Event-based agreement protocols for complex networks with time delays under pinning control

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Abstract

Most pinning synchronization schemes for complex networks with or without time delays reported in the literature require continuous state measurements and control updates, which are not suitable for implementation on digital platforms. In this paper, we first establish some pinning synchronization criteria for complex networks modeled by directed graphs with both internal and coupling delays based on an event-triggered mechanism. We also propose the design method for control gains under some mild assumptions. The piecewise constant nature of control law can help reduce the number of updates significantly, thereby saving energy resources. Moreover, by obtaining an explicit lower bound on the inter-event time interval, we prove that Zeno behavior does not exist in the evolution of each node of the complex network. At last, simulations are provided which verify the effectiveness of the proposed controllers.

Introduction

A great deal of complex systems in real world, such as large-scale sensor networks, neural networks, ecosystems, social systems, WWW, genetic regulatory networks, electrical power grids, and so on, can be modeled by complex networks with the nodes representing individuals in the system and the edges representing the connections among them. A key problem in the field of complex networks that receives particular interest is how to drive all the nodes with nonlinear self dynamics (corresponding to ṡ=f(t,s(t))) to reach a prescribed state with only local controllers on a small fraction of nodes, which is known as a pinning synchronization problem. In past few years, there has been extensive literature addressing the pinning synchronization problem by employing different methods from different viewpoints [1], [2], [3], [4], [5].

Among the vast amount of literature on pinning synchronization control of complex networks, most work assumes that the state feedbacks and control updates of network nodes are based on the continuous-time updating scheme, which poses a strict requirement on on-board processors and actuators. Taking into account the large scale of a complex network, it is undesirable to equip each node with such devices. Thus it is reasonable to design discrete-time updating algorithms with less requirement of hardware resources. A classical approach to design discrete algorithms is time-scheduled control (sampled control) whose information exchange and control actions are determined by a constant sampling period. Along this direction, several papers [6], [7], [8] have studied the sampled-data synchronization control of complex networks. However, time-scheduled control might be conservative in terms of the number of control updates, since the constant sampling period has to guarantee stability in a worst-case scenario. To overcome above drawbacks of time-scheduled scheme, event-scheduled scheme is introduced. In comparison with time-scheduled scheme, the sampling time in event-scheduled approach is determined by certain events instead of fixed sampling time in the time-scheduled scheme, thus the number of control actions as well as energy consumption over the network can be significantly reduced. However, compared to the time-scheduled control, event-scheduled scheme requires a more delicate controller design to guarantee system stability. Moreover, how to exclude Zeno behavior [9] in the design of event-based controller makes the analysis more challenging.

We note that event-triggered control has been widely studied in a relevant topic namely the consensus of multi-agent systems (MASs) [10], [11], [12], [13], [14]. In [10], the authors first propose a distributed, event-based controller design method for single integrator networks. After then, [11] provides a combinational measurement approach for event design to further reduce the amount of controller updates. The work in [12] uses a periodically event-detecting scheme to relax the strict requirement of continuous-time event monitoring. Moreover, the authors of [13] introduce a new time-dependent trigger function which does not contain any widely-used state information and further provides analysis for double integrator agent dynamics. However, all of the above mentioned work take the integrator dynamics, which means the motion of the agent is only determined by the information exchange with neighbors. Different from the consensus of MASs, the synchronization for the network with nonlinear node dynamics depends not only on the couplings among the nodes, but also the self-dynamics governing the evolution of each isolated node. Because of this, most of the analytical methods adopted for event-based consensus protocol cannot be extended directly to the synchronization analysis. Currently there is little work on solving the pinning synchronization problem of complex networks via an event-triggered viewpoint except the one in [15]. The objective of [15] is to introduce the event-triggered mechanism into pinning control of complex networks but without any consideration on time delays. Time delays are inevitable in real-world complex networks due to the finite speed of signal transmission and traffic congestion and may result in instability of the whole network system. Furthermore, the controller updates in event-based algorithms are discrete and asynchronous, with much less information used in comparison with those in continuous-time algorithms. However, this fact also causes event-triggered algorithms to be more sensitive to time delays. In conclusion, it is important to consider the delay effects in designing event-triggered algorithms for complex networks.

Motivated by above discussions, this paper aims to design an event-based controller to pin a complex network with both internal and coupling delays to a homogeneous state. Note that two main types of totally different delays have been considered in literature, namely coupling delays in the network [16], [17], [18], [19] and internal delays existed inside the node dynamics [20], [21], [22], respectively. In this paper, we take into account both coupling and internal delays for more general purpose, which also complicates the convergence analysis. The second aim of this paper is to exclude the Zeno behavior for each node completely. Different from [15] while their work focused on the stability conditions without any words for the inter-event situations, we provide a detailed analysis on the exclusion of Zeno behavior. Our result provides a strict avoidance of Zeno behavior for each node at any finite time via a modification on common state-dependent threshold with an exponentially decaying offset.

The rest of this paper is organized as follows. In Section 2, we present some background and provide the system model for later analysis. In Section 3, we provide detailed stability analysis of this event-driven system including the exclusion of Zeno behavior. In Section 4, an example is provided to verify the results of theoretical analysis. Finally, Section 5 includes a summary of this paper and indicates further research topics.

Section snippets

Notations

The notations used throughout this paper are standard. Rn denotes the n-dimensional Euclidean space. Rm×n denotes the set of m×n real matrices. If M is a vector or matrix, its transpose is denoted by MT. For a square matrix N, denote its inverse by N1, denote its determinant by det(N) and its i-th smallest eigenvalue is denoted by λi(N). For a real symmetric matrix A, let λmin(A) and λmax(A) be its minimum and maximum eigenvalues, respectively. The notation diag(x) denotes a (block) diagonal

Event-triggered pining controllability analysis

In this section, we will investigate the synchronization criteria for the pinning-controlled network model (5). First, we follow the same definition of state measurement errors presented in [15]ei(t)=xi(tki)xi(t),i=1,,Neij(t)={xj(tki)xj(t)ifthereisacouplingfromnodejtonodei[0,,0n]Totherwiseeis(t)={s(tki)s(t)ifnodeiispinned[0,,0n]Totherwisefor t[tki,tk+1i]. Here we need to define a new error vector e˜i(t)=[ei1T(t),,eiNT(t)]T for node i which consists of eij(t).

The isolated trigger

Simulation

In this section, we present a simulation example to verify the effectiveness of the theoretical results.

We consider a directed network consisting of five identical delayed Chua׳s circuits. Each node is described by Eq. (5). The nonlinear function f in Eq. (5) which is given by [27], [29] has the following form:f(t,xi(t),xi(tτ1))={10(xi2xi1g(xi1)),xi1xi2+xi3,19.53xi20.1636xi33.906sin(0.5(xi1τ1)),where g(xi1)=0.7831xi10.3247(|xi1+1||xi11|). According to [27], one can choose the values

Conclusions

In this paper, we have shown that a complex network with time delays can be pinned to a prescribed state with a discrete, event-based controller under some mild assumptions. The network topology is directed, which is more general for modeling complex network systems in real world. We further propose some criteria for designing control gains and also prove that Zeno behavior will not appear in this hybrid system. Future research topics include further relaxation of the conservative requirement

Acknowledgment

The work of Q. Liu and C. Yu was supported in part by the Australian Research Council through the Discovery Project DP160104500 and DP130103610, a Queen Elizabeth II Fellowship under grant DP110100538 and the National Natural Science Foundation of China (grant 61375072) and a China Scholarship Council PhD scholarship.

The work of J. Qin was supported in part by the National Natural Science Foundation of China (grant 61473269), in part by the Program for New Century Excellent Talents in

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