Adaptive Sliding Mode Controller Design for Projective Synchronization of Different Chaotic Systems with Uncertain Terms and External Bounded Disturbances

Abstract

Synchronization is very useful in many science and engineering areas. In practical application, it is general that there are unknown parameters, uncertain terms, and bounded external disturbances in the response system. In this paper, an adaptive sliding mode controller is proposed to realize the projective synchronization of two different dynamical systems with fully unknown parameters, uncertain terms, and bounded external disturbances. Based on the Lyapunov stability theory, it is proven that the proposed control scheme can make two different systems (driving system and response system) be globally asymptotically synchronized. The adaptive global projective synchronization of the Lorenz system and the Lü system is taken as an illustrative example to show the effectiveness of this proposed control method.

1. Introduction

The cooperative behavior of coupled nonlinear oscillators is of interest in connection with a wide variety of different phenomena in physics, engineering, biology, and economics. For example, systems of coupled nonlinear oscillators may be used to explain how different sectors of the economy adjust their individual commodity cycles relative to one another through the exchange of goods and capital units or via aggregate signals in the form of varying interest rates or raw materials prices. As part of the cooperative behaviors, the synchronization plays very important role in many applications. Synchronization occurs when oscillatory (or repetitive) systems via some kind of interaction adjust their behaviors relative to one another so as to attain a state where they work in unison [1], since the individual oscillators display chaotic dynamics in many cases and it is very important to analyze the synchronization of chaotic systems.

Synchronization of two coupled chaotic systems has attracted much attention for both theoretical studies and practical applications since the synchronization of the different chaotic systems was observed by Pecora et al. in 1997 [2]. Many investigations have been devoted to synchronization due to its potential application in many fields recently, such as security communication [3, 4], information processing [5, 6], and biological systems [7]. Many different synchronization strategies of different chaotic systems have been developed. Beside the generalized synchronization [8, 9], there are some other types of synchronization which are also very interesting and useful, like phase synchronization [10, 11], antiphase synchronization [12, 13], complete synchronization [14, 15], and lag synchronization [16, 17]. In 1999, a new synchronization observed by Mainieri and Rehacek in partially linear chaotic systems is called projective synchronization [18]. The dynamical behavior of projective synchronization refers to that two identical or different systems which are synchronized up to a constant scaling factor.

So far, great progress has been made in the research of projective synchronization among all types of synchronization because of its adjustable proportionality between the synchronized dynamical states [19, 20]. Xu et al. designed a control scheme to manipulate the scaling factor onto any desired value [21]. Wen realized full-state projective synchronization by using an observer-based control [22]. Sudheer and Sabir designed a coupling function for unidirectional coupling in identical and mismatched oscillators to realize function projective synchronization through open-plus-closed-loop coupling method [23]. In addition, Peng and Jiang realized generalized projective synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal [24]. When projective synchronization is applied to secure communications, the binary sequences can be extended to N-nary sequences in digital communication for achieving higher speed [25], which would exert a great influence on communication.

To achieve projective synchronization of identical or nonidentical chaotic systems with different initial conditions, many effective control methods have been proposed, such as linear and nonlinear feedback control [26], adaptive control [27], impulsive control [28], and sliding mode control [29]. With these control schemes, many dynamical systems, such as the Lorenz system, the Chen system, the Rössler system, and some other chaotic or hyperchaotic systems [30, 31], with known or unknown parameters, are all synchronized by various control methods based on Lyapunov stability theory. Sliding mode control is a variable structure control algorithm as it alters the dynamics of a nonlinear system by the application of a high-frequency switching control. In view of the simple structure, good robustness, and high reliability, sliding mode control is widely applied to motion control in engineering, especially used in determinacy system with an accurate mathematical model [32]. When the dynamical system has been precisely modeled, this control method is an effective way to realize the projective synchronization of various dynamical systems. Mou et al. and Yang et al. provided an adaptive sliding mode control method to achieve the generalized synchronization of integral-order and fractional-order chaotic dynamics with fully unknown parameters [33, 34]. However, the existing synchronizations were realized for only identical structure chaotic systems, and these methods cannot be directly applied to the systems with uncertain terms and disturbances which are the most cases in the applications.

In view of its great potential value in secure communication, the projective synchronization is increasingly becoming a very important research topic in the fields of synchronization and control of chaos. Since the system structure of the projective synchronization belongs to driving-response type (master-slave), the driving system and the response system synchronize up proportionally to a constant scaling factor by using a proper control scheme. Therefore, it is desirable to achieve the projective synchronization of nonidentical chaotic systems with unknown parameters, uncertain terms, and bounded external disturbances via adaptive sliding mode controller. This paper mainly focuses on the projective synchronization of nonidentical structure chaotic systems with fully unknown parameters, uncertain terms, and external bounded disturbances. Based on the Lyapunov stability theory, a kind of adaptive sliding mode controller is designed to achieve projective synchronization of two nonidentical structure chaotic systems. Here, the parameters of the response system are fully unknown, and there are uncertain terms and bounded external disturbances in the response system. Moreover, the complete projective synchronization and the antiphase projective synchronization can be realized by adjusting the scaling factor.

The remainder of this paper is organized as follows. Section 2 presents the general theory of projective synchronization. In Section 3, an adaptive sliding mode controller is proposed and the robust stability is analyzed. Synchronization analysis of the Lorenz system and the Lü system is given in Section 4, and finally, Sections 5 and 6 provide the numerical simulations and concluding remarks of this study, respectively.

2. General Theory of Projective Synchronization

The main problem discussed in this paper is projective synchronization, which will be implemented by driving-response scheme. The driving system can be described by the nonlinear differential equation 𝑥̇=𝑓𝑥,𝜙,𝑡, (1) where the observable state variable 𝑥∈𝑅^𝑛 is an 𝑛-dimensional column vector, 𝜙∈𝑅^𝑙  represents an 𝑙-dimensional parameter vector, and 𝑓(⋅)∈𝑅^𝑛 is a continuous or noncontinues nonlinear function vector.

The response system can be expressed as 𝑦̇=𝑔𝑦,𝜃,𝑡+𝑢(𝑡), (2) where 𝑦∈𝑅^𝑛  denotes the observable state variables of the response system (2), 𝜃∈𝑅^𝑚+1 represents an (𝑚+1)-dimensional known or unknown parameter vector, 𝑚 is the number of unknown parameters, 𝑔(⋅)∈𝑅^𝑛 is a function vector similar with 𝑓(⋅), and 𝑢(𝑡)∈𝑅^𝑛 is an 𝑛-dimensional control input vector which can be used to realize projective synchronization of the systems (1) and (2).

When the tracking error is defined as 𝑒_𝑖=𝑦_𝑖−𝜅𝑥_𝑖  (𝑖=1,2,…,𝑛), the synchronization error differential equation can be obtained as 𝑒̇=𝑦̇−𝜅𝑥̇=𝑔𝑦,𝜃,𝑡−𝜅𝑓𝑥,𝜙,𝑡+𝑢(𝑡). (3) Here, the parameter 𝜅 is a scaling factor, which can adjust the synchronization proportion of the systems (1) and (2).

In order to achieve the synchronization of the driving system (1) and the response system (2), the control input 𝑢(𝑡) needs to satisfy lim_𝑡→∞‖𝑒(𝑡)‖⟶0. (4) Using a proper control scheme, this kind of synchronization is called projective synchronization if lim_𝑡→∞‖𝑒(𝑡)‖→0 by varying the scaling factor 𝜅.

Projective synchronization includes several kinds of synchronization. Complete synchronization and antiphase synchronization are two special cases corresponding with scaling factor 𝜅=1 and −1, respectively.

In view of the structure of the systems (1) and (2), the projective synchronization can be divided into identical structure projective synchronization and nonidentical structure projective synchronization. In this paper, we focus on the projective synchronization of two nonidentical structure systems, which means that the driving system (1) and the response system (2) are not same, and the identical structure projective synchronization can be looked as a specific case of nonidentical structure projective synchronization.

3. Adaptive SMC Design and Robust Stability Analysis

In the most of the practical applications, the parameters of the response system are not known, and there are uncertain terms and external bounded disturbances in the response system. Therefore, it is necessary to consider this practical matter. In the following sections, this kind of response systems will be applied to the design and analysis of synchronization. This kind of response system can be described as 𝑦̇=𝑔𝑦,𝜃,𝑡+Δ𝑔𝑦,𝜃,𝑡+𝑑(𝑡)+𝑢(𝑡), (5) where parameter vector 𝜃∈𝑅^𝑚+1 is unknown, 𝑚 is the number of unknown parameters, Δ𝑔(𝑦,𝜃,𝑡) denotes uncertain and bounded function vector, and 𝑑(𝑡) is the unknown and bounded external disturbances, such as DC signal, AC signal, any kind of bounded noise signals, and chaotic signal and 𝑢(𝑡) represents the control input vector. Hence, Δ𝑔(𝑦,𝜃,𝑡) and 𝑑(𝑡) should meet the following conditions: ‖𝑑(𝑡)‖≤𝐷(𝑡)∈𝐿_∞,Δ𝑔𝑦,𝜃,𝑡≤𝐻𝑦,𝑡∈𝐿_∞, (6) where 𝐷(𝑡) and 𝐻(𝑦,𝑡) are two known upon bound of external disturbance and uncertain function, respectively, which are always set to be constants without loss of generality. In general, 𝑔(𝑦,𝜃,𝑡) can be described by the linear parameterization structure 𝑔𝑦,𝜃,𝑡=𝐺𝑦,𝑡𝜃, (7) where the elements of matrix 𝐺(𝑦,𝑡)∈𝑅^{𝑛×(𝑚+1)} and parameter vector 𝜃_𝑖 (𝑖=1,2,…,𝑛) can be one-order, high-order, or constant terms. Therefore, the response system (5) can be rewritten as 𝑦̇=𝐺𝑦,𝑡𝜃+Δ𝑔𝑦,𝜃,𝑡+𝑑(𝑡)+𝑢(𝑡). (8) Using the definition of tracking error, one can further obtain the synchronization error differential vector ̇̇̇(𝑡)(𝑡),𝑒=𝑦−𝜅𝑥=𝐺𝑦,𝑡𝜃+Δ𝑔𝑦,𝜃,𝑡+𝑑−𝜅𝑓𝑥,𝜙,𝑡+𝑢 (9) where 𝑒=[𝑒₁,𝑒₂,…,𝑒_𝑛]^𝑇.

Definition 1. For any initial conditions of the systems (1) and (8), the zero solution of the error system (9) is globally stable if the motion trajectory of the error dynamical system (9) satisfies |𝑒_𝑖|→0 (𝑖=1,2,…,𝑛) as 𝑡→+∞. The driving-response systems (1) and (8) are globally projective synchronization when the scaling factor 𝜅≠0. Specially, for 𝜅=1, it is called global complete synchronization, and it is called globally antiphase synchronization for 𝜅=−1.

We construct an 𝑛-dimensional sliding mode surface 𝑠=[𝑠₁,𝑠₂,…,𝑠_𝑛]^𝑇 with 𝑠_𝑖=𝜇_𝑖𝑒_𝑖 (𝑖=1,2,…,𝑛) and adopt a constant-velocity reaching law as a control scheme to realize globally projective synchronization of two nonidentical structure systems. Here, an 𝑛-dimensional matrix of gain adjustment Γ can be used to denote the parameters 𝜇_𝑖 (𝑖=1,2,…,𝑛), that is, Γ=diag(𝜇₁,𝜇₂,…,𝜇_𝑛). Further, the 𝑛-dimensional sliding mode surface 𝑠 can be represented as 𝑠=Γ𝑒.

Theorem 2. For the system (9), if the control input is designed as 𝑢(𝑡)=−𝐺𝑦,𝑡𝜃+𝜅𝑓𝑥,𝜙,𝑡−𝜌(𝑡)sgn(𝑠), (10) the zero solution of the error system (9) is globally stable when 𝜌(𝑡)=𝐻(𝑦,𝑡)+𝐷(𝑡)+𝜂  (𝜂>0). Here, 𝜃 is the estimated vector of the unknown parameter 𝜃 of the system (8), 𝜌(𝑡) denotes the constant-velocity of the trajectory inclining to the switch surface 𝑠=0, and the expression sgn(𝑠) is a sign function vector. In other words, the systems (1) and (8) can asymptotically achieve projective synchronization by using the control law (10) when 𝜌(𝑡)=𝐻(𝑦,𝑡)+𝐷(𝑡)+𝜂 (𝜂>0).

Proof. Construct the positive Lyapunov function 𝑉()=1𝑠,𝜃2𝑠^𝑇1𝑠+2𝜃^𝑇𝜃, (11) where 𝜃=𝜃−𝜃 is the estimation error vector of the parameter vector. Then, one can obtain ̇̇𝜃=−𝜃 as 𝜃 is a constant parameter vector for a certain response system. Taking time derivative of (11) along the solution of error dynamics (9) and control input (10), 𝑉̇(⋅)=𝑠^𝑇𝑠̇+𝜃^𝑇𝜃̇=𝑠^𝑇Γ𝑒̇+𝜃^𝑇̇𝜃=𝑠^𝑇Γ𝐺(𝑡)(𝑡+𝜃𝑦,𝑡𝜃+Δ𝑔𝑦,𝜃,𝑡+𝑑−𝜅𝑓𝑥,𝜙,𝑡+𝑢)^𝑇̇𝜃=𝑠^𝑇Γ𝐺(𝑡)(𝑡)(𝑠+𝜃𝑦,𝑡𝜃+Δ𝑔𝑦,𝜃,𝑡+𝑑−𝜌sgn)^𝑇̇𝜃=𝑠^𝑇Γ(𝑡)(𝑡)(𝑠+𝜃Δ𝑔𝑦,𝜃,𝑡+𝑑−𝜌sgn)_𝑇𝐺𝑦,𝑡^𝑇Γ^𝑇𝑠−𝜃^𝑇̇‌𝜃=𝑠^𝑇Γ(𝑡)−𝐻(𝑡)𝜈+𝜃Δ𝑔𝑦,𝜃,𝑡+𝑑𝑦,𝑡+𝐷+𝜂_𝑇𝐺𝑦,𝑡^𝑇Γ^𝑇̇𝑠−𝜃𝑠_𝑖𝑠≥0,𝑖=1,2,…,𝑛_𝑇Γ(𝑡)+𝐻(𝑡)𝜈+𝜃Δ𝑔𝑦,𝜃,𝑡+𝑑𝑦,𝑡+𝐷+𝜂_𝑇𝐺𝑦,𝑡^𝑇Γ^𝑇̇𝑠−𝜃𝑠_𝑖𝑠<0,𝑖=1,2,…,𝑛≤−𝜂^𝑇Γ𝜈+𝜃^𝑇𝐺𝑦,𝑡^𝑇Γ^𝑇̇𝑠−𝜃, (12) where 𝜈 is an 𝑛-dimensional vector with all entries equal to one, namely, 𝜈=[1,1,…,1]^𝑇. If the parameter identification update law is as follows: ̇‌𝜃=𝐺𝑦,𝑡^𝑇Γ^𝑇𝑠=𝐺𝑦,𝑡^𝑇Γ^𝑇Γ𝑒. (13) One can further obtain 𝑉̇(⋅)≤−𝜂𝑠^𝑇Γ𝜈=−𝜂^𝑛_𝑖=1𝜇_𝑖𝑠_𝑖≤0. (14) Then, 𝑉(⋅)𝑉̇(⋅)≥0,≤0⟶𝑠_𝑖∈𝐿_∞⟶𝑒_𝑖∈𝐿−−−−−→_∞^{𝑥𝑖∈𝐿∞}𝑦_𝑖∈𝐿_∞^{expression()14}̇−−−−−−−−−−−−→𝜃_𝑖∈𝐿_∞,𝑉(⋅)𝑉̇(⋅)≥0,≤0⟶𝜃∈𝐿_∞⟶𝜃⟹𝜃−−−−→^{𝜃∈𝐿∞}𝜃∈𝐿_∞,𝑥_𝑖,𝑦_𝑖,𝑠_𝑖,𝜃_𝑖,𝜌−−−−−−−−−−−−→^{expression(10)}𝑢_𝑖∈𝐿_{∞errorsystem(9)}−−−−−−−−−−−−→𝑒_𝑖̇∈𝐿_∞,𝑥̇_𝑖,𝑒̇−−−−−−−−→_𝑖^{̇𝑒𝑖=̇𝑦𝑖̇𝑥−𝜅𝑖}𝑦̇_𝑖∈𝐿_∞. (15)(16)(17)(18) In view of this, we define a new function 𝑓(⋅)=∑^𝑛_𝑖=1𝑘_𝑖𝑒²_𝑖, and the following equation (19) can be determined by derivative operation: 𝑓̇(⋅)=2^𝑛_𝑖=1𝑘_𝑖𝑒_𝑖𝑒_𝑖̇∈𝐿_∞. (19) As can be seen from (15) to (18), the estimated parameters, error variables, and state variables are bounded. Based on (15), (16), (19), and the Barbalat lemma, |𝑒_𝑖|→0 (𝑖=1,2,…,𝑛) as 𝑡→+∞, and the error dynamical system (9) is asymptotically stable at the origin. Therefore, this projective synchronization theorem was proved.

4. Projective Synchronization of the Lorenz System and the Lü System

In this section, the projective synchronization of the Lorenz system and the Lü system is used as an example to validate the proposed adaptive control technique in Section 3, where the Lorenz system is regarded as the driving system, and another with control inputs 𝑢(𝑡), unknown system parameters 𝜃, uncertain terms Δ𝑔(𝑦,𝜃,𝑡), and bounded external disturbances 𝑑(𝑡) acts as the response system.

The Lorenz system [35] is given by 𝑥̇₁=𝑎₁𝑥₂−𝑥₁,𝑥̇₂=𝑏₁𝑥₁−𝑥₂−𝑥₁𝑥₃,𝑥̇₃=𝑥₁𝑥₂−𝑐₁𝑥₃, (20) which has a typical butterfly chaotic attractor and captures lots of features of chaotic systems when the parameters 𝑎₁=10, 𝑏₁=28, and 𝑐₁=8/3. The Lü system [36] is a typical chaos anticontrol model, which is a chaotic system between the Lorenz system and the Chen system in the unified Lorenz system family. The nonlinear differential equations of the Lü system are 𝑦̇₁=𝑎₂𝑦₂−𝑦₁,𝑦̇₂=𝑏₂𝑦₂−𝑦₁𝑦₃,𝑦̇₃=𝑦₁𝑦₂−𝑐₂𝑦₃. (21) There exists a chaotic attractor in the Lü system when the parameters 𝑎₂=30, 𝑏₂=22, and 𝑐₂=3. In view of (8), the controlled Lü system (21) can be rearranged as 𝑦̇𝑦=𝑦̇¹₂𝑦̇₃𝜃=𝐺𝑦,𝑡¹𝜃₂𝜃₃𝜃₄+Δ𝑔₁𝑦,𝜃,𝑡Δ𝑔₂𝑦,𝜃,𝑡Δ𝑔₃+𝑑𝑦,𝜃,𝑡₁(𝑡)𝑑₂(𝑡)𝑑₃(𝑡)+𝑢₁(𝑡)𝑢₂(𝑡)𝑢₃(𝑡), (22) where 𝐺=𝑦𝑦,𝑡²−𝑦₁0000𝑦₂0−𝑦₁𝑦₃00−𝑦₃𝑦₁𝑦₂𝜃,𝜃=¹𝜃₂𝜃₃𝜃₄=𝑎₂𝑏₂𝑐₂1. (23)(24) Then, one can obtain the following error differential equations when the system (20) and (22) are the driven system and the response system, respectively: ̇𝑒̇𝑒=𝑒¹₂̇𝑒₃̇𝜃=𝐺𝑦,𝑡¹𝜃₂𝜃₃𝜃₄+Δ𝑔₁𝑦,𝜃,𝑡Δ𝑔₂𝑦,𝜃,𝑡Δ𝑔₃+𝑑𝑦,𝜃,𝑡₁(𝑡)𝑑₂(𝑡)𝑑₃(𝑡)+𝑢₁(𝑡)𝑢₂(𝑡)𝑢₃(𝑡)𝑎−𝜅¹𝑥₂−𝑥₁𝑏₁𝑥₁−𝑥₂−𝑥₁𝑥₃𝑥₁𝑥₂−𝑐₁𝑥₃. (25) One can construct a three-dimensional sliding mode surface 𝑠𝑠=𝑠¹₂𝑠₃=𝜇₁𝑒₁𝜇₂𝑒₂𝜇₃𝑒₃𝑒=Γ𝑒¹₂𝑒₃, (26) where the gain adjustment matrix Γ is 𝜇Γ=¹000𝜇₂000𝜇₃, (27) and adopt a constant-velocity reaching law as a control method to achieve projective synchronization. For the error system (24), when the control input can be designed as 𝑢(𝑡)=𝑢₁(𝑡)𝑢₂(𝑡)𝑢₃(𝑡)=−𝐺𝑦,𝑡𝜃₁𝜃₂𝜃₃𝜃₄𝑎+𝜅¹𝑥₂−𝑥₁𝑏₁𝑥₁−𝑥₂−𝑥₁𝑥₃𝑥₁𝑥₂−𝑐₁𝑥₃(𝑡)𝑠−𝜌sgn₁sgn𝑠₂sgn𝑠₃, (28) where 𝜃=[𝜃₁,𝜃₂,𝜃₃,𝜃₄]^𝑇 is the online estimation for unknown parameter vector 𝜃=[𝜃₁,𝜃₂,𝜃₃,𝜃₄]^𝑇 of the system (22), that is, 𝜃=𝜃₁𝜃₂𝜃₃𝜃₄=𝑎₂𝑏₂𝑐₂1, (29) hence, one can obtain ̇𝑒̇𝑒=𝑒¹₂̇𝑒₃̇=𝐺𝑦,𝑡𝜃+Δ𝑔₁𝑦,𝜃,𝑡Δ𝑔₂𝑦,𝜃,𝑡Δ𝑔₃+𝑑𝑦,𝜃,𝑡₁(𝑡)𝑑₂(𝑡)𝑑₃(𝑡)(𝑡)𝑠−𝜌sgn₁sgn𝑠₂sgn𝑠₃, (30) where 𝜃𝜃=¹𝜃₂𝜃₃𝜃₄−𝜃₁𝜃₂𝜃₃𝜃₄=𝑎₂−𝑎₂𝑏₂−𝑏₂𝑐₂−𝑐₂0=𝑎₂𝑏̃₂𝑐₂̃0. (31) Consequently, ̇𝜃₁̇𝜃₂̇𝜃₃̇𝜃₄̇=−𝜃₁̇‌𝜃₂̇‌𝜃₃̇‌𝜃₄⟶̇𝑎₂̇̃𝑏₂̇𝑐₂̃̇=−𝑎₂̇‌𝑏₂𝑐₂̇‌. (32) Construct the following positive Lyapunov function: 𝑉𝜃=1𝑠,2𝑠^𝑇1𝑠+2𝜃^𝑇𝜃. (33) Taking time derivative of (33) along the solution of error dynamics (25) and the control input (28), 𝑉̇(⋅)=𝑠^𝑇𝑠̇+𝜃^𝑇𝜃̇=𝑠^𝑇Γ𝑒̇+𝜃^𝑇𝜃̇=𝑠^𝑇𝑎Γ𝐺𝑦,𝑡²𝑏̃₂𝑐₂̃0+𝑠^𝑇ΓΔ𝑔₁𝑦,𝜃,𝑡Δ𝑔₂𝑦,𝜃,𝑡Δ𝑔₃+𝑑𝑦,𝜃,𝑡¹(𝑡)𝑑₂(𝑡)𝑑₃(𝑡)(𝑡)𝑠−𝜌sgn₁sgn𝑠₂sgn𝑠₃−𝑎₂𝑏̃₂𝑐₂̃0𝑎̇₂̇‌𝑏₂𝑐₂̇‌0=𝑎₂𝑏̃₂𝑐₂̃0𝐺^𝑇𝑦,𝑡Γ^𝑇̇𝑠−𝑎₂̇‌𝑏₂𝑐₂̇‌0+𝑠^𝑇ΓΔ𝑔₁𝑦,𝜃,𝑡Δ𝑔₂𝑦,𝜃,𝑡Δ𝑔₃+𝑑𝑦,𝜃,𝑡¹(𝑡)𝑑₂(𝑡)𝑑₃(𝑡)(𝑡)𝑠−𝜌sgn₁sgn𝑠₂sgn𝑠₃=𝑎₂𝑏̃₂𝑐₂̃0×𝑦₂−𝑦₁000𝑦₂000−𝑦₃0−𝑦₁𝑦₃𝑦₁𝑦₂×𝜇₁000𝜇₂000𝜇₃𝑠₁𝑠₂𝑠₃−𝑎̇₂̇‌𝑏₂𝑐₂̇‌0+𝑠^𝑇ΓΔ𝑔₁𝑦,𝜃,𝑡Δ𝑔₂𝑦,𝜃,𝑡Δ𝑔₃+𝑑𝑦,𝜃,𝑡¹(𝑡)𝑑₂(𝑡)𝑑₃(𝑡)(𝑡)𝑠−𝜌sgn₁sgn𝑠₂sgn𝑠₃=𝑎₂𝑏̃₂𝑐₂̃0×𝜇₁𝑠₁𝑦₂−𝑦₁−𝑎̇₂𝜇₂𝑠₂𝑦₂−̇𝑏₂−𝜇₃𝑠₃𝑦₃−𝑐₂̇−𝜇₂𝑠₂𝑦₁𝑦₃+𝜇₃𝑠₃𝑦₁𝑦₂+𝑠^𝑇ΓΔ𝑔₁𝑦,𝜃,𝑡Δ𝑔₂𝑦,𝜃,𝑡Δ𝑔₃+𝑑𝑦,𝜃,𝑡¹(𝑡)𝑑₂(𝑡)𝑑₃(𝑡)(𝑡)𝑠−𝜌sgn₁sgn𝑠₂sgn𝑠₃. (34) Here, the parameter update law is 𝑎̇‌₂=𝜇₁𝑠₁𝑦₂−𝑦₁=𝜇²₁𝑒₁𝑦₂−𝑦₁,̇𝑏₂=𝜇₂𝑠₂𝑦₂=𝜇²₂𝑒₂𝑦₂,𝑐₂̇=−𝜇₃𝑠₃𝑦₃=−𝜇²₃𝑒₃𝑦₃. (35) Using  𝜌(𝑡)=𝐻(𝑦,𝑡)+𝐷(𝑡)+𝜂 (𝜂>0) and (35), it can be concluded that𝑉̇(⋅)=𝑠^𝑇ΓΔ𝑔₁𝑦,𝜃,𝑡Δ𝑔₂𝑦,𝜃,𝑡Δ𝑔₃+𝑑𝑦,𝜃,𝑡¹(𝑡)𝑑₂(𝑡)𝑑₃(𝑡)−𝐻(𝑡)𝑠𝑦,𝑡+𝐷+𝜂sgn₁sgn𝑠₂sgn𝑠₃=𝑠^𝑇ΓΔ𝑔₁𝑦,𝜃,𝑡Δ𝑔₂𝑦,𝜃,𝑡Δ𝑔₃+𝑑𝑦,𝜃,𝑡¹(𝑡)𝑑₂(𝑡)𝑑₃(𝑡)−𝐻(𝑡)𝐻(𝑡)𝐻(𝑡)𝑠𝑦,𝑡+𝐷+𝜂𝑦,𝑡+𝐷+𝜂𝑦,𝑡+𝐷+𝜂^𝑖𝑠≥0,𝑖=1,2,3_𝑇ΓΔ𝑔₁𝑦,𝜃,𝑡Δ𝑔₂𝑦,𝜃,𝑡Δ𝑔₃+𝑑𝑦,𝜃,𝑡¹(𝑡)𝑑₂(𝑡)𝑑₃(𝑡)+𝐻(𝑡)𝐻(𝑡)𝐻(𝑡)𝑠𝑦,𝑡+𝐷+𝜂𝑦,𝑡+𝐷+𝜂𝑦,𝑡+𝐷+𝜂^𝑖𝜇<0,𝑖=1,2,3≤−𝜂₁𝑠₁(𝑡)+𝜇₂𝑠₂(𝑡)+𝜇₃𝑠₃(𝑡)≤0. (36)

The last part of this section is the proof of the projective synchronization of the Lorenz system and the Lü system using the control law (28) and the parameter update law (35). Hence, one can conclude that 𝑒₁, 𝑒₂, and 𝑒₃ tend to zero as 𝑡→+∞, 𝑎₂→𝑎₂, 𝑏₂→𝑏₂, and 𝑐₂→𝑐₂. Therefore, the projective synchronization of the Lorenz system and the Lü system with unknown parameters, uncertain terms, and bounded external disturbances is realized by the adaptive sliding mode controller (28). In view of (15) to (19), one can obtain 𝑒₁,𝑒₂,𝑒₃,𝑦₁,𝑦₂,𝑦₃,𝑎̇₂,̇𝑏₂,𝑐₂̇,𝑎₂,𝑏₂,𝑐₂,𝑢₁,𝑢₂,𝑢₃,𝑒₁̇,𝑒₂̇,𝑒₃̇,𝑦̇₁,𝑦̇₂,𝑦̇₃∈𝐿_∞. (37) The proof of the projective synchronization of the Lorenz system and the Lü system is thus completed.

5. Numerical Simulations

In this section, numerical simulations for the projective synchronization of the systems (20) and (22) are given to demonstrate the effectiveness and feasibility of the proposed controllers (28) and the online parameter update law (35). All the differential equations are solved by using the fourth-order Runge-Kutta method, and step size is set to 0.001.

In the simulations, parameters 𝑎₁, 𝑏₁, 𝑐₁ and the initial conditions of the Lorenz system are chosen as 10, 8⁄3, 28, and 𝑥₀=(2,−2,−5), respectively. Parameters 𝑎₂, 𝑏₂, 𝑐₂ and the initial conditions of the Lü system are set to 35, 7, 2.92, and 𝑦₀=(0,1,−2), respectively, and other parameters are Δ𝑔₁𝑦,𝜃,𝑡Δ𝑔₂𝑦,𝜃,𝑡Δ𝑔₃=(𝑡)()(),𝑑𝑦,𝜃,𝑡0.2sin0.5sin3𝑡0.3sin2𝑡₁(𝑡)𝑑₂(𝑡)𝑑₃(𝑡)=()()(),,(𝑡)0.5sin5𝑡0.3sin2𝑡0.1sin4𝑡Γ=200002000020𝜂=9,𝜅=1,𝜌=10. (38)

Figures 1(a), 2(a), and 3(a) display the synchronized state trajectories between the Lorenz system and the Lü system. From the evolution of the state variables, it can be found that the different trajectories from the initial values 𝑥₀ and 𝑦₀ are gradually close to each other and will overlap ultimately. Figures 1(b), 2(b), and 3(b) display that the errors converge to zero as time increases, which implies that the projective synchronization has been realized. Figure 4 shows that the relation curve is a line with a slop of 45°, which illustrates that the trajectories generated from the Lü system will equal that from the Lorenz system.

(a)

(b)

Figure 1 

Time series and synchronization error of trajectories of the Lorenz system and the Lü system: (a) 𝑥1(𝑦1) versus 𝑡; (b) 𝑒1 versus 𝑡.

(a)

(b)

Figure 2 

Time series and synchronization error of trajectories of the Lorenz system and the Lü system: (a) 𝑥2(𝑦2) versus 𝑡; (b) 𝑒2 versus 𝑡.

(a)

(b)

Figure 3 

Time series and synchronization error of trajectories of the Lorenz system and the Lü system: (a) 𝑥3(𝑦3) versus 𝑡; (b) 𝑒3 versus 𝑡.

Figure 4 

Synchronization behavior of the Lorenz system and the Lü system.

Figure 5 is the time series of control inputs 𝑢₁, 𝑢₂, and 𝑢₃, whose main role is to ensure that the response of the Lü system is the same as that of the Lorenz system with the lapse of time. Figure 6 is the projection of the completely synchronized attractors of the Lorenz system and the Lü system on phase plane. The trajectories of the driving system and the response system from different conditions will converge to an identical attractor. Figure 7 depicts the time series of the estimated parameter, which are frozen at 𝑎₂=35, 𝑏₂=7, and 𝑐₂=2.92, respectively.

Figure 5 

Time series of control input (28).

Figure 6 

Attractors of complete projective synchronization with different conditions between the Lorenz system and the Lü system.

Figure 7 

The estimated value of the unknown parameters 𝑎2, 𝑏2, and 𝑐2 in the Lü system.

When the scaling factor 𝜅=−1.2, using the control of inputs 𝑢₁, 𝑢₂, and 𝑢₃ and the parameter update law (35), the antiphase projection of synchronized attractors on 𝑥₁−𝑥₃  (𝑦₁−𝑦₃) plane and synchronization behavior of the Lorenz system and the Lü system are shown in Figures 8 and 9, respectively. It is obvious that the antiphase projective synchronization in proportion is obtained by varying the scaling factor 𝜅.

Figure 8 

Antiphase projective synchronization in proportion with the scaling factor 𝜅=−1.2.

Figure 9 

Synchronization behavior of the Lorenz system and the Lü system with the scaling factor 𝜅=−1.2.

6. Conclusions

In this paper, a projective synchronization theorem was proposed based on an adaptive sliding mode control algorithm when the parameters of the response system are unknown and there are uncertain terms and external bounded disturbances in the response system. As an application of this projective synchronization theorem, the projective synchronization of the Lorenz system and the Lü system with uncertain terms and external bounded disturbances was also analyzed. The globally projective synchronization was achieved by a sliding mode controller based on the adaptive technique. The numerical simulations demonstrated the effectiveness and feasibility of the proposed technique. For the proposed method, there are potential applications in higher speed secure digital communication. In the future, we will try to extend the proposed method to the consensus of multiagent systems such as the networked chaotic systems and complex networks.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (nos. 61174094 and 11202148), China/South Africa Research Cooperation Programme (nos. 78673 and CS06-L02), and South African National Research Foundation Incentive Grant (no. 81705).