# Synchronization of an Uncertain Fractional-Order Chaotic System via Backstepping Sliding Mode Control

Authors: Zhen Wang
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

Backstepping control approach combined with sliding mode control (SMC) technique is utilized to realize synchronization of uncertain fractional-order strict-feedback chaotic system. A backstepping SMC method is presented to compensate the uncertainty which occurs in the slave system. Moreover, the newly proposed control scheme is applied to implement synchronization of fractional-order Duffing-Holmes system. The simulation results demonstrate that the backstepping SMC method is robust against the modeling uncertainties and external disturbances.


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## Body

## 1. Introduction

Fractional calculus is a generalization of the ordinary integration and differentiation to noninteger-order case. In recent years, it has found widespread application in the fields of physics, applied mathematics, and engineering [1–3]. Moreover, it has been generally realized that fractional integrals and derivatives are more suitable to describe memory and hereditary properties of various materials and processes than its integer-order counterpart [4].Chaotic behavior has been observed in the laboratory in a variety of systems, including electrical circuits, lasers, chemical reactions, fluid dynamics, and analog computers [5]. Over the past few years, more and more researchers have turned their attention to chaotic dynamics of fractional-order system. It has been shown that chaotic behavior of an integer-order nonlinear system is preserved when the order becomes fractional [6]. As a consequence, a number of fractional-order chaotic systems have been proposed, including fractional-order variants of Chua’s system [7], fractional-order Chen system [8], fractional-order Lorenz, Rössler, and Liu systems [9–11].Synchronization control is one of the important research areas in chaos theory. During the past three decades, chaos synchronization has been attracting interest from researchers in various fields [12–14]. It should be pointed out that synchronization problem of fractional-order chaotic systems was first reported by Deng and Li [15]. Afterwards, different synchronization control methods have been successfully applied to chaos synchronization of factional order systems, such as sliding mode control [16, 17], adaptive fuzzy control [18, 19], nonlinear feedback control [20, 21], the open-plus-closed-loop control [22], observer-based control [23], and some other methods [24–26].Backstepping approach was first proposed in [27] for designing stabilizing controls for a special type of nonlinear dynamical systems, which is referred to as the strict-feedback system. The technique consists in a recursive procedure that skillfully constructs the Lyapunov function and designs the virtual control input [28–30]. When this recursive procedure terminates, a feedback design for the real control input results [31]. On the other hand, sliding mode control (SMC) is one of the popular strategies to deal with uncertain systems. The main feature of SMC is the robustness against parameter uncertainties and external disturbances. In this paper, combining the merits of backstepping control and SMC, a backstepping sliding mode controller is devised to implement chaos synchronization of uncertain fractional-order strict feedback chaotic system.The rest of this paper is organized as follows. In Section2, definition and lemma in fractional calculus are presented. In Section 3, the backstepping sliding mode controller is systematically designed. In Section 4, synchronization problem of uncertain fractional-order Duffing-Holmes system is handled by the proposed method. Some concluding remarks are made in Section 5.


## 2. Preliminaries

There are three most frequently used definitions for fractional derivatives [1], that is, Riemann-Liouville, Grünwald-Letnikov, and Caputo definitions. Caputo definition is widely used in engineering applications since it takes on the same form as for integer-order differential equations in the initial conditions. Therefore, the following sections are based on Caputo derivative.Definition 1 (see [1]).
The Caputo fractional derivative of orderα of a continuous function f:R+→R is defined as
(1)Dtαf(t)={1Γ(m-α)∫0t‍f(m)(τ)(t-τ)α-m+1dτ,m-1<α<m,dmdtmf(t),α=m,
where Γ is Γ-function, and
(2)Γ(z)=∫0∞‍e-ttz-1dt,Γ(z+1)=zΓ(z).It should be noticed that the fractional integral of orderα>0 is denoted by Dt-α.In what follows, we list several basic facts of fractional derivatives and integrals which will be used in the stability and stabilization analysis. The readers can refer to [1] for more details.Fact 1.
Forα=n, where n is an integer, the operation Dtnf(t) gives the same result as classical calculus of integer-order n. In particular, when α=1, the operation Dt1f(t) coincides with the ordinary derivative df(t)/dt.Fact 2.
Forα=0, the operation Dtαf(t) is the identity operation
(3)Dt0f(t)=f(t).Fact 3.
Similary to integer-order calculus, fractional differentiation and fractional integration are both linear operations(4)Dtα[af(t)+bg(t)]=aDtαf(t)+bDtαg(t).
where a and b are constants.Fact 4.
The additive law of exponents (semigroup property) holds(5)DtαDtβf(t)=DtβDtαf(t)=Dtα+βf(t).Fact 5.
Forα>0, the following equation holds:
(6)DtαDt-αf(t)=Dt0f(t)=f(t)
which means that the fractional differentiation operator is a left inverse to the fractional integration operator of the same order α.Lemma 2 (see [32]).
Letx=0 be an equilibrium point for either Caputo or RL fractional nonautonomous system
(7)Dtqx(t)=f(x,t),
where q∈(0,1) and f(x,t) satisfies the Lipschitz condition with Lipschitz constant l>0. Assume that there exits a Lyapunov function V(t,x(t)) satisfying
(8)α1∥x∥a≤V(t,x)≤α2∥x∥,V˙(t,x)≤-α3∥x∥,
where α1,α2,α3, and a are positive constants and ∥·∥ denotes an arbitrary norm. Then the equilibrium point of system (7) is Mittag-Leffler (asymptotically) stable.


## 3. Main Results

In synchronization task, there are two dynamical systems, which are, respectively, called master system and slave system. From the view point of control, the task is to design a controller which obtains signals from the master system to tune the behavior of the slave system. In this paper, we consider synchronization of fractional-order strict feedback system described by(9)Dtqx1=x2,Dtqx2=f(t,x1,x2),
which is regarded as master system and the slave system is defined as
(10)Dtqy1=y2,Dtqy2=f(t,y1,y2)+Δf(t,y1,y2)+d(t)+u(t),
where 0<q≤1, X=(x1,x2)T and Y=(y1,y2)T are the states of systems (9) and (10), Δf(t,y1,y2) denotes uncertainty, d(t) is the external disturbance, and u(t) is the control input.Defining the synchronization error asei=yi-xi(i=1,2), then the error system is given by
(11)Dtqe1=e2,Dtqe2=f(t,y1,y2)-f(t,x1,x2)+F(t,Y)+u(t),
where F(t,Y)=Δf(t,y1,y2)+d(t). Suppose γ is the upper bound of function F; that is, ∥F(t,Y)∥≤γ.Step 1.
We start with the scalar system(12)Dtqz1=e2,
where z1=e1, and e2 is a virtual controller selected as e2=α(z1).
Choose the candidate Lyapunov function as(13)V1=12z12>0;
then the time derivative of V1 along the trajectories of system (12) is
(14)V˙=z1z˙1=z1Dt1-q(Dtqz1)=z1Dt1-qe2.
Choose Dt1-qe2=-k1z1; that is, α(z1)=-k1Dtq-1z1, where k1 is a positive constant. Thus, we have V˙=-k1z12≤0.Step 2.
Letz2=e2-α(z1), so
(15)Dtqz2=Dtqe2-Dtqα(z1)Dtqz2=f(t,y1,y2)-f(t,x1,x2)+F(t,Y)Dtqz2=+u(t)+k1Dt2q-1z1.
Design the switching surface as(16)S=k2Dt-1z1+Dtq-1z2,
where k2 is the sliding surface parameter to be designed later.
According to SMC method, the condition which guarantees the trajectory of the system arrives at the sliding surface isSS˙<0, and when in the sliding mode, the switching surface and its derivative must satisfy the following conditions:
(17)S(t)=0,S˙(t)=0.
In view of (15), it means that
(18)S˙=k2z1+Dtqz2S˙=k2z1+f(t,Y)-f(t,X)+F(t,Y)+k1Dt2q-1z1+u(t)S˙=0.
Thus, the equivalent control ueq(t) in the case of uncertainty-free is calculated by
(19)ueq(t)=f(t,X)-f(t,Y)-k2z1-k1Dt2q-1z1,
To satisfy the sliding mode condition, the reaching law is chosen as(20)ur=-k3S-γsgn(S),
where k3 is a positive constant. Hence, the total control law is
(21)u=ueq+uru=f(t,X)-f(t,Y)-k2z1-k1Dt2q-1z1-k3S-γsgn(S).
Selecting(22)V2=|S|
as a composite Lyapunov function, we have
(23)V˙2=sgn(S)S˙V˙2=sgn(S)(k2z1+Dtqz2)V˙2=sgn(S)(k1Dt2q-1k2z1+f(t,Y)-f(t,X)V˙2=sgn(S)i+F(t,Y)+k1Dt2q-1z1+u(t)).
Substituting the control law (21) into the right hand side of (23), we have
(24)V˙2=sgn(S)[-k1Dt2q-1z1-k3S-γsgn(S)k2z1+f(t,Y)-f(t,X)+F(t,Y)+k1Dt2q-1z1+(-k1Dt2q-1z1-k3S-γsgn(S)f(t,X)-f(t,Y)-k2z1-k1Dt2q-1z1-k3S-γsgn(S))]V˙2=sgn(S)[F(t,Y)-k3S-γsgn(S)]V˙2≤γ-k3sgn(S)S-γsgn(S)sgn(S)V˙2=-k3|S|.
Hence, according to Lemma 2, system (15) is asymptotically stable.From the above two steps, we obtainlimt→∞zi=0(i=1,2). The properties z1=e1, z2=e2-α(z1)=e2+k1Dtq-1z1 imply that limt→∞ei=0(i=1,2). Thus, the global synchronization of systems (9) and (10) is achieved.


## 4. Applications

In order to demonstrate the effectiveness of the proposed control scheme, numerical simulations are made for synchronization of fractional-order Duffing-Holmes system. In the simulation, Adams-Bashforth-Moulton predictor-corrector algorithm is used, and the detailed descriptions of this algorithm are available in [33].


### 4.1. System Description

In [20], an integer-order chaotic Duffing-Holmes system has been studied. Let us consider the fractional-order case described by
(25)Dtqx1=x2,Dtqx2=x1-ax2-x13+bcost.
Initial conditions of master system and slave system are set as x1(0)=0.2, x2(0)=0.2 and y1(0)=0.1, y2(0)=-0.2, respectively.Phase portrait of the chaotic dynamics in system (25) when q=0.98 are shown in Figure 1. Parameters a and b are set as 0.25 and 0.3, respectively.Figure 1
Phase portrait of Duffing-Holmes system (25) with fractional-order q=0.98.The slave system is perturbed by the uncertainty and disturbance, which is described by(26)Dtqy1=y2,Dtqy2=y1-ay2-y13+bcost+Δf(t,Y)+d(t)+u,
where Δf(t,Y) and d(t) are chosen as 0.1sin(t)y12+y22 and 0.1sin(t), respectively.According to the scheme proposed in Section3, the control law is designed as follows:
(27)u=ae2-(1+k2)e1-k1Dt2q-1e1+y13u=-x13-γsgn(S)-k3S,
in which S=k2Dt-1z1+Dtq-1z2=k2Dt-1e1+Dtq-1e2+k1Dt2q-2e1. When the parameters are chosen as q=0.98, k1=k2=k3=2, γ=0.2, the simulation results are shown in Figures 2–5. Figures 2 and 3 show the synchronization performance between drive-response system (25)-(26) and the time response of the error system, respectively. The sliding surface and the control input u are displayed in Figures 4 and 5.Synchronization performance of fractional-order Duffing-Holmes systems (25) and (26).
(a)
(b)Synchronization error between fractional-order Duffing-Holmes systems (25) and (26).
(a)
(b)Figure 4
Switching surfaceS of the sliding mode controller (27).Figure 5
Control inputu of the slave system (26).


## 4.1. System Description

In [20], an integer-order chaotic Duffing-Holmes system has been studied. Let us consider the fractional-order case described by
(25)Dtqx1=x2,Dtqx2=x1-ax2-x13+bcost.
Initial conditions of master system and slave system are set as x1(0)=0.2, x2(0)=0.2 and y1(0)=0.1, y2(0)=-0.2, respectively.Phase portrait of the chaotic dynamics in system (25) when q=0.98 are shown in Figure 1. Parameters a and b are set as 0.25 and 0.3, respectively.Figure 1
Phase portrait of Duffing-Holmes system (25) with fractional-order q=0.98.The slave system is perturbed by the uncertainty and disturbance, which is described by(26)Dtqy1=y2,Dtqy2=y1-ay2-y13+bcost+Δf(t,Y)+d(t)+u,
where Δf(t,Y) and d(t) are chosen as 0.1sin(t)y12+y22 and 0.1sin(t), respectively.According to the scheme proposed in Section3, the control law is designed as follows:
(27)u=ae2-(1+k2)e1-k1Dt2q-1e1+y13u=-x13-γsgn(S)-k3S,
in which S=k2Dt-1z1+Dtq-1z2=k2Dt-1e1+Dtq-1e2+k1Dt2q-2e1. When the parameters are chosen as q=0.98, k1=k2=k3=2, γ=0.2, the simulation results are shown in Figures 2–5. Figures 2 and 3 show the synchronization performance between drive-response system (25)-(26) and the time response of the error system, respectively. The sliding surface and the control input u are displayed in Figures 4 and 5.Synchronization performance of fractional-order Duffing-Holmes systems (25) and (26).
(a)
(b)Synchronization error between fractional-order Duffing-Holmes systems (25) and (26).
(a)
(b)Figure 4
Switching surfaceS of the sliding mode controller (27).Figure 5
Control inputu of the slave system (26).


## 5. Conclusions

In this paper, the backstepping SMC method is developed for fractional-order strict-feedback chaotic system. The Lyapunov function, the virtual control, the switching surface, and the actual control are systematically designed, respectively. The proposed approach, which combines the merits of backstepping control and SMC, is robust against the modeling uncertainties and external disturbances. Numerical results demonstrate that the convergence speed of the synchronized error system is satisfactory. In addition, the strategy can be generalized to the investigation of synchronization of other fractional-order strict feedback chaotic or hyperchaotic systems.



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