The Dynamics Analysis And Control Methods Research Of Fractional Order Networks

Network science is an interdisciplinary field that combines graph theory,control theory and biology,Internet and other disciplines.There are an increasing number of systems in the real world,which could be described by a dynamical network.With the gradually recognization of fractional order calculus,people realize it is preferable to model systems by fractional order differential equations.Consequently,networks consist of nodes with fractional order dynamical behaviors have become a topic at present.Based on former study,this paper investigates some kinds of typical network systems with fractional order dynamical nodes.The stability is studied,influences of time-delay,impulsive,are taken into consideration.Furthermore,under impulsive control,event-triggered control and so on,the synchronization issues are discussed.Numerical results are given to check the theory results.The main results are as follows:1.Based on the definition of fractional order derivative operator,sufficient conditions of uniformly continuous function are derived,which is related with the fractional order derivative.Subsequently,combined with some properties and Barbalat's lemma,an important tool to analyze stability of fractional order time-delayed systems has been built.In the next,the fractional order Hopfield neural networks and BAM neural networks with both time-delay and impulsive effects are studied,several stability criteria are derived.Besides,for a fractional order complex dynamical network with coupled time-delay,under some adaptive pinning controllers,the cluster projective synchronization is investigated.2.The synchronization issue for fractional order complex dynamical networks and the consensus for fractional order multi-agent systems are both considered under impulsive strategy.At first,impulsive pinning controllers are designed for a general fractional order complex dynamical network,in which,just a small fraction of nodes will be controlled at every impulsive moment.Based on the relationship between the Mittag-Leffler function and exponential function,the exponential synchronization criteria are derived.Moreover,for fractional order multi-agent systems,both inner time-delay of every agent and transmission delay among agents are considered.A novel heterogeneous impulsive control protocol is applied,in which,impulsive gains among every impulsive moment and every agent are different.According to a new fractional order delay differential inequality,sufficient conditions for consensus are given.3.For the fractional order nonlinear multi-agent systems,the event-triggered control method is studied,in which,the information of agent will be updated after the predetermined event is triggered.At first,the leader-following fractional order multi-agent systems are considered,an event based on a monotone decreasing threshold function is designed.The information of every agent will be updated when the value of consensus error is higher than that of the threshold function.According to the properties of linear fractional order differential equation,the consensus criteria are derived,the Zeno behavior is proved to be avoided.In addition,for the leaderless case,an event based on the variation trend of consensus error is designed;the information of every agent will be updated when the ratio of consensus error at the present and consensus error at the last event,instant is higher than a constant one.According to the graph theory and some basic theories of fractional order differential equations,several consensus criteria are derived,then,based on the properties of the Mittag-Leffler function,the Zeno behavior is also proved to be excluded.4.The intermittent control method is considered for the synchronization problem of fractional order complex dynamical networks.At first,according to some properties of fractional order differential equation and the Mittag-Leffler function,a new piecewise linear fractional order differential inequality is built.Then,the static linear feedback controllers are considered,in addition,the adaptive intermittent control approach is studied.Based on the comparison principle of fractional order differential equations and theories of matrix,several synchronization criteria related with control width are derived.To sum up,this paper mainly studies networks which are made up of nodes with fractional order dynamic behavior.The dynamic characteristics for those networks have been investigated under the influence of time delay and impulsive.In addition,the synchronization and consensus problem under some control strategies also have been researched.Numerical results are given to verify the validity of the theoretical results.Finally,some conclusions of the thesis and prospects to be further studied are drawn.