# Sliding Mode Control and Modified Generalized Projective Synchronization of a New Fractional-Order Chaotic System

Authors: Junbiao Guan; Kaihua Wang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A new fractional-order chaotic system is addressed in this paper. By applying the continuous frequency distribution theory, the indirect Lyapunov stability of this system is investigated based on sliding mode control technique. The adaptive laws are designed to guarantee the stability of the system with the uncertainty and external disturbance. Moreover, the modified generalized projection synchronization (MGPS) of the fractional-order chaotic systems is discussed based on the stability theory of fractional-order system, which may provide potential applications in secure communication. Finally, some numerical simulations are presented to show the effectiveness of the theoretical results.


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## Body

## 1. Introduction

Though the concept of fractional calculus has been established more than three hundred years, its potential applications are fully carried out in recent decades, especially in the fields of control, engineering, and physics; see [1–3] and reference therein. It has been found that many systems in nature display fractional phenomena, such as electromagnetic waves, dielectric polarization, the chemotaxi behavior, quantitative finance, and evolution in complex media. Especially, the memristor, which was predicted as the missing circuit element [4], is more likely to be linked to fractional calculus due to its inherent features. Therefore, it is of considerable importance to study some aspects of fractional calculus.The stability of fractional differential equation (FDE) is one of the most important aspects in FDE’s application of control process [5, 6], and Lyapunov stability method provides an efficient way to analyze the stability of FDE without explicitly solving the differential equations [7]. There are two approaches with respect to Lyapunov stability, namely, direct one and indirect one. The former one is intuitive and specific, while the latter is based on the continuous frequency distribution theory, which appears more abstract but effective. Meanwhile, in order to stabilize the chaotic FDE, many control mechanisms are presented so far, among which sliding mode control (SMC) technique is extensively adopted as it can be utilized to improve the control performance criteria such as the robustness and fast time response [8–10].Another important aspect of FDE’s application of control process lies in synchronization of fractional-order chaotic systems [11, 12], which attracts increasing attention in recent years due to its potential applications in secure communication. There are many synchronization methods for fractional-order chaotic systems, including the Pecora and Carroll method, the one-way coupling method, feedback control method, the active control method, and the active sliding mode control method. Amongst these methods, it is worth mentioning that feedback control method serves as a good tool to synchronize the master (drive) and slave (response) systems up to a constant scaling factor, which is known as projective synchronization (PC) [13]. This synchronization scheme can be applied to secure communication easily because it can obtain faster communication with its proportional characteristics. Based on this, recently, Wu et al. proposed the modified generalized projective synchronization (MGPS) where the drive and response systems could be synchronized to a constant scaling matrix [14]. Compared with PC, the unpredictability of the scaling matrix in MGPS can greatly strengthen the security of communications.Motivated by the above discussions, in this paper, we are dedicated to the study of SMC of a new fractional-order chaotic system, together with its MGPS. The main purpose of this work is the integrated applications of the continuous frequency distribution theory and adaptive SMC technique to investigate indirect Lyapunov stability of a new fractional-order chaotic system. Moreover, the MGPS of the fractional-order chaotic systems is discussed based on the stability theory of fractional-order system, which may provide potential applications in secure communication. The remainder of this paper is organized as follows. In Section2, some basic definitions and lemmas are given. In Section 3, a new fractional-order chaotic system is presented and the stability of sliding mode dynamics of this system is studied based on the continuous frequency distribution theory. Adaptive sliding mode control of the fractional-order chaotic system with the model uncertainty and external disturbance is also investigated. In Section 4, MGPS of the fractional-order chaotic systems is discussed under the framework of stability theory of FDE. In Section 5, three numerical examples are given to show the effectiveness of the theoretical results. Finally, some concluding remarks are drawn in Section 6.


## 2. Preliminaries

In this section, we present some basic definitions and lemmas which will be useful throughout this paper.The commonly-used Riemann-Liouville definition of theαth-order derivative is given by
(1)Dαx(t)=ddtnJn-αx(t),α>0,
where n is the first integer which is not less than α, Jβ is the βth-order Riemann-Liouville integral operator as described by
(2)Jβx(t)=1Γ(β)∫0t‍x(τ)t-τ1-βdτ,0<β≤1,
where Γ(β)=∫0∞‍tβ-1e-tdt is the gamma function.Correspondingly, the Riemann-Liouville definition of theαth-order integral is given by
(3)D-αx(t)=1Γ(α)∫0t‍x(τ)t-τ1-αdτ,α>0.Besides the definitions of fractional calculus presented above, the following lemmas provided in [7] are necessary for the analysis in the next sections.Lemma 1.
Leth(t) be the impulse response of a linear system and let μ(ω) be the diffusive representation (or frequency weighting function) of the impulse response h(t), h(t) and μ(ω) verify the pseudo-Laplace transform definition
(4)h(t)=∫0∞‍μ(ω)e-ωtdω.Lemma 2.
Consider the nonlinear fractional-order differential equation (FDE)Dα(x)=f(x). Owing to the continuous frequency distributed model of the fractional integrator, the nonlinear system can be expressed as
(5)∂z(ω,t)∂t=-ωz(ω,t)+f(x(t)),x(t)=∫0∞‍μ(ω)z(ω,t)dω
with
(6)μ(ω)=sinαππω-n.


## 3. Sliding Mode Control of the New Fractional-Order Chaotic System

A new fractional-order chaotic system is described by(7)Dq1x(t)=ax(t)+dy(t)z(t)+gy2(t),Dq2y(t)=by(t)+ex(t)z(t)+hz(t),Dq3z(t)=cz(t)+fx(t)y(t),
where qi denotes fractional-order and 0<qi<1, (i=1,2,3), a,b,c,d,e,f,g,h∈R are parameters. If one of its Lyapunov exponents is positive, then the system exhibits chaotic dynamical behavior. Specially, system (7) is chaotic when a=-3, b=5, c=-10, d=1, e=-1, f=1, g=1, h=16, q1=0.995, q2=0.997, q3=0.998, as illustrated in Figure 1. When q1=q2=q3=1, the corresponding integer-order differential equation has been proposed and its dynamics have been studied in [15].Chaotic attractor of system (7). (a–c) Time responses of x,y, and z. (d) Phase portrait.
(a)
(b)
(c)
(d)In order to apply the nonlinear feedback control, we consider the controlled system described by(8)Dq1x(t)=ax(t)+dy(t)z(t)+gy2(t),Dq2y(t)=by(t)+ex(t)z(t)+hz(t)+u(t),Dq3zt=czt+fxtyt.Choose the switching surfaces(t) as
(9)s(t)=Dq2-1y(t)+∫0t‍ϕ(τ)dτ,
where ϕ(t) is described by ϕ(t)=(d+f)xz+gxy-(b-k)y with k>0.When reaching the switching surface, we have(10)s(t)=Dq2-1y(t)+∫0t‍ϕ(τ)dτ=0,s˙(t)=Dq2y(t)+ϕ(t)=0.Thus the corresponding control law is obtained(11)ut=-d+e+fxz-gxy-hz-ky.Hence we have(12)Dq1x(t)=ax(t)+dy(t)z(t)+gy2(t),Dq2y(t)=(b-k)y(t)-(d+f)x(t)z(t)-gx(t)y(t),Dq3zt=czt+fxtyt.Theorem 3.
The FDE sliding mode dynamics described in (12) are globally asymptotically stable if a<0, b-k<0, c<0.Proof.
Owing to the continuous frequency distributed model of the fractional integrator, the nonlinear system can be rewritten as(13)∂z1(ω,t)∂t=-ωz1(ω,t)+ax(t)+dy(t)z(t)+gy2(t),xt=∫0∞‍μ1ωz1ω,tdω,∂z2ω,t∂t=-ωz2(ω,t)+(b-k)y(t)-(d+f)x(t)z(t)-gx(t)y(t),yt=∫0∞‍μ2ωz2ω,tdω,∂z3(ω,t)∂t=-ωz3(ω,t)+cz(t)+fx(t)y(t),z(t)=∫0∞‍μ3(ω)z3(ω,t)dω,
where μi(ω)=sinqiπ/πω-qi, i=1,2,3.
Let us define two Lyapunov functions as follows.(1)
vi(ω,t) (i=1,2,3) are the monochromatic Lyapunov functions corresponding to the elementary frequency ω.(2)
Vi(t) (i=1,2,3) are the Lyapunov functions summing all the monochromatic vi(ω,t) with the weighting function μi(ω).
Letvi(ω,t)=1/2zi2(ω,t) and let Vi(t)=∫0∞‍μi(ω)zi(ω,t)dω, i=1,2,3. Note that vi(ω,t) (i=1,2,3) are positive, and μi(ω) (i=1,2,3) are positive for all ω. Thus Vi(t) (i=1,2,3) are positive Lyapunov functions. The time derivative of Vi(t) yields
(14)dVi(t)dt=∫0∞‍μiω∂viω,t∂tdω=∫0∞‍μiωzi∂zi∂tdω,i=1,2,3.
Combining (13) with (14), we have
(15)dV1(t)dt=∫0∞‍μ1(ω)∂v1(ω,t)∂tdω=∫0∞‍μ1(ω)z1[-ωz1+ax+dyz+gy2]dω=-∫0∞‍ωμ1ωz12dω+xax+dyz+gy2,dV2tdt=∫0∞‍μ2(ω)∂v2ω,t∂tdω=∫0∞‍μ2(ω)z2-ωz2+b-ky-(d+f)xz-gxydω=-∫0∞‍ωμ2(ω)z22dω+yb-ky-d+fxz-gxy,dV3(t)dt=∫0∞‍μ3ω∂v3ω,t∂tdω=∫0∞‍μ3(ω)z3[-ωz3+cz+fxy]dω=-∫0∞‍ωμ3(ω)z32dω+z(cz+fxy).
Let us defineV(t)=V1(t)+V2(t)+V3(t); then
(16)dV(t)dt=∑i=13‍dVi(t)dt=-∑i=13‍∫0∞‍ωμi(ω)zi2dω+[ax2+(b-k)y2+cz2].
Denote byWi=-∫0∞‍ωμi(ω)zi2dω (i=1,2,3) and W=∑i=13‍Wi. The frequency discretization of Wi gives
(17)Wi=-∑j=1J‍ωijμi(ωij)zi2(ωij,t)Δωij=∑j=1J‍ωijcijzi2(ωij,t),
where cij=-μi(ωij)Δωij<0. Hence Wi<0 and W=∑i=13Wi<0. On the other hand, if a<0, b-k<0, c<0, then ax2+(b-k)y2+cz2<0. According to the above analysis we have dV(t)/dt<0, which implies that system (12) is asymptotically stable.It is to be noted that the above slide mode control of the fractional-order chaotic system is conducted under the condition that no model uncertainty or external disturbance exists. But in reality, these factors are ubiquitous. In view of this, in what follows we consider this fractional-order chaotic system with the model uncertainty and external disturbance described by(18)Dq1x(t)=ax(t)+dy(t)z(t)+gy2(t),Dq2yt=byt+extzt+hzthhhhhhhh+Δh(x,y,z)+Δw(t)+u(t),Dq3z(t)=cz(t)+fx(t)y(t),
where Δh(x,y,z), Δw(t) are the model uncertainty and the external disturbance, respectively. Note that they are often bounded. Therefore we give the following assumption.Assumption 4.
The model uncertainty and the external disturbance are assumed to be bounded; that is, there exist unknown positive constantsψ, ω such that Δhx,y,z<ψ, Δwt<ω.In order to give an estimation of the two constantsψ and ω, we present the adaptive rules defined as follows:
(19)ψ~˙=λ1s,ω~˙=λ2s,
where ψ~, ω~ are estimations for ψ, ω, respectively, and λ1, λ2 are positive constants.The controller is designed as follows:(20)ut=-(d+e+f)xz-gxy-hz-ky-ρsign⁡(s)-(ψ~+ω~)sign⁡(s),
where ρ is a positive constant. Then we reach the following theorem.Theorem 5.
The FDE with the unknown bounded uncertainty and the external disturbance described in (18) will converge to the sliding surface s(t)=0 under the controller (20).Proof.
Selecting a Lyapunov candidate asV(t)=1/2s2+1/2λ1(ψ~-ψ)2+1/2λ2(ω~-ω)2 and taking the time derivative of both sides, one obtains
(21)V˙=ss˙+1λ1(ψ~-ψ)ψ~˙+1λ2(ω~-ω)ω~˙=s(Δh(x,y,z)+Δw(t))-s(ψ~+ω~)sign⁡(s)-sρsign⁡(s)+ψ~-ψs+ω~-ωs≤sψ+ω-sψ~+ω~-ρs+ψ~-ψs+ω~-ωs=-ρs≤0.
Integrating (21) from zero to t yields
(22)∫0t‍ρsdt≤V(0)-V(t),
which implies lim⁡t→∞s=0.


## 4. MGPS of the New Fractional-Order Chaotic Systems

In this section, we investigate MGPS of the new fractional-order chaotic systems. First of all, it is necessary to give the definition of MGPS [14].Definition 6.
The drive systemDα(Xm)=F(Xm) and the response system Dα(Xs)=F(Xs)+U are said to achieve modified general projection synchronization (MGPS) if there exists a controller U such that lim⁡t→∞Xs-ΛXm=0, where Λ=diag⁡(d1,d2,…,dn), di≠0,i=1,2,…,n and X=(x1,x2,…,xn), U=(u1,u2,…,un).We consider the drive and the response fractional-order chaotic systems described as follows:(23)DαxmDαymDαzm=a000bh00cxmymzm+dymzm+gym2exmzmfxmym,DαxsDαysDαzs=a000bh00cxsyszs+dyszs+gys2exszsfxsys+u1u2u3.Define the synchronization errors by(24)e1=xs-d1xm,e2=ys-d2ym,e3=zs-d3zm,
where Λ=(d1,d2,d3) and di (i=1,2,3) are the nonzero scaling factors.Then the error dynamics are obtained(25)Dαe1=ae1+dyszs+gys2-d1dymzm-d1gym2+u1,Dαe2=be2+hzs+exszs-d2hzm-d2exmzm+u2,Dαe3=ce3+fxsys-d3fxmym+u3.We design the controllersui (i=1,2,3) for the response system as follows:
(26)u1=-dyszs-gys2+d1dymzm+d1gym2-k1e1,u2=-hzs-exszs+d2hzm+d2exmzm-k2e2,u3=-fxsys+d3fxmym-k3e3,
where the control gains matrix K=diag⁡(k1,k2,k3).Thus the fractional-order error dynamics can be rewritten as follows:(27)Dαe1Dαe2Dαe3=a-k1000b-k2000c-k3e1e2e3=Ae,
where
(28)A=a-k1000b-k2000c-k3.According the stability theory of fractional-order system [16], straightforwardly we have the following theorem.Theorem 7.
Consider the drive and the response system described in (23), if the controllers in (26) are employed and argλA>απ/2 is satisfied, where λ(A) denotes the eigenvalue of the matrix A given in (28), then MGPS between the drive system and the response system is achieved.It is worth mentioning that the above MGPS can be easily utilized in secure communication. More specifically, the original signal can be modulated into the chaotic signal generated by the new fractional-order chaotic system and then forms a combined signal. The combined signal can be injected into one of three variables. The receiver can obtain and decode the combined signal via the MGPS and the scaling factor, respectively.


## 5. Three Illustrative Examples

In this section, we will perform some numerical examples to verify the theoretical analysis.Example 1.
Consider the following FDE:(29)Dq1x(t)=ax(t)+dy(t)z(t)+gy2(t),Dq2y(t)=by(t)+ex(t)z(t)+hz(t)+u(t),Dq3z(t)=cz(t)+fx(t)y(t),
with  a=-3, b=5, c=-10, d=1, e=-1, f=1, g=1, h=16, q1=0.995, q2=0.997, q3=0.998. If we choose the controller u(t)=-xz-xy-16z-6y, then from Theorem 3 we know that system (29) is asymptotically stable, as illustrated in Figure 2.Sliding mode dynamics of system (12) are globally stable. (a–c) Time responses of x, y, and z. (d) Phase portrait.
(a)
(b)
(c)
(d)Example 2.
Consider the following FDE with the model uncertainty and the external disturbance:(30)Dq1x(t)=ax(t)+dy(t)z(t)+gy2(t),Dq2yt=byt+extzt+hzthhhhhhhh+Δh(x,y,z)+Δw(t)+u(t),Dq3z(t)=cz(t)+fx(t)y(t),
where a=-3, b=5, c=-10, d=1, e=-1, f=1, g=1, h=16, q1=0.995, q2=0.997, q3=0.998, Δh(x,y,z)=0.1sin(y), Δw(t)=0.1sin(t). If we choose the controller u(t)=-xz-xy-16z-y-2sign⁡(s)-(ψ~+ω~)sign⁡(s) with the adaptive laws ψ~˙=1.12s, ω~˙=0.52s and the switching surface s(t)=Dq2-1y+∫0t‍[2xz+xy-4y]dτ, then according to Theorem 5 we know that system (30) converges to the sliding surface s=0, which is illustrated in Figure 3.Adaptive sliding mode control of system (30). (a) Time responses of x, y, and z. (b) Phase portrait. (c) Time response of s(t). (d) Time response of u(t).
(a)
(b)
(c)
(d)Example 3.
Consider the drive and the response system given by(31)DαxmDαymDαzm=a000bh00cxmymzm+dymzm+gym2exmzmfxmym,DαxsDαysDαzs=a000bh00cxsyszs+dyszs+gys2exszsfxsys+u1u2u3,
where a=-3, b=5, c=-10, d=1, e=-1, f=1, g=1, h=-16, α=0.995, d1=0.5, d2=-1.2, d3=3. The controllers ui (i=1,2,3) for the response system are designed as follows:
(32)u1=-yszs-ys2+0.5dymzm+0.5gym2-k1e1,u2=16zs+xszs+19.2zm+1.2xmzm-k2e2,u3=-xsys+3xmym-k3e3,
where the control gains matrix K=diag⁡(k1,k2,k3)=(0,6,0). According to Theorem 7 we know that MGPS between the drive system and the response system in (31) is achieved, as shown in Figure 4.MGPS of system (31). (a–c) Time responses of xm and xs, ym and ys, and zm and zs. (d) Time response of synchronization errors.
(a)
(b)
(c)
(d)


## 6. Concluding Remarks

A new fractional-order chaotic system has been addressed in this paper. The indirect Lyapunov stability of this system has been studied based on sliding mode control technique. Furthermore, the adaptive laws have been designed to guarantee the stability of this system with the uncertainty and external disturbance. By applying the stability theory of fractional-order system, MGPS of this chaotic systems has also been investigated, which may provide potential applications in secure communication. Finally, three illustrative numerical examples have been presented to verify the effectiveness of the theoretical results.



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