| Synchronization is the most important collective behavior of complex dynamic networks.The performance analysis and control of the synchronization behavior are two important aspects of the complex dynamic network research.In this field,it is very meaningful to explain the synchronization phenomenon in nature with math-ematical models and to design a synchronization mechanism for practical applica-tions.The Kuramoto model is one of the most important mathematical models that describe the complex dynamic networks,which has been widely used in physics,biology,and control theory and engineering,etc.As a coupled oscillator model,the Kuramoto model can approximately describe the dynamics of nonlinear net-work systems with coupled nodes.Most of the early work on the Kuramoto model only considered the all-to-all coupling case of nodes.However,in real situations,the coupling of the oscillators is affected by various factors,such as sudden interfer-ence and changes,which make the oscillators cannot maintain an all-to-all coupling,and further leads to the deterioration of the synchronization performance.Aiming at these problems,researchers recently introduce the hierarchical distributed control methods to drive the locally coupled Kuramoto models to synchronization.How-ever,most of the results can only obtain asymptotic synchronization,exponential synchronization,and finite time synchronization.The settling time of synchroniza-tion is affected by the initial states of the oscillators.In practice,the initial states of the oscillators from the network systems described by the Kuramoto model are difficult to be obtained.Moreover,the implementation of a distributed synchroniza-tion control needs a communication network to share the state information among the nodes.Each node calculates the control input based on its own states and the received information from its neighbors.When the communication bandwidth and computing power are limited,it is difficult to obtain real-time control,which also causes bad performance.In order to solve these difficulties,this paper introduces the event-triggered mechanism and fixed-time control method to locally coupled Kuramoto model synchronization problems.The proposed methods relax the con-ditions on initial states of oscillators for synchronization control and also improve the dynamic performance.Furthermore,it also reduces the requirements of com-munication and computation for implementation,which is conducive to a stable and fast synchronization of the network.This paper designs a distributed controller with a hierarchical structure for the locally coupled Kuramoto-oscillator network.The controllers of each node are con-nected to form a control network.At the same time,considering the speed of syn-chronization conditions and the consumption of communication resources,the fol-lowing main research work was carried out:(1)Aiming at the phase synchronization problem of the Kuramoto model os-cillator network,a distributed controller based on an event trigger mechanism is designed.The whole system has a layered structure,which is divided into the oscil-lator layer and the control layer.The coupling mode of the control layer is the same as that of the oscillator network.Through the proposed control structure and event triggering method,the fixed-time phase synchronization of the Kuramoto model os-cillator network is obtained.In this result,the upper bound of the synchronization settling time will not be affected by the initial state of the oscillators.Using the Lyapunov stability theory and algebraic graph theory,the boundary conditions of event triggering are derived.At the same time,an event-triggered control strategy that does not require continuous monitoring of the status of neighboring oscilla-tor nodes is studied,which further reduces the cost of status monitoring.Finally,simulation examples verify the validity of the theoretical results.(2)For the frequency synchronization problem of the Kuramoto model oscil-lator network,referring to the existing layered control structure in the phase syn-chronization control,event-triggered fixed-time frequency synchronization is fur-ther realized.By designing a new synchronization error system model,the order problem in frequency synchronization is transformed.A linear compensation term is added to the controller,which is used to compensate for the nonlinear coupling term which does not exist in the phase synchronization control.Then use Lyapunov stability theory and fixed time stability theorem to introduce new event-triggered boundary conditions.Also,in order to avoid the discontinuous and singular prob-lems caused by the sign function contained in the controller,the saturation func-tion is used as an alternative.Finally,the event-driven control method to avoid continuous communication is also introduced,and a comparison is carried through simulation experiments.(3)Due to the coupling of the actual oscillator network system is easy to change and the link in the networked control system is susceptible to interference,an event-driven fixed time phase agreement control strategy is proposed for the Kuramoto model oscillator network with time-varying topology.The hierarchical distributed network control structure is still used,but the topology of the control network can be different from the topology of the oscillator network.When the topological con-nectivity of the vibrator network changes within a certain range,it will not affect its fixed time synchronization results.Lyapunov stability theory analysis method is used to prove that the synchronization of the system will not be affected by the switching topology of the coupled oscillator.The specific conditions for the net-work connectivity that the control layer needs to meet are given.Using a similar control idea as above,an event-driven control method to avoid continuous monitor-ing of neighbor nodes is also given,saving network bandwidth resources.(4)A fixed-time event-driven phase synchronization control method is de-signed for the Kuramoto model coupled oscillator network with a starter.A two-layer network structure including an oscillator layer and a control layer is selected,in which the starter node is located in the control layer,and is connected to one or several nodes in the control network node.When designing the controller,it is necessary to consider the impact of the starter node on the topology of the coupled network,then reselect the synchronization error system and the candidate Lyapunov function.Besides,the performance of the system convergence when the sign func-tion in the measurement error function is replaced with the saturation function is theoretically analyzed.Using different trigger conditions by judging whether the synchronization error enters the boundary of the saturation function,a method that can reduce the conservativeness of the self-trigger condition is proposed.Mean-while,it is proved that the Zeno phenomenon can be avoided and event-driven rules that can avoid continuous monitoring.Finally,the fixed-time event-driven control of the Kuramoto oscillator network model based on the layered structure is summarized.The future research directions of the higher-order Kuramoto oscillator model,consideration of frustration,various interferences,and optimizing trigger conditions have prospected. |
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