1 Copyright © 2014 by ASME Proceedings of the ASME 2014 Dynamic Systems and Control Conference DSCC2014 October 22-24, 2014, San Antonio, TX, USA DSCC2014-6264 ROBUST ADAPTIVE SYNCHRONIZATION OF DYNAMIC NETWORKS WITH VARYING TIME DELAY COUPLING USING VARIBLE STRUCTURE CONTROL Vesna M. Ojleska School FEIT, SS Cyril & Methodius University Institute of Automation & Systems Engineering Karpos 2, MK-1000, Skopje, R. Macedonia vojleska@feit.ukim.edu.mk Dilek (Bilgin) Tükel Department of Control&Automation Engineering Dogus University, Acibadem, TR-34722 Istanbul, Turkey dtukel@dogus.edu.tr Georgi M. Dimirovski Department of Control&Automation Engineering Dogus University, Acibadem,Istanbul,Turkey School FEIT, SS Cyril & Methodius University Institute of Automation & Systems Engineering Karpos 2, MK-1000, Skopje, R. Macedonia gdimirovski@dogus.edu.tr ABSTRACT The synchronization problem for a class of delayed complex dynamical networks via employing variable structure control has been explored and a solution proposed. The synchronization controller guarantees the state of the dynamical network is globally asymptotically synchronized to arbitrary state. The switching surface has been designed via the left eigenvector function of the system, and assures the synchronization sliding mode possesses stability. The hitting condition and the adaptive law for estimating the unknown network parameters have been used for designing the controller hence the network state hits the switching manifold in finite time. Two illustrative examples along with the respective simulation results are given, which employ the designed variable structure controllers. INTRODUCTION Network structures have been subject of research for considerable time in mathematical systems and control science as well as in physics. Furthermore, it has been observed for some time that complex dynamic networks exist in all fields of science and humanities as well as in the nowadays networked technical and non-technical systems, such formations of moving objects and individuals or societal groups. Thus the latter systems have been studied extensively over the past couple of decades. As it is well-known, traditional networks are mathematically represented by a graph, e.g. a pair { } E P G , = in which P represents a set of N nodes (or vertices) N P P P , , , 2 1 L and E is a set of links (or arcs or edges) N L L L , , , 2 1 L each of which connects two elements in P . The well known chains, grids, lattices and fully connected graphs have been formulated to represent the so-called completely regular networks. (a) (b) Fig. 1: A random graph network (a), and a small-world network (b) [7] In due course of developments, the theory of random graphs (Figure 1-a) was first introduced by Paul Erdos and