On Synchronization Problems Of A Class Of Complex Dynamical Networks Composed Of Lur’e Systems

Generally speaking, a complex network is a large set of interconnected nodes, in which a node is a fundamental unit with specific contents. The recent decade has witnessed the birth of a new movement of interest and research on the study of complex networks throughout the world. At present, in the research on dynamic behavior of complex networks, the most attention is paid on the synchronization problem of the complex dynamical networks. Lur’e system is a typical nonlinear system and is applied to various fields of engineering. Recently, the synchronization for complex dynamical networks composed of Lur’e systems has attracted more and more attention from researchers. Based on the absolute stability theory and LMI method, the dissertation mainly investigates the synchronization problem of a class of compex dynamical networks composed of Lur’e systems by applying effective control strategy, and the main contributions are listed as follows:The synchronization problem for complex networks composed of general Lur’e systems is dealt with. Based on the Jordan canonical transformation method and a Lur’e-Postnikov function, the global synchronization criteria for complex networks with non-symmetric coupling are established and formulated as matrix inequality. If some free parameters are chosen, the matrix inequality can be translated into LMI, and the control gain matrix can be constructed via the feasible solutions of the LMI. Finally, a complex dynamical network composed of identical Chua’s circuit is adopted as a numerical example to demonstrate the effectiveness of the proposed results.Considering the time delay effect on the synchronization of complex dynamical networks, the synchronization problem for a class of delayed complex networks composed of Lur’e systems is investigated by applying feedback injections to a small fraction of nodes in the whole network. Based on the absolute stability theory, for the network with homogeneous coupling delay, both delay-independent and delay- dependent asymptotical stability criteria ensuring the network global synchronization are derived; for the network with heterogeneous delays, a delay-independent criterion ensuring the global synchronization of the whole network is derived. A complex dynamical network composed of identical Chua’s circuits is adopted as a numerical example to demonstrate the effectiveness of the proposed results. It is also shown that in some particular cases, only a single controller can achieve the control objective.The concept of synchronization manifold is introduced, and the synchronization problem of the network is transformed into the stability of the synchronization manifold. Based on the synchronization manifold, the synchronization of a class of complex dynamical networks composed of Lur’e systems is investigated. The assumption that the coupling configuration matrix is symmetric and irreducible is removed. Based on the absolute stability theory, some sufficient conditions are established to guarantee the global synchronization of complex networks with irreducible coupling matrix and reducible coupling matrix. A complex dynamical network composed of identical Chua’s circuits is adopted as a numerical example to demonstrate the effectiveness of the proposed results.In practical systems, uncertainty can be commonly encountered due to modeling inaccuracies and changes in the environment of the model. The synchronization problem for delayed complex dynamical networks composed of uncertain Lur’e systems is dealt with. Based on the absolute stability theory and a special decentralized control strategy, some delay-dependent synchronization criteria are derived such that the controlled complex dynamical network is synchronous for all admissible uncertainties. Finally, a corresponding numerical simulation demonstrates the effectiveness of the proposed results.Noise is ubiquitous in real-world systems, and it is practical and reasonable to take the noise phenomenon into account in synchronization problem. The problem of robust H∞synchronization for a class of complex dynamical networks with time-varying delay is dealt with. Each node of the network is a general Lur’e system subject to an energy bounded input noise. Based on the Lyapunov stability theory, a synchronization criterion formulated in the form of LMIs is obtained, under which the controlled network can be robustly stabilized onto an expected homogeneous state with a guaranteed H∞performance. The controller designed can be constructed via feasible solutions of LMIs. Finally, a dynamical network composed of identical Chua’s circuits is adopted as a numerical example to demonstrate the effectiveness of the proposed results.In practical engineering application, all state variables are rarely available from on-line measurement due to either the difficulties of measuring state directly or the economic and utilizing limitations of measuring equipment. This makes state feedback can not be physically realized. The robust H∞synchronization problem for a class of Lur’e type complex networks with disturbance is concerned with. The proposed observer-based control scheme is developed to ensure the asymptotic stability of the augmented system, to reconstruct the non-measurable state variables of each node, and to improve the H∞performance despite the external disturbance. Based on the Lyapunov stability theory, a synchronization criterion formulated as LMIs is obtained, under which the controlled network can be robustly stabilized onto a desired state with a guaranteed H∞performance. The controller and the observer gains can be given by the feasible solutions of LMIs. Finally, the effectiveness of the proposed control scheme is demonstrated by a numerical example through simulation.Lastly, the summary of the whole dissertation is given and the research directions in future are put forward.