Research On Fixed Time Synchronization Control In The Complex Networks With Time Delays

Complex network has become an important tool to describe the topology and understand the dynamical behaviors in the natural and artificial systems,such as power and transportation network,social relationship network,various protein metabolism systems and so on.As a typical dynamic behavior,synchronization is an important issue in complex networks.Different from the asymptotic or exponential synchronization whose synchronization error converges in infinite time,the fixed time synchronization has attracted a lot of attention because it can be realized in finite time and its settling time is only related to control parameters.Based on modern control theory and fixed time stability theory,this paper studies the fixed time synchronization in complex network.The main work of this paper is as follows:1.The fixed time generalized outer synchronization between two complex networks with varying time delays is studied.The auxiliary-system approach is used for the relations between the networks are unknown.By adding feedback interactions between the response and auxiliary networks,we propose two strategies: the controller with and without time delay.In the former,there is no requirement for the derivative of time delay.While the latter has more practical application when the time delay cannot be accurately known.The simulation results verify the effectiveness of the theorem.2.The fixed time leader following synchronization in complex networks with time-varying delays is studied.In general fixed time theory,the controller often contains sign function which will cause chattering.We propose the control strategy without sign function and give the corresponding convergence analysis of synchronization error.In addition,we construct a new function for the controller,and give its general expression under the fixed time method.Numerical simulation results show that the proposed controller not only has faster convergence time,but also can avoid chattering.3.To design the simple control for fixed time synchronization in complex networks is studied.The hyperbolic sine function is used to replace the two power function terms usually required in fixed time control,a simpler controller is designed in complex network with and without time delay respectively.Compared with most existing results,although there are fewer control parameters in the control,it has better convergence and tighter settling time.In addition,the controller does not contain sign function,which can avoid chattering.4.The fixed time synchronization in delayed networks via periodically intermittent control is studied.As a discontinuous control method,intermittent control is rarely applied to fixed time synchronization at present.In this paper,a fixed time stability theorem under periodic intermittent control is established and a fixed time expression is given.Compared with the existing literature,this result is more general and more suitable for practical networks.Then the intermittent control strategies with and without sign function,and the corresponding convergence analysis of synchronization error are given.The numerical simulation results verify the effectiveness of the proposed theorem.