On Synchronization of Pinning-Controlled Networks with Reducible and Asymmetric Coupling Matrix

Abstract

This paper investigates the synchronization of directed networks whose coupling matrices are reducible and asymmetrical by pinning-controlled schemes. A strong sufficient condition is obtained to guarantee that the synchronization of the kind of networks can be achieved. For the weakly connected network, a method is presented in detail to solve two challenging fundamental problems arising in pinning control of complex networks: 1) How many nodes should be pinned? 2) How large should the coupling strength be used in a fixed complex network to realize synchronization? Then, we show the answer to the question that why all the diagonal block matrices of Perron-Frobenius normal matrices should be pinned? Besides, we find out the relation between the Perron-Frobenius normal form of coupling matrix and the differences of two synchronization conditions for strongly connected networks and weakly connected ones with linear coupling configuration. Moreover, we propose adaptive feedback algorithms to make the coupling strength as small as possible. Finally, numerical simulations are given to verify our theoretical analysis.

Probabilistic synchronisation of pinning control

“…Comparing the two sides of (29) and noting that this equation must hold for any x t , it is then required that…”

“…As a typical collective behavior in nature, synchronization has been one of the key issues that is investigated in the literature. Over the past decade chaos control, stability (controllability) [16], [26], and synchronization [24], [25], [27], [29] of large scale dynamic systems have been some of the issues with wide interest due to their potential applications in power grid systems, biological systems, reaction diffusion systems and many others. Typically because of the large number of nodes in real world complex dynamical networks, it has been widely believed that it is impossible to add controllers to all nodes.…”

On Hybrid Adaptive and Pinning Consensus for Multiagent Networks

“…Theorem 11. Suppose that network (15) contains nodes and the coupling strength > 0; the prospective pinning consensus for network (15) with general connected topology can be achieved asymptotically if ( + 2 ) ≤ 2 holds, where are determined by the adaptive protocol (17) if node V is pinned; otherwise, = 0. is the diagonal elements of the Laplacian matrix .…”

“…It is not difficult to verify that the algebraic conditions in Theorem 11 are all satisfied. The state trajectories of the agents of network (15) under different cases are shown in Figure 5. From the results, it is obviously known that the consensus can be realized when the zero in-degree nodes are selected to be pinned.…”

“…Subsequently, Chen et al [14] investigated the pinning consensus of multiagent networks with strongly connected topology. In [15], the sufficient condition was proposed which can guarantee the directed network with reducible coupling matrix to realize the synchronization. In [16], the pinning global synchronization of both directed and undirected complex networks was discussed and some criteria were presented analytically.…”