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

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 [1–3], 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 [5–11]. 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 [13–21], nonlinear feedback control [22], backstepping [23, 24], fuzzy technique [25–27], and sliding mode control [28–30], 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 [28–30], 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: 𝑥̇=𝐴₁+Δ𝐴₁𝑥+𝐵₁+Δ𝐵₁𝑓₁(𝑥),(2.1) where 𝑥=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑇∈𝑅^𝑛 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 𝑦̇=𝐴₂+Δ𝐴₂𝑦+𝐵₂+Δ𝐵₂𝑓₂(𝑦)+𝑢,(2.2) where 𝑦=[𝑦₁,𝑦₂,…,𝑦_𝑛]^𝑇∈𝑅^𝑛 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),…,𝑢_𝑛(𝑡)]^𝑇∈𝑅^𝑛 is a control input vector to be designed.

Subtracting (2.1) from (2.2) yields the error dynamical system as follows: 𝑒̇=𝐴₂+Δ𝐴₂𝑦+𝐵₂+Δ𝐵₂𝑓₂(𝑦)−𝐴₁+Δ𝐴₁𝑥−𝐵₁+Δ𝐵₁𝑓₁(𝑥)+𝑢,(2.3) where 𝑒=𝑦−𝑥. 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 𝑁_𝐴1𝑖, that of Δ𝐴₂ is 𝑁_𝐴2𝑖, that of Δ𝐵₁ is 𝑁_𝐵1𝑖, and that of Δ𝐵₂ is 𝑁_𝐵2𝑖. Then (2.3) can be rewritten as 𝑒̇=𝐴₂𝑦+𝐵₂𝑓₂(𝑦)−𝐴₁𝑥−𝐵₁𝑓₁(𝑥)+𝑢+⎡⎢⎢⎢⎢⎢⎣^𝑁𝐴21_𝑖∑₌₁𝛿𝑎_21𝑖𝑦_{1𝑖𝑁𝐴2𝑛}⋮_𝑖∑₌₁𝛿𝑎_2𝑛𝑖𝑦_𝑛𝑖⎤⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎣^𝑁𝐵21_𝑖∑₌₁𝛿𝑏_21𝑖𝑓_{21𝑖𝑁𝐵2𝑛}⋮_𝑖∑₌₁𝛿𝑏_2𝑛𝑖𝑓_2𝑛𝑖⎤⎥⎥⎥⎥⎥⎦−⎡⎢⎢⎢⎢⎢⎣^𝑁𝐴11_𝑖∑₌₁𝛿𝑎_11𝑖𝑥_{1𝑖𝑁𝐴1𝑛}⋮_𝑖∑₌₁𝛿𝑎_1𝑛𝑖𝑥_𝑛𝑖⎤⎥⎥⎥⎥⎥⎦−⎡⎢⎢⎢⎢⎢⎣^𝑁𝐵11_𝑖∑₌₁𝛿𝑏_11𝑖𝑓_{11𝑖𝑁𝐵1𝑛}⋮_𝑖∑₌₁𝛿𝑏_1𝑛𝑖𝑓_1𝑛𝑖⎤⎥⎥⎥⎥⎥⎦,(2.4) where 𝛿𝑎_∗𝑗𝑖 are nonzero elements of the jth row of Δ𝐴_∗, 𝑦_𝑗𝑖 are corresponding elements of 𝑦, 𝛿𝑏_∗𝑗𝑖 are nonzero elements of the jth row of Δ𝐵_∗ and 𝑓_∗𝑗𝑖 are corresponding elements of 𝑓_∗, 𝑗=1,…,𝑛.

Assumption 2.1. The unknown parameters are norm-bounded, that is, ||𝛿𝑎_∗𝑗𝑖||≤𝑑_{𝑎∗𝑗𝑖},||𝛿𝑏_∗𝑗𝑖||≤𝑑_{𝑏∗𝑗𝑖},(2.5) where 𝑑_{𝑎∗𝑗𝑖} and 𝑑_{𝑏∗𝑗𝑖} 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 lim_𝑡→𝑇‖𝑒(𝑡)‖=0(2.6) and ‖𝑒(𝑡)‖≡0, if 𝑡≥𝑇, 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 𝑥̇=𝑓(𝑥),𝑓(0)=0,𝑥∈𝑅^𝑛,(2.7) where 𝑓∶𝐷→𝑅^𝑛 is continuous on an open neighborhood 𝐷∈𝑅^𝑛.

Suppose there exists a continuous differential positive-definite function 𝑉(𝑥)∶𝐷→𝑅, real numbers 𝑝>0, 0<𝜂<1, such that 𝑉̇‌(𝑥)+𝑝𝑉^𝜂(𝑥)≤0,∀𝑥∈𝐷.(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 𝑇𝑥₀≤𝑉^1−𝜂𝑥₀𝑝(1−𝜂).(2.9)

In addition, if 𝐷=𝑅^𝑛 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,…,a_n, and 0<q<2 are all real numbers, then the following inequality holds: ||a₁||^q+||a₂||^q+⋯+||a_n||^q≥a²₁+a²₂+⋯+a²_n^q/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 lim_𝑡→𝑇𝑒=0.(3.1)

Now we are ready to give the design steps.

Define Lyapunov function 𝑉=12𝑒^𝑇𝑒+12^𝑛_𝑗=1^{𝑁𝐴2𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛾⁻¹_{𝑎2𝑗𝑖}𝛿𝑎̃²_2𝑗𝑖+12^𝑛_𝑗=1^{𝑁𝐴1𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛾⁻¹_{𝑎1𝑗𝑖}𝛿𝑎̃²_1𝑗𝑖+12^𝑛_𝑗=1^{𝑁𝐵2𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛾⁻¹_{𝑏2𝑗𝑖}𝛿𝑏̃‌²_2𝑗𝑖+12^𝑛_𝑗=1^{𝑁𝐵1𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛾⁻¹_{𝑏1𝑗𝑖}𝛿𝑏̃‌²_1𝑗𝑖,(3.2) where 𝛿𝑎̃_2𝑗𝑖=𝛿𝑎̂_2𝑗𝑖−𝛿𝑎_2𝑗𝑖, 𝛿𝑎̃_1𝑗𝑖=𝛿𝑎̂_1𝑗𝑖−𝛿𝑎_1𝑗𝑖, 𝛿𝑏̃‌_2𝑗𝑖=𝛿𝑏̂‌_2𝑗𝑖−𝛿𝑏_2𝑗𝑖, 𝛿𝑏̃‌_1𝑗𝑖=𝛿𝑏̂‌_1𝑗𝑖−𝛿𝑏_1𝑗𝑖, and 𝛿𝑎̂_2𝑗𝑖, 𝛿𝑎̂_1𝑗𝑖, 𝛿𝑏̂‌_2𝑗𝑖, 𝛿𝑏̂‌_1𝑗𝑖 are estimation values of 𝛿𝑎_2𝑗𝑖, 𝛿𝑎_1𝑗𝑖, 𝛿𝑏_2𝑗𝑖, 𝛿𝑏_1𝑗𝑖, respectively, and 𝛾⁻¹_{𝑎2𝑗𝑖}, 𝛾⁻¹_{𝑎1𝑗𝑖}, 𝛾⁻¹_{𝑏2𝑗𝑖}, 𝛾⁻¹_{𝑏1𝑗𝑖}, 𝑘_𝑑𝑗 are constants greater than zero.

Taking the time derivative of (3.2) gives 𝑉̇‌=𝑒^𝑇𝑒̇+^𝑛_𝑗=1^{𝑁𝐴2𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛾⁻¹_{𝑎2𝑗𝑖}𝛿𝑎̃_2𝑗𝑖𝛿𝑎̇‌̂_2𝑗𝑖+^𝑛_𝑗=1^{𝑁𝐴1𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛾⁻¹_{𝑎1𝑗𝑖}𝛿𝑎̃_1𝑗𝑖𝛿𝑎̇‌̂_1𝑗𝑖+^𝑛_𝑗=1^{𝑁𝐵2𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛾⁻¹_{𝑏2𝑗𝑖}𝛿𝑏̃‌_2𝑗𝑖𝛿𝑏̇‌̂‌_2𝑗𝑖+^𝑛_𝑗=1^{𝑁𝐵1𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛾⁻¹_{𝑏1𝑗𝑖}𝛿𝑏̃‌_1𝑗𝑖𝛿𝑏̇‌̂‌_1𝑗𝑖.(3.3)

Design the control law as 𝑢=−𝐾𝑒−𝐾_𝐷𝑒̇+𝛼_𝑠−𝑐₁sgn𝑒₁||𝑒₁||^𝛼,…,𝑐_𝑛sgn𝑒_𝑛||𝑒_𝑛||^𝛼^𝑇−𝜇⎡⎢⎢⎣^𝑛_𝑗=1^{𝑁𝐴2𝑗}_𝑖=1||𝛿𝑎̂_2𝑗𝑖||+𝑑_{𝑎2𝑗𝑖}^1+𝛼+^𝑛_𝑗=1^{𝑁𝐴1𝑗}_𝑖=1||𝛿𝑎̂_1𝑗𝑖||+𝑑_{𝑎1𝑗𝑖}^1+𝛼+^𝑛_𝑗=1^{𝑁𝐵2𝑗}_𝑖=1||𝛿𝑏̂‌_2𝑗𝑖||+𝑑_{𝑏2𝑗𝑖}^1+𝛼+^𝑛_𝑗=1^{𝑁𝐵1𝑗}_𝑖=1||𝛿𝑏̂‌_1𝑗𝑖||+𝑑_{𝑏1𝑗𝑖}^1+𝛼⎤⎥⎥⎦𝛼_𝑒+⎡⎢⎣^𝑁𝐴11_𝑖=1𝛿𝑎̂_11𝑖𝑦_1𝑖,…,^{𝑁𝐴1𝑛}_𝑖=1𝛿𝑎̂_1𝑛𝑖𝑦_𝑛𝑖⎤⎥⎦^𝑇−⎡⎢⎣^𝑁𝐵21_𝑖=1𝛿𝑏̂‌_21𝑖𝑓_21𝑖,…,^{𝑁𝐴1𝑛}_𝑖=1𝛿𝑏̂‌_2𝑛𝑖𝑓_2𝑛𝑖⎤⎥⎦^𝑇−⎡⎢⎣^𝑁𝐴21_𝑖=1𝛿𝑎̂_21𝑖𝑦_1𝑖,…,^{𝑁𝐴2𝑛}_𝑖=1𝛿𝑎̂_2𝑛𝑖𝑦_𝑛𝑖⎤⎥⎦^𝑇+⎡⎢⎣^𝑁𝐵11_𝑖=1𝛿𝑏̂‌_11𝑖𝑓_11𝑖,…,^{𝑁𝐵1𝑛}_𝑖=1𝛿𝑏̂‌_1𝑛𝑖𝑓_1𝑛𝑖⎤⎥⎦^𝑇,(3.4) where 𝐾=diag{𝑘₁,𝑘₂,…,𝑘_𝑛}, 𝑘_𝑖>0, 𝐾_𝐷=diag{𝑘_𝑑1,𝑘_𝑑2,…,𝑘_𝑑𝑛}, 𝑘_𝑑𝑖>0, and 𝑐_𝑖>0 are constants, 0<𝛼<1 is a constant, 𝛼_𝑠=[^{𝛼𝑠1…𝛼𝑠𝑛}]^𝑇 and 𝛼_𝑠𝑖 is given in (3.5) 𝛼_𝑠𝑖=⎧⎪⎪⎨⎪⎪⎩−𝜕𝑉_𝐴/𝜕𝑒_𝑖𝑓_𝐴𝑖+𝐾_𝐴𝑖(𝜕𝑉_𝐴)/(𝜕𝑒_𝑖)𝑓_𝐴𝑖²+(𝜕𝑉_𝐴)/(𝜕𝑒_𝑖)⁴𝜕𝑉_𝐴/𝜕𝑒_𝑖,if𝜕𝑉_𝐴𝜕𝑒_𝑖≠00,if𝜕𝑉_𝐴𝜕𝑒_𝑖=0,𝑖=1,…,𝑛,(3.5) where 𝑉_𝐴=(1/2)𝑒^𝑇𝑒, 𝑓_𝐴=−𝐴₂𝑦−𝐵₂𝑓₂(𝑦)+𝐴₁𝑥+𝐵₁𝑓₁(𝑥) and 𝐾_𝐴𝑖>0. And 𝛼_𝑒=[^{𝛼𝑒1…𝛼𝑒𝑛}]^𝑇 and 𝛼_𝑒𝑖 is given in 𝛼_𝑒𝑖=⎧⎪⎨⎪⎩1𝑒_𝑖,if||𝑒_𝑖||≥𝑒_𝜎sgn𝑒_𝑖𝑒_𝜎,if||𝑒_𝑖||<𝑒_𝜎,(3.6) where 𝑒_𝜎>0 is a small constant.

Substituting (3.4) into (2.4) gives 𝑒̇=−𝐾𝐼+𝐾_𝐷⁻¹𝑒−⎡⎢⎢⎢⎢⎢⎣^𝑁𝐴21_𝑖∑₌₁1+𝑘_𝑑1⁻¹𝛿𝑎̃_21𝑖𝑦_{1𝑖𝑁𝐴2𝑛}⋮_𝑖∑₌₁1+𝑘_𝑑𝑛⁻¹𝛿𝑎̃_2𝑛𝑖𝑦_𝑛𝑖⎤⎥⎥⎥⎥⎥⎦−⎡⎢⎢⎢⎢⎢⎣^𝑁𝐵21_𝑖∑₌₁1+𝑘_𝑑1⁻¹𝛿𝑏̃‌_21𝑖𝑓_{21𝑖𝑁𝐵2𝑛}⋮_𝑖∑₌₁1+𝑘_𝑑𝑛⁻¹𝛿𝑏̃‌_2𝑛𝑖𝑓_2𝑛𝑖⎤⎥⎥⎥⎥⎥⎦+𝑓_𝐴+𝛼_𝑠+⎡⎢⎢⎢⎢⎢⎣^𝑁𝐴11_𝑖∑₌₁1+𝑘_𝑑1⁻¹𝛿𝑎̃_11𝑖𝑥_{1𝑖𝑁𝐴1𝑛}⋮_𝑖∑₌₁1+𝑘_𝑑𝑛⁻¹𝛿𝑎̃_1𝑛𝑖𝑥_𝑛𝑖⎤⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎣^𝑁𝐵11_𝑖∑₌₁1+𝑘_𝑑1⁻¹𝛿𝑏̃‌_11𝑖𝑓_{11𝑖𝑁𝐵1𝑛}⋮_𝑖∑₌₁1+𝑘_𝑑𝑛⁻¹𝛿𝑏̃‌_1𝑛𝑖𝑓_1𝑛𝑖⎤⎥⎥⎥⎥⎥⎦−𝜇𝐼+𝐾_𝐷⁻¹⎡⎢⎢⎣^𝑛_𝑗=1^{𝑁𝐴2𝑗}_𝑖=1||𝛿𝑎̂_2𝑗𝑖||+𝑑_{𝑎2𝑗𝑖}^1+𝛼+^𝑛_𝑗=1^{𝑁𝐴1𝑗}_𝑖=1||𝛿𝑎̂_1𝑗𝑖||+𝑑_{𝑎1𝑗𝑖}^1+𝛼+^𝑛_𝑗=1^{𝑁𝐵2𝑗}_𝑖=1||𝛿𝑏̂‌_2𝑗𝑖||+𝑑_{𝑏2𝑗𝑖}^1+𝛼+^𝑛_𝑗=1^{𝑁𝐵1𝑗}_𝑖=1||𝛿𝑏̂‌_1𝑗𝑖||+𝑑_{𝑏1𝑗𝑖}^1+𝛼⎤⎥⎥⎦𝛼_𝑒−𝐼+𝐾_𝐷⁻¹𝑐₁sgn𝑒₁||𝑒₁||^𝛼,…,𝑐_𝑛sgn𝑒_𝑛||𝑒_𝑛||^𝛼^𝑇.(3.7)

Case 1  ( |e_i|≥e_𝜎). Substituting (3.7) into (3.3) yields 𝑉̇‌≤−𝑒^𝑇𝐾𝐼+𝐾_𝐷⁻¹𝑒−^𝑛_𝑖=1𝑐_𝑖1+𝑘_𝑑𝑖⁻¹||𝑒_𝑖||^1+𝛼−𝜇𝑘_𝑑⎡⎢⎢⎣^𝑛_𝑗=1^{𝑁𝐴2𝑗}_𝑖=1||𝛿𝑎̂_2𝑗𝑖||+𝑑_{𝑎2𝑗𝑖}^1+𝛼+^𝑛_𝑗=1^{𝑁𝐴1𝑗}_𝑖=1||𝛿𝑎̂_1𝑗𝑖||+𝑑_{𝑎1𝑗𝑖}^1+𝛼+^𝑛_𝑗=1^{𝑁𝐵2𝑗}_𝑖=1||𝛿𝑏̂‌_2𝑗𝑖||+𝑑_{𝑏2𝑗𝑖}^1+𝛼+^𝑛_𝑗=1^{𝑁𝐵1𝑗}_𝑖=1||𝛿𝑏̂‌_1𝑗𝑖||+𝑑_{𝑏1𝑗𝑖}^1+𝛼⎤⎥⎥⎦−^𝑛_𝑗=1𝑒^{𝑗𝑁𝐴2𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛿𝑎̃_2𝑗𝑖𝑦_𝑗𝑖+^𝑛_𝑗=1𝑒^{𝑗𝑁𝐴1𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛿𝑎̃_1𝑗𝑖𝑥_𝑗𝑖−^𝑛_𝑗=1𝑒^{𝑗𝑁𝐵2𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛿𝑏̃‌_2𝑗𝑖𝑓_2𝑗𝑖+^𝑛_𝑗=1𝑒^{𝑗𝑁𝐵1𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛿𝑏̃‌_1𝑗𝑖𝑓_1𝑗𝑖+^𝑛_𝑗=1^{𝑁𝐴2𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛾⁻¹_{𝑎2𝑗𝑖}𝛿𝑎̃_2𝑗𝑖𝛿𝑎̇‌̂_2𝑗𝑖+^𝑛_𝑗=1^{𝑁𝐴1𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛾⁻¹_{𝑎1𝑗𝑖}𝛿𝑎̃_1𝑗𝑖𝛿𝑎̇‌̂_1𝑗𝑖+^𝑛_𝑗=1^{𝑁𝐵2𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛾⁻¹_{𝑏2𝑗𝑖}𝛿𝑏̃‌_2𝑗𝑖𝛿𝑏̇‌̂‌_2𝑗𝑖+^𝑛_𝑗=1^{𝑁𝐵1𝑗}_𝑖=11+𝑘_𝑑𝑗⁻¹𝛾⁻¹_{𝑏1𝑗𝑖}𝛿𝑏̃‌_1𝑗𝑖𝛿𝑏̇‌̂‌_1𝑗𝑖,(3.8) where 𝑘_𝑑=min{(1+𝑘_𝑑1)⁻¹,(1+𝑘_𝑑2)⁻¹,…,(1+𝑘_𝑑𝑛)⁻¹}. Choosing the updating law as 𝛿𝑎̇‌̂_2𝑗𝑖=𝛾_{𝑎2𝑗𝑖}𝑒_𝑗𝑦_𝑗𝑖,if||𝑎̂_2𝑗𝑖||<𝑑_{𝑎2𝑗𝑖}0,otherwise,𝛿𝑎̇‌̂_1𝑗𝑖=−𝛾_{𝑎1𝑗𝑖}𝑒_𝑗𝑥_𝑗𝑖,if||𝑎̂_1𝑗𝑖||<𝑑_{𝑎1𝑗𝑖}0,otherwise,𝛿𝑏̇‌̂‌_2𝑗𝑖=𝛾_{𝑏2𝑗𝑖}𝑒_𝑗𝑓_2𝑗𝑖,if||𝑏̂‌_2𝑗𝑖||<𝑑_{𝑏2𝑗𝑖}0,otherwise,𝛿𝑏̇‌̂‌_1𝑗𝑖=−𝛾_{𝑏1𝑗𝑖}𝑒_𝑗𝑓_1𝑗𝑖,if||𝑏̂‌_1𝑗𝑖||<𝑑_{𝑏1𝑗𝑖}0,otherwise.(3.9)

Substituting (3.9) into (3.8) yields 𝑉̇‌≤−𝑒^𝑇𝐾𝐼+𝐾_𝐷⁻¹𝑒−^𝑛_𝑖=1𝑐_𝑖1+𝑘_𝑑𝑖⁻¹||𝑒_𝑖||^1+𝛼−𝜇𝑘_𝑑⎡⎢⎢⎣^𝑛_𝑗=1^{𝑁𝐴2𝑗}_𝑖=1||𝛿𝑎̂_2𝑗𝑖||+𝑑_{𝑎2𝑗𝑖}^1+𝛼+^𝑛_𝑗=1^{𝑁𝐴1𝑗}_𝑖=1||𝛿𝑎̂_1𝑗𝑖||+𝑑_{𝑎1𝑗𝑖}^1+𝛼+^𝑛_𝑗=1^{𝑁𝐵2𝑗}_𝑖=1||𝛿𝑏̂‌_2𝑗𝑖||+𝑑_{𝑏2𝑗𝑖}^1+𝛼+^𝑛_𝑗=1^{𝑁𝐵1𝑗}_𝑖=1||𝛿𝑏̂‌_1𝑗𝑖||+𝑑_{𝑏1𝑗𝑖}^1+𝛼⎤⎥⎥⎦.(3.10)

Since ||𝛿𝑎̂_2𝑗𝑖−𝛿𝑎_2𝑗𝑖||≤||𝛿𝑎̂_2𝑗𝑖||+||𝛿𝑎_2𝑗𝑖||≤||𝛿𝑎̂_2𝑗𝑖||+𝑑_{𝑎2𝑗𝑖},||𝛿𝑎̂_1𝑗𝑖−𝛿𝑎_1𝑗𝑖||≤||𝛿𝑎̂_1𝑗𝑖||+||𝛿𝑎_1𝑗𝑖||≤||𝛿𝑎̂_1𝑗𝑖||+𝑑_{𝑎1𝑗𝑖},||𝛿𝑏̂‌_2𝑗𝑖−𝛿𝑏_2𝑗𝑖||≤||𝛿𝑏̂‌_2𝑗𝑖||+||𝛿𝑏_2𝑗𝑖||≤||𝛿𝑏̂‌_2𝑗𝑖||+𝑑_{𝑏2𝑗𝑖},||𝛿𝑏̂‌_1𝑗𝑖−𝛿𝑏_1𝑗𝑖||≤||𝛿𝑏̂‌_1𝑗𝑖||+||𝛿𝑏_1𝑗𝑖||≤||𝛿𝑏̂‌_1𝑗𝑖||+𝑑_{𝑏1𝑗𝑖}(3.11) hold, one can conclude that −(|𝛿𝑎̂_2𝑗𝑖|+𝑑_{𝑎2𝑗𝑖})^1+𝛼≤−|𝛿𝑎̂_2𝑗𝑖−𝛿𝑎_2𝑗𝑖|^1+𝛼, −||𝛿𝑎̂_1𝑗𝑖||+𝑑_{𝑎1𝑗𝑖}^1+𝛼≤−||𝛿𝑎̂_1𝑗𝑖−𝛿𝑎_1𝑗𝑖||^1+𝛼,−||𝛿𝑏̂‌_2𝑗𝑖||+𝑑_{𝑏2𝑗𝑖}^1+𝛼≤−||𝛿𝑏̂‌_2𝑗𝑖−𝛿𝑏_2𝑗𝑖||^1+𝛼,(3.12) and −(|𝛿𝑏̂‌_1𝑗𝑖|+𝑑_{𝑏1𝑗𝑖})^1+𝛼≤−|𝛿𝑏̂‌_1𝑗𝑖−𝛿𝑏_1𝑗𝑖|^1+𝛼. Therefore, the inequality (3.10) can be rewritten as 𝑉̇‌≤−𝑒^𝑇𝐾𝐼+𝐾_𝐷⁻¹𝑒−^𝑛_𝑖=1𝑐_𝑖1+𝑘_𝑑𝑖⁻¹||𝑒_𝑖||^1+𝛼−𝜇𝑘_𝑑⎡⎢⎢⎣^𝑛_𝑗=1^{𝑁𝐴2𝑗}_𝑖=1||𝛿𝑎̂_2𝑗𝑖−𝛿𝑎_2𝑗𝑖||^1+𝛼+^𝑛_𝑗=1^{𝑁𝐴1𝑗}_𝑖=1||𝛿𝑎̂_1𝑗𝑖−𝛿𝑎_1𝑗𝑖||^1+𝛼+^𝑛_𝑗=1^{𝑁𝐵2𝑗}_𝑖=1||𝛿𝑏̂‌_2𝑗𝑖−𝛿𝑏_2𝑗𝑖||^1+𝛼+^𝑛_𝑗=1^{𝑁𝐵1𝑗}_𝑖=1||𝛿𝑏̂‌_1𝑗𝑖−𝛿𝑏_1𝑗𝑖||^1+𝛼⎤⎥⎥⎦≤−𝑐_𝜇⎡⎢⎢⎣^𝑛_𝑖=1||𝑒_𝑖||²+^𝑛_𝑗=1^{𝑁𝐴2𝑗}_𝑖=1||𝛿𝑎̂_2𝑗𝑖−𝛿𝑎_2𝑗𝑖||²+^𝑛_𝑗=1^{𝑁𝐴1𝑗}_𝑖=1||𝛿𝑎̂_1𝑗𝑖−𝛿𝑎_1𝑗𝑖||²+^𝑛_𝑗=1^{𝑁𝐵2𝑗}_𝑖=1||𝛿𝑏̂‌_2𝑗𝑖−𝛿𝑏_2𝑗𝑖||²+^𝑛_𝑗=1^{𝑁𝐵1𝑗}_𝑖=1||𝛿𝑏̂‌_1𝑗𝑖−𝛿𝑏_1𝑗𝑖||²⎤⎥⎥⎦^(1+𝛼)/2≤−𝑐_𝜇𝑉^(1+𝛼)/2,(3.13) where 𝑐_𝜇=min{𝑐_𝑖(1+𝑘_𝑑𝑖)⁻¹,𝑘_𝑖(1+𝑘_𝑑𝑖)⁻¹,𝜇𝑘_𝑑,𝑖=1,…,𝑛}. According to Lemma 2.3, 𝑒→𝐵_𝑒𝜎 in a finite time, where 𝐵_𝑒𝜎≜{𝑒||𝑒_𝑖|≤𝑒_𝜎,𝑖=1,…,𝑛}.

Case 2  (|𝑒_i|<𝑒_𝜎). Using (3.3)–(3.7) and (3.9), it is easy to show that 𝑉̇‌≤−𝑒^𝑇𝐾𝐼+𝐾_𝐷⁻¹𝑒−^𝑛_𝑖=1𝑐_𝑖1+𝑘_𝑑𝑖⁻¹||𝑒_𝑖||^1+𝛼(3.14) holds. According to Barbalat’s lemma [37], we can conclude that 𝑒→0 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 𝐵_𝑒𝜎 in finite time, e→0 as 𝑡→∞, and 𝛿𝑎̃_2𝑗𝑖, 𝛿𝑎̃_1𝑗𝑖, 𝛿𝑏̃‌_2𝑗𝑖, and 𝛿𝑏̃‌_1𝑗𝑖 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: ⎡⎢⎢⎢⎢⎣𝑥̇₁𝑥̇₂𝑥̇₃⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣𝑎𝑥₂−𝑥₁𝑏𝑥₁−𝑐𝑥₁𝑥₃−𝑔𝑥₃+ℎ𝑥²₁⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣−𝑎𝑎0𝑏0000−𝑔⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝑥₁𝑥₂𝑥₃⎤⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎣0000−𝑐000ℎ⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣0𝑥₁𝑥₃𝑥²₁⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣−𝑎₀𝑎₀0𝑏₀0000−𝑔₀⎤⎥⎥⎥⎥_{𝐴1}⎦⎡⎢⎢⎢⎢⎣𝑥₁𝑥₂𝑥₂⎤⎥⎥⎥⎥_{𝑥}⎦+⎡⎢⎢⎢⎢⎣−𝛿𝑎₀𝛿𝑎₀0𝛿𝑏₀0000−𝛿𝑔₀⎤⎥⎥⎥⎥_{Δ𝐴1}⎦⎡⎢⎢⎢⎢⎣𝑥₁𝑥₂𝑥₃⎤⎥⎥⎥⎥_{𝑥}⎦+⎡⎢⎢⎢⎢⎣0000−𝑐₀000ℎ₀⎤⎥⎥⎥⎥_{𝐵1}⎦⎡⎢⎢⎢⎢⎣0𝑥₁𝑥₃𝑥²₁⎤⎥⎥⎥⎥_{𝑓1(𝑥)}⎦+⎡⎢⎢⎢⎢⎣0000−𝛿𝑐₀000𝛿ℎ₀⎤⎥⎥⎥⎥_{Δ𝐵1}⎦⎡⎢⎢⎢⎢⎣0𝑥₁𝑥₃𝑥²₁⎤⎥⎥⎥⎥_{𝑓1(𝑥)}⎦,(4.1) where 𝑎=𝑎₀+𝛿𝑎₀, 𝑏=𝑏₀+𝛿𝑏₀, 𝑐=𝑐₀+𝛿𝑐₀, 𝑔=𝑔₀+𝛿𝑔₀, ℎ=ℎ₀+ℎ𝑔₀, 𝑎₀=8, 𝛿𝑎₀=2, 𝑏₀=35, 𝛿𝑏₀=5, 𝑐₀=0.7, 𝛿𝑐₀=0.3, 𝑔₀=2.0, 𝛿𝑔₀=0.5, ℎ₀=0.8, and 𝛿ℎ₀=0.2.

The slave system is given with ⎡⎢⎢⎢⎢⎣𝑦₁̇𝑦₂̇𝑦₃̇⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣𝑎₁𝑦₂−𝑦₁+𝑦₂𝑦₃𝑏₁𝑦₂−𝑐₁𝑦₁𝑦₃𝑔₁𝑦₂−ℎ₁𝑦₃⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣−𝑎₁𝑎₁00𝑏₁00𝑔₁−ℎ₁⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝑦₁𝑦₂𝑦₃⎤⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎣𝑎₁000−𝑐₁0000⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝑦₂𝑦₃𝑦₁𝑦₃0⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣−𝑎₁₀𝑎₁₀00𝑏₁₀00𝑔₁₀−ℎ₁₀⎤⎥⎥⎥⎥_{𝐴2}⎦⎡⎢⎢⎢⎢⎣𝑦₁𝑦₂𝑦₃⎤⎥⎥⎥⎥_{𝑦}⎦+⎡⎢⎢⎢⎢⎣−𝛿𝑎₁₀𝛿𝑎₁₀00𝛿𝑏₁₀00𝛿𝑔₁₀−𝛿ℎ₁₀⎤⎥⎥⎥⎥_{Δ𝐴2}⎦⎡⎢⎢⎢⎢⎣𝑦₁𝑦₂𝑦₃⎤⎥⎥⎥⎥_{𝑦}⎦+⎡⎢⎢⎢⎢⎣𝑎₁₀000−𝑐₁₀0000⎤⎥⎥⎥⎥_{𝐵2}⎦⎡⎢⎢⎢⎢⎣𝑦₂𝑦₃𝑦₁𝑦₃0⎤⎥⎥⎥⎥_{𝑓2(𝑦)}⎦+⎡⎢⎢⎢⎢⎣𝛿𝑎₁₀000−𝛿𝑐₁₀0000⎤⎥⎥⎥⎥_{Δ𝐵2}⎦⎡⎢⎢⎢⎢⎣𝑦₂𝑦₃𝑦₁𝑦₃0⎤⎥⎥⎥⎥_{𝑓2(𝑦)}⎦+⎡⎢⎢⎢⎢⎣𝑢₁𝑢₂𝑢₃⎤⎥⎥⎥⎥⎦,(4.2) where 𝑎₁=𝑎₁₀+𝛿𝑎₁₀, 𝑏₁=𝑏₁₀+𝛿𝑏₁₀, 𝑐₁=𝑐₁₀+𝛿𝑐₁₀, 𝑔₁=𝑔₁₀+𝛿𝑔₁₀, ℎ₁=ℎ₁₀+ℎ𝑔₁₀, 𝑎₁₀=0.8, 𝛿𝑎₁₀=0.2, 𝑏₁₀=2.0, 𝛿𝑏₁₀=0.5, 𝑐₁₀=0.7, 𝛿𝑐₁₀=0.3, 𝑔₁₀=0.7, 𝛿𝑔₁₀=0.3, ℎ₁₀=3.0, 𝛿ℎ₁₀=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 𝛿𝑎₀̂=0, 𝛿𝑏̂‌₀=0, 𝛿𝑐₀̂=0, 𝛿𝑔₀̂=0, 𝛿ℎ₀=0, 𝛿𝑎̂₁₀=0, 𝛿𝑏̂‌₁₀=0, 𝛿𝑐̂₁₀=0, 𝛿𝑔̂₁₀=0, and 𝛿ℎ₁₀=0.

According to Remark 3.2, the control law (3.4) is modified as follows: 𝑢=−𝐾𝑒−𝐾_𝐷𝑒̇+𝛼_𝑠−𝐶_𝐴2−𝐶_𝐵2+𝐶_𝐴1+𝐶_𝐵1−𝜇2||𝛿𝑎₀̂||+𝑑_𝑎0^1+𝛼+||𝛿𝑏̂‌₀||+𝑑_𝑏0^1+𝛼+||𝛿𝑐₀̂||+𝑑_𝑐0^1+𝛼+||𝛿𝑔₀̂||+𝑑_𝑔0^1+𝛼+|||𝛿ℎ₀|||+𝑑_ℎ0^1+𝛼+3||𝛿𝑎̂₁₀||+𝑑_𝑎10^1+𝛼+||𝛿𝑏̂‌₁₀||+𝑑_𝑏10^1+𝛼+||𝛿𝑐̂₁₀||+𝑑_𝑐10^1+𝛼+||𝛿𝑔̂₁₀||+𝑑_𝑔10^1+𝛼+|||𝛿ℎ₁₀|||+𝑑_ℎ10^1+𝛼𝛼_𝑒−𝑐₁tanh𝜀𝑒₁||𝑒₁||^𝛼,…,𝑐_𝑛tanh𝜀𝑒_𝑛||𝑒_𝑛||^𝛼^𝑇,(4.3) where 𝐶_𝐴2=𝛿𝑎̂₁₀𝑦₂−𝑦₁,𝛿𝑏̂‌₁₀𝑦₂,𝛿𝑔̂₁₀𝑦₂−𝛿ℎ₁₀𝑦₃^𝑇,𝐶_𝐵2=𝛿𝑎̂₁₀𝑦₂𝑦₃,−𝛿𝑐̂₁₀𝑦₁𝑦₃,0^𝑇,𝐶_𝐴1=𝛿𝑎₀̂𝑥₂−𝑥₁,𝛿𝑏̂‌₀𝑥₁,−𝛿𝑔₀̂𝑥₃^𝑇,𝐶_𝐵1=0,−𝛿𝑐₀̂𝑥₁𝑥₃,𝛿ℎ₀𝑥²₁^𝑇,𝐾=diag{70,54,30},𝐾_𝐷=diag{0.93,0.75,0.1},𝜇=diag{1,0.2,0.01},𝐾_𝐴=diag{2.3,2.1,2.3}𝑑_𝑎0=2,𝑑_𝑏0=10,𝑑_𝑐0=1,𝑑_𝑔0=1,𝑑_ℎ0=2,𝑑_𝑎10=1,𝑑_𝑏10=2,𝑑_𝑐10=2,𝑑_𝑔10=2,𝑑_ℎ10=2,𝜀=40,𝑐₁=1,𝑐₂=3,𝑐₃=1.(4.4)

Choosing the updating law as 𝛿𝑎̇‌₀̂=0.30𝑒₁𝑥₁−𝑥₂,if||𝑎₀̂||<𝑑_𝑎00,otherwise,𝛿𝑏̇‌̂‌₀=−0.90𝑒₂𝑥₁,if||𝑏̂‌₀||<𝑑_𝑏00,otherwise,𝛿𝑐̇‌₀̂=−0.001𝑒₂𝑥₁𝑥₃,if||𝑐₀̂||<𝑑_𝑐00,otherwise,𝛿𝑔̇‌₀̂=0.002𝑒₃𝑥₃,if||𝑔₀̂||<𝑑_𝑔00,otherwise,𝛿̇‌ℎ₀=⎧⎨⎩−0.018𝑒₃𝑥²₁,if|||ℎ₀|||<𝑑_ℎ00,otherwise,𝛿𝑎̇‌̂₁₀=0.0006𝑒₁−𝑦₁+𝑦₂+𝑦₂𝑦₃,if||𝑎̂₁₀||<𝑑_𝑎100,otherwise,𝛿𝑏̇‌̂‌₁₀=−0.08𝑒₂𝑦₂,if||𝑏̂‌₁₀||<𝑑_𝑏100,otherwise,𝛿𝑐̇‌̂₁₀=−0.0015𝑒₂𝑦₁𝑦₃,if||𝑐̂₁₀||<𝑑_𝑐100,otherwise,𝛿𝑔̇‌₀̂=0.1𝑒₃𝑦₂,if||𝑔̂₁₀||<𝑑_𝑔100,otherwise,𝛿̇‌ℎ₁₀=⎧⎨⎩−0.017𝑒₃𝑦₃,if|||ℎ₁₀|||<𝑑_ℎ100,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 𝑎₀̂, 𝑏̂‌₀, 𝑐₀̂, 𝑔₀̂, and ℎ₀ are depicted in Figure 6. The time responses of parameter estimations 𝑎̂₁₀, 𝑏̂‌₁₀, 𝑐̂₁₀, 𝑔̂₁₀, and ℎ₁₀ are depicted in Figure 7.

Figure 1 

Chaotic behavior of the master chaotic system under the proposed parameters.

Figure 2 

Chaotic behavior of the slave chaotic system under the proposed parameters.

Figure 3 

Synchronization error 𝑒1.

Figure 4 

Synchronization error 𝑒2.

Figure 5 

Synchronization error 𝑒3.

Figure 6 

Curves of parameter estimations for the master system.

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.