Synchronization And Learning Control For Some Complex Dynamical Networks

With the development of system science, complexity of systems has drawn many scholars’ attentions from mathematics, physics, biology, control science, social sciences, economics, and management and so on. The analysis and control of complex systems are the practical and important problems to be faced with. Complex network theory is an important research topic of system science, in which the model of a complex system is a network consisting of a great deal of nodes and connections between every two nodes. Up to now, complex networks have become the new research direction of systems. A complex dynamical network is a network in which each node is a dynamic system. Synchronization phenomenon is common in network systems. The synchronizability analysis and controlled synchronization are two important aspects of complex network research. Based on the control theory and by adding controllers into complex dynamical networks, the state of each node becomes identical with each other, which is the synchronization control of complex dynamical networks. In order to achieve the synchronization of complex dynamical networks, various control methods are proposed, such as feedback control, adaptive control, pinning control, and impulsive control. On the other hand, a complex dynamical network is a complex system with network connection, which makes the tracking control for the state of each node become a new problem need to be solved.The synchronization control and tracking control of complex dynamical networks are mentioned in this paper. Some network models are studied, mainly including complex dynamical networks with time-varying topological structures, nonlinearly coupled complex dynamical networks, time-varying delays coupled complex dynamical networks, complex networks with stochastic perturbations and with non-identical dynamical nodes, etc. Using adaptive control and learning control, we obtain some criteria and algorithms to realize network synchronization. The synchronization control problems with unknown control directions are studied in this paper, too. An adaptive iterative learning control scheme is proposed for complex dynamical networks with non-identical dynamical nodes, and then the state of each node in the network tracks the disired trajectory in the iteration domain. According to the theoretical analysis and proposed schemes, some numerical simulations are given. There are six chapters in this paper. The background, research actuality and research ideas of complex networks are mentioned in Chapter 1. The synchronization and control approaches of complex dynamical networks are presented from Chapter 2 to Chapter 5, which are the main results of this paper. In Chapter 6, we pointed out some existing problems and the future research direction. The main contributions of our work can be summarized as follows:1. The synchronization control algorithm for a class of time-varying complex dynamical networks is obtained by using the adaptive control and learning control method. Based on the Lyapunov stability theory, the appropriate adaptive controller and the periodic adaptive laws are designed to achieve the completely synchronization of the network. Numerical simulations are given in view of the design scheme.2. The stochastic perturbations are discussed in the control problems of time-varying complex dynamical networks. Applying the stochastic differential equation theory, adaptive control and feedback control, we obtain the criteria of synchronization in the mean square. The detailed theoretical analysis and numerical simulations are given to testify the availability of the presented method.3. The controllers are designed for the tracking problems of unknown complex dynamical networks with non-identical nodes. The networks with time delayed coupling and without time delayed coupling are investigated. Applying the signal replacement technique and remodeling the error equations of complex dynamical networks, we combine all unknown terms into a periodically time-varying vector, and then estimate it by a periodic adaptive learning mechanism. The state of each node can track the desired trajectory. When the desired trajectory is identical, the network is said to be synchronized. A sufficient condition is given by constructing a Lyapunov-Krasovskii functional, and simulation examples are given to show the availability of the designed method.4. The control direction problem of complex dynamical networks is studied in the synchronization control. Using Nussbaum-type function to deal with the control direction of the network, we design the adaptive controllers for complex dynamical networks with unknown control direction, and obtain the sufficient conditions of synchronization. The Lyapunov stability theory is used for the theoretical proofs. We give some simulation examples and then verify the effectiveness of the presented approach.5. An adaptive learning control scheme is proposed for complex dynamical networks with repetitive operation over a fixed time interval. By updating the control input iteratively, the network improves its control performance. By designing learning laws for unknown time-varying parameters and coupling strength, the state of each node in complex dynamical networks tracks the reference signal in the iteration domain. The composite energy function is used for the theoretical proofs. The validity of the presented method is shown by a simulation example.6. At last, we sum up the main results of this paper. Furthermore, we point out some existing problems and the future work.