Robust Adaptive Sliding Mode Control for Generalized Function Projective Synchronization of Different Chaotic Systems with Unknown Parameters

Abstract

When the parameters of both drive and response systems are all unknown, an adaptive sliding mode controller, strongly robust to exotic perturbations, is designed for realizing generalized function projective synchronization. Sliding mode surface is given and the controlled system is asymptotically stable on this surface with the passage of time. Based on the adaptation laws and Lyapunov stability theory, an adaptive sliding controller is designed to ensure the occurrence of the sliding motion. Finally, numerical simulations are presented to verify the effectiveness and robustness of the proposed method even when both drive and response systems are perturbed with external disturbances.

1. Introduction

Chaos synchronization has been a hot topic since the pioneering work of Pecora and Carroll [1]. Various methods and techniques have been proposed for the control and synchronization of chaotic systems. In the past, most synchronizing techniques, such as linear and nonlinear feedback control [2–4], adaptive control [5, 6], sliding mode control [7, 8], and the liner matrix inequality (LMI) technique [9], have been designed for realizing synchronization of chaotic system.

Amongst all kinds of chaos synchronization, projective synchronization (PS), which was first reported by Mainieri and Rehacek [10], has been extensively investigated in the recent years because it can obtain faster communication with its proportional feature. this means that the drive and response systems can synchronize up to a scaling factor. Subsequently, Chen et al. [11] put forward another new projective synchronization, which is called function projective synchronization (FPS). Function projective synchronization is that the chaotic signals of the drive and response systems can synchronize up to a scaling function factor. Recently, some researchers [12, 13] have proposed a new synchronization scheme, called generalized function projective synchronization (GFPS), which is an extension of FPS. GFPS is more generalized and can give information transmission more security in secure communication. Wang and Fan [12] and Li and Zhao [13] have, respectively, used the state observer and adaptive controller to realize GFPS of a class of chaotic systems. It is a pity that they have not considered the influence of external perturbations.

In practice, the uncertainties and external disturbances affect the whole dynamics of the systems. So Yang and Ou [14] have discussed complete synchronization and antisynchronization of chaotic gyros with external perturbations using adaptive sliding mode control. But this method is not universal due to the discussion for specific chaotic system. Xiang and Chen [15] and Aghababa and Akbari [16] have all proposed sliding mode control method to synchronize two different chaotic systems with disturbances. Unfortunately, they [15, 16] only realize complete synchronization, and the parameters of the system in [14–16] are all known.

So, in this paper, we will extend adaptive sliding mode control method to realize GFPS of a class of chaotic systems with unknown parameters. According to two major stages of sliding mode design, we firstly give sliding mode surface, and the controlled system is robust to external perturbations on this surface. Secondly, adaptive sliding mode controller and the update laws of unknown parameters are given in order to make the system reach and subsequently stay on the sliding surface from any initial values. Finally two different chaotic systems are illustrated to verify the effectiveness of the proposed method, and it can be found that the designed controller has stronger robustness when both drive and response systems are all perturbed with external disturbances.

2. Description of Chaotic System

Consider a class of chaotic systems with unknown parameters which can be described as 𝑥̇₁=𝐹₁𝑥₁,𝑥₂,…,𝑥_𝑛+𝑓₁₁𝑥₁,𝑥₂,…,𝑥_𝑛𝑝₁₁+𝑓₁₂𝑥₁,𝑥₂,…,𝑥_𝑛𝑝₁₂+⋯+𝑓_1𝑚𝑥₁,𝑥₂,…,𝑥_𝑛𝑝_1𝑚,𝑥̇₂=𝐹₂𝑥₁,𝑥₂,…,𝑥_𝑛+𝑓₂₁𝑥₁,𝑥₂,…,𝑥_𝑛𝑝₂₁+𝑓₂₂𝑥₁,𝑥₂,…,𝑥_𝑛𝑝₂₂+⋯+𝑓_2𝑚𝑥₁,𝑥₂,…,𝑥_𝑛𝑝_2𝑚,⋮𝑥̇_𝑛=𝐹_𝑛𝑥₁,𝑥₂,…,𝑥_𝑛+𝑓_𝑛1𝑥₁,𝑥₂,…,𝑥_𝑛𝑝_𝑛1+𝑓_𝑛2𝑥₁,𝑥₂,…,𝑥_𝑛𝑝_𝑛2+⋯+𝑓_𝑛𝑚𝑥₁,𝑥₂,…,𝑥_𝑛𝑝_𝑛𝑚, (1)

where 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝑅^𝑛 denotes the state vector, 𝐹_𝑖,𝑓_𝑖𝑗(𝑖=1,2,…,𝑛;𝑗=1,2,…,𝑚) are nonlinear functions, and 𝑝_𝑖𝑗(𝑖=1,2,…,𝑛;𝑗=1,2,…,𝑚) are unknown parameters.

Take system (1) as the drive system; the response system, different from the system (1), with a controller 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇∈𝑅^𝑛, is as follows: 𝑦̇₁=𝐺₁𝑦₁,𝑦₂,…,𝑦_𝑛+𝑔₁₁𝑦₁,𝑦₂,…,𝑦_𝑛𝑞₁₁+𝑔₁₂𝑦₁,𝑦₂,…,𝑦_𝑛𝑞₁₂+⋯+𝑔_1𝑘𝑦₁,𝑦₂,…,𝑦_𝑛𝑞_1𝑘+𝑢₁,𝑦̇₂=𝐺₂𝑦₁,𝑦₂,…,𝑦_𝑛+𝑔₂₁𝑦₁,𝑦₂,…,𝑦_𝑛𝑞₂₁+𝑔₂₂𝑦₁,𝑦₂,…,𝑦_𝑛𝑞₂₂+⋯+𝑔_2𝑘𝑦₁,𝑦₂,…,𝑦_𝑛𝑞_2𝑘+𝑢₂,⋮𝑦̇_𝑛=𝐺_𝑛𝑦₁,𝑦₂,…,𝑦_𝑛+𝑔_𝑛1𝑦₁,𝑦₂,…,𝑦_𝑛𝑞_𝑛1+𝑔_𝑛2𝑦₁,𝑦₂,…,𝑦_𝑛𝑞_𝑛2+⋯+𝑔_𝑛𝑘𝑦₁,𝑦₂,…,𝑦_𝑛𝑞_𝑛𝑘+𝑢_𝑛, (2)

where 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇∈𝑅^𝑛 is the state vector of the system (2), 𝐺_𝑖,𝑔_𝑖𝑙(𝑖=1,2,…,𝑛;𝑙=1,2,…,𝑘) are nonlinear functions, and 𝑞_𝑖𝑙(𝑖=1,2,…,𝑛;𝑙=1,2,…,𝑘) are unknown parameters of the system (2).

Define the error between the drive and response systems as 𝑒₁=𝑦₁−ℎ₁𝑥₁,𝑥₂,…,𝑥_𝑛𝑥₁,𝑒₂=𝑦₂−ℎ₂𝑥₁,𝑥₂,…,𝑥_𝑛𝑥₂,⋮𝑒_𝑛=𝑦_𝑛−ℎ_𝑛𝑥₁,𝑥₂,…,𝑥_𝑛𝑥_𝑛, (3)

where the error vector is 𝑒=(𝑒₁,𝑒₂,…,𝑒_𝑛)^𝑇; ℎ(𝑥)=diag(ℎ₁(𝑥),ℎ₂(𝑥),…,ℎ_𝑛(𝑥)), ℎ_𝑖(𝑥)∶𝑅^𝑛→𝑅(𝑖=1,2,…,𝑛) are called scaling function factors, which compose the scaling function matrix ℎ(𝑥).

And suppose 𝐽=𝑑(ℎ(𝑥)𝑥)/𝑑𝑥=diag(𝐽₁,𝐽₂,…,𝐽_𝑛); the synchronization error can be obtained in the following compact form: 𝑒_𝑖=𝐺_𝑖𝑦+^𝑘_𝑙=1𝑞_𝑖𝑙𝑔_𝑖𝑙𝑦−𝐽_𝑖𝐹_𝑖(𝑥)−𝐽_𝑖^𝑚_𝑗=1𝑝_𝑖𝑗𝑓_𝑖𝑗(𝑥)+𝑢_𝑖. (4)

We say that generalized function projective synchronization (GFPS) with respect to the scaling function matrix ℎ(𝑥) is realized between the drive system (1) and response system (2) if a controller is designed such that the error system (4) can be achieved in the sense that lim_𝑡→∞‖𝑒(𝑡)‖=lim_𝑡→∞𝑦(𝑡)−ℎ(𝑥)𝑥(𝑡)=0. (5)

From the definition of GFPS, we know that the synchronization between two different chaotic systems with unknown parameters is that a controller U and a corresponding parameter update law are chosen to make the error system asymptotically converge to zero.

Remark 1. For the scaling function matrix ℎ(𝑥), if ℎ₁(𝑥)=ℎ₂(𝑥)=⋯=ℎ_𝑛(𝑥), GFPS is simplified to FPS. If ℎ(𝑥)=ℎ=diag(ℎ₁,ℎ₂,…,ℎ_𝑛) are all real constants, then FPS becomes PS. In particular, if ℎ(𝑥)=diag(−1,−1,…,−1), FPS problem is antisynchronization, and if ℎ(𝑥)=𝐼=diag(1,1,…,1), FPS turns to complete synchronization. So we can flexibly choose the scaling function matrix ℎ(𝑥) for the actual requirement in engineering.

3. Design of the Adaptive Sliding Mode Controller

As we know, all systems are inevitably influenced by exotic perturbations. For a chaotic system, small error may lead to serious deterioration of synchronic performance because of the Butterfly Effect. Among many control methods, sliding mode control is a robust control method which has many interesting features such as fast response, low sensitivity to external disturbances, robustness to the plant uncertainties, and easy realization. So, in this paper, we use sliding mode control method to design a controller to realize GFPS.

Generally speaking, sliding mode design involves two major stages; the first step is selecting a switching surface such that the sliding motion is asymptotically stable and has a prescribed performance. And, in this paper, we define switching surface as 𝑠_𝑖=𝜆_𝑖𝑒_𝑖,(𝑖=1,2,…,𝑛), (6)

where the sliding surface parameters 𝜆_𝑖(𝑖=1,2,…,𝑛) are positive constants.

Secondly, a controller is designed to guarantee that the system can reach and subsequently stay on the sliding surface. We adopt an adaptive sliding mode controller as 𝑢_𝑖=𝐽_𝑖𝐹_𝑖(𝑥)−𝐺_𝑖𝑦−𝜉_𝑖tanh𝜂_𝑖𝑠_𝑖+𝐽_𝑖^𝑚_𝑗=1𝑝_𝑖𝑗𝑓_𝑖𝑗(𝑥)−^𝑘_𝑙=1𝑞_𝑖𝑙𝑔_𝑖𝑙𝑦,(𝑖=1,2,…,𝑛), (7)

where 𝜉_𝑖 and 𝜂_𝑖 are adaptation coefficients which tune the gain and steepness of the tanh function, respectively. The parameters 𝑝_𝑖𝑗 and 𝑞_𝑖𝑙 are estimates of parameter values 𝑝_𝑖𝑗 and 𝑞_𝑖𝑙. And their update laws are in the following: 𝜉̇_𝑖=−𝛼_𝑖𝑠_𝑖𝑒_𝑖,𝜉_𝑖(0)=𝜉_𝑖0𝜂̇_𝑖=−𝛽_𝑖𝑠_𝑖𝑒_𝑖,𝜂_𝑖(0)=𝜂_𝑖0𝑝̇_𝑖𝑗=−𝜆_𝑖𝐽_𝑖𝑓_𝑖𝑗𝑠_𝑖,𝑞𝑖=1,2,…,𝑛;𝑗=1,2,…,𝑚_𝑖𝑙=𝜆_𝑖𝑔_𝑖𝑙𝑠_𝑖,(𝑖=1,2,…,𝑛;𝑙=1,2,…,𝑘), (8)

where 𝛼_𝑖, 𝛽_𝑖, 𝜉_𝑖0, and 𝜂_𝑖0(𝑖=1,2,…,𝑛) are all positive constants, and 𝜉_𝑖0 and 𝜂_𝑖0 are the initial values of 𝜉_𝑖 and 𝜂_𝑖(𝑖=1,2,…,𝑛).

Theorem 2. For the given continuous differential scaling matrix function ℎ(𝑥), if the parameters Ψ_𝑖(𝑖=1,2,…,𝑛) are greater than or equal to zero, where Ψ_𝑖=(𝜆_𝑖𝜉_𝑖|tanh(𝜂_𝑖𝑠_𝑖)|+𝜉_𝑖𝛼_𝑖|𝑒_𝑖|+𝜂_𝑖𝛽_𝑖|𝑒_𝑖|)(𝑖=1,2,…,𝑛), the error system will converge to the sliding surface 𝑠(𝑡)=0 when one chooses the sliding mode controller (7) and the parameter update law (8).

Proof. Choose the Lyapunov function as 1𝑉=2^𝑛_𝑖=1𝑠²_𝑖+^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑗−𝑝_𝑖𝑗²+^𝑛𝑘_𝑖=1𝑙=1𝑞_𝑖𝑙−𝑞_𝑖𝑙²+^𝑛_𝑖=1𝜉²_𝑖+^𝑛_𝑖=1𝜂²_𝑖. (9)
Taking derivative of the Lyapunov function with respect to time, one has 𝑉̇=^𝑛_𝑖=1𝑠_𝑖𝑠_𝑖̇+^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑗−𝑝_𝑖𝑗𝑝̇_𝑖𝑗+^𝑛𝑘_𝑖=1𝑙=1𝑞_𝑖𝑙−𝑞_𝑖𝑙𝑞̇_𝑖𝑙+^𝑛_𝑖=1𝜉_𝑖𝜉̇_𝑖+^𝑛_𝑖=1𝜂_𝑖𝜂̇_𝑖=^𝑛_𝑖=1𝜆_𝑖𝑠_𝑖𝐺_𝑖𝑦+^𝑘_𝑙=1𝑞_𝑖𝑙𝑔_𝑖𝑙𝑦−𝐽_𝑖𝐹_𝑖(𝑥)−𝐽_𝑖^𝑚_𝑗=1𝑝_𝑖𝑗𝑓_𝑖𝑗(𝑥)+𝑢_𝑖−^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑗−𝑝_𝑖𝑗𝜆_𝑖𝐽_𝑖𝑠_𝑖𝑓_𝑖𝑗(𝑥)+^𝑛𝑘_𝑖=1𝑙=1𝑞_𝑖𝑙−𝑞_𝑖𝑙𝜆_𝑖𝑠_𝑖𝑔_𝑖𝑙𝑦−^𝑛_𝑖=1𝜉_𝑖𝛼_𝑖𝑠_𝑖𝑒_𝑖−^𝑛_𝑖=1𝜂_𝑖𝛽_𝑖𝑠_𝑖𝑒_𝑖=^𝑛_𝑖=1𝜆_𝑖𝑠_𝑖^𝑘_𝑙=1𝑞_𝑖𝑙𝑔_𝑖𝑙𝑦−𝐽_𝑖^𝑚_𝑗=1𝑝_𝑖𝑗𝑓_𝑖𝑗(𝑥)−𝜉_𝑖tanh𝜂_𝑖𝑠_𝑖+𝐽_𝑖^𝑚_𝑗=1𝑝_𝑖𝑗𝑓_𝑖𝑗(𝑥)−^𝑘_𝑙=1𝑞_𝑖𝑙𝑔_𝑖𝑙𝑦−^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑗−𝑝_𝑖𝑗𝜆_𝑖𝐽_𝑖𝑠_𝑖𝑓_𝑖𝑗(𝑥)+^𝑛𝑘_𝑖=1𝑙=1𝑞_𝑖𝑙−𝑞_𝑖𝑙𝜆_𝑖𝑠_𝑖𝑔_𝑖𝑙𝑦−^𝑛_𝑖=1𝜉_𝑖𝛼_𝑖𝑠_𝑖𝑒_𝑖−^𝑛_𝑖=1𝜂_𝑖𝛽_𝑖𝑠_𝑖𝑒_𝑖=−^𝑛_𝑖=1𝜆_𝑖𝑠_𝑖𝜉_𝑖tanh𝜂_𝑖𝑠_𝑖−^𝑛_𝑖=1𝜉_𝑖𝛼_𝑖𝑠_𝑖𝑒_𝑖−^𝑛_𝑖=1𝜂_𝑖𝛽_𝑖𝑠_𝑖𝑒_𝑖≤−^𝑛_𝑖=1𝜆_𝑖𝜉_𝑖tanh𝜂_𝑖𝑠_𝑖+𝜉_𝑖𝛼_𝑖𝑒_𝑖+𝜂_𝑖𝛽_𝑖𝑒_𝑖𝑠_𝑖=−^𝑛_𝑖=1Ψ_𝑖𝑠_𝑖, (10)
where Ψ_𝑖=(𝜆_𝑖𝜉_𝑖|tanh(𝜂_𝑖𝑠_𝑖)|+𝜉_𝑖𝛼_𝑖|𝑒_𝑖|+𝜂_𝑖𝛽_𝑖|𝑒_𝑖|)(𝑖=1,2,…,𝑛).
For a given error system, if we deduce that Ψ_𝑖 is greater than or equal to zero, that is, (𝜆_𝑖𝜉_𝑖|tanh(𝜂_𝑖𝑠_𝑖)|+𝜉_𝑖𝛼_𝑖|𝑒_𝑖|+𝜂_𝑖𝛽_𝑖|𝑒_𝑖|)≥0(𝑖=1,2,…,𝑛), the following conclusions will be derived: 𝑉̇≤−^𝑛_𝑖=1Ψ_𝑖𝑠_𝑖=−𝜔(𝑡)≤0, (11)
where 𝜔(𝑡)=∑^𝑛_𝑖=1Ψ_𝑖|𝑠_𝑖|. Integrating (11) from zero to 𝑡 yields 𝑉(0)≥𝑉(𝑡)+^𝑡₀𝜔(𝑠)𝑑𝑠≥^𝑡₀𝜔(𝑠)𝑑𝑠, (12)
as 𝑡 approach infinite, and 𝑉(0) is positive and bounded, so lim_𝑡→∞∫^𝑡₀𝜔(𝑠)𝑑𝑠 exits and is finite. By Barbalat lemma, we have lim_𝑡→∞𝜔(𝑡)=lim^𝑛_{𝑡→∞𝑖=1}Ψ_𝑖𝑠_𝑖=0. (13)
Now, one will ask, “how can we conclude Ψ_𝑖≥0(𝑖=1,2,…,𝑛)?” and from theirs expressions, we can see that it is hard to conclude Ψ_𝑖≥0(𝑖=1,2,…,𝑛) by means of formula derivation. But we can get it with computer simulation, which will be seen in the following concrete example.
If the parameters Ψ_𝑖(𝑖=1,2,…,𝑛) are greater than or equal to zero, the sliding mode controller and the parameter update law are chosen by (7) and (8). this concludes that 𝑠_𝑖(𝑖=1,2,…,𝑛) are equal to zero when the system is operating on the sliding surface, and (6) is satisfied, so GFPS of the drive system (1) and response system (2) can be achieved as time goes on.

Remark 3. As we know, the conventional sliding mode controllers often use the sign function [17, 18], but the discontinuity of the sign function causes the chattering. In order to avoid the chattering, the discontinuous sign function is replaced by the continuous tanh function with the adaptive gain and steepness. As in [15, 16], the function is used as an approximate of the sign function.

4. Numerical Simulations

In this section, we take two different systems as the drive and response systems, respectively, to validate the robustness and effectiveness of the proposed method.

The drive system is the Lorenz system, which is the following mathematical expression: 𝑥̇₁=𝑝₁𝑥₂−𝑥₁,𝑥̇₂=𝑝₂𝑥₁−𝑥₁𝑥₃−𝑥₂,𝑥̇₃=−𝑝₃𝑥₃+𝑥₁𝑥₂, (14)

where 𝑥=(𝑥₁,𝑥₂,𝑥₃)^𝑇 is the state vector of the system (14), and 𝑝=(𝑝₁,𝑝₂,𝑝₃)^𝑇 is the parameter vector of (14). When these parameters are taken as 𝑝₁=10, 𝑝₂=28, and 𝑝₃=8/3, the system is chaotic.

The Genesio-Tesi (GT) system is chosen as the response system, and the expression with a controller is 𝑦̇₁=𝑦₂+𝑢₁,𝑦̇₂=𝑦₃+𝑢₂,𝑦̇₃=−𝑞₁𝑦₁−𝑞₂𝑦₂−𝑞₃𝑦₃+𝑦²₁+𝑢₃, (15)

where 𝑦=(𝑦₁,𝑦₂,𝑦₃)^𝑇 is the state vector of the system (15) and 𝑞=(𝑞₁,𝑞₂,𝑞₃)^𝑇 is the parameter vector of (15). When these parameters are taken as 𝑞₁=1.0, 𝑞₂=1.1, and 𝑞₃=0.44, the system is chaotic.

In numerical simulation, the scaling function matrix is chosen as ℎ₁𝑥₁=𝑣₁₁sin𝑣₁₂𝑥₁+𝑣₁₃+𝑣₁₄,ℎ₂𝑥₂=𝑣₂₁sin𝑣₂₂𝑥₂+𝑣₂₃+𝑣₂₄,ℎ₃𝑥₃=𝑣₃₁sin𝑣₃₂𝑥₃+𝑣₃₃+𝑣₃₄, (16)

and then the derivative of ℎ_𝑖(𝑖=1,2,3) can be derived as 𝐽₁=𝑣₁₁sin𝑣₁₂𝑥₁+𝑣₁₃+𝑣₁₁𝑣₁₂𝑥₁cos𝑣₁₂𝑥₁+𝑣₁₃+𝑣₁₄,𝐽₂=𝑣₂₁sin𝑣₂₂𝑥₂+𝑣₂₃+𝑣₂₁𝑣₂₂𝑥₂cos𝑣₂₂𝑥₂+𝑣₂₃+𝑣₂₄,𝐽₃=𝑣₃₁sin𝑣₃₂𝑥₃+𝑣₃₃+𝑣₃₁𝑣₃₂𝑥₃cos𝑣₃₂𝑥₃+𝑣₃₃+𝑣₃₄, (17)

where the parameters 𝑣₁₁=0.05, 𝑣₁₂=0.1, 𝑣₁₃=−0.2, 𝑣₁₄=1.0, 𝑣₂₁=0.02, 𝑣₂₂=−0.2, 𝑣₂₃=0.3,𝑣₂₄=1.0, 𝑣₃₁=0.03, 𝑣₃₂=0.3, 𝑣₃₃=−0.5, and 𝑣₃₄=1.0.

Now, the parameters Ψ₁, Ψ₂, and Ψ₃ of the error system are firstly simulated with the help of the computer. The sliding surface parameters 𝜆_𝑖(𝑖=1,2,…,𝑛) are all selected as 1.5. Vectors [3,−4,2.6] and [0.1,0.3,0] are selected as the initial conditions of the Lorenz and GT systems, respectively. And vectors [8,10,5] and [0.6,0.7,10] are selected as the initial values of the adaptation vector parameters 𝑝 and 𝑞. And the initial values of adaptation coefficients are chosen as 𝜉₀=[10,10,10] and 𝜂₀=[0.01,0.01,0.01]. The corresponding parameters 𝛼_𝑖,𝛽_𝑖(𝑖=1,2,3) are chosen as 𝛼₁=1.0, 𝛼₂=1.1, 𝛼₃=0.8, 𝛽₁=0.8, 𝛽₂=0.67, and 𝛽₃=0.78.

Figure 1 gives time evolution of the parameters Ψ₁, Ψ₂, and Ψ₃ of the error system. It shows that Ψ₁, Ψ₂, and Ψ₃ are greater than zero in the first few seconds and Ψ₁, Ψ₂, and Ψ₃ become zero rapidly as time goes on. According to Theorem 2, the control inputs are obtained by (7) 𝑢₁=−𝑦₂−𝜉₁tanh𝜂₁𝑠₁+𝐽₁𝑝₁𝑥₂−𝑥₁,𝑢₂=−𝑦₃−𝜉₂tanh𝜂₂𝑠₂−𝐽₂𝑥₁𝑥₃+𝑥₂+𝐽₂𝑝₂𝑥₁,𝑢₃=−𝑦²₁+𝐽₃𝑥₁𝑥₂−𝜉₃tanh𝜂₃𝑠₃+𝑞₁𝑦₁+𝑞₂𝑦₂+𝑞₃𝑦₃−𝐽₃𝑝₃𝑥₃, (18)

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Figure 1 

Time evolution of the parameters Ψ1, Ψ2, and Ψ3 of the error system.

where the parameter 𝑝=(𝑝₁,𝑝₂,𝑝₃)^𝑇and 𝑞=(𝑞₁,𝑞₂,𝑞₃)^𝑇; their update laws can be obtained by (8). Thus, GFPS of the Lorenz system (14) and response system (15) can be realized.

Figure 2 shows time evolution of GFPS errors of Lorenz and GT systems, without exotic perturbation. We can see that, with the adaptive sliding mode controller (18), the GFPS errors converge to zero rapidly, which verifies that our proposed method is feasible. Further, Figure 3 gives time evolution of the controller, 𝑢₁, 𝑢₂, and 𝑢₃. It shows that the chattering phenomenon does not really occur by the use of the continuous tanh function with the adaptive gain and steepness.

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Figure 2 

Time evolution of GFPS errors of Lorenz and GT system without exotic perturbation.

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Figure 3 

Time evolution of the controller 𝑢1, 𝑢2, and 𝑢3.

Figures 4 and 5 illustrate time evolution of the parameters 𝑝₁, 𝑝₂, 𝑝₃, 𝑝₃, 𝑞₁, 𝑞₂, and 𝑞₃ of the error system. They show that these parameters do not converge to the parameters 𝑝₁=10, 𝑝₂=28, 𝑝₃=8/3, 𝑞₁=1.0, 𝑞₂=1.1, and 𝑞₃=0.44 of the drive (14) and response systems (15) as time 𝑡 goes to infinity. So the unknown parameters cannot be estimated accurately by means of the parameters 𝑝₁, 𝑝₂, 𝑝₃, 𝑝₃, 𝑞₁, 𝑞₂, and 𝑞₃. In addition, we can derive from (8) that the update laws of 𝑝 and 𝑞 will converge to zero when the error signals tend to zero, so, as time goes on, the parameters 𝑝 and 𝑞 converge to some constant and do not necessarily tend to 𝑝 and 𝑞.

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Figure 4 

Time evolution of the parameters 𝑝1, 𝑝2, and 𝑝3 of the error system.

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Figure 5 

Time evolution of the parameters 𝑞1, 𝑞2, and 𝑞3 of the error system.

Figure 6 gives time evolution of the controlled signals 𝑦₁, 𝑦₂, and 𝑦₃ of the response system (15). It illustrates that, with the controller, the response system is still chaotic.

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Figure 6 

Time evolution of the controlled signals 𝑦1, 𝑦2, and 𝑦3 of the response system (15).

To further verify the robustness of the controller, it is assumed that the drive and response systems are influenced by exotic perturbations. It is assumed that the state vector of the system (14) is perturbed by 𝛿₁[sin5𝑡,cos7𝑡,sin4𝑡] and the state vector of the system (15) is perturbed by 𝛿₂[cos2𝑡,sin6𝑡,sin5𝑡], where 𝛿₁ and 𝛿₂ are the amplitudes of the exotic perturbations. When 𝛿₁=𝛿₂=0, that is, the systems (14) and (15) are all undisturbed by external factors, from Figure 1, we still can realize GFPS.

Further, the amplitudes are selected as 𝛿₁=𝛿₂=1.0. Figure 7 shows time evolution of GFPS errors of Lorenz and GT systems with the amplitude 1.0. We can see that the GFPS errors converge to zero rapidly, which implies our proposed method is robust. Secondly, the amplitudes are selected as 𝛿₁=𝛿₂=2.5. Figure 8 shows time evolution of GFPS errors of Lorenz and GT systems with the amplitude 2.5.

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Figure 7 

Time evolution of GFPS errors of Lorenz and GT system with the amplitude 1.0.

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Figure 8 

Time evolution of GFPS errors of Lorenz and GT system with the amplitude 2.5.

It shows that the GFPS errors 𝑒₁ and 𝑒₂ still converge to zero and 𝑒₃ slightly fluctuates around zero. Figure 9 gives time evolution of GFPS errors of Lorenz and GT systems with the amplitude 3.0. It implies that the GFPS errors 𝑒₁ and 𝑒₂ still converge to zero and 𝑒₃ fluctuates around zero. The fluctuation of 𝑒₃ around zero in Figure 9 becomes a little greater than that in Figure 8. Figure 10 illustrates a comparison of time evolution of the error 𝑒₃ with different amplitudes, which further show the greater amplitude of the exotic perturbation, the greater fluctuation around zero of the GFPS errors.

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Figure 9 

Time evolution of GFPS errors of Lorenz and GT system with the amplitude 3.0.

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Figure 10 

Time evolution of the error 𝑒3 with different amplitude.

All these results imply that our proposed method is strongly robust to exotic perturbations, especially the errors 𝑒₁ and 𝑒₂; they still converge to zero when the amplitude of the exotic perturbation is 3. So the method is effective.

5. Conclusions

Because the sliding mode control has some interesting features such as fast response, low sensitivity to external disturbances, robustness to the plant uncertainties, and easy realization, in this paper, we use this method to realize generalized function projective synchronization of two different chaotic systems with unknown parameters. According to two major stages of sliding mode design, we give the sliding mode surface and adaptive sliding controller. Finally, numerical simulations of two different chaotic systems are presented to show that the proposed method is effective and robust to exotic perturbations even when both drive and response systems are perturbed with external disturbances.

Acknowledgment

This Project is supported by the National Natural Science Foundation of China (Grant nos. 11002114, 60901076, 11102157, 11202035, and 11272258).