Synchronization Analysis And Control For Switching Complex Networks Based On M-Matrix Approach

Complex network is a new discipline which studies the group behavior of network system composed of a large number of nodes and their interaction. It studies the network’s group behavior from a holistic perspective, which is affect-ed by individual interaction. As this kind of research method can predict various overall behavior of network system better, it has received extensive attention of scholars both at home and abroad. Synchronization behavior of complex net-works can well explain many common phenomena in the natural world and the field of engineering. Among dynamical behaviors of complex network, synchro-nization is one of the most important topics. However, in the earlier study of synchronization in complex networks, most work focused on the analysis of the rules topology network synchronization features. As the deep understanding of the structural properties of real complex networks, more and more attention has been focused on the effects of time-varying topology structure and switching topology structure on network dynamical behavior, especially the influence on synchronous behavior. The research on synchronization of complex networks with time-varying topology structure and switching topology structure has become an important subject in the area of complex networks.Some new complex networks with switching topology are established, and the synchronization dynamic are studied in this thesis. Firstly, the synchroniza-tion problem of complex networks with delay coupled and switching topology is investigated by establishing a new impulsive delay differential inequality. Sec-ondly, based on M-matrix approach, the synchronization problem of complex networks with impulse and Markovian switching is investigated. Thirdly, based on M-matrix approach, the robust synchronization problem of a complex network model with randomly uncertain parameters and Markovian switching topologies is studied. Finally, by using adaptive method, the synchronization problem of complex network with Markovian switching and adaptive local coupling strength is investigated. Based on Lyapunov stability theory, M-matrix approach and stochastic delay differential equation theory, combining with impulsive control, state feedback control and adaptive control, some synchronization criteria are obtained to guarantee the networks achieve synchronization. The conservation for our proposed criteria is lower than that in the existing literatures.The main contents and innovation points of this thesis are as follows:(1) The synchronization problem is investigated for a class of complex dy-namical network with switching topology and coupled delay. Firstly, a new impulsive delay differential inequality is established which is different from the existing impulsive delay differential inequality. It established an inequality to de-termine the value of impulse which ensures unstable impulsive system to obtain stable. Secondly, according to Lyapunov stability theory and a new impulsive delay differential inequality, a simple controller is designed and some exponential synchronization criteria of switching complex networks are obtained. Thirdly, a simulation is used in Chua’s circuit network, and the simulation results indicate that the designing methods of impulsive controller are effective.(2) A complex network model with impulse and Markovian switching topol-ogy is presented as well as its synchronization analysis. Compared with the existing literatures, the considered network model contains κ modes which are switching with Markov processes and stochastic switch and impulse are applied as well. Based on Lyapunov stability theory, M-matrix approach and stochas-tic delay differential equation theory, some exponential criteria are obtained and maximum impulsive gain is evaluated respectively with un-synchronization im-pulse and synchronization impulse. Moreover, the effectiveness and conservation of the results are verified by two simulations. In the first simulation, the t-wo models of network notes are two stable systems and the simulation results demonstrate that the established network has strong ability of resisting impulsive disturbing. In the second simulation, the two models of notes are Lorenz chaot-ic system and Rossler chaotic system. The results indicate that the designed controller in this thesis can exert unstable network to achieve synchronization effectively. (3) A complex network model with random uncertain parameters and Marko-vian switching topologies is presented and of which synchronization problem is studied. Different from the existing uncertain models of complex networks, the mode of the networks switches randomly according to a Markov process, and the uncertain parameters are random Bernoulli distributed variables. Based on Lyapunov stability theory, M-matrix approach and stochastic delay differential equation theory, synchronization criteria are obtained respectively under the sit-uations which the transition rate matrix is entirely known or partly known. The results are simple and available. Moreover, the designed controller depends on the mode judged by the Markov chain, which is helpful for reducing the energy to control network. In addition, the simulation results indicate that the method is available and effective. In the first simulation, transition rate matrix is known, and the control is only exerted on the first mode. The simulation results demon-strate that the mode-dependent feedback controller is effective. In the second simulation that transition rate matrix is partly known and four different simu-lations are involved by using four different transition rate matrices. The results reveal, although different transition rate matrices have great impact on the w-hole behavior of the network, the networks is still synchronous by the designed controller.(4) A complex network model with Markovian switching and adaptive cou-pling strength is proposed and its synchronization problem is studied. Compared with the existing literatures, the considered network model can not only switch randomly by the rules from Markov chain, but also be self-adaptive with its con-nection weights. Due to the self adaptiveness and random switching of connection weights, the methods in the existing literatures can’t analyze synchronization of this kind of network effectively and can’t design self-adaptive controller. To solve this problem, a new network model is proposed in this paper. The syn-chronization criteria are obtained by using Lyapunov stability theory, adaptive control scheme and M-matrix approach, as well as designing suitable self-adaptive update laws. It is verified at last by the simulations that the results are effective.