Synchronization Analysis Of Several Classes Of Complex Dynamical Networks With Time Delay Couplings

As one of the most important collective behaviors of complex dynamical networks, synchronization has received tremendous attention in the past decade, and a lot of results have been reported in the literature. However, the complexity of a complex dynamical network that both come from the structure of the network topology and the node dynamics makes the analysis of synchronization of complex dynamical networks very hard, and a lot of problems remain open and thus deserve more attention.This dissertation focuses on the synchronization problem of several classes of complex dynamical networks with time delay couplings. The switched system theory and adaptive control method are used to investigate synchronization of these networks. The analysis and control of synchronization of several particular classes of complex dynamical networks are solved. Frist, for undirected networks with switching topology, synchronization criteria are developed, and the switching signals that can synchronize the network are identified. Then, for directed networks with fixed topology, the design methods of decentralized controllers are provided, and with the proposed controller, the controlled network can achieve synchronization globally. The main methods used in this dissertation include the average dwell time method, the model-based average dwell time method, the single Lyapunov function method, the convex combination technique and the decentralized adaptive control method.The main contributions of this dissertation are as follows.1. Based on the condition of simultaneously diagonalizable, the synchronization problem of a class of complex dynamical networks with time-invariant delay and switching topology is investigated. With the assumption that all the outer coupling matrices are simultaneously diagonalizable, a synchronization criterion is given by using the stability theory of switched system and the topology structure of an undirected network provided that all sub-networks are self-synchronizing. A class of switching signals under which such a network can achieve synchronization is identified with the help of the average dwell time method. This result is then extended into a more general case where not all sub-network are self-synchronizing. By constraining the ratio of the total active time between the self-synchronizing sub-networks and non-synchronizing sub-networks, a corresponding synchronization criterion as well as a class of synchronizing switching signals is given.2. The synchronization problem of a class of undirected complex dynamical networks with time-varying delay couplings and switching topology is discussed. First, the model-based average dwell time method is extended into the analysis of the stability of linear switched systems with time-varying delay. Under the assumption that all subsystems are stable, a stability criterion for such a switched system is developed. For each subsystem, according to its own characteristics, an average dwell time condition is constructed under which the entire switched system is stable. Then, the proposed results are applied to analyzing the synchronization problem of an undirected network with time-varying delay couplings and switching topology. Without the assumption that all outer coupling matrices are simultaneously diagonalizable, a sufficient condition on synchronization of the network that consists of self-synchronizing sub-networks is obtained. A class of switching signals is identified as well by using the model-based average dwell time method.3. Synchronization analysis is also carried out by using the convex combination technique and the single Lyapunov function method for the case that all sub-networks of an undirected switched network with time-varying delay couplings are non-synchronizing. A full dimensional synchronization condition is explored and a synchronizing switching signal is designed. Considering the fact that a complex dynamical network usually has a large number of nodes which will lead to a high dimensional condition that may be hard to check, a lower dimensional synchronization condition is established with the assumption that all the outer coupling matrices are simultaneously diagonalizable. The corresponding synchronizing switching signal is also identified.4. The controlled synchronization of a direct dynamical network with time-invariant delay couplings is studied. By using the Lyapunov stability theory and the property of the network structure, a novel delay-dependent decentralized controller is designed. With the proposed decentralized controller, the controlled network can achieve synchronization exponentially with a given convergence rate.5. Synchronization of nonlinearly coupled complex dynamical networks with directed topology and time-invariant delay couplings is studied. Via Lyapunov-Krasovskii stability theory and the network topology, global synchronization of such networks is investigated. Under the assumption that coupling coefficients are known, a family of delay-independent decentralized nonlinear feedback controllers are designed to globally synchronize the network. When coupling coefficients are unavailable, an adaptive mechanism is introduced to synthesize a family of delay-independent decentralized adaptive controllers that guarantee the global synchronization of the uncertain network.The conclusions and the some future works are given at the end of the dissertation.