# Sliding Mode Control of the Fractional-Order Unified Chaotic System

Authors: Jian Yuan; Bao Shi; Xiaoyun Zeng; Wenqiang Ji; Tetie Pan
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper deals with robust synchronization of the fractional-order unified chaotic systems. Firstly, control design for synchronization of nominal systems is proposed via fractional sliding mode technique. Then, systematic uncertainties and external disturbances are considered in the fractional-order unified chaotic systems, and adaptive sliding mode control is designed for the synchronization issue. Finally, numerical simulations are carried out to verify the effectiveness of the two proposed control techniques.


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## Body

## 1. Introduction

Even though the theory of fractional calculus dates back to the end of the 17th century, the subject only really came to life over the last few decades [1]. The most significant advantage of fractional calculus is that it provides a powerful instrument of describing memory and hereditary properties of different substances [2]. In particular, fractional differential equations, as the basic theory for fractional-order control [3], have become a powerful tool in describing the dynamics of complex systems and gained great development very recently [4–6].One of the most important areas of application is the fractional-order chaotic systems, which have wide potential applications in engineering. Since Hartley et al. firstly discovered chaotic phenomenon in fractional dynamics systems [7], there has emerged great interest in this novel and promising topic. On one hand, more and more fractional nonlinear systems which exhibit chaos have been discovered, and their chaotic behaviors have been studied with numerical simulations, such as the fractional-order Chua circuit [8], the fractional-order Van der Pol oscillator [9–11], the fractional-order Lorenz system [12, 13], the fractional-order Chen system [14–16], the fractional-order Lü system [17], the fractional-order Liu system [18], the fractional-order Rössler system [19, 20], the fractional-order Arneodo system [21], the fractional-order Lotka-Volterra system [22, 23], the fractional-order financial system [24, 25], and the discrete fractional logistic map [26]. On the other hand, control and synchronization of fractional-order dynamical systems have been attracting growing investigations. Linear-state feedback control approach has been designed in [14, 27–35], nonlinear feedback control in [36–40], fractional PID control in [41, 42], and open-plus-closed-loop control in [43]. To tackle with modeling inaccuracies and external noises which are unavoidable in the real-world application, fractional-order sliding mode control methodology has been established in [44–51].In this paper, we investigate robust synchronization of the fractional-order unified chaotic systems. We firstly propose controllers to synchronize the nominal systems via fractional sliding mode technique. Secondly, we consider systematic uncertainties and external disturbances in the fractional-order unified chaotic systems and establish adaptive sliding mode control for synchronization of the uncertain systems.The rest of this paper is organized as follows. Section2 presents some basic definitions and theorems about fractional calculus and fractional-order dynamical system. Section 3 describes the general form of fractional-order unified chaotic system and presents our main objective in this paper. Section 4 proposes the sliding mode control design for synchronization of nominal systems and adaptive sliding mode control design for the uncertain system. Numerical simulations are presented to show the effectiveness of the proposed schemes in Section 5. Finally, this paper is concluded in Section 6.


## 2. Preliminaries

Definition 1.
The most important function used in fractional calculus is Euler’s Gamma function, which is defined as(1)Γ(n)=∫0∞‍tn-1e-tdt.Definition 2.
Another important function is a two-parameter function of the Mittag-Leffler type defined as(2)Eα,β(z)=∑k=0∞‍zkΓ(αk+β),α>0,β>0.
Fractional calculus is a generalization of integration and differentiation to noninteger-order fundamental operatorDatα, where a and t are the bounds of the operation and a∈ℝ. The continuous integrodifferential operator is defined as
(3)Datα={dαdtα,α>0,1,α=0,∫at(dτ)α,α<0.
The three most frequently used definitions for the general fractional calculus are the Grünwald-Letnikov definition, the Riemann-Liouville definition, and the Caputo definition [2, 23, 52, 53].Definition 3.
The Grünwald-Letnikov derivative definition of orderα is described as
(4)Datαf(t)=limh→01hα∑j=0∞‍(-1)j(αj)f(t-jh).
For binomial coefficients calculation, we can use the relation between Euler’s Gamma function and factorial defined as(5)(αj)=α!j!(α-j)!=Γ(α+1)Γ(j+1)Γ(α-j+1)
for
(6)(α0)=1.Definition 4.
The Riemann-Liouville derivative definition of orderα is described as
(7)Datαf(t)=1Γ(n-α)dndtn∫at‍f(τ)dτ(t-τ)α-n+1,n-1<α<n.
However, applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which containf(a), f′(a), and so forth. Unfortunately, the Riemann-Liouville approach fails to meet this practical need. It is M. Caputo who solved this conflict.Definition 5.
The Caputo definition of fractional derivative can be written as(8)Datαf(t)=1Γ(n-α)∫at‍f(n)(τ)dτ(t-τ)α-n+1,n-1<α<n.
In the following, we use the Caputo approach to describe the fractional chaotic systems and the Grünwald-Letnikov approach to propose numerical simulations. To simplify the notation, we denote the fractional-order derivative asDα instead of D0tα in this paper.Lemma 6 (see [22]).
Consider the following commensurate fractional-order dynamics system:(9)Dαx=f(x),
where 0<α≤1 and x∈ℝn. The equilibrium points of system (9) are calculated by solving the following equation:
(10)f(x)=0.
These points are locally asymptotically stable if all eigenvalues λi of the Jacobian matrix J=∂f/∂x evaluated at the equilibrium points satisfy
(11)|arg(λ)|>απ2.Lemma 7 (see [22]).
Consider the followingn-dimensional linear fractional-order dynamics system:
(12)Dq1x1=a11x1+a12x2+⋯+a1nxn,Dq2x2=a21x1+a22x2+⋯+a2nxn,⋯Dqnxn=an1x1+an2x2+⋯+annxn,
where all αi’s are rational numbers between 0 and 1. Assume M be the lowest common multiple of the denominators ui’s of αi’s, where αi=vi/ui, (ui,vi)=1, and ui,vi∈Z+, for i=1,2,…,n. Define
(13)Δ(λ)=(λMα1-a11-a12⋯-a1n-a21λMα2-a22⋯-a2n⋮⋮⋱⋮-an1-an2⋯λMαn-ann).
Then, the zero solution of system (12) is globally asymptotically stable in the Lyapunov sense if all roots of the equation det(Δ(λ))=0 satisfy |arg(λ)|>π/2M.Lemma 8 (see [53]).
Assume that there exists a scalar functionV of the state x with continuous first-order derivative such that the following are given: (i)
V(x) is positive definite,(ii)
V˙(x) is negative definite,(iii)
V(x)→∞ as ∥x∥→∞.Then, the equilibrium at the origin is globally asymptotically stable.


## 3. Problem Formulation

In [54], Lü et al. have considered a kind of chaotic systems and pointed out that these systems can be described in a unified form as follows:
(14)x˙1=(25α+10)(x2-x1),x˙2=(28-35α)x1-x1x3+(29α-1)x2,x˙3=x1x2-(8+α)x33,
where x1, x2, and x3 are state variables and α∈[0,1] is the system parameter. Lü et al. [54] call system (14) a unified chaotic system because it is chaotic for any α∈[0,1]. When α∈[0,0.8), system (14) is called a the generalized Lorenz chaotic system. When α=0.8, it is called the Lü chaotic system. And it is called the generalized Chen chaotic system when α∈(0.8,1].The fractional-order unified chaotic system has been firstly introduced and studied in [55] and reads as
(15)Dq1x1=(25α+10)(x2-x1),Dq2x2=(28-35α)x1-x1x3+(29α-1)x2,Dq3x3=x1x2-(8+α)x33,
where q1,q2,q3∈(0,1] is the fractional order.System (15) is considered as the drive (master) system and the response (slave) system is a controlled system as follows:
(16)Dq1y1=(25α+10)(y2-y1)+u1,Dq2y2=(28-35α)y1-y1y3+(29α-1)y2+u2,Dq3y3=y1y2-(8+α)y33+u3.Let us define the state errors between the response system (16) and the drive system (15) as e1=y1-x1, e2=y2-x2, and e3=y3-x3.By subtracting (15) from (16), one can get the following error dynamical system:
(17)Dq1e1=(25α+10)(e2-e1)+u1,Dq2e2=(28-35α)e1+e1e3-e1y3-e3y1+(29α-1)e2+u2,Dq3e3=-e1e2+e1y2+e2y1-(8+α)e33+u3.
Our main objective in this paper is to investigate the synchronization issue for the fractional-order unified chaotic system (15). It is clear that the synchronization of systems (15) and (16) is equivalent to the stabilization of the error dynamical system (17).


## 4. Synchronization Design

In the following, the sliding mode control technique, which can maintain low sensitivity to unmodeled dynamics and external disturbances, is applied to establish an effective control law to guarantee the synchronization of the drive system (15) and the response system (16). Two major steps are involved in the sliding mode control design: firstly, constructing an appropriate sliding surface on which the desired system dynamics is stable and, secondly, developing a suitable control law such that the sliding condition is attained.


### 4.1. Synchronization of the Nominal System

In this subsection, let us firstly consider a simple case: the nominal fractional-order unified chaotic system; that is, the system contains no systematic uncertainties or external disturbances. The design procedure is elaborated in the rest part of this subsection.


#### 4.1.1. Sliding Surfaces Design

In order to achieve the stability of system (17), three sliding surfaces S1, S2, and S3 are introduced as
(18)si(t)=(Dqi+λi)∫0t‍ei(τ)dτ,i=1,2,3,
the time derivative of which becomes
(19)s˙i(t)=Dqiei(t)+λiei(t),i=1,2,3.As long as system (17) operates on the sliding surfaces, it satisfies si=0 and s˙i=0, i=1,2,3, which yields the following sliding mode dynamics:
(20)Dq1e1=-λ1e1,Dq2e2=-λ2e2,Dq3e3=-λ3e3.By using Lemmas6 or 7, system (20) is asymptotically stable. As a result, the sliding mode surfaces (18) we have just constructed are appropriate for the control design.


#### 4.1.2. Control Laws Design

Step 1.
Choose the first control Lyapunov function(21)V1(t)=12s12.
Taking time derivative gives
(22)V˙1(t)=s1s˙1=s1(Dq1e1+λ1e1).
Substituting the first state equation of (17) into (22), one has
(23)V˙1(t)=s1[(25α+10)(e2-e1)+u1+λ1e1].
Therefore, by designing the first control law as
(24)u1=-(25α+10)(e2-e1)-λ1e1-k1sgn(s1),
where k1>0 and
(25)sgn(s)={1,s>0,0,s=0,-1,s<0,
then, (23) becomes
(26)V˙1(t)=-k1|s1|.Step 2.
Choose the second control Lyapunov function(27)V2(t)=12s22.
By taking its derivative with respect to time yields
(28)V˙2(t)=s2s˙2=s2(Dq2e2+λ2e2).
Substituting the second state equation of (17) into (28), one has
(29)V˙2(t)=s2[(28-35α)e1+e1e3-e1y3-e3y1V˙2(t)+(29α-1)e2+u2+λ2e2].
Therefore, by designing the second control law as
(30)u2=-(28-35α)e1-e1e3+e1y3+e3y1-(29α-1)e2-λ2e2-k2sgn(s3),
where k2>0.
Equation (29) becomes
(31)V˙2(t)=-k2|s2|.Step 3.
Choose the third control Lyapunov function(32)V3(t)=12s32.
Its time derivative is given by
(33)V˙3(t)=s3s˙3=s3(Dq3e3+λ3e3).
Substituting the third state equation of (17) into (33), one has
(34)V˙3(t)=s3[-e1e2+e1y2+e2y1-(8+α)e33+u3+λ3e3].
We are, then, in the position to design the third control law as follows:
(35)u3=e1e2-e1y2-e2y1+(8+α)e33-λ3e3-k3sgn(s3),
where k3>0.
With this choice, (34) can be rewritten as
(36)V˙3(t)=-k3|s3|.Step 4.
Finally, we gather the above three control functions as(37)V(s1,s2,s3)=12s12+12s22+12s32.
It is clear from (26), (31), and (36) that
(38)V˙(s1,s2,s3)=-(k1|s1|+k2|s2|+k3|s3|).
There exists some k>0 such that
(39)k1|s1|+k2|s2|+k3|s3|>k∥s∥,
where
(40)∥s∥=s12+s22+s32.
The resulting derivative of V is
(41)V˙(s1,s2,s3)<-k∥s∥.
In terms of Lemma8, the Lyapunov function (37) provides the proof of globally asymptotical stability with the control laws (24), (30), and (35).


### 4.2. Synchronization of the Uncertain System

In this subsection, we will proceed to study the synchronization of the fractional-order unified chaotic system in the presence of systematic uncertainties and external disturbances which can be hardly ignored in the real-world application. It is assumed that systematic uncertaintiesΔf1, Δf2, and Δf3 and external disturbances d1, d2, and d3 are all bounded; that is, |Δfi|<ρi and |di(t)|<θi, where ρi and θi are unknown positive constants, i=1,2,3. Let us denote that ρ^i is an estimate of ρi, while θ^i is an estimate of θi. Since ρi and θi are unknown, our task in this subsection is fulfilled with an adaptive controller consisting of control laws and update laws to obtain ρ^i and θ^i, i=1,2,3.The uncertain fractional-order unified chaotic system can be described as(42)Dq1e1=(25α+10)(e2-e1)+Δf1+d1+u1,Dq2e2=(28-35α)e1+e1e3-e1y3-e3y1+(29α-1)e2+Δf2+d2+u2,Dq3e3=-e1e2+e1y2+e2y1-(8+α)e33+Δf3+d3+u3.The adaptive sliding mode design of system (42) consists of four steps which are elaborated as follows.Step 1.
Consider the first control Lyapunov function(43)V1(t)=12[s12+1μ1(ρ^1-ρ1)2+1γ1(θ^1-θ1)2],
the time derivative of which becomes
(44)V˙1(t)=s1s˙1+1μ1(ρ^1-ρ1)ρ^˙1+1γ1(θ^1-θ1)θ^˙1=s1(Dq1e1+λ1e1)+1μ1(ρ^1-ρ1)ρ^˙1+1γ1(θ^1-θ1)θ^˙1.
Substituting the first state equation of (17) into (44), one has
(45)V˙1(t)=s1[(25α+10)(e2-e1)+Δf1+d1+u1+λ1e1]+1μ1(ρ^1-ρ1)ρ^˙1+1γ1(θ^1-θ1)θ^˙1.
By designing the first control law and adaptive law as
(46)u1=-(25α+10)(e2-e1)-λ1e1-(ρ^1+θ^1+k1)sgn(s1),(47)ρ^˙1=μ1|s1|,θ^˙1=γ1|s1|,
then, (45) becomes
(48)V˙1=s[(Δf1+d1)-(ρ^1+θ^1+k1)sgn(s1)]+(ρ^1-ρ1)|s1|+(θ^1-θ1)|s1|≤(|Δf1|+|d1|)|s1|-(ρ^1+θ^1+k1)|s1|+(ρ^1-ρ1)|s1|+(θ^1-θ1)|s1|<(ρ1+θ1)|s1|-(ρ^1+θ^1+k1)|s1|+(ρ^1-ρ1)|s1|+(θ^1-θ1)|s1|=-k1|s1|.Step 2.
Choose the second control Lyapunov function(49)V2(t)=12[s22+1μ2(ρ^2-ρ2)2+1γ2(θ^2-θ2)2]
whose derivative is
(50)V˙2(t)=s2s˙2+1μ2(ρ^2-ρ2)ρ^˙2+1γ2(θ^2-θ2)θ^˙2=s2(Dq2e2+λ2e2)+1μ2(ρ^2-ρ2)ρ^˙2+1γ1(θ^2-θ2)θ^˙2.
Substituting the second state equation of (17) into (50), one has
(51)V˙2(t)=s2[(28-35α)e1+e1e3-e1y3iiii-e3y1+(29α-1)e2]+s2[Δf2+d2+u2+λ2e2]+1μ2(ρ^2-ρ2)ρ^˙2+1γ2(θ^2-θ2)θ^˙2.
We choose the control law and the adaptive law
(52)u2=-(28-35α)e1-e1e3+e1y3+e3y1-(29α-1)e2-λ2e2-(ρ^2+θ^2+k2)sgn(s2),(53)ρ^˙2=μ2|s2|,(54)θ^˙2=γ2|s2|.
With this choice, (51) becomes
(55)V˙2(t)<-k2|s2|.Step 3.
Choose the third Lyapunov function(56)V3(t)=12[s32+1μ3(ρ^3-ρ3)2+1γ3(θ^3-θ3)2].
Taking time derivative gives
(57)V˙3(t)=s3s˙3+1μ3(ρ^3-ρ3)ρ^˙3+1γ3(θ^3-θ3)θ^˙3=s3(Dq3e3+λ3e3)+1μ3(ρ^3-ρ3)ρ^˙3+1γ3(θ^3-θ3)θ^˙3.
Substituting the third state equation of (17) into (57), one has
(58)V˙3(t)=s3[-e1e2+e1y2+e2y1-(8+α)e33]+s3[Δf3+d3+u3+λ3e3]+1μ3(ρ^3-ρ3)ρ^˙3+1γ3(θ^3-θ3)θ^˙3.
We choose the third control law and adaptive law
(59)u3=e1e2-e1y2-e2y1+(8+α)e33-λ3e3-(ρ^3+θ^3+k3)sgn(s3),(60)ρ^˙3=μ3|s3|,θ^˙3=γ3|s3|.
Then, the resulting derivative of V3 is
(61)V˙3(t)<-k3|s3|.Step 4.
Finally, we gather the above three control functions as(62)V(t)=12∑i=13‍si2+∑i=13‍12μi(ρ^i-ρi)2+∑i=13‍12γi(θ^i-θi)2.
It is clear from (48), (55), and (61) that
(63)V˙(t)=-(k1|s1|+k2|s2|+k3|s3|).
There exists some k>0 such that
(64)k1|s1|+k2|s2|+k3|s3|>k∥s∥.
The resulting derivative of V˙ is
(65)V˙(s1,s2,s3)<-k∥s∥,
where
(66)∥s∥=s12+s22+s32.
By using Lemma8, this Lyapunov function provides the proof of globally asymptotical stability with the control laws (46), (52), and (59) and adaptive laws (47), (53), and (60).


## 4.1. Synchronization of the Nominal System

In this subsection, let us firstly consider a simple case: the nominal fractional-order unified chaotic system; that is, the system contains no systematic uncertainties or external disturbances. The design procedure is elaborated in the rest part of this subsection.


### 4.1.1. Sliding Surfaces Design

In order to achieve the stability of system (17), three sliding surfaces S1, S2, and S3 are introduced as
(18)si(t)=(Dqi+λi)∫0t‍ei(τ)dτ,i=1,2,3,
the time derivative of which becomes
(19)s˙i(t)=Dqiei(t)+λiei(t),i=1,2,3.As long as system (17) operates on the sliding surfaces, it satisfies si=0 and s˙i=0, i=1,2,3, which yields the following sliding mode dynamics:
(20)Dq1e1=-λ1e1,Dq2e2=-λ2e2,Dq3e3=-λ3e3.By using Lemmas6 or 7, system (20) is asymptotically stable. As a result, the sliding mode surfaces (18) we have just constructed are appropriate for the control design.


### 4.1.2. Control Laws Design

Step 1.
Choose the first control Lyapunov function(21)V1(t)=12s12.
Taking time derivative gives
(22)V˙1(t)=s1s˙1=s1(Dq1e1+λ1e1).
Substituting the first state equation of (17) into (22), one has
(23)V˙1(t)=s1[(25α+10)(e2-e1)+u1+λ1e1].
Therefore, by designing the first control law as
(24)u1=-(25α+10)(e2-e1)-λ1e1-k1sgn(s1),
where k1>0 and
(25)sgn(s)={1,s>0,0,s=0,-1,s<0,
then, (23) becomes
(26)V˙1(t)=-k1|s1|.Step 2.
Choose the second control Lyapunov function(27)V2(t)=12s22.
By taking its derivative with respect to time yields
(28)V˙2(t)=s2s˙2=s2(Dq2e2+λ2e2).
Substituting the second state equation of (17) into (28), one has
(29)V˙2(t)=s2[(28-35α)e1+e1e3-e1y3-e3y1V˙2(t)+(29α-1)e2+u2+λ2e2].
Therefore, by designing the second control law as
(30)u2=-(28-35α)e1-e1e3+e1y3+e3y1-(29α-1)e2-λ2e2-k2sgn(s3),
where k2>0.
Equation (29) becomes
(31)V˙2(t)=-k2|s2|.Step 3.
Choose the third control Lyapunov function(32)V3(t)=12s32.
Its time derivative is given by
(33)V˙3(t)=s3s˙3=s3(Dq3e3+λ3e3).
Substituting the third state equation of (17) into (33), one has
(34)V˙3(t)=s3[-e1e2+e1y2+e2y1-(8+α)e33+u3+λ3e3].
We are, then, in the position to design the third control law as follows:
(35)u3=e1e2-e1y2-e2y1+(8+α)e33-λ3e3-k3sgn(s3),
where k3>0.
With this choice, (34) can be rewritten as
(36)V˙3(t)=-k3|s3|.Step 4.
Finally, we gather the above three control functions as(37)V(s1,s2,s3)=12s12+12s22+12s32.
It is clear from (26), (31), and (36) that
(38)V˙(s1,s2,s3)=-(k1|s1|+k2|s2|+k3|s3|).
There exists some k>0 such that
(39)k1|s1|+k2|s2|+k3|s3|>k∥s∥,
where
(40)∥s∥=s12+s22+s32.
The resulting derivative of V is
(41)V˙(s1,s2,s3)<-k∥s∥.
In terms of Lemma8, the Lyapunov function (37) provides the proof of globally asymptotical stability with the control laws (24), (30), and (35).


## 4.1.1. Sliding Surfaces Design

In order to achieve the stability of system (17), three sliding surfaces S1, S2, and S3 are introduced as
(18)si(t)=(Dqi+λi)∫0t‍ei(τ)dτ,i=1,2,3,
the time derivative of which becomes
(19)s˙i(t)=Dqiei(t)+λiei(t),i=1,2,3.As long as system (17) operates on the sliding surfaces, it satisfies si=0 and s˙i=0, i=1,2,3, which yields the following sliding mode dynamics:
(20)Dq1e1=-λ1e1,Dq2e2=-λ2e2,Dq3e3=-λ3e3.By using Lemmas6 or 7, system (20) is asymptotically stable. As a result, the sliding mode surfaces (18) we have just constructed are appropriate for the control design.


## 4.1.2. Control Laws Design

Step 1.
Choose the first control Lyapunov function(21)V1(t)=12s12.
Taking time derivative gives
(22)V˙1(t)=s1s˙1=s1(Dq1e1+λ1e1).
Substituting the first state equation of (17) into (22), one has
(23)V˙1(t)=s1[(25α+10)(e2-e1)+u1+λ1e1].
Therefore, by designing the first control law as
(24)u1=-(25α+10)(e2-e1)-λ1e1-k1sgn(s1),
where k1>0 and
(25)sgn(s)={1,s>0,0,s=0,-1,s<0,
then, (23) becomes
(26)V˙1(t)=-k1|s1|.Step 2.
Choose the second control Lyapunov function(27)V2(t)=12s22.
By taking its derivative with respect to time yields
(28)V˙2(t)=s2s˙2=s2(Dq2e2+λ2e2).
Substituting the second state equation of (17) into (28), one has
(29)V˙2(t)=s2[(28-35α)e1+e1e3-e1y3-e3y1V˙2(t)+(29α-1)e2+u2+λ2e2].
Therefore, by designing the second control law as
(30)u2=-(28-35α)e1-e1e3+e1y3+e3y1-(29α-1)e2-λ2e2-k2sgn(s3),
where k2>0.
Equation (29) becomes
(31)V˙2(t)=-k2|s2|.Step 3.
Choose the third control Lyapunov function(32)V3(t)=12s32.
Its time derivative is given by
(33)V˙3(t)=s3s˙3=s3(Dq3e3+λ3e3).
Substituting the third state equation of (17) into (33), one has
(34)V˙3(t)=s3[-e1e2+e1y2+e2y1-(8+α)e33+u3+λ3e3].
We are, then, in the position to design the third control law as follows:
(35)u3=e1e2-e1y2-e2y1+(8+α)e33-λ3e3-k3sgn(s3),
where k3>0.
With this choice, (34) can be rewritten as
(36)V˙3(t)=-k3|s3|.Step 4.
Finally, we gather the above three control functions as(37)V(s1,s2,s3)=12s12+12s22+12s32.
It is clear from (26), (31), and (36) that
(38)V˙(s1,s2,s3)=-(k1|s1|+k2|s2|+k3|s3|).
There exists some k>0 such that
(39)k1|s1|+k2|s2|+k3|s3|>k∥s∥,
where
(40)∥s∥=s12+s22+s32.
The resulting derivative of V is
(41)V˙(s1,s2,s3)<-k∥s∥.
In terms of Lemma8, the Lyapunov function (37) provides the proof of globally asymptotical stability with the control laws (24), (30), and (35).


## 4.2. Synchronization of the Uncertain System

In this subsection, we will proceed to study the synchronization of the fractional-order unified chaotic system in the presence of systematic uncertainties and external disturbances which can be hardly ignored in the real-world application. It is assumed that systematic uncertaintiesΔf1, Δf2, and Δf3 and external disturbances d1, d2, and d3 are all bounded; that is, |Δfi|<ρi and |di(t)|<θi, where ρi and θi are unknown positive constants, i=1,2,3. Let us denote that ρ^i is an estimate of ρi, while θ^i is an estimate of θi. Since ρi and θi are unknown, our task in this subsection is fulfilled with an adaptive controller consisting of control laws and update laws to obtain ρ^i and θ^i, i=1,2,3.The uncertain fractional-order unified chaotic system can be described as(42)Dq1e1=(25α+10)(e2-e1)+Δf1+d1+u1,Dq2e2=(28-35α)e1+e1e3-e1y3-e3y1+(29α-1)e2+Δf2+d2+u2,Dq3e3=-e1e2+e1y2+e2y1-(8+α)e33+Δf3+d3+u3.The adaptive sliding mode design of system (42) consists of four steps which are elaborated as follows.Step 1.
Consider the first control Lyapunov function(43)V1(t)=12[s12+1μ1(ρ^1-ρ1)2+1γ1(θ^1-θ1)2],
the time derivative of which becomes
(44)V˙1(t)=s1s˙1+1μ1(ρ^1-ρ1)ρ^˙1+1γ1(θ^1-θ1)θ^˙1=s1(Dq1e1+λ1e1)+1μ1(ρ^1-ρ1)ρ^˙1+1γ1(θ^1-θ1)θ^˙1.
Substituting the first state equation of (17) into (44), one has
(45)V˙1(t)=s1[(25α+10)(e2-e1)+Δf1+d1+u1+λ1e1]+1μ1(ρ^1-ρ1)ρ^˙1+1γ1(θ^1-θ1)θ^˙1.
By designing the first control law and adaptive law as
(46)u1=-(25α+10)(e2-e1)-λ1e1-(ρ^1+θ^1+k1)sgn(s1),(47)ρ^˙1=μ1|s1|,θ^˙1=γ1|s1|,
then, (45) becomes
(48)V˙1=s[(Δf1+d1)-(ρ^1+θ^1+k1)sgn(s1)]+(ρ^1-ρ1)|s1|+(θ^1-θ1)|s1|≤(|Δf1|+|d1|)|s1|-(ρ^1+θ^1+k1)|s1|+(ρ^1-ρ1)|s1|+(θ^1-θ1)|s1|<(ρ1+θ1)|s1|-(ρ^1+θ^1+k1)|s1|+(ρ^1-ρ1)|s1|+(θ^1-θ1)|s1|=-k1|s1|.Step 2.
Choose the second control Lyapunov function(49)V2(t)=12[s22+1μ2(ρ^2-ρ2)2+1γ2(θ^2-θ2)2]
whose derivative is
(50)V˙2(t)=s2s˙2+1μ2(ρ^2-ρ2)ρ^˙2+1γ2(θ^2-θ2)θ^˙2=s2(Dq2e2+λ2e2)+1μ2(ρ^2-ρ2)ρ^˙2+1γ1(θ^2-θ2)θ^˙2.
Substituting the second state equation of (17) into (50), one has
(51)V˙2(t)=s2[(28-35α)e1+e1e3-e1y3iiii-e3y1+(29α-1)e2]+s2[Δf2+d2+u2+λ2e2]+1μ2(ρ^2-ρ2)ρ^˙2+1γ2(θ^2-θ2)θ^˙2.
We choose the control law and the adaptive law
(52)u2=-(28-35α)e1-e1e3+e1y3+e3y1-(29α-1)e2-λ2e2-(ρ^2+θ^2+k2)sgn(s2),(53)ρ^˙2=μ2|s2|,(54)θ^˙2=γ2|s2|.
With this choice, (51) becomes
(55)V˙2(t)<-k2|s2|.Step 3.
Choose the third Lyapunov function(56)V3(t)=12[s32+1μ3(ρ^3-ρ3)2+1γ3(θ^3-θ3)2].
Taking time derivative gives
(57)V˙3(t)=s3s˙3+1μ3(ρ^3-ρ3)ρ^˙3+1γ3(θ^3-θ3)θ^˙3=s3(Dq3e3+λ3e3)+1μ3(ρ^3-ρ3)ρ^˙3+1γ3(θ^3-θ3)θ^˙3.
Substituting the third state equation of (17) into (57), one has
(58)V˙3(t)=s3[-e1e2+e1y2+e2y1-(8+α)e33]+s3[Δf3+d3+u3+λ3e3]+1μ3(ρ^3-ρ3)ρ^˙3+1γ3(θ^3-θ3)θ^˙3.
We choose the third control law and adaptive law
(59)u3=e1e2-e1y2-e2y1+(8+α)e33-λ3e3-(ρ^3+θ^3+k3)sgn(s3),(60)ρ^˙3=μ3|s3|,θ^˙3=γ3|s3|.
Then, the resulting derivative of V3 is
(61)V˙3(t)<-k3|s3|.Step 4.
Finally, we gather the above three control functions as(62)V(t)=12∑i=13‍si2+∑i=13‍12μi(ρ^i-ρi)2+∑i=13‍12γi(θ^i-θi)2.
It is clear from (48), (55), and (61) that
(63)V˙(t)=-(k1|s1|+k2|s2|+k3|s3|).
There exists some k>0 such that
(64)k1|s1|+k2|s2|+k3|s3|>k∥s∥.
The resulting derivative of V˙ is
(65)V˙(s1,s2,s3)<-k∥s∥,
where
(66)∥s∥=s12+s22+s32.
By using Lemma8, this Lyapunov function provides the proof of globally asymptotical stability with the control laws (46), (52), and (59) and adaptive laws (47), (53), and (60).


## 5. Numerical Simulations

### 5.1. Chaotic Behaviors of Fractional-Order Chaotic System

In [55], the authors have provided us with numerical methods of fractional calculus. In [56], the chaotic behaviors of the fractional-order unified system were numerically investigated, where it is found that the lowest order to exhibit chaos is 2.76.The chaotic behaviors are presented in Figures1 and 2 with fractional orders of q1=0.93, q2=0.94, and q3=0.95 and the initial conditions of (x1(0),x2(0),x3(0))=(-1,-2,1). The numerical algorithm is based on the following Grünwald-Letnikov’s definition:
(67)x(tk)=(25α+10)(x2(tk-1)-x1(tk-1))hq1-∑j=2k‍cj(q1)x1(tk-j),x2(tk)=[(28-35α)x1(tk)-x1(tk)x3(tk-1)+(29α-1)x2(tk-1)]hq2-∑j=2k‍cj(q2)x2(tk-j),x3(tk)=[x1(tk)x2(tk)-α+83x3(tk-1)]hq3-∑j=2k‍cj(q3)x3(tk-j),
where Tsim is the simulation time, k=1,2,…,N, for N=[Tsim/h].Chaotic trajectories withα=0.5.
(a)
(b)
(c)
(d)Chaotic trajectories withα=1.
(a)
(b)
(c)
(d)


### 5.2. Simulations of Synchronization of the Nominal System

In this subsection, numerical simulations are presented to demonstrate the effectiveness of the proposed sliding model control in Section4.1. In the numerical simulations, the fractional orders are chosen as q1=0.93, q2=0.94, and q3=0.95. The initial conditions of the drive system (15) and the response system (16) are chosen as (x1(0),x2(0),x3(0))=(-1,-2,1) and (y1(0),y2(0),y3(0))=(4,-4,4), respectively. Parameters in (18) are chosen as λ1=λ2=λ3=4. Gains of the control inputs in (24), (30), and (35) are chosen as k1=k2=k2=1.Whenα=0.5 and α=1, numerical simulations of synchronization of system (15) are presented in Figures 3, 4, 5, 6, 7, and 8 with control inputs (24), (30), and (35). For interpretations of the references to colors in these figure legends, the reader is referred to the web version of this paper.Synchronization of determined fractional-order unified chaotic system withα=0.5 (blue line represents the trajectories of the drive system, while red line represents the trajectories of the response system).
(a)
(b)
(c)Figure 4
Synchronization errors states withα=0.5 (red line represents the first error state e1, blue line represents the second error state e2, and black line represents the third error state e3).Figure 5
Sliding surfaces states withα=0.5 (red line represents the first sliding surface state s1, blue line represents the second sliding surface state s2, and black line represents the third sliding surface state s3).Synchronization of determined fractional-order unified chaotic system withα=1 (blue line represents the trajectories of the drive system, while red line represents the trajectories of the response system).
(a)
(b)
(c)Figure 7
Synchronization errors states withα=1 (red line represents the first error state e1, blue line represents the second error state e2, and black line represents the third error state e3).Figure 8
Sliding surfaces states withα=1 (red line represents the first sliding surface state s1, blue line represents the second sliding surface state s2, and black line represents the third sliding surface state s3).


### 5.3. Simulations of Synchronization of the Uncertain System

In this subsection, numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive sliding model control in Section4.2. In the numerical simulations, the fractional orders are always chosen as q1=0.93, q2=0.94, and q3=0.95. The initial conditions of the drive system (15) and the response system (16) are also chosen as (x1(0),x2(0),x3(0))=(-1,-2,1) and (y1(0),y2(0),y3(0))=(4,-4,4), respectively. Parameters in (18) are chosen as λ1=λ2=λ3=20. Gains of the control laws (46), (52), and (59) are chosen as k1=k2=k2=0.5. Gains of the adaptive laws (47), (53), and (60) are chosen as μ1=γ1=μ2=γ2=μ3=γ3=0.1. Systematic uncertainties and external disturbances are assumed to be Δf1=0, d1=-3cos(π(t-0.1)), Δf2=-0.6sin(2(y1-x1)), d2=5sin(t), Δf3=0, and d3=2cos(t-0.1).Whenα=0.5 and α=1, numerical simulations are presented in Figures 9, 10, 11, 12, 13, and 14 with control inputs (46), (52), and (59) and adaptive laws (47), (53), and (60). For interpretations of the references to colors in these figure legends, the reader is referred to the web version of this paper.Synchronization of fractional-order unified chaotic system in the presence of systematic uncertainties and external disturbances withα=0.5 (blue line represents the trajectories of the drive system, while red line represents the trajectories of the response system).
(a)
(b)
(c)Figure 10
Synchronization errors states in the presence of systematic uncertainties and external disturbances withα=0.5 (red line represents the first error state e1, blue line represents the second error state e2, and black line represents the third error state e3).Figure 11
Sliding surfaces states in the presence of systematic uncertainties and external disturbances withα=0.5 (red line represents the first sliding surface state s1, blue line represents the second sliding surface state s2, and black line represents the third sliding surface state s3).Synchronization of fractional-order unified chaotic system in the presence of systematic uncertainties and external disturbances withα=1 (blue line represents the trajectories of the drive system, while red line represents the trajectories of the response system).
(a)
(b)
(c)Figure 13
Synchronization errors states in the presence of systematic uncertainties and external disturbances withα=1 (red line represents the first error state e1, blue line represents the second error state e2, and black line represents the third error state e3).Figure 14
Sliding surfaces states in the presence of systematic uncertainties and external disturbances withα=1 (red line represents the first sliding surface state s1, blue line represents the second sliding surface state s2, and black line represents the third sliding surface state s3).


## 5.1. Chaotic Behaviors of Fractional-Order Chaotic System

In [55], the authors have provided us with numerical methods of fractional calculus. In [56], the chaotic behaviors of the fractional-order unified system were numerically investigated, where it is found that the lowest order to exhibit chaos is 2.76.The chaotic behaviors are presented in Figures1 and 2 with fractional orders of q1=0.93, q2=0.94, and q3=0.95 and the initial conditions of (x1(0),x2(0),x3(0))=(-1,-2,1). The numerical algorithm is based on the following Grünwald-Letnikov’s definition:
(67)x(tk)=(25α+10)(x2(tk-1)-x1(tk-1))hq1-∑j=2k‍cj(q1)x1(tk-j),x2(tk)=[(28-35α)x1(tk)-x1(tk)x3(tk-1)+(29α-1)x2(tk-1)]hq2-∑j=2k‍cj(q2)x2(tk-j),x3(tk)=[x1(tk)x2(tk)-α+83x3(tk-1)]hq3-∑j=2k‍cj(q3)x3(tk-j),
where Tsim is the simulation time, k=1,2,…,N, for N=[Tsim/h].Chaotic trajectories withα=0.5.
(a)
(b)
(c)
(d)Chaotic trajectories withα=1.
(a)
(b)
(c)
(d)


## 5.2. Simulations of Synchronization of the Nominal System

In this subsection, numerical simulations are presented to demonstrate the effectiveness of the proposed sliding model control in Section4.1. In the numerical simulations, the fractional orders are chosen as q1=0.93, q2=0.94, and q3=0.95. The initial conditions of the drive system (15) and the response system (16) are chosen as (x1(0),x2(0),x3(0))=(-1,-2,1) and (y1(0),y2(0),y3(0))=(4,-4,4), respectively. Parameters in (18) are chosen as λ1=λ2=λ3=4. Gains of the control inputs in (24), (30), and (35) are chosen as k1=k2=k2=1.Whenα=0.5 and α=1, numerical simulations of synchronization of system (15) are presented in Figures 3, 4, 5, 6, 7, and 8 with control inputs (24), (30), and (35). For interpretations of the references to colors in these figure legends, the reader is referred to the web version of this paper.Synchronization of determined fractional-order unified chaotic system withα=0.5 (blue line represents the trajectories of the drive system, while red line represents the trajectories of the response system).
(a)
(b)
(c)Figure 4
Synchronization errors states withα=0.5 (red line represents the first error state e1, blue line represents the second error state e2, and black line represents the third error state e3).Figure 5
Sliding surfaces states withα=0.5 (red line represents the first sliding surface state s1, blue line represents the second sliding surface state s2, and black line represents the third sliding surface state s3).Synchronization of determined fractional-order unified chaotic system withα=1 (blue line represents the trajectories of the drive system, while red line represents the trajectories of the response system).
(a)
(b)
(c)Figure 7
Synchronization errors states withα=1 (red line represents the first error state e1, blue line represents the second error state e2, and black line represents the third error state e3).Figure 8
Sliding surfaces states withα=1 (red line represents the first sliding surface state s1, blue line represents the second sliding surface state s2, and black line represents the third sliding surface state s3).


## 5.3. Simulations of Synchronization of the Uncertain System

In this subsection, numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive sliding model control in Section4.2. In the numerical simulations, the fractional orders are always chosen as q1=0.93, q2=0.94, and q3=0.95. The initial conditions of the drive system (15) and the response system (16) are also chosen as (x1(0),x2(0),x3(0))=(-1,-2,1) and (y1(0),y2(0),y3(0))=(4,-4,4), respectively. Parameters in (18) are chosen as λ1=λ2=λ3=20. Gains of the control laws (46), (52), and (59) are chosen as k1=k2=k2=0.5. Gains of the adaptive laws (47), (53), and (60) are chosen as μ1=γ1=μ2=γ2=μ3=γ3=0.1. Systematic uncertainties and external disturbances are assumed to be Δf1=0, d1=-3cos(π(t-0.1)), Δf2=-0.6sin(2(y1-x1)), d2=5sin(t), Δf3=0, and d3=2cos(t-0.1).Whenα=0.5 and α=1, numerical simulations are presented in Figures 9, 10, 11, 12, 13, and 14 with control inputs (46), (52), and (59) and adaptive laws (47), (53), and (60). For interpretations of the references to colors in these figure legends, the reader is referred to the web version of this paper.Synchronization of fractional-order unified chaotic system in the presence of systematic uncertainties and external disturbances withα=0.5 (blue line represents the trajectories of the drive system, while red line represents the trajectories of the response system).
(a)
(b)
(c)Figure 10
Synchronization errors states in the presence of systematic uncertainties and external disturbances withα=0.5 (red line represents the first error state e1, blue line represents the second error state e2, and black line represents the third error state e3).Figure 11
Sliding surfaces states in the presence of systematic uncertainties and external disturbances withα=0.5 (red line represents the first sliding surface state s1, blue line represents the second sliding surface state s2, and black line represents the third sliding surface state s3).Synchronization of fractional-order unified chaotic system in the presence of systematic uncertainties and external disturbances withα=1 (blue line represents the trajectories of the drive system, while red line represents the trajectories of the response system).
(a)
(b)
(c)Figure 13
Synchronization errors states in the presence of systematic uncertainties and external disturbances withα=1 (red line represents the first error state e1, blue line represents the second error state e2, and black line represents the third error state e3).Figure 14
Sliding surfaces states in the presence of systematic uncertainties and external disturbances withα=1 (red line represents the first sliding surface state s1, blue line represents the second sliding surface state s2, and black line represents the third sliding surface state s3).


## 6. Conclusions

This work is concerned with robust synchronization of the fractional-order unified chaotic system. The sliding mode control technique was applied to propose the control design of nominal system and adaptive sliding mode control scheme was designed to develop the control laws and adaptive laws for uncertain system with systematic uncertainties and external disturbances whose bounds are unknown. Numerical simulations were presented to demonstrate the effectiveness of the two kinds of techniques.



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*Source: 397504-2013-11-25.xml*