Pinning controllers and fully equipped controllers were, respectively, established by direct
and indirect adaptive methods in [11,12] to compensate for the effe cts of deteriorated
networks. The papers indica ted that fully equipped control can obtain the asymptotic
synchronization results of networks, while pinning control can ensure the synchronization
errors reduced as small as desired in the presence of the network deterioration and
uncertainties. It should be mentioned that the above studies are developed for the linear
coupled networks. The recent paper [13] utilized a pinning adaptive control method to deal
with the network deteriorat ion problem for nonlinear coupled networks without
considering the delay in couplings. In this pa per, the sliding mode control method is
proposed to dispose the deteriorated networks of chaotic systems with nonlinear and
delayed couplings.
As we know, the asymptotic synchronization problem is more challenging in the presence of
unknown time-varying delays. The existing studies of delayed dynamical networks were
involved in coupling delays, delayed nodes, decaying function, distributed delays, etc.
In practice, the delay of a signal is always bounded, since the signal channel is considered
as disconnection if the delay goes into infinity. Based on that, lots of works
[2–9,14–16,20,21,23] dealt with the delays under some conditions, such as constant status,
known upper delay bounds, or the delay derivative less than 1. Most of the existing works
constructed controllers via LMI methods and Lyapunov functions. Actually, according to
the bounded property of delays, one can indicate that the error between the delayed signal
and the original signal is also bounded. Thus, the synchronization problem against delays
can always be solved based on this bounded status. Motivated by that, the delay can be
disposed as the way of dealing with the uncertainty and disturbance. Then, the sidling
mode method and adaptive compensation method [12] are also suitable for dealing with
the bounded delays without further assumptions on delays.
In this paper, the robust FTC and synchronization problem is studied for a class of
chaotic systems with signal attenuations, network deterioration, nonlinear couplings and
time de lays. A novel adaptive sliding mode control strategy is developed to achieve
asymptotic synchronization of systems when the network topology, coupling strength,
network faults and time-delays are assumed to be unknown. Based on this proposed
method, an integral sliding manifold is developed for chaos synchronization. An adaptive
law is designed to estimate the control gain. Then, adaptive sliding mode controllers are
constructed relying on the updated control gain and an integral gain. Based on Lyapunov
stability theory, it is indicated that the adaptive closed-loop chaotic system can be
guaranteed to be asymptotic synchronization in the presence of faulted and perturbed
couplings.
The robust FTC and synchronization problem formulation is described in Section 2.In
Section 3 adaptive sliding mode controllers are developed to obtain our main results.
A simulation example is provided in Section 4 to verify our theoretical results. Finally,
conclusions are given in Section 5.
2. Model description and preliminaries
We first introduce our notation. R stands for the set of real numbers. For a real vector
v, v
T
and JvJ represent its transpose and Euclid norm, respectively. The symbol of jj
denotes its absolute value.
X.-Z. Jin, G.-H. Yang / Journal of the Franklin Institute 350 (2013) 1206–12201208