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Available online at www.sciencedirect.com

Journal of the Franklin Institute 350 (2013) 1206–1220

Adaptive sliding mode fault-tolerant control for

nonlinearly chaotic systems against network faults

and time-delays

Xiao-Zheng Jin

, Guang-Hong Yang

b,c

Key Laboratory of Manufacturing Industrial Integrated Automation, Shenyang University,

Shenyang 110044, China

College of Information Science and Engineering, Northeastern University, Shenyang 110819, China

State Key Laboratory of Integrated Automation for Process Industries, Northeastern University,

Shenyang 110819, China

Received 19 July 2012; received in revised form 25 January 2013; accepted 23 February 2013

Available online 6 March 2013

Abstract

The robust fault-tolerant control (FTC) problem against network faults and time-delays of a class of

nonlinearly coupled chaotic systems is addressed in this paper. Two kinds of faults on networks, that is,

network deterioration and signal attenuations, and a perturbation caused by general nonlinear

couplings and time-varying bounded delays are considered in the FTC chaotic system designs. A

sliding mode control strategy is proposed, guaranteeing the compensation of the faulted and perturbed

couplings by means of adaptive estimations of unknown controller parameters. It is shown that,

through Lyapunov stability theory, the proposed adaptive sliding mode controllers are successful in

ensuring the achievement of asymptotic synchronization of chaotic systems even in the case of faulted

and perturbed couplings. The proposed controller is tested by simulation on a uniﬁed chaotic system.

www.elsevier.com/locate/jfranklin

http://dx.doi.org/10.1016/j.jfranklin.2013.02.015

This work is supported by National Nature Science Foundation (Grant nos. 61104029, 60974043, 61273148,

61273155), and the National 973 Programme of China (Grant no. 2009CB320604), the Funds of Doctoral

Programme of Ministry of Education, China (Grant no. 20100042110027), the funds for the Author of National

Excellent Doctoral Dissertation of People’s Republic of China (Grant no. 201157), the Fundamental Research

Funds for the Central Universities (No. N110804001), the Natural Science Foundation of Liaoning Province

(Grant no. 201202156), the Scientiﬁc Research Foundation for Doctor of Liaoning Province of China (Grant no.

20121040).

Corresponding author. Tel.: þ862462269901.

E-mail addresses: jin445118@163.com (X.-Z. Jin), yangguanghong@ise.neu.edu.cn (G.-H. Yang).

1. Introduct ion

During the past several years, considerable attention has been paid to the robust

synchronization analysis and synthesis problems of dynamical chaotic systems (please see

[1] and references therein). As various unexpected factors will disturb the systems in

practical applications, many efforts have been made to obtain the synchronization of

systems in the cases of time-delays [2–9], deteriorated couplings [10–13], nonlinear

couplings [14–22], uncertainties and disturbances [23–31], etc.

For the sake of eliminating the effects of those factors for synchronization of dynamical

chaotic systems, the controlled synchronization has become a rather signiﬁcant topic

nowadays. There are several approache s to obtain con troller gains such as linear matrix

inequality (LMI) approach [2–5,16,11–13,18–21], adaptive approach [6,11–13,18–21],

backstepping approach [25], slide-mode approach [26–35] . As the ability of robustness and

insensitivity to withstand certa in types external disturbances and model uncertainties, the

slide-mode control method has been implemented successfully by many studi es to solve the

robust synchronization problem of chaotic systems [26–31]. Indeed, some of studies

designed slide-mode controllers relying on adaptive technique, which has properties of

quick and automatic response for estimating of unknown parameters at each instant. An

adaptive sliding-mode observer design method was propo sed to synchronize chaotic

systems with known bound perturbations in [26]. The adaptive sliding- mode technique was

also utilized in [27] to ensure that the slave chao tic systems with uncertainties can

synchronize with the master system accurately. The paper [28] presented an ad aptive

terminal sliding mode control method for synchronization subject to input nonlinearities

between two identical attractors.

It should be noted that the fault -tolerant control (FTC) problem has not been seriously

considered in the existing ch aotic system designs. In the traditional designs of FTC

systems, actuator and sensor faults have been usually addressed in the linear or nonl inear

multi-inputs and multi-outputs systems (see for example [36–38]). However, in the

networked chaotic systems, beyond considering the actuator and sensor faults, an especial

FTC design for the faults of coupled networks should be rigorously attended. There are

several network faults in the couplings such as random link failures (like package loss) [39],

network deterioration [10–13] and the fault of losing effectiveness in the process of

networked transmission which named as signal attenuations [40,41]. Note that the above

mentioned network faults are broadly existed in secure communications, formation of

unmanned aerial vehicl e (UAV) team, the world wide web (WWW). However, few efforts

have been made to consider the synchronization behavior of networked chaotic systems

with signal attenuations and network deterioration. The related studies about signal

attenuations can be found in the designs of large scale interconnected systems. Lossy

interconnection links (signal attenuations) were considered in [40] of distributed systems by

using LMI-based methods. The recent paper [41] proposed an indirect adaptive method to

deal with the asymptotic synchronization problem of master–slave large-scale systems with

bias actuators and network attenuations of communications. But the method proposed in

[41] utilizes large computational resource for estimating many control parameters by using

adaptive laws. Thus, more effectively methods for asymptotic synchronization of chaotic

systems against signal attenuations need to be further studied.

For the case of network deterioration, the recent paper [10] proposed a discrete-time

consensus protocol to deal with the convergence speed problem of switching random networks.

X.-Z. Jin, G.-H. Yang / Journal of the Franklin Institute 350 (2013) 1206–1220 1207

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

and JvJ represent its transpose and Euclid norm, respectively. The symbol of jj

denotes its absolute value.

X.-Z. Jin, G.-H. Yang / Journal of the Franklin Institute 350 (2013) 1206–12201208

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