Event-based discrete-time multi-agent consensus over signed digraphs with communication delays

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Abstract

The ability to ensure the desired performance of the cooperative-antagonistic multi-agent networks (MANs) in the presence of communication constraints is an important task in many applications of real systems. In this paper, under the proposed event-triggered condition (ETC), different types of consensus are obtained under different network topology. We concentrates on the event-based bipartite consensus. It is shown that under the proposed ETC (i) the addressed cooperative-antagonistic network with arbitrary communication delays reaches bipartite consensus provided that the network is balanced; (ii) the network model reaches zero if the network is unbalanced. Further, to avoid the continuously verifying the triggering condition, a self-triggered algorithm is proposed for realizing the bipartite consensus of the network model. A numerical example is given to illustrate the effectiveness of the theoretical results.

Introduction

The consensus of multi-agent networks (MANs) has been a hot topic due to its wide applications (see, e.g., [1], [2], [3], [4], [5], [6]). Up to now, a large number of related results have been obtained to solve consensus problem. The common feature of these works lies that consensus is achieved through collaboration. However, it can be found that both collaborative and adversary relationships exist in many real world MANs. Social network, which is represented as signed graphs with negative edges weights is a typical example of MANs with antagonistic interactions [7], [8], [9], [10], [11], [12].

In [13], by investigating the Laplacian matrix of signed graphs, it is proved that bipartite consensus of MANs can be achieved, i.e, all agents can converge to a consensus value which has the same modulus and different sign. In [14], it is proved that interval bipartite consensus instead of bipartite consensus will be achieved if the network topology only containing a spanning tree for the signed networks. In [15], [16], the bipartite multi-agent consensus problems with directed and undirected communication topologies respectively are studied for high-order systems. In [17], [18], bipartite consensus/synchroniztion of cooperative-competitive complex networks under switching topologies are investigated.

In MANs, the key factor to realize expected collective behaviors is the successful and efficient communication among agents. Since the bandwidth of the communication channels are always limited, it is unavoidable to impose the energy requirement to the MANs. To satisfy the energy saving constraint of the dynamical systems, a new control method, namely, event-triggered control(ETC) (see, e.g., [19]) was proposed in recent years. ETC can be defined as the control signals are kept unchanged until the certain condition is violated and then the control signal will be updated. Compared with other control scheme, ETC can greatly decreasing the system’s unnecessary execution behavior while maintaining acceptable levels of the performance of MANs, i.e., ETC can save energy as well as resources of MANs. Owing to its great potential for applications, up to now, there are a lot of papers on distributed cooperative control of networked systems under event-triggered mechanism [21], [22], [23], [24], [25], [26], [27]. In [19], centralised and distributed triggering conditions to insure the network consensus are proposed, respectively. Event-based synchronization of complex networks was studied in [24]. In [28], event-based bipartite consensus on signed networks is investigated. To void verifying the event-triggered condition at each time instant, self-triggered communication schemes in MANs is proposed recently, which means the agents broadcast information only at precomputed triggering time [19]. Compared with ETC, under self-triggered control (STC), the agents in MANs do not need to verify the triggering condition for each time instants [19], [20], [21]. It is claimed in [20] that STC has high robustness against various uncertainties and disturbances.

It can be observed that communication delays are often encountered in spreading information through complex networks [30], [31], [32], [33], [34], [35], [36], [37], [38]. The existence of time delays may cause poor performance of the dynamical systems and ignoring them may lead to incorrect analysis conclusions. Hence, communication delays should be considered in the design of the event-based consensus protocol. In [39], a discrete-time event-triggered scheme is proposed for studying multi-agent consensus problem with communication delays. In [40], a comparison principle method is proposed to deal with the event-based state estimation problem with time-varying delays.

Although much effort has been made to solve the multi-agent consensus problem under event-based mechanism, to the best of our knowledge, bipartite multi-agent consensus with signed digraphs and communication delays has not been investigated. A signed graph is normally classified as balanced and unbalanced graph (see next section for mathematical definitions). The main challenges of this paper are mainly twofold: (i) how to solve the bipartite consensus problem of MANs in the event-based setting with communication delays? (ii) how to analyze dynamical behavior of the MANs under the unbalanced topology and obtain the corresponding consensus results? This paper mainly concentrates on a unified approach to study the effect of communication delays and event-based communication strategy on discrete-time MANs over signed digraphs. The main contributions of this paper are given as follows:

  • A new bipartite consensus protocol considering event-based communication and time delay effect simultaneously is proposed. Under the balanced network topology, some event-based conditions are given to ensure the bipartite consensus.

  • To avoid the continuously verifying the triggering condition and realize the bipartite consensus of the signed multi-agent network, a distributed self-triggered algorithm is proposed.

  • For the unbalanced network topology, based on the Lyapunov functional technique, it is strictly proved that all the agents in the signed network reach zero under the proposed event-triggered condition.

The rest of this paper is organized as follows. In Section 2, some preliminaries on signed digraph are summarized. In Sections 3 and 4, bipartite consensus results under the event-trigger and self-trigger protocol protocol are discussed in detail. In Section 5, a simulation example is presented to demonstrate the correctness of theoretical results.

Notation: | · | denotes absolute value of real numbers. Matrix A=[aij]N×N, |A|=[|aij|]N×N. τ=max{τij,i,jN};Υτ={τ,τ+1,,0.}; Let X{ψ:ΥτR and ψ is bounded} with the norm defined by ψ=maxsΥτ|ψ(s)|.

Section snippets

Preliminaries and problem formulation

In this section, some preliminaries on signed digraphs will be given firstly and event-based bipartite consensus protocol for the discrete-time delayed MANs will be presented then.

Distributed event-based bipartite consensus

In this section, we will give the distributed event-based bipartite consensus criteria for the considered signed network model. We always assume that the network topology of the signed digraph is strongly connected in this paper. Let ξ={ξ1,ξ2,.,ξN} be the normalized left eigenvector of matrix |A| with respect to the eigenvalue 1. From Perron-Frobenius theorem (see [41]), we obtain that i=1Nξi=1 and ξi > 0.

Theorem 1

Consider the multi-agent system (1) with arbitrary finite communication delay τij under

Self-triggered approach

In Theorem 1, we have proved that the proposed event-based protocol is effective to realize the bipartite consensus of the network model. However, the triggering condition needs to be continuously verified for each agents. In this section, we aim to solve this difficult problem by designing a self-triggered algorithm, i.e., the next update time is precomputed based on predictions using the received data. Under the proposed self-triggered algorithm, the signal remains unchanged until next

Numerical example

Example 1

Consider a signed multi-agent network with structurally balanced topology and structurally unbalanced topology, respectively (see Fig. 1). Set σ=0.9 in event-trigger condition (4) and the distinct communication delays are as follows:Γ=(τij)6×6=(020000003000100002002030000001000200).

According to Theorem 1, one can easily conclude that under the proposed event-triggered condition (4),

  • (i)

    the system with communication delays will achieve bipartite consensus when the network topology is shown in Fig. 1

Conclusion and discussions

In this paper, event-based bipartite consensus of discrete-time multi-agent system was studied. Communication delays and event-based strategy were considered in the network model. The proposed event-triggered protocol not only avoids the continuous communication between agents but also provides distributed method to transmit the information in the presence of time delays. It is shown that under the proposed event-triggered condition the bipartite consensus can be asymptotically achieved if the

Acknowledgments

This work was jointly supported by Research Grants Council of the Hong Kong Special Administrative Region, China (CityU 11200717), the National Natural Science Foundation of China under Grants 61503115, 61573102, the Fundamental Research Funds for the Central Universities under Grants JZ2017HGTB0188 and the Natural Science Foundation of Jiangsu Province of China under Grant BK20170019.

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