigation using multiagent adaptive dynamic programming

    IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans

    (2003)
  • C.-S. Wu et al.

    Optimizing medical data quality based on multiagent web service framework

    IEEE Transactions on Information Technology in Biomedicine

    (2012)
  • C.-Q. Ma et al.

    Necessary and sufficient conditions for consensusability of linear multi-agent systems

    IEEE Transactions on Automatic Control

    (2010)
  • Cited by (4)

    Rong Fang is currently a Master with the College of Automation and the College of Artificial Intelligence, Nanjing University of Posts and Telecommunications, Nanjing. Her current research interests include the coordination control of multi-agent systems and uncertain networked systems.

    Xiaoling Wang received the Ph.D. degree in control science and engineering from Shanghai Jiao Tong University, Shanghai, China, in 2017. In 2015–2017, she was a Research Assistant with the City University of Hong Kong and the University of Hong Kong. Since 2017, she has joined Nanjing University of Posts and Telecommunications. Her research interests include robust control of uncertain multi-agent systems and coordinated control of nonlinear networked systems.

    Housheng Su received his B.S. degree in automatic control and his M.S. degree in control theory and control engineering from Wuhan University of Technology, Wuhan, China, in 2002 and 2005, respectively, and his Ph.D. degree in control theory and control engineering from Shanghai Jiao Tong University, Shanghai, China, in 2008. From December 2008 to January 2010, he was a Postdoctoral researcher with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong. Since November 2014, he has been a full professor with the School of Automation, Huazhong University of Science and Technology, Wuhan, China. His research interests lie in the areas of multi-agent coordination control theory and its applications to autonomous robotics and mobile sensor networks.

    Min Xiao received the B.S. degree in mathematics and the M.S. degree in fundamental mathematics from Nanjing Normal University, Nanjing, China, in 1998 and 2001, respectively, and the Ph.D. degree in applied mathematics from Southeast University, Nanjing, in 2007. He was a Post-Doctoral Researcher or a Visiting Researcher with Southeast University, the City University of Hong Kong, Hong Kong, and Western Sydney University, Sydney, NSW, Australia. He is currently a Professor with the College of Automation and the College of Artificial Intelligence, Nanjing University of Posts and Telecommunications, Nanjing. His current research interests include memristor-based dynamical systems, fractional-order systems, networked control systems, control of bifurcation, intelligent logistics systems, and cyber–physical systems.

    View full text