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

Authors: Jui-sheng Lin; Neng-Sheng Pai; Her-Terng Yau
Journal: Mathematical Problems in Engineering
 (2009)

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## 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.


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## Body

## 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(2.1)dxmdt=f(xm,t),(2.2)dxsdt=g(xs,t)+ϕ(u(xm,xs,t)),
wherexm=[x1y1z1⋯]T∈Rn, xs=[x2y2z2⋯]∈Rn,  f,g∈Cr[R+×Rn,Rn], u∈Cr[R+×Rn×Rn,Rn], 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(xm,xs,t))is the nonlinear control input attached in the response system. Ifforallxm(t0),xs(t0)∈Rn, limt→∞|xsi(t)-αixmi(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 αiare equal to 1. Further, if all values αiare 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:(2.3)dxdt=-yz+ax,dydt=xz+by,dzdt=(13)xy+cz,
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.The phase plane trajectories of Chen-Lee chaotic system witha=5, b=-10, c=-3.8.
(a)(b)(c)(d)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(2.4)dx1dt=-y1z1+ax1,dy1dt=x1z1+by1,dz1dt=(13)x1y1+cz1.
Response system is(2.5)dx2dt=-y2z2+ax2+ϕ1(u1),dy2dt=x2z2+by2+ϕ2(u2),dz2dt=(13)x2y2+cz2+ϕ3(u3).
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:(2.6)de1dt=-e2e3-α3e2z1-α2e3y1+(α1-α2α3)y1z1+ae1+ϕ(u1),de2dt=e1e3+α3e1z1+α1e3x1+(α1α3-α2)x1z1+be2+ϕ2(u2),de3dt=13(e1e2+α2e1y1+α1e2x1)+13(α1α2-α3)x1y1+ce3+ϕ3(u3).
The ϕi(ui(t))∈C1(Rn→R) is a continues nonlinear function with ϕi(0)=0, and ui(t)→ϕi(ui(t))is inside sector [ςi,ρi](i=1,2,3), that is,(2.7)ςiui2≤uiϕi(ui)≤ρiui2,
where ςi and ρiare nonzero positive constants. A nonlinear function ϕi(ui(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(2.8)Si=ei+∫0tλiei(τ)dτ,i=1,2,3,
where Si(t)∈R and λi is the design parameters which can be determined later. For the existence of the sliding mode [19], it is necessary and sufficient that(2.9)Si=ei+∫0tλiei(τ)dτ=0,i=1,2,3,dSidt=deidt+λiei=0,i=1,2,3.
Therefore, the following sliding mode dynamics can be obtained as(2.10)deidt=-λiei,i=1,2,3.
Obviously, if the design parameters λi>0,i=1,2,3, the stability of (2.10) is surely guaranteed, that is limt→∞ei(t)→0. Thus, the response system will be derived to drive system by designing the appropriate signal control inputsui(t),i=1,2,3.


## 3. Description of Variable Structure Control Law with Input Nonlinearity

We choose a control law of the form(3.1)ui=-γiηisign(Si),γi>1ςi,i=1,2,3,
where
(3.2)η1=|-e2e3-α3e2z1-α2e3y1+(α1-α2α3)y1z1+ae1+λ1e1|,η2=|e1e3+α3e1z1+α1e3x1+(α1α3-α2)x1z1+be2+λ2e2|,η3=|13(e1e2+α2e1y1+α1e2x1)+13(α1α2-α3)x1y1+ce3+λ3e3|.
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 ui(t) is given by (3.1) fori=1,2,3,4.Proof.
Letting the Lyapunov function of the system beV=(1/2)(S12+S22+S32), then its derivative with respect to time is
(3.3)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+α1e2x1)+13(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)+|S3||13(e1e2+α2e1y1+α1e2x1)+13(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|,
where
(3.4)uiϕi(ui)≥ςiui2⇒-γηisigu(Si)ϕ(ui)≥ςiγ2ηi2sign2(Si)⇒-γηi|Si|Siϕ(ui)≥ςiγηi|Si||Si|,fori=1,2,3.⇒-Siϕ(ui)≥ςiγiηi|Si|⇒Siϕ(ui)≤-ςiγiηi|Si|.
Therefore, if
(3.5)γi>1ςi,fori=1,2,3,
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(4.1)ϕi(ui(t))=[0.7+0.2·sin(ui(t))]ui(t),i=1,2,3.
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.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.
(a)(b)(c)Figure 5
Synchronization errorse1,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.



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