Elsevier

Neurocomputing

Volume 379, 28 February 2020, Pages 214-226
Neurocomputing

Fixed-time event-triggered synchronization of a multilayer Kuramoto-oscillator network

https://doi.org/10.1016/j.neucom.2019.10.040Get rights and content

Abstract

This paper investigates the synchronization problem of the Kuramoto-oscillator network with non-identical oscillators. The fixed-time event-triggered synchronization control strategies are developed for phase agreement and frequency synchronization under both continuous and intermittent communication. With the developed fixed-time controller, the synchronization can be achieved within a pre-defined time for any initial phase of each oscillator. The event-triggered mechanism avoids continuous controller update and data transmission, which significantly saves the computation and communication resources. Furthermore, theoretical analysis shows that the fixed-time convergence can be guaranteed and the Zeno behavior is avoided by the proposed methods. The numerical simulations of each situation also verify the effectiveness of the proposed synchronization control strategies.

Introduction

Synchronization is a common but important phenomenon in human society and nature environment [1], [2], [3], which can be easily observed in many different scenarios, such as circadian rhythm of all living beings [4], [5], neural activities in brains [6], [7], and flocking behaviors of animals [8]. Over the past decades, the synchronization of complex networked systems has attracted wide attention from researches in multiple fields, including biology, chemistry, and engineering [9], [10], [11], [12], [13]. Among the existing complex network models, the Kuramoto-oscillator network model (Kuramoto model) is an important basic coupled oscillator model. Especially, the unification of neuron signals in biology and the synchronization of the power grids can be modeled by the Kuramoto model and its variants [14], [15], [16]. The Kuramoto model consists of finite oscillators which are coupled with each other by periodic sinusoidal functions, and the oscillators could have different intrinsic natural frequencies [17]. The original Kuramoto model was firstly proposed by Winfree [18] and then was significantly extended by Kuramoto [19], [20]. In the past decades, the Kuramoto model has received wide attention in the control community.

For the generalized Kuramoto model, the synchronization control problem is a meaningful and important research topic. By now, scientists and researchers have been devoted to exploring the control method while considering various factors that could affect the synchronization [16], [21]. These factors include network topology [22], [23], oscillator dynamics [24], initial phase [25], coupling strength [26], communication delay [27] etc. For the topology factor, the phase locking phenomenon of the Kuramoto oscillators coupled with a chain or ring topology was discussed in [22] and [23]. Moreover, the graph theory was taken into consideration, such that the coupling topology can be regular or arbitrary [24], [28]. Chopra et al. [25] revealed that the Kuramoto-oscillator network can achieve exponential synchronization when the initial phase diameter is within π/2, and the coupling strength should exceed a certain threshold. In order to guarantee the emergence of synchronization for various coupling feedback laws, the adaptive coupling strength was considered to obtain synchronization of the Kuramoto-oscillator network in [26]. In [27], the sufficient conditions for both frequency synchronization and phase agreement were obtained with heterogeneous communication delays. Although there are plenty of results about the synchronization of Kuramoto-oscillator network, few concerned about how to achieve a faster convergence rate.

Build on top of the aforementioned results, researchers have been further trying to accelerate the synchronization process, because a higher synchronization rate means better dynamic performance in practical applications. To improve this, the finite-time and fixed-time control methods were applied owing to their faster convergence rate and disturbance rejection ability [29], [30], [31]. In the latest results of neural network synchronization problem, accelerating the synchronization process also attracts lots of attentions, and flourish literatures are published. For an instance, the finite-time and fixed-time problems were both analyzed in [32], [33], [34], some complicated problems such as time-delay and impulsive effects are also considered. Particularly, the fixed-time control method can achieve synchronization within the desired time without any limitation on the range of the initial phase, which is a significant improvement for the Kuramoto model. This is because the traditional control methods can only achieve synchronization asymptotically while also have limits on the initial phase. However, in practical applications of most Kuramoto models, the initial phase usually follows a distribution instead of known exactly. Although the finite-time and fixed-time synchronization problems of multi-agent systems have been extensively studied [35], [36], [37], only a few theoretical results are related to the Kuramoto model. In [38], the finite-time synchronization of the Kuramoto-oscillator network was studied. The convergence time has an upper bound with the initial phase lie in a half-circle. A novel multilayer control algorithm was developed in [39] to achieve finite/fixed-time synchronization of the Kuramoto-oscillator network. Besides, if the topology of the control layer is not the same as the Kuramoto-oscillator layer, the connectivity between control nodes may be sparser. This kind of multilayer scheme has been presented in recent work [40], [41], [42]. The above-mentioned researches are all time-triggered. That is to say, the controllers receive the information from sensors and generate control input signals continuously. This operation usually takes up a lot of resources of computation and data transmission.

Recently, it is popular to introduce the event-triggered mechanism in synchronization control of complex network systems in order to save the computation and communication resources [43], [44], [45], [46], [47]. Under the event-triggered scheme, the controller is only updated when a certain condition is triggered, which effectively avoid continuous sensing and computation. In addition, the consumption of communication can be significantly reduced if each node only transmits local states to neighbors when necessary [48], [49]. For instance, the power grid which can be modeled with the Kuramoto-oscillator network has practical limits on data transmission capacity [39], [50], [51]. However, the researchers and engineers prefer the states of all nodes back to synchronization expeditiously with least resource consumption after perturbations. Consequently, it is critical to develop a fast and resource-saving synchronization approach for the generalized Kuramoto-oscillator network.

In this paper, we develop an event-triggered based fixed-time synchronization control scheme for a generalized Kuramoto-oscillator network with non-identical oscillators. To the best of the authors’ knowledge, this architecture has not been considered and investigated yet. Inspired by the multilayer scheme presented in [39], [40], we represent the network system using a multilayer structure. Different from them, the same topology for all layers are used. Based on this structure, we investigated the phase agreement and frequency synchronization problems of the Kuramoto-oscillator network under both continuous and intermittent communication. The distributed fixed-time and event-triggered controllers are designed for each situation. The theoretical proof of the fixed-time convergence for each situation is given by Lyapunov analysis. Moreover, the simulation results show that the proposed method can achieve fixed-time synchronization with less controller updating and information transmission. The contributions of this paper are summarized as follows:

  • 1.

    A set of fixed-time event-triggered synchronization control strategies for the Kuramoto-oscillator network model are developed. It is the first time that the Kuramoto model being investigated under both fixed-time and event-triggered control schemes.

  • 2.

    With the fixed-time control scheme, the developed strategies could achieve phase agreement and frequency synchronization within an arbitrary pre-defined time, while not requires any limitation on the initial phases of the oscillators.

  • 3.

    The event-triggered mechanism is adopted in the synchronization strategies, which avoids periodically updates of the controllers. Different from the classical event-triggered controller, our improved event-triggered method could anticipate the next triggered instant without real-time states of neighbor nodes. This improvement effectively avoids continuous data transmitting, which further saves the communication and computation resources.

The remainder of this paper is outlined as follows. In Section 2, some important notations, definitions, basics of the graph theory, as well as the multilayer Kuramoto-oscillator model are given. In Section 3, the event-triggered fixed-time phase agreement control scheme are presented. Then, the frequency synchronization problem is investigated in Section 4. Section 5 provides simulations to verify the proposed control methods. The conclusion is drawn in Section 6.

Section snippets

Notations

The following notations are used throughout the paper. Let R,R+,RN,RN×N be the one-dimensional real space, one-dimensional positive real space, N-dimensional real vector space, N × N real matrix space, respectively. Notation 1N denotes a N × 1 column vector with all entries are 1, and IN denotes the identity matrix of dimension N. Let vector x=[x1,x2,,xN]T, where xiR, and xη=[x1η,x2η,,xNη]TRN, where ηR. In addition, sign( · ) denotes the signum function:sign(x)={1,x>00,x=01,x<0.

Algebraic graph theory

Consider a

Fixed-time Event-triggered Non-identical Kuramoto-oscillator Phase Agreement

In this section, two types of event-triggered control approaches are proposed to achieve phase agreement for Kuramoto-oscillator network in a fixed time. These two controllers are designed for continuous communication and intermittent communication, respectively.

Fixed-time event-triggered non-identical Kuramoto-oscillator Frequency Synchronization

In this section, the distributed controllers are designed to make the Kuramoto-oscillator network obtain frequency synchronization. Similar to Section 3, two types of controller are proposed, and they are under event-triggered strategy with continuous communication and intermittent communication respectively.

Simulation results

In this section, simulation results are laid out to demonstrate the availability and effectiveness of the proposed distributed control scheme. The whole control structure is shown in Fig. 1.

The upper layer is the network of controllers, and the lower layer is the Kuramoto-oscillator network layer. It can be seen that the two layers have the same topology. The five nodes of the Kuramoto-oscillator network with non-identical natural frequency are linked by an undirected graph. The graph applied

Conclusion

The phase agreement and frequency synchronization problems of a generalized Kuramoto-oscillator network model have been addressed in this paper. A set of fixed-time event-triggered control strategies with a multilayer structure have been proposed. The upper bound of the settling time for synchronization can be designed in advance for any initial states of the oscillators. The improved event-triggered methods could anticipate the next triggered time, which has successfully avoided continuous

Declaration of Competing Interest

The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Jia Sun received the B.S. degree in automation from the School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, China, in 2015, where she is currently pursuing the Ph.D. degree in control science and engineering with the School of Automation and Electrical Engineering. She has been a visiting Ph.D. student with the Department of Electrical, Computer, and Biomedical Engineering, University of Rhode Island, Kingston, RI, USA, from 2017 to 2018. Her

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    Jia Sun received the B.S. degree in automation from the School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, China, in 2015, where she is currently pursuing the Ph.D. degree in control science and engineering with the School of Automation and Electrical Engineering. She has been a visiting Ph.D. student with the Department of Electrical, Computer, and Biomedical Engineering, University of Rhode Island, Kingston, RI, USA, from 2017 to 2018. Her current research interests include quadrotor unmanned aerial vehicle, multi-agent systems, robust nonlinear control, and reinforcement learning.

    Jian Liu received his B.S degree from the School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, China, in 2015. He is a Ph.D. candidate in School of Automation and Electrical Engineering, University of Science and Technology Beijing. From September 2017 to September 2018, He was a joint training student with the Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA. His current research interests include multi-agent systems, nonlinear control, event-triggered control.

    Yuanda Wang received the B.S. degree in automation from the Nanjing University of Information Science and Technology, Nanjing, China, in 2014. He is currently pursuing the Ph.D. degree in control science and engineering with the School of Automation, Southeast University, Nanjing, China. He has been a visiting Ph.D. student with the Department of Electrical, Computer, and Biomedical Engineering, University of Rhode Island, Kingston, RI, USA, from 2016 to 2018. His current research interests include deep reinforcement learning, neural networks, and multi-agent systems.

    Yao Yu received her B.S. degree from Department of Control Science and Engineering, Huazhong University of Science and Technology in 2004, the M.S. and Ph.D. degrees from Department of Automation, Tsinghua University in 2010. She was a postdoc with Tsinghua University. Currently, she is an Associate Professor with School of Automation and Electrical Engineering, University of Science and Technology Beijing. Her current research interests include nonlinear control, robust control and time-delay systems.

    Changyin Sun received his B.S. degree from College of Mathematics, Sichuan University, Chengdu, China, in 1996, and the M.S. and Ph.D. degrees in electrical engineering from Southeast University, Nanjing, China, in 2001 and 2003, respectively. He is a distinguished Professor with School of Automation, Southeast University, Nanjing, China. His current research interests include intelligent control, flight control, pattern recognition, and optimal theory. Prof. Sun is an Associate Editor of the IEEE Transactions on Neural Networks and Learning Systems.

    This work is supported by National Natural Science Foundation of China (Grant nos. 61520106009, 61533008, 61703037).

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