# The Robust Control and Synchronization of a Class of Fractional-Order Chaotic Systems with External Disturbances via a Single Output

Authors: Runzi Luo; Haipeng Su
Journal: Complexity
 (2018)

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## Abstract

This paper investigates the stabilization and synchronization of a class of fractional-order chaotic systems which are affected by external disturbances. The chaotic systems are assumed that only a single output can be used to design the controller. In order to design the proper controller, some observer systems are proposed. By using the observer systems some sufficient conditions for achieving chaos control and synchronization of fractional-order chaotic systems are derived. Numerical examples are presented by taking the fractional-order generalized Lorenz chaotic system as an example to show the feasibility and validity of the proposed method.


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## Body

## 1. Introduction

Chaos control and synchronization have attracted a great deal of attention since the innovative works proposed by Huber, Pecora, and Carroll in 1990 [1]. Nowadays, owing to their potential applications in many areas such as in chemical reactions, power converters, information processing, and secure communications, various types of synchronization phenomena have been discovered, such as complete synchronization [2], combination synchronization [3], and equal combination synchronization [4].Fractional-order calculus is a branch of mathematics that deals with derivatives and integrals of non-integer orders. It has been shown that the models presented by fractional-order systems are more adequate than that described by integer order systems. Many systems such as viscoelastic systems, dielectric polarization, and electromagnetic waves [5] are known to display fractional-order dynamics. In recent years, a number of fractional-order chaotic systems have been investigated, such as the fractional-order economical system [6] and the fractional-order Lorenz system [7].Similarly to integer order chaotic systems, the control and synchronization of fractional-order chaotic systems has become an active research field [8–15]. It is not difficult to see that in papers [8–15] the authors have used all state variables to design controllers. However, in the real situation it is well known that only part of the variables can be used in many nonlinear systems. Therefore, it is necessary to investigate the control and synchronization of fractional-order chaotic system with a single output.Motivated by the above discussion, in this paper we consider the stabilization and synchronization of a class of fractional-order chaotic systems via a single output. Some sufficient conditions for achieving chaos control and synchronization of fractional-order chaotic systems are derived via the observer systems. The fractional-order generalized Lorenz chaotic system is taken as an example to show the feasibility and validity of the proposed method.The rest of the paper is organized as follows. In Section2, we introduce some preliminaries, including some definitions, lemmas, and the general form description of a class of fractional-order chaotic systems. The control and synchronization schemes of a class of fractional-order chaotic systems via a single output are presented in Sections 3 and 4, respectively. In Section 5, numerical simulations results are shown. Some conclusions are drawn in Section 6.


## 2. Preliminaries and System Description

### 2.1. Fractional-Order Integral and Derivative

In this subsection, some basic definitions with respect to Caputo’s fractional derivative are introduced. Also, some useful lemmas are proposed.Definition 1 (see [16]).
Caputo’s fractional derivative of functionf(t)∈Cn([t0,∞),R) is defined by (1)Dt0αft=1Γ1-α∫t0tf′τt-ταdτ,where 0<α≤1.Definition 2 (see [16]).
The Mittag-Leffler function with two parameters is given by(2)Eα,βz=∑p=0∞zpΓpα+β,where α>0,β>0 and z∈C.
Ifβ=1, then Eα,1(z)=Eα(z)=∑p=0∞‍zp/Γ(pα+1).
LetL(f(t)) denote the Laplace transform of a function f(t). Based on the definition of Laplace transform: F(s)=L(f(t))=∫0∞‍e-stf(t)dt, we have (3)LD0αft=sαFs-sα-1f0,0<α≤1.The Laplace transform of Mittag-Leffler function with two parameters is (4)Ltβ-1Eα,β-λtα=sα-βsα+λ,Res>λ1/α.Lemma 3 (see [17]).
Ifλ>0, then the origin of system (5)Dαxt=-λxt,0<α≤1is globally asymptotically stable.Lemma 4 (see [18]).
Ifλ>0 and limt→∞η(t)=0, then the origin of system (6)Dαxt=-λxt+ηt,0<α<1is globally asymptotically stable.
Ifx(t)≥0, then based on Lemma 4 we can easily get the following Corollary 5.Corollary 5.
Supposex(t)≥0. If λ>0 and limt→∞η(t)=0, then the origin of system (7)Dαxt≤-λxt+ηt,0<α<1is globally asymptotically stable.Lemma 6.
Consider the following two systems:(8)Dαx=atx+bt,0<α<1.and(9)Dαy=a¯y+Bt,0<α<1.where x∈R,y∈R,a(t)∈R,b(t)∈R and a(t)≤a¯<0,bt≤B(t), and then we have (10)x≤y+Eαa¯tαx0-y0,where x(0),y(0) are initial values of x and y, respectively.Proof.
Based on system (8), we obtain(11)Dαx=Dαxsignx=signxDαx=signxatx+bt≤a¯x+Bt.The Laplace transform of (11) gives(12)sαxs-sα-1x0≤a¯xs+Bs,s∈R.The Laplace transform of (9) is (13)sαys-sα-1y0=a¯ys+Bs.Subtracting (13) from (12) one has (14)sαxs-ys-sα-1x0-y0≤a¯xs-ys.The above inequality is equivalent to (15)xs-ys≤sα-a¯-1x0-y0sα-1.Taking the inverse Laplace transform of (15) yields (16)xt-yt≤x0-y0Eαa¯tα.It implies that (17)xt≤yt+x0-y0Eαa¯tα.This concludes the proof of Lemma 6.Remark 7.
System (9) is called as the observer of system (8).


### 2.2. System Description

In this paper we consider the class of chaotic systems which are described by(18)Dαx1=atx1+a1x2+a2x22,Dαx2=b1x1+b2x2+b3x3+b4x1x3+dt,Dαx3=ctx3+c1x1x2+c2x22,xout=x2,where x=(x1,x2,x3)T∈R3×1 is the state vector of system (18) and α∈(0,1) is the order of fractional derivatives. a(t)(<0),c(t)(<0) and aj,cj,bi(i=1,⋯,4,j=1,2) are system’s parameters. d(t) represents the model uncertainty or the external disturbance. xout is the measured output signal which can be used to design the controller.


## 2.1. Fractional-Order Integral and Derivative

In this subsection, some basic definitions with respect to Caputo’s fractional derivative are introduced. Also, some useful lemmas are proposed.Definition 1 (see [16]).
Caputo’s fractional derivative of functionf(t)∈Cn([t0,∞),R) is defined by (1)Dt0αft=1Γ1-α∫t0tf′τt-ταdτ,where 0<α≤1.Definition 2 (see [16]).
The Mittag-Leffler function with two parameters is given by(2)Eα,βz=∑p=0∞zpΓpα+β,where α>0,β>0 and z∈C.
Ifβ=1, then Eα,1(z)=Eα(z)=∑p=0∞‍zp/Γ(pα+1).
LetL(f(t)) denote the Laplace transform of a function f(t). Based on the definition of Laplace transform: F(s)=L(f(t))=∫0∞‍e-stf(t)dt, we have (3)LD0αft=sαFs-sα-1f0,0<α≤1.The Laplace transform of Mittag-Leffler function with two parameters is (4)Ltβ-1Eα,β-λtα=sα-βsα+λ,Res>λ1/α.Lemma 3 (see [17]).
Ifλ>0, then the origin of system (5)Dαxt=-λxt,0<α≤1is globally asymptotically stable.Lemma 4 (see [18]).
Ifλ>0 and limt→∞η(t)=0, then the origin of system (6)Dαxt=-λxt+ηt,0<α<1is globally asymptotically stable.
Ifx(t)≥0, then based on Lemma 4 we can easily get the following Corollary 5.Corollary 5.
Supposex(t)≥0. If λ>0 and limt→∞η(t)=0, then the origin of system (7)Dαxt≤-λxt+ηt,0<α<1is globally asymptotically stable.Lemma 6.
Consider the following two systems:(8)Dαx=atx+bt,0<α<1.and(9)Dαy=a¯y+Bt,0<α<1.where x∈R,y∈R,a(t)∈R,b(t)∈R and a(t)≤a¯<0,bt≤B(t), and then we have (10)x≤y+Eαa¯tαx0-y0,where x(0),y(0) are initial values of x and y, respectively.Proof.
Based on system (8), we obtain(11)Dαx=Dαxsignx=signxDαx=signxatx+bt≤a¯x+Bt.The Laplace transform of (11) gives(12)sαxs-sα-1x0≤a¯xs+Bs,s∈R.The Laplace transform of (9) is (13)sαys-sα-1y0=a¯ys+Bs.Subtracting (13) from (12) one has (14)sαxs-ys-sα-1x0-y0≤a¯xs-ys.The above inequality is equivalent to (15)xs-ys≤sα-a¯-1x0-y0sα-1.Taking the inverse Laplace transform of (15) yields (16)xt-yt≤x0-y0Eαa¯tα.It implies that (17)xt≤yt+x0-y0Eαa¯tα.This concludes the proof of Lemma 6.Remark 7.
System (9) is called as the observer of system (8).


## 2.2. System Description

In this paper we consider the class of chaotic systems which are described by(18)Dαx1=atx1+a1x2+a2x22,Dαx2=b1x1+b2x2+b3x3+b4x1x3+dt,Dαx3=ctx3+c1x1x2+c2x22,xout=x2,where x=(x1,x2,x3)T∈R3×1 is the state vector of system (18) and α∈(0,1) is the order of fractional derivatives. a(t)(<0),c(t)(<0) and aj,cj,bi(i=1,⋯,4,j=1,2) are system’s parameters. d(t) represents the model uncertainty or the external disturbance. xout is the measured output signal which can be used to design the controller.


## 3. The Control Scheme

In this section, the stabilization of system (18) is investigated. In order to force the states of system (18) to its origin, the control input u is added to the second state equation. Thus, system (18) can be rewritten as(19)Dαx1=atx1+a1x2+a2x22,Dαx2=b1x1+b2x2+b3x3+b4x1x3+dt+u,Dαx3=ctx3+c1x1x2+c2x22,xout=x2,where u is a controller to be designed later.Now, some Assumptions are introduced.Assumption 8.
Supposed(t) is bounded which means that there exists constants δ>0 such that dt≤δ.Assumption 9.
Suppose there exist two constantsa¯,c¯ such that a(t)≤a¯<0,c(t)≤c¯<0.
Sincea1x2+a2x22≤a1x2+a2x22 and a(t)≤a¯, by Lemma 6 the observer of system (20)Dαx1=atx1+a1x2+a2x22is (21)Dαx^1=a¯x^1+a1x2+a2x22.Furthermore, by Lemma 6 we have (22)x1≤x^1+x10-x^10Eαa¯tα.Thus, we get (23)c1x1x2≤c1x1x2≤c1x2x^1+Eαa¯tαx10-x^10.In view of c1x1x2+c2x22≤c1x1x2+c2x22≤c1x2(x^1+Eα(a¯tα)(x10-x^1(0)))+c2x22 and c(t)≤c¯, then by Lemma 6 we know the observer of system (24)Dαx3=ctx3+c1x1x2+c2x22is (25)Dαx^3=c¯x^3+c1x2x^1+Eαa¯tαx10-x^10+c2x22.
Thus, in order to design proper controlleru, we can propose the following observer system for variables x1 and x3: (26)Dαx^1=a¯x^1+a1x2+a2x22,Dαx^3=c¯x^3+c1x2x^1+Eαa¯tαx10-x^10+c2x22,where x^1 and x^3 are the estimated values x1 and x3, respectively.Theorem 10.
Suppose Assumptions8 and 9 are satisfied. For system (19), if we choose u such that (27)u=-b1x^1+b3x^3+b4x^1x^3+δ+kx2+1signx2-b2x2,then the origin of system (19) is stable in the sense of limt→∞x1=limt→∞x2=limt→∞x3=0, where x^1,x^3 are defined by (26) and k>0.Proof.
The proof of Theorem10 is divided into two steps. In the first step, we shall show that limt→∞x2=0.
To this purpose, let us consider the following Lyapunov function candidate:(28)V=12x22.The time derivative of (28) is (29)DαV≤x2Dαx2=x2b1x1+b2x2+b3x3+b4x1x3+dt+u≤b1x1+b3x3+b4x1x3+dtx2+b2x2+ux2.By Lemma 6, we obtain (30)x1≤x^1+ρ1x,x3≤x^3+ρ3x,where ρ1x=Eα(a¯tα)(x10-x^1(0)),ρ3x=Eα(c¯tα)(x30-x^3(0)).
Substituting inequality (30) into (29), we have (31)DαV≤b1x^1+ρ1x+b3x^3+ρ3x+b4x^1+ρ1xx^3+ρ3x+dtx2+b2x2+ux2=b1x^1+b3x^3+b4x^1x^3+dtx2+b2x2+ux2+b1ρ1x+b3ρ3x+b4ρ1xx^3+b4ρ3xx^1+b4ρ1xρ3xx2.
By using (27), we have(32)DαV≤-1-b1ρ1x-b3ρ3x-b4ρ1xx^3-b4ρ3xx^1-b4ρ1xρ3x+kx2x2.Note that limt→∞ρ1x=limt→∞ρ3x=0. Therefore, we have limt→∞(b1ρ1x+b3ρ3x+b4ρ1xx^3+b4ρ3xx^1+b4ρ1xρ3x)=0. Thus, there exists a finite time t0 such that when t≥t0 we get b1ρ1x+b3ρ3x+b4ρ1xx^3+b4ρ3xx^1+b4ρ1xρ3x-1<0. Thus, when t≥t0 we obtain(33)DαV≤-kx22=-2kV.
It should be noted thatV≥0 and k>0. Thus, based on Lemma 3 we have limt→∞x2=0.
Now, in the second step we prove thatlimt→∞x1=limt→∞x3=0. By using the results obtained in the proof of step 1, it is obvious that limt→∞(a1x2+a2x22)=0. From the first equation of system (19) we derive that (34)Dαx1=atx1+a1x2+a2x22.Since a(t)<0, according to Lemma 4 we know that limt→∞x1=0. In the same way we have limt→∞x3=0. This completes the proof of Theorem 10.Remark 11.
The constant 1 in controller (27) is designed to eliminate the affect caused by b1ρ1x+b3ρ3x+b4ρ1xx^3+b4ρ3xx^1+b4ρ1xρ3x. Note that limt→∞(b1ρ1x+b3ρ3x+b4ρ1xx^3+b4ρ3xx^1+b4ρ1xρ3x)=0, and therefore the constant 1 can be replaced by any positive number. Since DαV≤-2kV, thus the factor k in controller (27) determines the speed of convergence. In general, the larger the number k the faster the rate of the convergence.


## 4. The Synchronization Scheme

In this section the synchronization scheme of a class of fractional-order chaotic systems is presented via the observer based method.Suppose system (18) is the drive system; in order to synchronize system (18) the corresponding response system with controller u is constructed as(35)Dαy1=aty1+a1y2+a2y22,Dαy2=b1y1+b2y2+b3y3+b4y1y3+d1t+u,Dαy3=cty3+c1y1y2+c2y22,yout=y2,where y=(y1,y2,y3)T∈R3×1 is the state vector of system (35), α∈(0,1) is the order of fractional derivatives. d1(t) represents the model uncertainty or the external disturbance. yout is the measured output signal which can be used to design the controller. u is the controller to be designed later.Let us define the synchronization error ase=y-x, then the dynamics of synchronization error between systems (18) and (35) can be described by(36)Dαe1=ate1+a1y2+a2y22-a1x2+a2x22,Dαe2=b1y1+b2y2+b3y3+b4y1y3+d1t-b1x1+b2x2+b3x3+b4x1x3+dt+u,Dαe3=cte3+c1y1y2+c2y22-c1x1x2+c2x22,Assumption 12.
Supposed(t),d1(t) are all bounded which means that there exists constants δ>0 such that d1t-dt≤δ.
The objective of the current synchronization problem is to design an appropriate control signalu such that, for any initial conditions of the drive and response systems, the synchronization errors converge to zero. For this end, similar to system (26) we proposed the following observer for variables x1,x3 and y1,y3: (37)Dαx^1=a¯x^1+a1x2+a2x22,Dαx^3=c¯x^3+c1x2x^1+Eαa¯tαx10-x^10+c2x22,Dαy^1=a¯y^1+a1y2+a2y22,Dαy^3=c¯y^3+c1y2y^1+Eαa¯tαy10-y^10+c2y22,where x^1,x^3,y^1,y^3 are the estimated values x1,x3 and y1,y3, respectively.Theorem 13.
Suppose Assumptions9 and 12 are satisfied. If we choose u such that (38)u=-b1y^1+x^1+b3y^3+x^3+b4y^1y^3+x^1x^3+δ+ke2+1signe2-b2e2,then the synchronization between drive system (18) and response system (35) will occur in the sense of limt→∞e1=limt→∞e2=limt→∞e3=0, where x^1,x^3,y^1,y^3 are defined by (37) and k>0.Proof.
The proof of Theorem13 is similar to that of Theorem 10. In the first step, we shall show that (39)limt→∞e2=0.
To this purpose, let us consider the following Lyapunov function candidate:(40)V=12e22.The time derivative of (40) is (41)DαV≤e2Dαe2=e2b1y1+b2y2+b3y3+b4y1y3+d1t-b1x1+b2x2+b3x3+b4x1x3+dt+u≤b1y1+x1+b3y3+x3+b4y1y3+x1x3+d1t-dte2+b2e2+ue2. By Lemma 6, we obtain (42)x1≤x^1+ρx1,x3≤x^3+ρx3,y1≤y^1+ρy1,y3≤y^3+ρy3,where ρx1=Eα(a¯tα)(x10-x^1(0)),ρx3=Eα(c¯tα)(x30-x^3(0)),ρy1=Eα(a¯tα)(y10-y^1(0)),ρy3=Eα(c¯tα)(y30-y^3(0)).
Substituting inequality (42) into (41), we have (43)DαV≤b1y^1+ρy1+x^1+ρx1+b3y^3+ρy3+x^3+ρx3+d1t-dt+b4y^1+ρy1y^3+ρy3+x^1+ρx1x^3+ρx3e2+b2e2+ue2=b1y^1+x^1+b3y^3+x^3+d1t-dt+b4y^1y^3+x^1x^3e2+b2e2+ue2+b1ρy1+ρx1+b3ρy3+ρx3+b4y^1ρy3+ρy1y^3+ρy1ρy3+x^1ρx3+ρx1x^3+ρx1ρx3e2.By using (38), we have(44)DαV≤-1-b1ρy1+ρx1-b3ρy3+ρx3-b4y^1ρy3+ρy1y^3+ρy1ρy3+x^1ρx3+ρx1x^3+ρx1ρx3+ke2e2.The following proof is similar to that of Theorem 10 and omitted here. This ends the proof of Theorem 13.


## 5. Numerical Simulations

In this section we take the fractional-order generalized Lorenz chaotic systems as an example to verify and demonstrate the effectiveness of the proposed control scheme.The integer-order generalized Lorenz chaotic systems can be described as [19](45)x˙1=10+2529θx2-x1,x˙2=28-3529θx1+θ-1x2-x1x3,x˙3=-83-187θx3+x1x2,where x=(x1,x2,x3)T∈R3×1 is the state vector of system (32) and θ is the system parameter which satisfies θ∈{θ∣-232<θ<-11.6}∪{θ∣θ>-11.6}. It is well known that the systems (32) display chaotic behavior for each 0≤θ≤29 [19].Base on system (45), the fractional-order generalized Lorenz chaotic systems are given as(46)Dαx1=10+2529θx2-x1,Dαx2=28-3529θx1+θ-1x2-x1x3+dt,Dαx3=-83-187θx3+x1x2.where d(t) represents the model uncertainty or the external disturbance. The chaos attractor with α=0.995,d(t)=0,θ=2 is shown in Figure 1.Figure 1
The chaotic trajectories of system (35) with α=0.995,d(t)=0,θ=2 and x1(0)=4,x2(0)=-3,x3(0)=2.By comparing system (46) with system (18), it yields that a(t)=-(10+25/29θ),a1=(10+25/29θ),b1=(28-35/29θ),b2=θ-1,b4=-1,c(t)=-8/3-1/87θ,c1=1,a2=b3=c2=0. Since systems (32) display chaotic behavior for each 0≤θ≤29, so we suppose 0≤θ≤29 in systems (46). Thus, it is easy to see that a¯=-(10+25/29θ),c¯=-8/3-1/87θ. In the simulation process, we take α=0.995,θ=2 such that system (46) is chaotic.Example 1.
The control of the fractional-order generalized Lorenz chaotic systems.
The controlled system, based on system (46), is given as (47)Dαx1=10+2529θx2-x1,Dαx2=28-3529θx1+θ-1x2-x1x3+dt+u,Dαx3=-83-187θx3+x1x2.where u is the controller to be designed later.
Letdt=2sin⁡(t); then we get δ=2,a¯=-(10+25/29),c¯=(-8/3-1/87). The parameter k is taken as k=3. Suppose the observer system is (26) and the controller u is selected as (27); then according to Theorem 10 we know that the origin of system (47) is stable in the sense of limt→∞x1=limt→∞x2=limt→∞x3=0. The simulation results with x1(0)=4,x2(0)=-3,x3(0)=2 and x^1(0)=x^3(0)=1 are shown in Figures 2 and 3. Figure 2 is the time response of states x1,x2,x3 of system (47) and Figure 3 is the time response of states x^1,x^3 of system (26). From Figure 2 it is easy to see that although there is disturbance in system (47) the states x1,x2,x3 of system (47) approach 0 as t→+∞, respectively, which means that the origin of system (47) is asymptotic stable. Figure 3 shows that limt→∞x^1=limt→∞x^3=0 which means that in this case x^1 and x^3 can recover the information of x1 and x3.Figure 2
The time response of statesx1,x2,x3 of system (47) with controller (27).Figure 3
The time response of statesx^1,x^3 of system (26) with x^1(0)=1,x^3(0)=1.Example 2.
The synchronization between two identical fractional-order generalized Lorenz chaotic systems.
Let system (46) be the drive system; then the corresponding response system is given as(48)Dαy1=10+2529θy2-y1,Dαy2=28-3529θy1+θ-1y2-y1y3+d1t+u,Dαy3=-83-187θy3+y1y2.where u is the controller to be designed later and d1(t) is the model uncertainty or the external disturbance.In our simulation, we taked(t)=2sin⁡t,d1(t)=2cos⁡(t), thus dt-d1t≤4 which means that δ=4. The parameter k is taken as k=30. Suppose the observer system is (37) and the controller u is designed as (38); then by Theorem 13 we know the synchronization between the drive system (46) and the response system (48) will be achieved. Numerical simulations with x(0)=(4,-3,2)T,y(0)=(2,1,6)T, and x^1(0)=x^3(0)=y^1(0)=y^3(0)=1 are shown in Figure 4. Figure 4 is the time evolution of errors e1,e2,e3 between system (46) and system (48) with controller (38). From Figure 4 it is easy to see that although there are disturbances both in the drive system and the response system, the synchronization errors converge quickly to zero which means that the synchronization between system (46) and response system (48) is reached.Figure 4
The time response of synchronization errorse1,e2,e3 between system (46) and response system (48).


## 6. Conclusions

The observer-chaos-based stabilization and synchronization of a class of fractional-order chaotic systems with a single output are investigated in this paper. In the literature, there are some papers that consider the stabilization and synchronization of chaotic systems via the observer -based method. There are two main differences between our paper and the published papers:(1) it is easy to see that in the published paper the constructed observer system can exactly recover the information of the unavailable states. One can use the recovery states to design the controller easily. However, the observer system presented in this paper cannot recover the information of the unavailable states exactly; it just provides the upper bound of the unavailable states. Thus, to design the proper controller is a more difficult task. (2) In the literature, most of the papers that concerned the observer-based scheme do not consider the effects of external disturbances. However, the external disturbances are taken into consideration in this paper. By using the observer system, some sufficient conditions for achieving chaos control and synchronization of fractional-order chaotic systems are derived. The fractional-order generalized Lorenz chaotic system is taken as an example to show the feasibility of the designed method.



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