ABSTRACT
In this article, we search for polynomial Lyapunov functions beyond the quadratic form to investigate the synchronization problems of nonlinearly coupled complex networks. First, with a relaxed assumption than the quadratic condition, a synchronization criterion is established for nonlinearly coupled networks with asymmetric coupling matrices. Compared with the existing synchronization criteria, our results are less conservative and have a wider application. Second, the synchronization problem for polynomial networks is characterized as the sum-of-squares (SOS) optimization one. In this way, polynomial Lyapunov functions can be obtained efficiently with SOS programming tools. Furthermore, it is shown that the local synchronization of certain nonpolynomial networks can also be analyzed by using the SOS optimization method through the Taylor series expansion. Finally, three numerical examples are presented to verify the effectiveness and less conservatism of our analytical results.
Subject(s)
Algorithms , Neural Networks, ComputerABSTRACT
We in this paper propose an invariant set based distributed control protocol for synchronization of discrete-time heterogeneous multiagent systems. Starting with the assumption that the distributed control input will vanish once a multiagent system achieves synchronization, we attain an easily verifiable method for the nonexistence of synchronous trajectories through characterizing the vector fields of agents. Then, we introduce an invariant set to analyze the limit behaviors of all the synchronous trajectories. Afterwards, based on the assumption that the above invariant set can be characterized by the graph of a function, we design a distributed control protocol to transform the heterogeneous system into an equivalent one, which is composed of two lower dimensional systems. Moreover, for this equivalent system, we provide a synchronization criterion via constructing corresponding Lyapunov-type functions for these two lower dimensional systems, arriving at a synchronization criterion for the original heterogeneous system. Especially, we further improve the applicability of this synchronization criterion by using multiple Lyapunov-type functions. Finally, three examples are presented to demonstrate the validity of the corresponding theoretical results.
ABSTRACT
In this paper, we study synchronization of heterogeneous linear networks with distinct inner coupling matrices. Firstly, for synchronous networks, we show that any synchronous trajectory will converge to a corresponding synchronous state. Then, we provide an invariant set, which can be exactly obtained by solving linear equations and then used for characterizing synchronous states. Afterwards, we use inner coupling matrices and node dynamics to successively decompose the original network into a new network, composed of the external part and the internal part. Moreover, this new network can be proved to synchronize to the above invariant set by constructing the corresponding desired Lyapunov-like functions for the internal part and the external part respectively. In particular, this result still holds if the coupling strength is disturbed slightly. Finally, examples with numerical simulations are given to illustrate the validity and applicability of our theoretical results.
