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Journal of Applied Mathematics
Volume 2012, Article ID 607491, 16 pages
http://dx.doi.org/10.1155/2012/607491
Research Article

Robust Adaptive Finite-Time Synchronization of Two Different Chaotic Systems with Parameter Uncertainties

Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai, Shandong 264001, China

Received 10 April 2012; Revised 20 July 2012; Accepted 22 July 2012

Academic Editor: Laurent Gosse

Copyright © 2012 Yun-An Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with the finite-time synchronization problem for two different chaotic systems with parameter uncertainties. Using finite-time control approach and robust control method, an adaptive synchronization scheme is proposed to make the synchronization errors of the systems with parameter uncertainties zero in a finite time. On the basis of Lyapunov stability theory, appropriate adaptive laws are derived to deal with the unknown parameters of the systems. And the convergence of the parameter errors is guaranteed in a finite time. The proposed method can be applied to a variety of chaos systems. Numerical simulations are given to demonstrate the efficiency of the proposed control scheme.

1. Introduction

In the past few decades, chaos synchronization has gained much attention from various fields [13], since Pecora and carroll [4] introduced a method to synchronize two identical chaotic systems with different initial conditions in 1990. Most of the works on chaos synchronization have focused on two identical chaotic systems [511]. However, in many real world applications, there are no exactly two identical chaotic systems. Therefore, the problem of chaos synchronization between two different chaotic systems with uncertainties is an important research issue [12]. Different synchronization control methods for two different chaotic systems, such as adaptive control [1321], nonlinear feedback control [22], backstepping [23, 24], fuzzy technique [2527], and sliding mode control [2830], have been proposed to solve the synchronization problem.

Since some systems’ parameters cannot be exactly known in advance, many efforts have been devoted to adaptive synchronization. In [18, 31], Huang discussed the synchronizations between Lorenz-Stenflo (LS) system and CYQY system, and between LS system and hyperchaotic Chen system with fully uncertain parameters. Wang et al. [15] designed a general adaptive robust controller and parameter update laws which made the drive-response systems with different structures asymptotically synchronized. In [16], the sufficient conditions for achieving synchronization between generalized Henon-Heiles system and hyperchaotic Chen system with unknown parameters were derived based on Lyapunov stability theory. A new adaptive synchronization scheme by pragmatical asymptotically stability theorem was proposed for two different uncertain chaotic systems [17], but the unknown signals were used in the controller. Chaos synchronization between two different chaotic systems with uncertainties in both master and slave chaotic systems remains a challenging problem [30].

Most methods only guarantee the asymptotic stability of the synchronization error dynamics, namely, the trajectories of the slave system approach the trajectories of the master system as 𝑡. From a practical point of view, however, it is more valuable that the synchronization objective is realized in a finite time [28]. In recent years, some researchers have applied finite-time control techniques, such as nonsingular terminal sliding mode control method [32], CLF-based method [33, 34], sliding mode control method [2830], and the finite-time stability theory-based method [28, 35, 36], to realize synchronization.

Compared with the existing results in the literature, there are three advantages which make our approach attractive. First, based on the finite-time control technique, adaptive control, and robust control, a new synchronization method is presented for a wide class of nonlinear systems. Second, it guarantees that all the errors are driven to zero in a finite time even for the systems with parameter uncertainties. Third, it guarantees that all the parameter errors converge to zero in a finite time.

In this paper, an adaptive finite-time synchronization scheme is proposed for a class of chaotic systems. The rest of the paper is organized as follows. In Section 2, we introduce the chaotic systems considered in this paper and preliminary lemmas. In Section 3, the proposed finite-time controller is designed to synchronize two different chaotic systems. We give the simulation results and the conclusions in Sections 4 and 5, respectively.

2. System Description

Consider the following master chaotic system: ̇𝑥=𝐴1+Δ𝐴1𝑥+𝐵1+Δ𝐵1𝑓1(𝑥),(2.1) where 𝑥=[𝑥1,𝑥2,,𝑥𝑛]𝑇𝑅𝑛 denotes a state vector, 𝑓1 is a nonlinear continuous vector function, 𝐴1 and 𝐵1 are 𝑛×𝑛 nominal coefficient matrices, Δ𝐴1 and Δ𝐵1 are unknown parts of 𝑛×𝑛 coefficient matrices.

The slave system is given with ̇𝑦=𝐴2+Δ𝐴2𝑦+𝐵2+Δ𝐵2𝑓2(𝑦)+𝑢,(2.2) where 𝑦=[𝑦1,𝑦2,,𝑦𝑛]𝑇𝑅𝑛 denotes a state vector, 𝑓2 is a nonlinear continuous vector function, 𝐴2 and 𝐵2 are 𝑛×𝑛 nominal coefficient matrices, Δ𝐴2 and Δ𝐵2 are unknown parts of 𝑛×𝑛 coefficient matrices, and 𝑢=[𝑢1(𝑡),𝑢2(𝑡),,𝑢𝑛(𝑡)]𝑇𝑅𝑛 is a control input vector to be designed.

Subtracting (2.1) from (2.2) yields the error dynamical system as follows: ̇𝑒=𝐴2+Δ𝐴2𝑦+𝐵2+Δ𝐵2𝑓2(𝑦)𝐴1+Δ𝐴1𝑥𝐵1+Δ𝐵1𝑓1(𝑥)+𝑢,(2.3) where 𝑒=𝑦𝑥. Note that only a part of elements of the coefficient matrices unknown, without loss of generality, we assume that the number of the unknown elements of the 𝑖th row of Δ𝐴1 is 𝑁𝐴1𝑖, that of Δ𝐴2 is 𝑁𝐴2𝑖, that of Δ𝐵1 is 𝑁𝐵1𝑖, and that of Δ𝐵2 is 𝑁𝐵2𝑖. Then (2.3) can be rewritten as ̇𝑒=𝐴2𝑦+𝐵2𝑓2(𝑦)𝐴1𝑥𝐵1𝑓1(𝑥)+𝑢+𝑁𝐴21𝑖=1𝛿𝑎21𝑖𝑦1𝑖𝑁𝐴2𝑛𝑖=1𝛿𝑎2𝑛𝑖𝑦𝑛𝑖+𝑁𝐵21𝑖=1𝛿𝑏21𝑖𝑓21𝑖𝑁𝐵2𝑛𝑖=1𝛿𝑏2𝑛𝑖𝑓2𝑛𝑖𝑁𝐴11𝑖=1𝛿𝑎11𝑖𝑥1𝑖𝑁𝐴1𝑛𝑖=1𝛿𝑎1𝑛𝑖𝑥𝑛𝑖𝑁𝐵11𝑖=1𝛿𝑏11𝑖𝑓11𝑖𝑁𝐵1𝑛𝑖=1𝛿𝑏1𝑛𝑖𝑓1𝑛𝑖,(2.4) where 𝛿𝑎𝑗𝑖 are nonzero elements of the jth row of Δ𝐴, 𝑦𝑗𝑖 are corresponding elements of 𝑦, 𝛿𝑏𝑗𝑖 are nonzero elements of the jth row of Δ𝐵 and 𝑓𝑗𝑖 are corresponding elements of 𝑓, 𝑗=1,,𝑛.

Assumption 2.1. The unknown parameters are norm-bounded, that is, ||𝛿𝑎𝑗𝑖||𝑑𝑎𝑗𝑖,||𝛿𝑏𝑗𝑖||𝑑𝑏𝑗𝑖,(2.5) where 𝑑𝑎𝑗𝑖 and 𝑑𝑏𝑗𝑖 are known positive constants.

Definition 2.2 (see [28]). Consider the master and slave chaotic systems described by (2.1) and (2.2), respectively. If there exists a constant 𝑇=𝑇(𝑒(0))>0, such that lim𝑡𝑇𝑒(𝑡)=0(2.6) and 𝑒(𝑡)0, if 𝑡𝑇, then the chaos synchronization between the systems (2.1) and (2.2) is achieved in a finite time.

Lemma 2.3 (see [28]). Consider the system ̇𝑥=𝑓(𝑥),𝑓(0)=0,𝑥𝑅𝑛,(2.7) where 𝑓𝐷𝑅𝑛 is continuous on an open neighborhood 𝐷𝑅𝑛.

Suppose there exists a continuous differential positive-definite function 𝑉(𝑥)𝐷𝑅, real numbers 𝑝>0, 0<𝜂<1, such that ̇𝑉(𝑥)+𝑝𝑉𝜂(𝑥)0,𝑥𝐷.(2.8)

Then, the origin of system (2.7) is a locally finite-time stable equilibrium, and the settling time, depending on the initial state x(0)=x0, satisfies 𝑇𝑥0𝑉1𝜂𝑥0𝑝(1𝜂).(2.9)

In addition, if 𝐷=𝑅𝑛 and 𝑉(𝑥) is also radially unbounded (i.e., 𝑉(𝑥)+ as 𝑥+), then the origin is a globally finite-time stable equilibrium of system (2.7).

Lemma 2.4 (see [28]). Suppose a1,a2,,an, and 0<q<2 are all real numbers, then the following inequality holds: ||a1||q+||a2||q++||an||qa21+a22++a2nq/2.(2.10)

3. Synchronization of Two Different Chaotic Systems with Parameter Uncertainties

Consider two different chaotic systems (2.1) and (2.2) from different initial states. The aim of controller design is to determine appropriate 𝑢 such that lim𝑡𝑇𝑒=0.(3.1)

Now we are ready to give the design steps.

Define Lyapunov function 𝑉=12𝑒𝑇𝑒+12𝑛𝑗=1𝑁𝐴2𝑗𝑖=11+𝑘𝑑𝑗1𝛾1𝑎2𝑗𝑖𝛿̃𝑎22𝑗𝑖+12𝑛𝑗=1𝑁𝐴1𝑗𝑖=11+𝑘𝑑𝑗1𝛾1𝑎1𝑗𝑖𝛿̃𝑎21𝑗𝑖+12𝑛𝑗=1𝑁𝐵2𝑗𝑖=11+𝑘𝑑𝑗1𝛾1𝑏2𝑗𝑖𝛿̃𝑏22𝑗𝑖+12𝑛𝑗=1𝑁𝐵1𝑗𝑖=11+𝑘𝑑𝑗1𝛾1𝑏1𝑗𝑖𝛿̃𝑏21𝑗𝑖,(3.2) where 𝛿̃𝑎2𝑗𝑖=𝛿̂𝑎2𝑗𝑖𝛿𝑎2𝑗𝑖, 𝛿̃𝑎1𝑗𝑖=𝛿̂𝑎1𝑗𝑖𝛿𝑎1𝑗𝑖, 𝛿̃𝑏2𝑗𝑖=𝛿̂𝑏2𝑗𝑖𝛿𝑏2𝑗𝑖, 𝛿̃𝑏1𝑗𝑖=𝛿̂𝑏1𝑗𝑖𝛿𝑏1𝑗𝑖, and 𝛿̂𝑎2𝑗𝑖, 𝛿̂𝑎1𝑗𝑖, 𝛿̂𝑏2𝑗𝑖, 𝛿̂𝑏1𝑗𝑖 are estimation values of 𝛿𝑎2𝑗𝑖, 𝛿𝑎1𝑗𝑖, 𝛿𝑏2𝑗𝑖, 𝛿𝑏1𝑗𝑖, respectively, and 𝛾1𝑎2𝑗𝑖, 𝛾1𝑎1𝑗𝑖, 𝛾1𝑏2𝑗𝑖, 𝛾1𝑏1𝑗𝑖, 𝑘𝑑𝑗 are constants greater than zero.

Taking the time derivative of (3.2) gives ̇𝑉=𝑒𝑇̇𝑒+𝑛𝑗=1𝑁𝐴2𝑗𝑖=11+𝑘𝑑𝑗1𝛾1𝑎2𝑗𝑖𝛿̃𝑎2𝑗𝑖𝛿̇̂𝑎2𝑗𝑖+𝑛𝑗=1𝑁𝐴1𝑗𝑖=11+𝑘𝑑𝑗1𝛾1𝑎1𝑗𝑖𝛿̃𝑎1𝑗𝑖𝛿̇̂𝑎1𝑗𝑖+𝑛𝑗=1𝑁𝐵2𝑗𝑖=11+𝑘𝑑𝑗1𝛾1𝑏2𝑗𝑖𝛿̃𝑏2𝑗𝑖𝛿̇̂𝑏2𝑗𝑖+𝑛𝑗=1𝑁𝐵1𝑗𝑖=11+𝑘𝑑𝑗1𝛾1𝑏1𝑗𝑖𝛿̃𝑏1𝑗𝑖𝛿̇̂𝑏1𝑗𝑖.(3.3)

Design the control law as 𝑢=𝐾𝑒𝐾𝐷̇𝑒+𝛼𝑠𝑐1sgn𝑒1||𝑒1||𝛼,,𝑐𝑛sgn𝑒𝑛||𝑒𝑛||𝛼𝑇𝜇𝑛𝑗=1𝑁𝐴2𝑗𝑖=1||𝛿̂𝑎2𝑗𝑖||+𝑑𝑎2𝑗𝑖1+𝛼+𝑛𝑗=1𝑁𝐴1𝑗𝑖=1||𝛿̂𝑎1𝑗𝑖||+𝑑𝑎1𝑗𝑖1+𝛼+𝑛𝑗=1𝑁𝐵2𝑗𝑖=1||𝛿̂𝑏2𝑗𝑖||+𝑑𝑏2𝑗𝑖1+𝛼+𝑛𝑗=1𝑁𝐵1𝑗𝑖=1||𝛿̂𝑏1𝑗𝑖||+𝑑𝑏1𝑗𝑖1+𝛼𝛼𝑒+𝑁𝐴11𝑖=1𝛿̂𝑎11𝑖𝑦1𝑖,,𝑁𝐴1𝑛𝑖=1𝛿̂𝑎1𝑛𝑖𝑦𝑛𝑖𝑇𝑁𝐵21𝑖=1𝛿̂𝑏21𝑖𝑓21𝑖,,𝑁𝐴1𝑛𝑖=1𝛿̂𝑏2𝑛𝑖𝑓2𝑛𝑖𝑇𝑁𝐴21𝑖=1𝛿̂𝑎21𝑖𝑦1𝑖,,𝑁𝐴2𝑛𝑖=1𝛿̂𝑎2𝑛𝑖𝑦𝑛𝑖𝑇+𝑁𝐵11𝑖=1𝛿̂𝑏11𝑖𝑓11𝑖,,𝑁𝐵1𝑛𝑖=1𝛿̂𝑏1𝑛𝑖𝑓1𝑛𝑖𝑇,(3.4) where 𝐾=diag{𝑘1,𝑘2,,𝑘𝑛}, 𝑘𝑖>0, 𝐾𝐷=diag{𝑘𝑑1,𝑘𝑑2,,𝑘𝑑𝑛}, 𝑘𝑑𝑖>0, and 𝑐𝑖>0 are constants, 0<𝛼<1 is a constant, 𝛼𝑠=[𝛼𝑠1𝛼𝑠𝑛]𝑇 and 𝛼𝑠𝑖 is given in (3.5) 𝛼𝑠𝑖=𝜕𝑉𝐴/𝜕𝑒𝑖𝑓𝐴𝑖+𝐾𝐴𝑖(𝜕𝑉𝐴)/(𝜕𝑒𝑖)𝑓𝐴𝑖2+(𝜕𝑉𝐴)/(𝜕𝑒𝑖)4𝜕𝑉𝐴/𝜕𝑒𝑖,if𝜕𝑉𝐴𝜕𝑒𝑖00,if𝜕𝑉𝐴𝜕𝑒𝑖=0,𝑖=1,,𝑛,(3.5) where 𝑉𝐴=(1/2)𝑒𝑇𝑒, 𝑓𝐴=𝐴2𝑦𝐵2𝑓2(𝑦)+𝐴1𝑥+𝐵1𝑓1(𝑥) and 𝐾𝐴𝑖>0. And 𝛼𝑒=[𝛼𝑒1𝛼𝑒𝑛]𝑇 and 𝛼𝑒𝑖 is given in 𝛼𝑒𝑖=1𝑒𝑖,if||𝑒𝑖||𝑒𝜎sgn𝑒𝑖𝑒𝜎,if||𝑒𝑖||<𝑒𝜎,(3.6) where 𝑒𝜎>0 is a small constant.

Substituting (3.4) into (2.4) gives ̇𝑒=𝐾𝐼+𝐾𝐷1𝑒𝑁𝐴21𝑖=11+𝑘𝑑11𝛿̃𝑎21𝑖𝑦1𝑖𝑁𝐴2𝑛𝑖=11+𝑘𝑑𝑛1𝛿̃𝑎2𝑛𝑖𝑦𝑛𝑖𝑁𝐵21𝑖=11+𝑘𝑑11𝛿̃𝑏21𝑖𝑓21𝑖𝑁𝐵2𝑛𝑖=11+𝑘𝑑𝑛1𝛿̃𝑏2𝑛𝑖𝑓2𝑛𝑖+𝑓𝐴+𝛼𝑠+𝑁𝐴11𝑖=11+𝑘𝑑11𝛿̃𝑎11𝑖𝑥1𝑖𝑁𝐴1𝑛𝑖=11+𝑘𝑑𝑛1𝛿̃𝑎1𝑛𝑖𝑥𝑛𝑖+𝑁𝐵11𝑖=11+𝑘𝑑11𝛿̃𝑏11𝑖𝑓11𝑖𝑁𝐵1𝑛𝑖=11+𝑘𝑑𝑛1𝛿̃𝑏1𝑛𝑖𝑓1𝑛𝑖𝜇𝐼+𝐾𝐷1𝑛𝑗=1𝑁𝐴2𝑗𝑖=1||𝛿̂𝑎2𝑗𝑖||+𝑑𝑎2𝑗𝑖1+𝛼+𝑛𝑗=1𝑁𝐴1𝑗𝑖=1||𝛿̂𝑎1𝑗𝑖||+𝑑𝑎1𝑗𝑖1+𝛼+𝑛𝑗=1𝑁𝐵2𝑗𝑖=1||𝛿̂𝑏2𝑗𝑖||+𝑑𝑏2𝑗𝑖1+𝛼+𝑛𝑗=1𝑁𝐵1𝑗𝑖=1||𝛿̂𝑏1𝑗𝑖||+𝑑𝑏1𝑗𝑖1+𝛼𝛼𝑒𝐼+𝐾𝐷1𝑐1sgn𝑒1||𝑒1||𝛼,,𝑐𝑛sgn𝑒𝑛||𝑒𝑛||𝛼𝑇.(3.7)

Case 1  ( |ei|e𝜎). Substituting (3.7) into (3.3) yields ̇𝑉𝑒𝑇𝐾𝐼+𝐾𝐷1𝑒𝑛𝑖=1𝑐𝑖1+𝑘𝑑𝑖1||𝑒𝑖||1+𝛼𝜇𝑘𝑑𝑛𝑗=1𝑁𝐴2𝑗𝑖=1||𝛿̂𝑎2𝑗𝑖||+𝑑𝑎2𝑗𝑖1+𝛼+𝑛𝑗=1𝑁𝐴1𝑗𝑖=1||𝛿̂𝑎1𝑗𝑖||+𝑑𝑎1𝑗𝑖1+𝛼+𝑛𝑗=1𝑁𝐵2𝑗𝑖=1||𝛿̂𝑏2𝑗𝑖||+𝑑𝑏2𝑗𝑖1+𝛼+𝑛𝑗=1𝑁𝐵1𝑗𝑖=1||𝛿̂𝑏1𝑗𝑖||+𝑑𝑏1𝑗𝑖1+𝛼𝑛𝑗=1𝑒𝑗𝑁𝐴2𝑗𝑖=11+𝑘𝑑𝑗1𝛿̃𝑎2𝑗𝑖𝑦𝑗𝑖+𝑛𝑗=1𝑒𝑗𝑁𝐴1𝑗𝑖=11+𝑘𝑑𝑗1𝛿̃𝑎1𝑗𝑖𝑥𝑗𝑖𝑛𝑗=1𝑒𝑗𝑁𝐵2𝑗𝑖=11+𝑘𝑑𝑗1𝛿̃𝑏2𝑗𝑖𝑓2𝑗𝑖+𝑛𝑗=1𝑒𝑗𝑁𝐵1𝑗𝑖=11+𝑘𝑑𝑗1𝛿̃𝑏1𝑗𝑖𝑓1𝑗𝑖+𝑛𝑗=1𝑁𝐴2𝑗𝑖=11+𝑘𝑑𝑗1𝛾1𝑎2𝑗𝑖𝛿̃𝑎2𝑗𝑖𝛿̇̂𝑎2𝑗𝑖+𝑛𝑗=1𝑁𝐴1𝑗𝑖=11+𝑘𝑑𝑗1𝛾1𝑎1𝑗𝑖𝛿̃𝑎1𝑗𝑖𝛿̇̂𝑎1𝑗𝑖+𝑛𝑗=1𝑁𝐵2𝑗𝑖=11+𝑘𝑑𝑗1𝛾1𝑏2𝑗𝑖𝛿̃𝑏2𝑗𝑖𝛿̇̂𝑏2𝑗𝑖+𝑛𝑗=1𝑁𝐵1𝑗𝑖=11+𝑘𝑑𝑗1𝛾1𝑏1𝑗𝑖𝛿̃𝑏1𝑗𝑖𝛿̇̂𝑏1𝑗𝑖,(3.8) where 𝑘𝑑=min{(1+𝑘𝑑1)1,(1+𝑘𝑑2)1,,(1+𝑘𝑑𝑛)1}. Choosing the updating law as 𝛿̇̂𝑎2𝑗𝑖=𝛾𝑎2𝑗𝑖𝑒𝑗𝑦𝑗𝑖,if||̂𝑎2𝑗𝑖||<𝑑𝑎2𝑗𝑖0,otherwise,𝛿̇̂𝑎1𝑗𝑖=𝛾𝑎1𝑗𝑖𝑒𝑗𝑥𝑗𝑖,if||̂𝑎1𝑗𝑖||<𝑑𝑎1𝑗𝑖0,otherwise,𝛿̇̂𝑏2𝑗𝑖=𝛾𝑏2𝑗𝑖𝑒𝑗𝑓2𝑗𝑖,if||̂𝑏2𝑗𝑖||<𝑑𝑏2𝑗𝑖0,otherwise,𝛿̇̂𝑏1𝑗𝑖=𝛾𝑏1𝑗𝑖𝑒𝑗𝑓1𝑗𝑖,if||̂𝑏1𝑗𝑖||<𝑑𝑏1𝑗𝑖0,otherwise.(3.9)

Substituting (3.9) into (3.8) yields ̇𝑉𝑒𝑇𝐾𝐼+𝐾𝐷1𝑒𝑛𝑖=1𝑐𝑖1+𝑘𝑑𝑖1||𝑒𝑖||1+𝛼𝜇𝑘𝑑𝑛𝑗=1𝑁𝐴2𝑗𝑖=1||𝛿̂𝑎2𝑗𝑖||+𝑑𝑎2𝑗𝑖1+𝛼+𝑛𝑗=1𝑁𝐴1𝑗𝑖=1||𝛿̂𝑎1𝑗𝑖||+𝑑𝑎1𝑗𝑖1+𝛼+𝑛𝑗=1𝑁𝐵2𝑗𝑖=1||𝛿̂𝑏2𝑗𝑖||+𝑑𝑏2𝑗𝑖1+𝛼+𝑛𝑗=1𝑁𝐵1𝑗𝑖=1||𝛿̂𝑏1𝑗𝑖||+𝑑𝑏1𝑗𝑖1+𝛼.(3.10)

Since ||𝛿̂𝑎2𝑗𝑖𝛿𝑎2𝑗𝑖||||𝛿̂𝑎2𝑗𝑖||+||𝛿𝑎2𝑗𝑖||||𝛿̂𝑎2𝑗𝑖||+𝑑𝑎2𝑗𝑖,||𝛿̂𝑎1𝑗𝑖𝛿𝑎1𝑗𝑖||||𝛿̂𝑎1𝑗𝑖||+||𝛿𝑎1𝑗𝑖||||𝛿̂𝑎1𝑗𝑖||+𝑑𝑎1𝑗𝑖,||𝛿̂𝑏2𝑗𝑖𝛿𝑏2𝑗𝑖||||𝛿̂𝑏2𝑗𝑖||+||𝛿𝑏2𝑗𝑖||||𝛿̂𝑏2𝑗𝑖||+𝑑𝑏2𝑗𝑖,||𝛿̂𝑏1𝑗𝑖𝛿𝑏1𝑗𝑖||||𝛿̂𝑏1𝑗𝑖||+||𝛿𝑏1𝑗𝑖||||𝛿̂𝑏1𝑗𝑖||+𝑑𝑏1𝑗𝑖(3.11) hold, one can conclude that (|𝛿̂𝑎2𝑗𝑖|+𝑑𝑎2𝑗𝑖)1+𝛼|𝛿̂𝑎2𝑗𝑖𝛿𝑎2𝑗𝑖|1+𝛼, ||𝛿̂𝑎1𝑗𝑖||+𝑑𝑎1𝑗𝑖1+𝛼||𝛿̂𝑎1𝑗𝑖𝛿𝑎1𝑗𝑖||1+𝛼,||𝛿̂𝑏2𝑗𝑖||+𝑑𝑏2𝑗𝑖1+𝛼||𝛿̂𝑏2𝑗𝑖𝛿𝑏2𝑗𝑖||1+𝛼,(3.12) and (|𝛿̂𝑏1𝑗𝑖|+𝑑𝑏1𝑗𝑖)1+𝛼|𝛿̂𝑏1𝑗𝑖𝛿𝑏1𝑗𝑖|1+𝛼. Therefore, the inequality (3.10) can be rewritten as ̇𝑉𝑒𝑇𝐾𝐼+𝐾𝐷1𝑒𝑛𝑖=1𝑐𝑖1+𝑘𝑑𝑖1||𝑒𝑖||1+𝛼𝜇𝑘𝑑𝑛𝑗=1𝑁𝐴2𝑗𝑖=1||𝛿̂𝑎2𝑗𝑖𝛿𝑎2𝑗𝑖||1+𝛼+𝑛𝑗=1𝑁𝐴1𝑗𝑖=1||𝛿̂𝑎1𝑗𝑖𝛿𝑎1𝑗𝑖||1+𝛼+𝑛𝑗=1𝑁𝐵2𝑗𝑖=1||𝛿̂𝑏2𝑗𝑖𝛿𝑏2𝑗𝑖||1+𝛼+𝑛𝑗=1𝑁𝐵1𝑗𝑖=1||𝛿̂𝑏1𝑗𝑖𝛿𝑏1𝑗𝑖||1+𝛼𝑐𝜇𝑛𝑖=1||𝑒𝑖||2+𝑛𝑗=1𝑁𝐴2𝑗𝑖=1||𝛿̂𝑎2𝑗𝑖𝛿𝑎2𝑗𝑖||2+𝑛𝑗=1𝑁𝐴1𝑗𝑖=1||𝛿̂𝑎1𝑗𝑖𝛿𝑎1𝑗𝑖||2+𝑛𝑗=1𝑁𝐵2𝑗𝑖=1||𝛿̂𝑏2𝑗𝑖𝛿𝑏2𝑗𝑖||2+𝑛𝑗=1𝑁𝐵1𝑗𝑖=1||𝛿̂𝑏1𝑗𝑖𝛿𝑏1𝑗𝑖||2(1+𝛼)/2𝑐𝜇𝑉(1+𝛼)/2,(3.13) where 𝑐𝜇=min{𝑐𝑖(1+𝑘𝑑𝑖)1,𝑘𝑖(1+𝑘𝑑𝑖)1,𝜇𝑘𝑑,𝑖=1,,𝑛}. According to Lemma 2.3, 𝑒𝐵𝑒𝜎 in a finite time, where 𝐵𝑒𝜎{𝑒||𝑒𝑖|𝑒𝜎,𝑖=1,,𝑛}.

Case 2  (|𝑒i|<𝑒𝜎). Using (3.3)–(3.7) and (3.9), it is easy to show that ̇𝑉𝑒𝑇𝐾𝐼+𝐾𝐷1𝑒𝑛𝑖=1𝑐𝑖1+𝑘𝑑𝑖1||𝑒𝑖||1+𝛼(3.14) holds. According to Barbalat’s lemma [37], we can conclude that 𝑒0 as 𝑡.

From the discussion above, we have the following result.

Theorem 3.1. For the systems (2.1) and (2.2), under Assumption 2.1, if the control law is designed as (3.4), updating laws are chosen as (3.9), then e will converge to 𝐵𝑒𝜎 in finite time, e0 as 𝑡, and 𝛿̃𝑎2𝑗𝑖, 𝛿̃𝑎1𝑗𝑖, 𝛿̃𝑏2𝑗𝑖, and 𝛿̃𝑏1𝑗𝑖 remain bounded.

Remark 3.2. Since the control signal (3.4) contains the discontinuous sign functions, as a hard switcher, it may cause undesirable chattering. In order to avoid the chattering, the “sgn” function can be replaced by a continuous function (tanh) to remove discontinuity.

4. Numerical Simulation

In this section, we present numerical results to verify the proposed synchronization approach.

Consider the following master chaotic system: ̇𝑥1̇𝑥2̇𝑥3=𝑎𝑥2𝑥1𝑏𝑥1𝑐𝑥1𝑥3𝑔𝑥3+𝑥21=𝑎𝑎0𝑏0000𝑔𝑥1𝑥2𝑥3+0000𝑐0000𝑥1𝑥3𝑥21=𝑎0𝑎00𝑏00000𝑔0𝐴1𝑥1𝑥2𝑥2𝑥+𝛿𝑎0𝛿𝑎00𝛿𝑏00000𝛿𝑔0Δ𝐴1𝑥1𝑥2𝑥3𝑥+0000𝑐00000𝐵10𝑥1𝑥3𝑥21𝑓1(𝑥)+0000𝛿𝑐0000𝛿0Δ𝐵10𝑥1𝑥3𝑥21𝑓1(𝑥),(4.1) where 𝑎=𝑎0+𝛿𝑎0, 𝑏=𝑏0+𝛿𝑏0, 𝑐=𝑐0+𝛿𝑐0, 𝑔=𝑔0+𝛿𝑔0, =0+𝑔0, 𝑎0=8, 𝛿𝑎0=2, 𝑏0=35, 𝛿𝑏0=5, 𝑐0=0.7, 𝛿𝑐0=0.3, 𝑔0=2.0, 𝛿𝑔0=0.5, 0=0.8, and 𝛿0=0.2.

The slave system is given with ̇𝑦1̇𝑦2̇𝑦3=𝑎1𝑦2𝑦1+𝑦2𝑦3𝑏1𝑦2𝑐1𝑦1𝑦3𝑔1𝑦21𝑦3=𝑎1𝑎100𝑏100𝑔11𝑦1𝑦2𝑦3+𝑎1000𝑐10000𝑦2𝑦3𝑦1𝑦30=𝑎10𝑎1000𝑏1000𝑔1010𝐴2𝑦1𝑦2𝑦3𝑦+𝛿𝑎10𝛿𝑎1000𝛿𝑏1000𝛿𝑔10𝛿10Δ𝐴2𝑦1𝑦2𝑦3𝑦+𝑎10000𝑐100000𝐵2𝑦2𝑦3𝑦1𝑦30𝑓2(𝑦)+𝛿𝑎10000𝛿𝑐100000Δ𝐵2𝑦2𝑦3𝑦1𝑦30𝑓2(𝑦)+𝑢1𝑢2𝑢3,(4.2) where 𝑎1=𝑎10+𝛿𝑎10, 𝑏1=𝑏10+𝛿𝑏10, 𝑐1=𝑐10+𝛿𝑐10, 𝑔1=𝑔10+𝛿𝑔10, 1=10+𝑔10, 𝑎10=0.8, 𝛿𝑎10=0.2, 𝑏10=2.0, 𝛿𝑏10=0.5, 𝑐10=0.7, 𝛿𝑐10=0.3, 𝑔10=0.7, 𝛿𝑔10=0.3, 10=3.0, 𝛿10=1.0. 𝑥(0)=1.8, 𝑦(0)=1.2, and 𝑧(0)=1.5.

The initial states in master system (4.1) are 𝑥1(0)=1.8, 𝑥2(0)=1.2, 𝑥3(0)=1.5. The initial states in slave system (4.2) are 𝑦1(0)=1.5, 𝑦2(0)=1.2, 𝑦3(0)=1.1. The initial parameter estimation values of the systems (2.1) and (2.2) are 𝛿̂𝑎0=0, 𝛿̂𝑏0=0, 𝛿̂𝑐0=0, 𝛿̂𝑔0=0, 𝛿0=0, 𝛿̂𝑎10=0, 𝛿̂𝑏10=0, 𝛿̂𝑐10=0, 𝛿̂𝑔10=0, and 𝛿10=0.

According to Remark 3.2, the control law (3.4) is modified as follows: 𝑢=𝐾𝑒𝐾𝐷̇𝑒+𝛼𝑠𝐶𝐴2𝐶𝐵2+𝐶𝐴1+𝐶𝐵1𝜇2||𝛿̂𝑎0||+𝑑𝑎01+𝛼+||𝛿̂𝑏0||+𝑑𝑏01+𝛼+||𝛿̂𝑐0||+𝑑𝑐01+𝛼+||𝛿̂𝑔0||+𝑑𝑔01+𝛼+|||𝛿0|||+𝑑01+𝛼+3||𝛿̂𝑎10||+𝑑𝑎101+𝛼+||𝛿̂𝑏10||+𝑑𝑏101+𝛼+||𝛿̂𝑐10||+𝑑𝑐101+𝛼+||𝛿̂𝑔10||+𝑑𝑔101+𝛼+|||𝛿10|||+𝑑101+𝛼𝛼𝑒𝑐1tanh𝜀𝑒1||𝑒1||𝛼,,𝑐𝑛tanh𝜀𝑒𝑛||𝑒𝑛||𝛼𝑇,(4.3) where 𝐶𝐴2=𝛿̂𝑎10𝑦2𝑦1,𝛿̂𝑏10𝑦2,𝛿̂𝑔10𝑦2𝛿10𝑦3𝑇,𝐶𝐵2=𝛿̂𝑎10𝑦2𝑦3,𝛿̂𝑐10𝑦1𝑦3,0𝑇,𝐶𝐴1=𝛿̂𝑎0𝑥2𝑥1,𝛿̂𝑏0𝑥1,𝛿̂𝑔0𝑥3𝑇,𝐶𝐵1=0,𝛿̂𝑐0𝑥1𝑥3,𝛿0𝑥21𝑇,𝐾=diag{70,54,30},𝐾𝐷=diag{0.93,0.75,0.1},𝜇=diag{1,0.2,0.01},𝐾𝐴=diag{2.3,2.1,2.3}𝑑𝑎0=2,𝑑𝑏0=10,𝑑𝑐0=1,𝑑𝑔0=1,𝑑0=2,𝑑𝑎10=1,𝑑𝑏10=2,𝑑𝑐10=2,𝑑𝑔10=2,𝑑10=2,𝜀=40,𝑐1=1,𝑐2=3,𝑐3=1.(4.4)

Choosing the updating law as 𝛿̇̂𝑎0=0.30𝑒1𝑥1𝑥2,if||̂𝑎0||<𝑑𝑎00,otherwise,𝛿̇̂𝑏0=0.90𝑒2𝑥1,if||̂𝑏0||<𝑑𝑏00,otherwise,𝛿̇̂𝑐0=0.001𝑒2𝑥1𝑥3,if||̂𝑐0||<𝑑𝑐00,otherwise,𝛿̇̂𝑔0=0.002𝑒3𝑥3,if||̂𝑔0||<𝑑𝑔00,otherwise,𝛿̇0=0.018𝑒3𝑥21,if|||0|||<𝑑00,otherwise,𝛿̇̂𝑎10=0.0006𝑒1𝑦1+𝑦2+𝑦2𝑦3,if||̂𝑎10||<𝑑𝑎100,otherwise,𝛿̇̂𝑏10=0.08𝑒2𝑦2,if||̂𝑏10||<𝑑𝑏100,otherwise,𝛿̇̂𝑐10=0.0015𝑒2𝑦1𝑦3,if||̂𝑐10||<𝑑𝑐100,otherwise,𝛿̇̂𝑔0=0.1𝑒3𝑦2,if||̂𝑔10||<𝑑𝑔100,otherwise,𝛿̇10=0.017𝑒3𝑦3,if|||10|||<𝑑100,otherwise.(4.5)

Chaotic behavior of the master chaotic system under the proposed parameters is shown in Figure 1. Chaotic behavior of the slave chaotic system under the proposed parameters is shown in Figure 2. From Figures 1 and 2, we know that the two systems are still chaotic under adopted uncertain parameters. The synchronization errors between two different chaotic systems are illustrated in Figures 3, 4, and 5, where the control inputs are activated at 𝑡=1s. One can see that the synchronization errors converge to the zero in a finite time, which implies that the chaos synchronization between the two different chaotic systems is realized. The time responses of parameter estimations ̂𝑎0, ̂𝑏0, ̂𝑐0, ̂𝑔0, and 0 are depicted in Figure 6. The time responses of parameter estimations ̂𝑎10, ̂𝑏10, ̂𝑐10, ̂𝑔10, and 10 are depicted in Figure 7.

607491.fig.001
Figure 1: Chaotic behavior of the master chaotic system under the proposed parameters.
607491.fig.002
Figure 2: Chaotic behavior of the slave chaotic system under the proposed parameters.
607491.fig.003
Figure 3: Synchronization error 𝑒1.
607491.fig.004
Figure 4: Synchronization error 𝑒2.
607491.fig.005
Figure 5: Synchronization error 𝑒3.
607491.fig.006
Figure 6: Curves of parameter estimations for the master system.
607491.fig.007
Figure 7: Curves of parameter estimations for the slave system.

According to the simulations, it has been shown that the proposed control algorithm provides stable behavior when using online adaptive laws. The control performance is satisfactory and the chattering phenomenon has been successfully improved by using tanh functions. In addition, it is easy to see that the parameter estimation values approach their real values in a finite time.

5. Conclusions

In this paper, we have studied chaos synchronization of two different chaotic systems with parameter uncertainties. The two different chaotic systems with parameter uncertainties are synchronized via robust adaptive control based on the Lyapunov stability theory and finite-time theory. The proposed method can be applied to a variety of chaos systems. It guarantees that all the error states are driven to zero in a finite time. Numerical simulations are given to show the proposed synchronization approach works well for synchronizing two different chaotic systems in a finite time, even when the parameters of both the master and slave systems are unknown.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant no. 60674090.

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