Robust Controller Design for Modified Projective Synchronization of Chen-Lee Chaotic Systems with Nonlinear Inputs

Abstract

This study demonstrates the modified projective synchronization in Chen-Lee chaotic system. The variable structure control technology is used to design the synchronization controller with input nonlinearity. Based on Lyapunov stability theory, a nonlinear controller and some generic sufficient conditions can be obtained to guarantee the modified projective synchronization, including synchronization, antisynchronization, and projective synchronization in spite of the input nonlinearity. The numerical simulation results show that the synchronization and antisynchronization can coexist in Chen-Lee chaotic systems. It demonstrates the validity and feasibility of the proposed controller.

1. Introduction

Chaos is very interesting nonlinear phenomenon and has been intensively studied in the last three decades [1–3]. A fundamental characteristic of a chaotic system is its extreme sensitivity to initial conditions; that is, small differences in the initial state can lead to extraordinary differences in the system state. Since the ideal of synchronizing two identical chaotic systems from different initial conditions was introduced by Carroll and Pecora in 1990, chaos synchronization has received increasing attention, especially in secure communication [4, 5]. Many methods have been presented for the synchronization of chaotic system [6–11]. However most of these methods are concentrated on studying complete synchronization (CS). In the practical applications, CS only occurs at a certain point in the parameter space, and it is difficult to achieve CS except under ideal conditions. Recently, thus a more general form of synchronization scheme, called generalized synchronization (GS), has been extensively investigated [12–16], where the drive and response system could be synchronized up to a scaling factor𝛼. More recently, Li [17] considers a new GS method, called modified projective synchronization (MPS), where the responses of the synchronized dynamical states synchronize up to a constant scaling matrix. However, most of research efforts mentioned above have concentrated on the linear control input. Moreover, when the controller is realized in practical physical systems, due to physical limitations of actuators, the nonlinearities in control input do exist. The presence of nonlinearities in control input may cause serious influence of system performance and decrease the system response. Besides, the nonlinearity in control input may cause the chaotic system perturbed to unpredictable results because the chaotic system is very sensitive to any system parameters. Therefore, its effect cannot be ignored in analysis of control design and realization for chaos synchronization. Thus the derivation of controller with input nonlinearity for chaos synchronization is an important problem.

In this paper, the problem of chaos synchronization to Chen-Lee system with input nonlinearity is considered. For MPS of the system, a variable structure control scheme has been proposed. The technique requires two stages. The first stage is to select stable sliding surfaces for the desired dynamics, and the second stage is to design a switching control law to achieve the stable sliding surfaces. Then, the chaos synchronization of the system is proved by the Lypapunov stability theory. Finally, numerical simulation is carried to confirm the validity of the proposed theoretical approach.

2. Description of the Problem

Consider two chaotic systems given by

𝑑𝑥_𝑚𝑑𝑡=𝑓(𝑥_𝑚,𝑡),𝑑𝑥_𝑠𝑑𝑡=𝑔(𝑥_𝑠,𝑡)+𝜙(𝑢(𝑥_𝑚,𝑥_𝑠,𝑡)), (2.1)(2.2) where𝑥_𝑚=[𝑥₁𝑦₁𝑧₁⋯]^𝑇∈𝑅^𝑛, 𝑥_𝑠=[𝑥₂𝑦₂𝑧₂⋯]∈𝑅^𝑛, 𝑓,𝑔∈𝐶^𝑟[𝑅⁺×𝑅^𝑛,𝑅^𝑛], 𝑢∈𝐶^𝑟[𝑅⁺×𝑅^𝑛×𝑅^𝑛,𝑅^𝑛], and 𝑟≥1, 𝑅⁺is the set of nonnegative numbers. Assume that (2.1) is the drive system (master system), (2.2) is the response system (slave system), and 𝜙(𝑢(𝑥_𝑚,𝑥_𝑠,𝑡))is the nonlinear control input attached in the response system. Ifforall𝑥_𝑚(𝑡₀),𝑥_𝑠(𝑡₀)∈𝑅^𝑛, lim_𝑡→∞|𝑥_𝑠𝑖(𝑡)−𝛼_𝑖𝑥_𝑚𝑖(𝑡)|=0, for 𝑖=1,2,…,𝑛, then the response and drive systems are said to be in modified projective synchronization (MPS). In particular, the drive-response systems achieve complete synchronization when all values of 𝛼_𝑖are equal to 1. Further, if all values 𝛼_𝑖are equal to −1, then two systems are said to be anti-synchronization.

In this paper, our purpose is to achieve the MPS of two identical Chen-Lee chaotic systems by using variable structure control with input nonlinearity. Chen and Lee reported a new chaotic system [18] in 2004, which now called the Chen-Lee system. The system is described by the following nonlinear differential equations and is denoted as the following system:

𝑑𝑥𝑑𝑡=−𝑦𝑧+𝑎𝑥,𝑑𝑦𝑑𝑡=𝑥𝑧+𝑏𝑦,𝑑𝑧1𝑑𝑡=(3)𝑥𝑦+𝑐𝑧, (2.3) where x, y, and z are state variables, and a, b, and c are three system parameters. When(𝑎,𝑏,𝑐)=(5,−10,−3.8), system (2.3) demonstrates a complex attractor as shown in Figure 1. A positive maximum Lyapunov exponent is shown in Figure 2. It shows that the Chen-Lee system is in the state of chaotic motion at this condition.

(a)

(b)

(c)

(d)

Figure 1 

The phase plane trajectories of Chen-Lee chaotic system with 𝑎=5, 𝑏=−10, 𝑐=−3.8.

Figure 2 

The maximum Lyapunov exponent of Chen-Lee chaotic system plotted as a function of the number of driving cycles with𝑎=5, 𝑏=−10, 𝑐=−3.8.

For the Chen-Lee chaotic system, the drive and response systems are defined as follows.

Drive system is

𝑑𝑥₁𝑑𝑡=−𝑦₁𝑧₁+𝑎𝑥₁,𝑑𝑦₁𝑑𝑡=𝑥₁𝑧₁+𝑏𝑦₁,𝑑𝑧₁1𝑑𝑡=(3)𝑥₁𝑦₁+𝑐𝑧₁. (2.4) Response system is

𝑑𝑥₂𝑑𝑡=−𝑦₂𝑧₂+𝑎𝑥₂+𝜙₁(𝑢₁),𝑑𝑦₂𝑑𝑡=𝑥₂𝑧₂+𝑏𝑦₂+𝜙₂(𝑢₂),𝑑𝑧₂1𝑑𝑡=(3)𝑥₂𝑦₂+𝑐𝑧₂+𝜙₃(𝑢₃). (2.5) where 𝜙₁(𝑢₁),𝜙₂(𝑢₂),and𝜙₃(𝑢₃)are the nonlinear control inputs attached in the slave system. Let the synchronization error vector state be𝑒=[𝑒₁𝑒₂𝑒₃]^𝑇=[𝑥₂−𝛼₁𝑥₁𝑦₂−𝛼₂𝑦₁𝑧₂−𝛼₃𝑧₁]. Substitution (2.4) and (2.5) into the error state, the error dynamic equations can be obtained as follows:

𝑑𝑒₁𝑑𝑡=−𝑒₂𝑒₃−𝛼₃𝑒₂𝑧₁−𝛼₂𝑒₃𝑦₁+(𝛼₁−𝛼₂𝛼₃)𝑦₁𝑧₁+𝑎𝑒₁+𝜙(𝑢₁),𝑑𝑒₂𝑑𝑡=𝑒₁𝑒₃+𝛼₃𝑒₁𝑧₁+𝛼₁𝑒₃𝑥₁+(𝛼₁𝛼₃−𝛼₂)𝑥₁𝑧₁+𝑏𝑒₂+𝜙₂(𝑢₂),𝑑𝑒₃=1𝑑𝑡3(𝑒₁𝑒₂+𝛼₂𝑒₁𝑦₁+𝛼₁𝑒₂𝑥₁1)+3(𝛼₁𝛼₂−𝛼₃)𝑥₁𝑦₁+𝑐𝑒₃+𝜙₃(𝑢₃). (2.6) The 𝜙_𝑖(𝑢_𝑖(𝑡))∈𝐶¹(𝑅^𝑛→𝑅) is a continues nonlinear function with 𝜙_𝑖(0)=0, and 𝑢_𝑖(𝑡)→𝜙_𝑖(𝑢_𝑖(𝑡))is inside sector [𝜍_𝑖,𝜌_𝑖](𝑖=1,2,3), that is,

𝜍_𝑖𝑢²_𝑖≤𝑢_𝑖𝜙_𝑖(𝑢_𝑖)≤𝜌_𝑖𝑢²_𝑖, (2.7) where 𝜍_𝑖 and 𝜌_𝑖are nonzero positive constants. A nonlinear function 𝜙_𝑖(𝑢_𝑖(𝑡))is shown in Figure 3.

Figure 3 

A scalar nonlinear function𝜙𝑖(𝑢𝑖(𝑡))inside sector [𝜍𝑖,𝜌𝑖].

Now, the sliding surfaces suitable for the application can be defined as

𝑆_𝑖=𝑒_𝑖+^𝑡₀𝜆_𝑖𝑒_𝑖(𝜏)𝑑𝜏,𝑖=1,2,3, (2.8) where 𝑆_𝑖(𝑡)∈𝑅 and 𝜆_𝑖 is the design parameters which can be determined later. For the existence of the sliding mode [19], it is necessary and sufficient that

𝑆_𝑖=𝑒_𝑖+^𝑡₀𝜆_𝑖𝑒_𝑖(𝜏)𝑑𝜏=0,𝑖=1,2,3,𝑑𝑆_𝑖=𝑑𝑡𝑑𝑒_𝑖𝑑𝑡+𝜆_𝑖𝑒_𝑖=0,𝑖=1,2,3. (2.9) Therefore, the following sliding mode dynamics can be obtained as

𝑑𝑒_𝑖𝑑𝑡=−𝜆_𝑖𝑒_𝑖,𝑖=1,2,3. (2.10) Obviously, if the design parameters 𝜆_𝑖>0,𝑖=1,2,3, the stability of (2.10) is surely guaranteed, that is lim_𝑡→∞𝑒_𝑖(𝑡)→0. Thus, the response system will be derived to drive system by designing the appropriate signal control inputs𝑢_𝑖(𝑡),𝑖=1,2,3.

3. Description of Variable Structure Control Law with Input Nonlinearity

We choose a control law of the form

𝑢_𝑖=−𝛾_𝑖𝜂_𝑖sign(𝑆_𝑖),𝛾_𝑖>1𝜍_𝑖,𝑖=1,2,3, (3.1) where 𝜂₁=|−𝑒₂𝑒₃−𝛼₃𝑒₂𝑧₁−𝛼₂𝑒₃𝑦₁+(𝛼₁−𝛼₂𝛼₃)𝑦₁𝑧₁+𝑎𝑒₁+𝜆₁𝑒₁𝜂|,₂=|𝑒₁𝑒₃+𝛼₃𝑒₁𝑧₁+𝛼₁𝑒₃𝑥₁+(𝛼₁𝛼₃−𝛼₂)𝑥₁𝑧₁+𝑏𝑒₂+𝜆₂𝑒₂𝜂|,₃1=|3(𝑒₁𝑒₂+𝛼₂𝑒₁𝑦₁+𝛼₁𝑒₂𝑥₁1)+3(𝛼₁𝛼₂−𝛼₃)𝑥₁𝑦₁+𝑐𝑒₃+𝜆₃𝑒₃|. (3.2) Based on the control law (3.1), the reaching condition 𝑠(𝑡)(𝑑𝑠(𝑡)/𝑑𝑡)<0 is guaranteed in the following theorem; that is, the proposed scheme (3.1) will derive the system (2.6) with nonlinear inputs onto the sliding mode𝑠(𝑡)=0.

Theorem 3.1. Consider the error dynamics system (2.6) with input nonlinearities. The hitting condition of the sliding mode is satisfied, if the control 𝑢_𝑖(𝑡) is given by (3.1) for𝑖=1,2,3,4.

Proof. Letting the Lyapunov function of the system be 𝑉=(1/2)(𝑆₁²+𝑆₂²+𝑆₃²), then its derivative with respect to time is 𝑑𝑉𝑑𝑡=𝑆₁𝑑𝑆₁𝑑𝑡+𝑆₂𝑑𝑆₂𝑑𝑡+𝑆₃𝑑𝑆₃𝑑𝑡=𝑆₁(𝑑𝑒₁𝑑𝑡+𝜆₁𝑒₁)+𝑆₂(𝑑𝑒₂𝑑𝑡+𝜆₂𝑒₂)+𝑆₃(𝑑𝑒₃𝑑𝑡+𝜆₃𝑒₃)=𝑆₁(−𝑒₂𝑒₃−𝛼₃𝑒₂𝑧₁−𝛼₂𝑒₃𝑦₁+(𝛼₁−𝛼₂𝛼₃)𝑦₁𝑧₁+𝑎𝑒₁+𝜙(𝑢₁)+𝜆₁𝑒₁)+𝑆₂(𝑒₁𝑒₃+𝛼₃𝑒₁𝑧₁+𝛼₁𝑒₃𝑥₁+(𝛼₁𝛼₃−𝛼₂)𝑥₁𝑧₁+𝑏𝑒₂+𝜙₂(𝑢₂)+𝜆₂𝑒₂)+𝑆₃(13(𝑒₁𝑒₂+𝛼₂𝑒₁𝑦₁+𝛼₁𝑒₂𝑥₁1)+3(𝛼₁𝛼₂−𝛼₃)𝑥₁𝑦₁+𝑐𝑒₃+𝜙₃(𝑢₃)+𝜆₃𝑒₃)≤|𝑆₁||−𝑒₂𝑒₃−𝛼₃𝑒₂𝑧₁−𝛼₂𝑒₃𝑦₁+(𝛼₁−𝛼₂𝛼₃)𝑦₁𝑧₁+𝑎𝑒₁+𝜆₁𝑒₁|+𝑆₁𝜙₁(𝑢₁)+|𝑆₂||𝑒₁𝑒₃+𝛼₃𝑒₁𝑧₁+𝛼₁𝑒₃𝑥₁+(𝛼₁𝛼₃−𝛼₂)𝑥₁𝑧₁+𝑏𝑒₂+𝜆₂𝑒₂|+𝑆₂𝜙₂(𝑢₂)+|𝑆₃1||3(𝑒₁𝑒₂+𝛼₂𝑒₁𝑦₁+𝛼₁𝑒₂𝑥₁1)+3(𝛼₁𝛼₂−𝛼₃)𝑥₁𝑦₁+𝑐𝑒₃+𝜆₃𝑒₃|+𝑆₃𝜙₃(𝑢₃)≤𝜂₁|𝑆₁|−𝜍₁𝛾₁𝜂₁|𝑆₁|+𝜂₂|𝑆₂|−𝜍₂𝛾₂𝜂₂|𝑆₂|𝜂₂|𝑆₂|−𝜍₃𝛾₃𝜂₃|𝑆₃|≤(1−𝛾₁𝜍₁)𝜂₁|𝑆₁|+(1−𝛾₂𝜍₂)𝜂₂|𝑆₂|+(1−𝛾₃𝜍₃)𝜂₃|𝑆₃|, (3.3) where 𝑢_𝑖𝜙_𝑖(𝑢_𝑖)≥𝜍_𝑖𝑢²_𝑖⇒−𝛾𝜂_𝑖sig𝑢(𝑆_𝑖)𝜙(𝑢_𝑖)≥𝜍_𝑖𝛾²𝜂²_𝑖sign²(𝑆_𝑖)⇒−𝛾𝜂_𝑖|𝑆_𝑖|𝑆_𝑖𝜙(𝑢_𝑖)≥𝜍_𝑖𝛾𝜂_𝑖|𝑆_𝑖||𝑆_𝑖|,for𝑖=1,2,3.⇒−𝑆_𝑖𝜙(𝑢_𝑖)≥𝜍_𝑖𝛾_𝑖𝜂_𝑖|𝑆_𝑖|⇒𝑆_𝑖𝜙(𝑢_𝑖)≤−𝜍_𝑖𝛾_𝑖𝜂_𝑖|𝑆_𝑖|. (3.4) Therefore, if 𝛾_𝑖>1𝜍_𝑖,for𝑖=1,2,3, (3.5) then 𝑑𝑉/𝑑𝑡<0, confirming the presence of reaching condition. Thus the proof is achieved completely.

4. Numerical Studies

In this simulation, the 4th-order Runge-Kutta algorithm was used to solve the sets of differential equations related to the drive and response systems with a time grid of 0.0001. The initial values of drive and response Chen-Lee chaotic system are [𝑥₁(0)𝑦₁(0)𝑧₁(0)]=[0.20.20.2], [𝑥₂(0)𝑦₂(0)𝑧₂(0)]=[101010]. In the synchronization example, we selected 𝜆₁=𝜆₂=𝜆₃=3to result in stable sliding modes and the nonlinear inputs are defined as

𝜙_𝑖(𝑢_𝑖(𝑡))=[0.7+0.2⋅sin(𝑢_𝑖(𝑡))]𝑢_𝑖(𝑡),𝑖=1,2,3. (4.1) Furthermore, it is assumed that the slope of nonlinear sectors in these three synchronization examples is 𝜍₁=𝜍₂=𝜍₃=0.5 and 𝜌₁=𝜌₂=𝜌₃=0.9, and the parameters 𝛾₁=𝛾₂=𝛾₃=5 are selected to satisfy the condition (3.1). The time responses of controlled drive-response Chen-Lee systems are shown in Figures 4(a)–4(c). It can be see that the response system synchronizes with the drive system in spite of input nonlinearity. Obviously, the synchronization errors converge asymptotically to zero after the control is active at time 𝑡=10 seconds in Figure 5.

(a)

(b)

(c)

Figure 4 

The time history of MPS of controlled drive (𝑥1,𝑥2,𝑥3) and response (𝑦1,𝑦2,𝑦3) Chen-Lee chaotic systems: (a) 𝑥1,𝑦1 versus time t; (b) 𝑥2,𝑦2 versus time t; (c) 𝑥3,𝑦3 versus time t. The control is active at 𝑡=10 seconds.

Figure 5 

Synchronization errors 𝑒1,𝑒2,𝑒3 versus time t. The control is active at 𝑡=10 seconds.

5. Conclusions

In this paper, we investigate the hybrid projective synchronization of controlled Chen-Lee chaotic system with input nonlinearity. Based on Lyapunov stability theorem, an effective control method for synchronizing two identical Chen-Lee chaotic systems with different initial conditions has been proposed using variable structure design. The proposed nonlinear control enables stabilization of synchronization error dynamics to zeros asymptotically in spite of input nonlinearity. Numerical simulation results are presented to show that the synchronization and antisynchronization coexist by the proposed synchronization technique. The main feature of this approach is that it gives the flexibility to construct a control law so that the control strategy can be easily extended to any other types of chaotic systems.

Acknowledgment

The financial support of this research by the National Science Council of Taiwan, under Grant no. NSC96-2221-E-269-010-MY2 is greatly appreciated.