Elsevier

Neurocomputing

Volume 339, 28 April 2019, Pages 1-9
Neurocomputing

Brief papers
η(t)-consensus of multi-agent systems with directed graphs via event-triggered principles

https://doi.org/10.1016/j.neucom.2019.01.061Get rights and content

Abstract

In this paper, we investigate η(t)-consensus problem of the multi-agent system with directed graph via event-triggered principles. The η(t)-consensus is defined as the difference between agents’ states less than a certain preassigned time-dependent bound function η(t). This concept can be regarded as a generalization of the existing concepts such as quasi-consensus/synchronization, where the state differences are requested below a certain constant bound, and exponential synchronization, where the state differences converge to zero exponentially, and more generally, μ-consensus, where the state differences converge to zero with diverse convergence ratios and patterns. The event-triggering principles are asynchronous for each agent and rely on the bound function η(t), as well as its derivative, the agent’s own state information and some algebraic information of the network, to determine the next trigger time. We prove that with a loose assumption of the bound function, if the graph is strongly connected graph or generally possesses a spanning tree, then the proposed principles realize η(t)-consensus of the multi-agent system, and moreover, the Zeno behavior is excluded. We give several numerical examples to illustrate the effectiveness of the theoretical results.

Introduction

Consensus is a classic problem in the coordination of multi-agent systems (MAS) in many fields such as computer science, management science and distributed computing [1], which is defined that the states of agents in a networked dynamical system reach an agreement. A protocol in the form of single-integrator agent with linear dynamical system can be written asx˙i(t)=jNi(xj(t)xi(t))i=1,,m.where xi(t)R is the state of agent i and Ni is the neighborhood set of agent i. The last decade has witnessed increasing interests in pursuing sufficient conditions that guarantee consensus in the system under diverse restrictions, for instance, undirected communication graph [2], directed graph [3], stochastic switching topology [3], [4] and time-delays [5].

Rather than continuous feedback that is efficient but could be expensive for real-world application, in recent decades, event-based control was proposed and showed a good capability to significantly reduce the frequency of information exchange between agents but maintain control performance [6], [7], [8], in particular compared with the periodic sampling strategy. This basic idea of event-triggering control is sampling and updating information based on whether certain measurement exceeds a threshold. As a result of these benefits, the event-trigger principles have been applied to diverse scenarios [9], [10] such as wireless communication technology [11], [12]. Among them, decentralized and distributed event-trigger principles were widely utilized in large-scale systems [13], [14], [15], [16], in particular, the distributed networked control systems [17]. Recently, a lot of literature was engaged in improving the performance of event-trigger principles. Towards distributed principle, Yi et al. [18] proposed event-trigger rules without global information of the communication graph; comparing with continuously monitoring events [19], [20] requires monitoring neighborhood information at discrete time points, which results in to less communication load; by periodic sampling, hybrid event-trigger rules in [21], [22] possessed lower bound of triggering interval, which excluded the Zeno behavior and reduced control updates; the novel self-trigger rule in [11] was been proved more efficient since the principle only collects data at triggered time; besides, there were widely discussions about the practical implementation issues of event-trigger principles. These surveys [23], [24] systematically reviewed the different mechanisms of these event-trigger principles and introduced their applications in various areas.

More related to the present paper, other than the majority of the existing papers concerned with undirected graphs, Yang et al. [25] gave the distributed principles for consensus of the system with directed graph structure; Yi et al. [26] discussed the consensus problem via the push-based (broadcasting) monitoring strategy with directed graph and gave event-trigger and self-trigger principles; Yi et al. [27] considered the pull-based (observation) monitoring strategy; On the other hand, besides exact-consensus that means the state differences between all agents of the system converging to zeros, in applications, there always arise the scenario that the differences between states of agents are less than a preassigned constant bound, known as quasi-consensus [28]. It should be highlighted that a conservative definition of consensus (quasi-consensus) implies less trigger event count than exact consensus. Furthermore, as illustrated in [29], consensus convergence rate is relevant to trigger event count, namely, higher convergence rate requires more trigger event count. Therefore, there is a trade-off between consensus performance and the load of triggering events.

Inspired by these works, towards a regulation of the balance between consensus performance in terms of preciseness and convergence rate and event triggering load, we propose an extended definition of η(t)-consensus that generalizes the quasi-consensus of system [30], by introducing a time-dependent bound, named bound function, η(t). Besides, this definition includes the exact-consensus by taking η(t) as a function that converges to zero. In addition, the convergence rates and patterns of exact-consensus can be formulated by this bound function, including exponential consensus, power-rate consensus [31], and could achieve other styles such as fixed-time consensus [32], [33]. Moreover, we utilizing the event-based monitoring strategy to propose the asynchronous event-trigger principles in the system with directed graph structure and the Zeno behaviors can be excluded.

Notation. ‖ · ‖ stands for the Euclidean norm of vector and the induced matrix norm. For any symmetry matrix X of m-dimensions, without loss of generality, we denote and sort its eigenvalues (counting the multiplicities) as λ1(X) ≤ λ2(X) ≤ ⋅⋅⋅ ≤ λm(X). Xij represents the element in the ith row and jth column of matrix X. Boldface 1=[1,,1] be the column vector whose dimension is m with all components equal to one. diag[x] is a diagonal matrix whose diagonal elements are corresponding to the components of vector x.

Section snippets

Problem formulation and preliminaries

In this paper, we consider the following linearly MAS with piecewise constant interactions as follows:dxi(t)dt=j=1mLij(xj(tkj(t)j)xi(tki(t)i)),tkj(t)jt<tkj(t)+1jfor i=1,,m, where L=[Lij]i,j=1mRm×m is the Laplacian matrix of the underlying directed graph G={V,E}, where V represents the agents’ set and EV × V is a directed edge set, in the term that if there is a directed edge from agent j to agent i, then Lij < 0; otherwise Lij=0. The diagonals of L are defined as Lii=j=1,jimLij. We

η(t)-consensus analysis

In this section, we provide an event-trigger rule for the next event trigger’s time that guarantees the system (1) achieved η(t)-consensus. We can rewrite the system (1) in the following form:x˙(t)=Lx^(t)where x(t)=[x1(t),,xm(t)] and x^(t)=[x1(tk1(t)1),,xm(tkm(t)m)]. Furthermore, set the matrices U=Ξξξ and Q=ΞLLΞ, which has UL=ΞL and for each xRm with x1, it holdsλ2(L^)xxxL^xλm(L^)xx,xQxλm(Q)xx

We choose the auxiliary function as:V(t)=(x(t)1·x¯(t))Ξ(x(t)1·x¯(t))=x(t)Ux(t)

Reducible graph with spanning tree

If the graph is reducible, with proper permutation, the Laplacian matrix L of the reducible graph can be written in the following Perron–Frobenius form [34]:L=[L1,1L1,2L1,K0L2,2L2,K00LK,K]where Lk,k represents the kth strongly connected component (SCC) of graph G, denoted as SCCk with the number of agents in such component nk. If G has a spanning tree, then for each k < K, there exists some j > k such that Lk,j ≠ 0. The result in Theorem 1 can be extended to the reducible graph by

Fixed-time η(t)-consensus

In many real-world applications, consensus is requested to be reached within a preassigned time-length but Theorem 1 only gives the result of asymptotic behavior. Hence, the fixed-time η(t)-consensus is considered in system (1) through adjusting the coupling strength c in following coupled system,dxi(t)dt=cj=1mLij(xj(tkj(t)j)xi(tki(t)i)),tkj(t)jt<tkj(t)+1jfor i=1,,m.

Theorem 4

Suppose that directed graph G is strongly connected, η(t) is a bounded, positive and differentiable function, and there exists

Numerical simulation

In this section, we present three numerical simulations to illustrate the main results. In these examples, the graph structure is shown in Fig. 1 with corresponding Laplacian matrix L=[1100121000111113]. It can be seen that the hypothesis in Theorem 1 are satisfied for all η(t) in following examples.

Direct calculations give λ2(L^)=0.1784, ξmax=0.5, ξmin=0.25, λm(Q)=0.8238. We set the initial state x(0)=[0.6346,0.7374,0.8311,0.2004] in all experiments, the upper bound of derivative of xi

Conclusions

In this paper, we proposed a general definition of η(t)-consensus that includes exact-consensus, in particular μ-consensus, and quasi-consensus in the previous literatures, and provided an asynchronous event-trigger principles to guarantee η(t)-consensus of the linear MAS with directed graph structures either strong connected or possessing spanning tree. We also proved exclusion of Zeno behaviors and extended to the fixed-time η(t)-consensus. Numerical experiments were presented to verify the

Acknowledgments

This work is jointly supported by the Natural Science Foundation of China under Grant No. 61673119 and Shanghai Municipal Science and Technology Major Project (No. 2018SHZDZX01) and ZHANGJIANG LAB.

Zongzong Lin received the B.S. degree in mathematics from Zhejiang Normal University, Zhejiang, China, in 2014. He is currently pursuing the Ph.D. degree in applied mathematics with the Fudan University, Shanghai, China. His current research interests include dynamical systems, complex networks and neural networks.

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    • A hybrid protocol for the average consensus of multi-agent systems with impulse time window

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      The communication edges based on the directed graphs are directional, and the communication edges based on the undirected graphs can be regarded as bidirectional. The authors in [9] mainly studied the consensus problem of MASs based on communication of undirected graph, and the authors in [10] mainly studied a MAS based on a directed graph as a communication method. The communication network topologies of MASs can also be divided into fixed topologies and switched topologies according to whether changes occur during the evolution of the entire MASs.

    Zongzong Lin received the B.S. degree in mathematics from Zhejiang Normal University, Zhejiang, China, in 2014. He is currently pursuing the Ph.D. degree in applied mathematics with the Fudan University, Shanghai, China. His current research interests include dynamical systems, complex networks and neural networks.

    Wenlian Lu received the B.S. degree in mathematics and the Ph.D. degree in applied mathematics from Fudan University, Shanghai, China, in 2000 and 2005, respectively. He was a postdoc fellow at the Max Planck Institute for Mathematics in the Science, Leipzig, Germany from 2005 to 2007, and a Marie-Curie International Incoming Research Fellow with the Department of Computer Sciences, University of Warwick, Coventry, UK from 2012 to 2014. He is currently a Professor with the School of Mathematical Sciences and the Institute for Science and Technology of Brain-Inspired AI, Fudan University. His current research interests include neural networks, computational systems biology, nonlinear dynamical systems, and complex systems. He has served as an Associated Editor for the IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS since 2013 and Neurocomputing from 2010 to 2015.

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