Bounded synchronization of coupled Kuramoto oscillators with phase lags via distributed impulsive control☆
Introduction
In recent years, extensive interest has been devoted to consensus and synchronization problems for complex dynamical systems. The Kuramoto model [1] is regarded as one of the most representative models to characterize the synchronization of coupled oscillators [2], [3], [4]. It is broadly applied in various fields such as chemistry, biology, engineering and mathematics [5], [6], [7], [8], [9], [10], [11], [12], [13]. Especially, unification of brain signals [9] and the synchronization of the power grids which focus on the transient stability problems [4], [5], [6] are the typical applications of the model. Moreover, the power grids modeled as structure-preserving and network-reduced power system models can be referred to as coupled oscillators networks [14]. Therefore, considerable attention is being paid on this area. In the study of the Kuraomoto oscillators, synchronization is one of the most important topics. For example, an exact solution is given [15] for first-order synchronization transition in a generalized Kuramoto model. The synchronization condition is given [16] related to the size of the attractive and random coupling terms. The relationship of the critical coupling and synchronization is explored [17]. Although there are abundant good results about the synchronization of oscillators, but few of them discuss the synchronization with phase lags. Sometimes, the phase lags between oscillators need to be taken into consideration in practical problems, such as the lags between generators in power grids [14], and the lags between neurons [9].
Phase lag which occurs naturally in complex network models [18], [19], [20], [21] due to the delay of communication and the limit of the inherent structure was first introduced in Kuramoto model [22] to model the synchronized oscillators in which the common frequency differs from the average of the natural frequencies. Phase lag is equivalent in synchronized systems to delay couplings which occur in biological and other systems [23], and plays a significant role in the dynamics of the system and the properties of synchronized states. But the phase lag may be random with a boundary and very hard to control. As a result, the synchronization control problem for the Kuramoto model with random phase lags is of great significance.
The synchronization control problem for the generalized Kuramoto model is a new and urgent challenge topic. Recently, many progresses have been made in this topic. For example, Wang and Doyle [24] investigate the Kuramoto oscillator model in the presence of a pacemaker, providing both analytical results and simulation study. Using potential functions, Dong and Xue [25] signed gradient form of the Kuramoto model and give the finite-time synchronization conditions for identical and non-identical oscillators under some circumstances. However, these works only apply for the generalized Kuramoto model without phase lags. Very recently, Lohe [26] investigates the Sakaguchi–Kuramoto model with phase lag and treats it as a control strategy. In the study of the networks, bounded synchronization has also been studied [27]. For example, distributed impulsive control strategies to ensure quasi-synchronization has been given in [28], [29], [30]. Distributed impulsive control strategies [18], [28], [31] may be helpful to solve this question. What is more, these control strategies mentioned above all require continuous control which may result in much large amounts of communication bandwidth. Compared with the above continuous control strategies, the impulsive control strategy which only works at impulsive instants has advantages in some aspects, such as less occupation of communication bandwidth and simpler implementation [31], [32], [33].
Inspired by above-mentioned works, this paper investigates the problem of synchronization in the Kuramoto model with phase lags via distributed impulsive control. Some criteria are derived to ensure the bounded synchronization of coupled Kuramoto oscillators with phase lags. Each distributed controller in the proposed scheme is allowed to use information from different nodes. Compared with other control strategies, impulsive excitation reacts quickly and saves more energy. What is more, in order to ensure global synchronization of the entire network, impulsive control strategies only need the nodes information at certain discrete time, which can reduce cost and are more applicable in practical useFig. 1, Fig. 2, Fig. 3, Fig. 4.
The paper is organized as follows. In Section 2, the Kuramoto model with phase lags is introduced and some necessary mathematical preliminaries are given. In Section 3, an impulsive distributed control strategy is developed and bounded synchronization criteria for coupled Kuramoto oscillators with phase lags are derived. In Section 4, some numerical simulations are given. In Section 5, brief concluding remarks and future works are stated.
Section snippets
Model description and mathematical preliminaries
In this section, we briefly introduce the general Kuramoto model, and some mathematical preliminaries.
Distributed impulsive synchronization of Kuramoto model with phase delays
In this section, the distributed impulsive control strategies are developed and some bounded synchronization criteria are derived for coupled Kuramoto oscillators with phase lags.
Illustrative examples
In this section, we present numerical simulations to illustrate the results in Section 3. Take the following networked Kuramoto oscillators with 8 nodes as an example, which is described bywhere , the overall coupling strength κ=0.1, the initial condition is chosen randomly from .
Conclusions
This paper studies the synchronization of the Kuramoto model with phase lags. A distributed impulsive control strategy is designed to ensure bounded synchronization of the system. Different from existing control schemes in the literature, each distributed controller in the proposed scheme is allowed to use information from different nodes and uses sampled information rather than continuous information which saves energy. For coupled Kuramoto oscillators with phase lags, bounded synchronization
Acknowledgment
We acknowledge help with experiments and simulations from Shengwei Mei and Shaowei Huang of the Department of Electrical Engineering, Tsinghua University.
Wen-Yi Zhang received the B.Eng. degree from the College of Automation in Huazhong University of Science and Technology,Wuhan, China, in 2014. Currently, he is pursuing Master's degree at the College of Automation, Huazhong University of Science and Technology,Wuhan, China and is an exchange student in the Department of Electrical Engineering, Tsinghua University, Beijing. His research interests include transient stability of power system and synchronization of complex dynamical networks.
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Wen-Yi Zhang received the B.Eng. degree from the College of Automation in Huazhong University of Science and Technology,Wuhan, China, in 2014. Currently, he is pursuing Master's degree at the College of Automation, Huazhong University of Science and Technology,Wuhan, China and is an exchange student in the Department of Electrical Engineering, Tsinghua University, Beijing. His research interests include transient stability of power system and synchronization of complex dynamical networks.
Chao Yang received the B.Eng. degree in Mechanical Manufacture and Automation from China Agricultural University, Beijing, China, 2005. Currently, he is working towards the Ph.D. degree at the College of Automation, Huazhong University of Science and Technology, Wuhan. His research interests include technology of smart grid and complex dynamical networks.
Zhi-Hong Guan is a Huazhong Leading Professor in the College of Automation, Huazhong University of Science and Technology, Wuhan, China. He received his Ph.D. degree in Control Theory and Applications from the South China University of Technology, Guangzhou, China, in 1994. His research interests include complex systems and complex networks, impulsive and hybrid control systems, networked control systems, multi-agent systems, networked robotic systems, complex smart grids and genetic regulatory networks.
Zhi-Wei Liu (M'14) received the B.S. degree in information management and information system from Southwest Jiaotong University, Chengdu, China, in 2004, and the Ph.D. degree in control science and engineering from the Huazhong University of Science and Technology, Wuhan, China, in 2011.From 2011 to 2013, he was a Post-Doctoral Research Fellow with the School of Power and Mechanical Engineering, Wuhan University, Wuhan, China, where he is currently an Associate Professor with the Department of Automation, School of Power and Mechanical Engineering. He has held visiting positions at the City University of Hong Kong, Hong Kong, and RMIT University, Melbourne, VIC, Australia. His research interests include cooperative control and optimization of distributed network systems.
Ming Chi received Ph.D. degree in Control Theory and Control Engineering from Huazhong University of Science and Technology, Wuhan, China, in 2013. He is currently a Lecturer in the College of Automation, Huazhong University of Science and Technology, Wuhan, China. His research interests include networked control systems and multi-agent systems.
Gui-Lin Zheng was born in 1963, IEEE Senior Member, a Professor of Wuhan University, devoted to the field of smart power grid and digital smart transducers as well as automation project and equipment study.