# Robust Control and Synchronization of 3-D Uncertain Fractional-Order Chaotic Systems with External Disturbances via Adding One Power Integrator Control

Authors: Runzi Luo; Meichun Huang; Haipeng Su
Journal: Complexity
 (2019)

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## Abstract

This paper investigates the control and synchronization of a class of 3-D uncertain fractional-order chaotic systems with external disturbances. The adding one power integrator control scheme, which is the generalization of the traditional backstepping method, is used to investigate the global stability of the control and synchronization manifold. As a result, several criteria for chaos control and synchronization are obtained. Compared with the previous results, the presented strategies can not only be applied to a class of strict-feedback systems but also be applied to more general class of fractional-order chaotic systems. In addition, the proposed controllers are robust against uncertain parameters and external disturbances. To validate the effectiveness of the proposed criteria, two illustrative examples are given.


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## Body

## 1. Introduction

Chaotic systems, which are characterized by possessing at least one positive Lyapunov exponent, are a special case of nonlinear deterministic systems with unpredictable behaviors. In general, chaotic systems can be classified into two categories: integer-order and fractional-order. Due to its potential applications in secure communications, biological systems [1] and so on, the control and synchronization of integer-order chaotic systems have been extensively studied for more than twenty years. Recently, chaos control and synchronization of fractional-order chaotic systems have attracted extensive attention of researchers and many results have been proposed by employing different control methods such as backstepping control method [2], sliding mode control method [3, 4], PID control approach [5], periodically intermittent control strategy [6], adaptive fuzzy backstepping control [7], and adaptive fuzzy prescribed performance control [8]. The backstepping method is a systematic design approach. By designing virtual controllers and partial Lyapunov functions step by step one can finally derive a common Lyapunov function for the overall system. The main advantage of this method is that it can guarantee the global stability, tracking, and transient performance of nonlinear systems [9]. Various excellent backstepping strategies for the control and synchronization of fractional-order chaotic system have been investigated and proposed in recent studies. For example, the control fractional-order ferroresonance system, which can show chaotic phenomenon, was considered in paper [10] by using the adaptive backstepping control. Papers [11, 12] investigated the stabilization and synchronization of a class of fractional-order chaotic systems via backstepping approach. Paper [13] proposed a new chaotic system with no equilibrium point and discussed its fractional-order backstepping control. The stabilization problem of a class of fractional-order chaotic systems with unknown parameters by using adaptive backstepping strategy was considered in [14]. However, this schemes presented in papers [10–14] can only be used for nonlinear systems with strict-feedback structure. In addition, the external perturbations have not been taken into account here. Since in practical situations the external disturbance unavoidably exists and it may destroy the stabilization of chaotic systems, it is necessary to consider the effect of external disturbances in designing an adaptive controller.In 2000, paper [15] presented a new control method: adding a power integrator control. Similar to the traditional backstepping method, this new control method is also a systematic design approach and recursive Lyapunov-based scheme. In order to obtain a common Lyapunov function for the overall system, one needs to design power integrators and partial Lyapunov functions step by step. Since the power integrator is the generalization of the virtual controller, thus adding a power integrator control can be viewed as the generalization of the traditional backstepping method. The remarkable superiorities of adding a power integrator control are its robustness to parameter uncertainties and good transient response of nonlinear systems. Therefore, the adding a power integrator control method is applicable for the control and synchronization problem of chaotic system.Motivated by the above discussions this paper considers the control and synchronization of a class of 3-D fractional-order chaotic systems. This chaotic system with uncertain parameters is assumed to be affected by external disturbances. By using the adding one power integrator control scheme, several criteria for chaos control and synchronization are presented. The theoretical results are validated by two numerical simulations.Compared with the mentioned prior papers [10–14], there are two advantages which make our control scheme attractive and meaningful. (a) The schemes presented in papers [10–14] are only valid for a class of strict-feedback chaotic systems. However, the technique proposed in this paper can be applied for any 3-D fractional-order chaotic systems. (b) The controllers constructed in papers [10–14] are under the same assumption that systems are free from external perturbations. However, the external perturbations are taken into consideration in our paper.The subsequent sections of this paper are organized as follows: the system description and related preliminaries are introduced in Section2. The control and synchronization schemes are presented in Sections 3 and 4, respectively. Some illustrative examples to verify the effectiveness of the obtained theoretical result are given in Section 5. Finally, conclusions are drawn in Section 6.


## 2. Preliminaries and System Description

Definition 1 (see [16]).
The Caputo’s fractional derivative of orderα of function f(t) is(1)Dt0αft=1Γn-α∫t0tfnτt-τα-n+1dτ, where t≥t0,n∈N+,n-1<α≤n and(2)fnτ=dnfτdτn.
When0<α≤1, then the Caputo fractional derivative of order α of function f(t) reduces to(3)Dt0αft=1Γ1-α∫t0tf′τt-ταdτ.Lemma 2 (see [17]).
Supposex(t)∈R is a derivable function, then(4)12Dt0αx2t≤xtDt0αxt,∀α∈0,1,t0≥0.
In the sequel, for the sake of simplicity, we useDαx to denote D0αx.
The fractional-order chaotic system which is considered in this paper is(5)Dαx1=f11x+a1g11x+d11,Dαx2=f12x+b1g12x+d12,Dαx3=f13x+c1g13x+d13, where 0<α≤1, x=(x1,x2,x3)T∈R3×1 is the state vector of the system (5), f1i(x),g1i(x) are all known continuous functions, i=1,2,3. a1,b1,c1 are unknown parameters, d11,d12,d13 are bounded external disturbances.Assumption 3.
Suppose there exist known constantsd¯1i>0 such that d1i≤d¯1i,i=1,2,3.


## 3. The Control Scheme

The control scheme of system (5) is presented in this section.Based on system (5), the controlled system is rewritten as(6)Dαx1=f11x+a1g11x+d11+u1,Dαx2=f12x+b1g12x+d12+u2,Dαx3=f13x+c1g13x+d13+u3,where u1,u2,u3 are controllers.The purpose of this section is to find suitable controllersu1,u2,u3, such that the equilibrium point (0,0,0) of system (6) is asymptotically stable.Let us introduce some new variables:ξ1=x1, x2∗=-α1x1, ξ2=x2-x2∗, x3∗=-α2ξ2, ξ3=x3-x3∗, where α1>0,α2>0.By using these new variables, the controllersu1,u2,u3 in system (6) are designed as(7)u1=x2-f11x-a¯1g11x+β1,u2=x3-f12x-b¯1g12x+β2,u3=-α6ξ3-f13x-c¯1g13x+β3, where a¯1,b¯1,c¯1 are the estimated values of a1,b1,c1, respectively, β1=-d¯11sign(x1+α2ξ2+α1α2ξ3), β2=-d¯12sign(ξ2+α2ξ3), β3=-d¯13sign(ξ3), α6>0 is a constant.After plugging controllers (7) into system (6), one derives(8)Dαx1=x2+a1-a¯1g11x+d11+β1,Dαx2=x3+b1-b¯1g12x+d12+β2,Dαx3=-α6ξ3+c1-c¯1g13x+d13+β3,Obviously, the control of system (6) is transformed into the stability problem of equilibrium point (0,0,0) of system (8). To solve the stability problem of system (8), we propose the following Theorem 4.Theorem 4.
Suppose the update laws ofa¯1,b¯1,c¯1 are designed as(9)Dαa¯1=ξ1+α2ξ2+α1α2ξ3g11x,Dαb¯1=ξ2+α2ξ3g12x,Dαc¯1=ξ3g13x. If there exist constants αi>0(i=1,2,3,4,5,6), such that the following inequalities are satisfied:(10)-α1+α3+α5<0,α1-α2+-α12+124α3+α4<0,-α6+α2+α1α2-α12+124α4+α14α224α5<0,then the equilibrium point of system (8) is asymptotically stable.Proof.
We use the adding one power integrator control scheme to prove Theorem4. Suppose the first power integrator is(11)V1=∫0x1s-0ds=12x12. Clearly, V1 can be viewed as a Lyapunov function.
Now, by using Lemma2 we can calculate the derivative of V1 and derive that (12)DαV1≤x1Dαx1=x1x2+a-a¯1g11x+β1+d11=x1x2-x2∗+x1x2∗+a-a¯1x1g11x+x1β1+d11.
Note thatξ1=x1,x2∗=-α1x1=-α1ξ1,ξ2=x2-x2∗, then we get(13)DαV1≤ξ1ξ2-α1ξ12+a-a¯1x1g11x+x1β1+d11.
The second power integrator is constructed as(14)W1=∫x2∗x2s-x2∗ds=12x2-x2∗2, we have (15)W˙1≤x2-x2∗Dαx2-x2-x2∗Dαx2∗≤ξ2x3+b-b¯1g12x+β2+d12-ξ2Dα-α1x1=ξ2x3+ξ2b-b¯1g12x+ξ2β2+d12+α1ξ2x2+a-a¯1g11x+β1+d11=ξ2x3+ξ2b-b¯1g12x+α1ξ2x2-x2∗+α1ξ2x2∗+a-a¯1α1ξ2g11x+ξ2β2+d12+α1ξ2β1+d11=ξ2x3+α1ξ22-α22ξ1ξ2+ξ2b-b¯1g12x+a-a¯1α1ξ2g11x+ξ2β2+d12+α1ξ2β1+d11. Choosing V2=V1+W1, we obtain (16)DαV2=DαV1+DαW1≤-α1ξ12+ξ1ξ2+a-a¯1x1g11x+x1β1+d11+ξ2x3+α1ξ22-α12ξ1ξ2+ξ2b-b¯1g12x+a-a¯1α1ξ2g11x+ξ2β2+d12+α1ξ2β1+d11=-α1ξ12+α1ξ22+1-α12ξ1ξ2+ξ2x3+a-a¯1x1g11x+α1ξ2g11x+b-b¯1ξ2g12x+x1+α1ξ2β1+d11+ξ2β2+d12=-α1ξ12+α1ξ22+1-α12ξ1ξ2+ξ2x3-x3∗+ξ2x3∗+a-a¯1x1g11x+α1ξ2g11x+b-b¯1ξ2g12x+x1+α1ξ2β1+d11+ξ2β2+d12.
Sincex3∗=-α2ξ2,ξ3=x3-x3∗, we get (17)DαV2≤-α1ξ12+α1ξ22-α2ξ22+1-α12ξ1ξ2+ξ2ξ3+a-a¯1x1g11x+α1ξ2g11x+b-b¯1ξ2g12x+x1+α1ξ2β1+d11+ξ2β2+d12.
The third power integrator is similarly defined as(18)W2=∫x3∗x3s-x3∗ds=12x3-x3∗2,then we derive (19)DαW2=x3-x3∗Dαx3-x3-x3∗Dαx3∗≤ξ3-α6ξ3+c-c¯1g13x+β3+d13-ξ3Dα-α2ξ2=ξ3-α6ξ3+c-c¯1g13x+β3+d13+α2ξ3Dαx2-x2∗=ξ3-α6ξ3+c-c¯1g13x+β3+d13+α2ξ3x2+α1x1=ξ3-α6ξ3+c-c¯1g13x+β3+d13+α2ξ3x3+b-b¯1g12x+β2+d12+α1α2ξ3x2+a-a¯1g11x+β1+d11=ξ3-α6ξ3+c-c¯1ξ3g13x+α2ξ3x3+α1α2ξ3x2+b-b¯1α2ξ3g12x+α1α2ξ3a-a¯1g11x+ξ3β3+d13+α2ξ3β2+d12+α1α2ξ3β1+d11=ξ3-α6ξ3+α2ξ3x3-x3∗+α2ξ3x3∗+α1α2ξ3x2-x2∗+α1α2ξ3x2∗+c-c¯1ξ3g13x+b-b¯1α2ξ3g12x+a-a¯1α1α2ξ3g11x+ξ3β3+d13+α2ξ3β2+d12+α1α2ξ3β1+d11=ξ3-α6ξ3+α2ξ32+α2ξ3-α2ξ2+α1α2ξ3ξ2+α1α2ξ3-α1ξ1+c-c¯1ξ3g13x+b-b¯1α2ξ3g12x+a-a¯1α1α2ξ3g11x+ξ3β3+d13+α2ξ3β2+d12+α1α2ξ3β1+d11=ξ3-α6ξ3+α2ξ32+α1α2-α22ξ2ξ3-α12α2ξ1ξ3+c-c¯1ξ3g13x+b-b¯1α2ξ3g12x+a-a¯1α1α2ξ3g11x+ξ3β3+d13+α2ξ3β2+d12+α1α2ξ3β1+d11.
Now, the final Lyapunov function is chosen as(20)V3=V2+W2+12a1-a¯12+12b1-b¯12+12c1-c¯12.
The derivative ofV3 is (21)DαV3≤DαV2+DαW2-a1-a¯1Dαa¯1-b1-b¯1Dαb¯1-c1-c¯1Dαc¯1≤-α1ξ12+α1ξ22-α2ξ22+1-α12ξ1ξ2+ξ2ξ3+a-a¯1x1g11x+α1ξ2g11x+b-b¯1ξ2g12x+ξ3-α6ξ3+α2ξ32+α1α2-α22ξ2ξ3-α12α2ξ1ξ3+a-a¯1α1α2ξ3g11x+b-b¯1α2ξ3g12x+c-c¯1ξ3g13x+x1+α1ξ2+α1α2ξ3β1+d11+ξ2+α2ξ3β2+d12+ξ3β3+d13-a1-a¯1Dαa¯1-b1-b¯1Dαb¯1-c1-c¯1Dαc¯1.
Note that(22)x1+α1ξ2+α1α2ξ3β1+d11≤0,ξ2+α2ξ3β2+d12≤0,ξ3β3+d13≤0.We get (23)DαV3≤-α1ξ12+α1ξ22-α2ξ22+1-α12ξ1ξ2+ξ2ξ3+ξ3-α6ξ3+α2ξ32+α1α2-α22ξ2ξ3-α12α2ξ1ξ3+a-a¯1x1+α1ξ2+α1α2ξ3g11x+b-b¯1ξ2+α2ξ3g12x+c-c¯1ξ3g13x-a1-a¯1Dαa¯1-b1-b¯1Dαb¯1-c1-c¯1Dαc¯1.
SubstitutingDαa¯1,Dαb¯1,Dαc¯1 into DαV3 yields (24)DαV3≤-α1ξ12+α1-α2ξ22+1-α12ξ1ξ2+α2ξ32+α1α2-α22+1ξ2ξ3-α12α2ξ1ξ3+ξ3-α6ξ3. By using the inequality: ab≤ϵa2+b2/4ϵ(ϵ>0), we have (25)DαV3≤-α1ξ12+α1-α2ξ22+α2ξ32+α3ξ12+1-α1224α3ξ22+α4ξ22+1+α1α2-α1224α4ξ32+α5ξ12+α14α224α5ξ32+ξ3-α6ξ3=-α1+α3+α5ξ12+α1-α2+1-α1224α3+α4ξ22+-α6+α2+1+α1α2-α1224α4+α14α224α5ξ32.
In view of that(26)-α1+α3+α5<0,α1-α2+-α12+124α3+α4<0,-α6+α2+α1α2-α12+124α4+α14α224α5<0,we obtain DαV3<0, which implies that limt→∞ξ1=limt→∞ξ2=limt→∞ξ3=0. According to the definition of ξ1,ξ2,ξ3 we conclude that limt→∞x1=limt→∞x2=limt→∞x3=0. The proof of Theorem 4 is completed.Remark 5.
From the first inequality of (10), it is easy to see that for any given values α3 and α5 we can choose proper α1 such that -α1+α3+α5<0. Based on the second inequality of (10) one can know that for any given value α1 and α4 we can take suitable α2 such that α1-α2+(-α12+1)2/4α3+α4<0. Similarly, according to the third inequality of (10) one can derive that for any given values α1,α2,α4 and α5 one can select appropriate α6 such that -α6+α2+(α1α2-α12+1)2/4α4+α14α22/4α5<0. Therefore, there are many feasible solutions of equation (10). For example, α1=2,α2=8,α3=α4=α5=1/2,α6=586 is one of the feasible solutions. In addition, from (25) it is not difficult to know that the smaller the values of -α1+α3+α5,α1-α2+(-α12+1)2/4α3+α4 and -α6+α2+(α1α2-α12+1)2/4α4+α14α22/4α5, the faster the convergence speed of V3.Remark 6.
In Theorem4 we assume that each equation in system (6) has one uncertain parameter. However, in practical application each equation may have more than one uncertain parameter. In fact, Theorem 4 can be easily modified to deal with the case which has more than one uncertain parameter. For example, if the first equation of system (6) has two uncertain parameters: (27)Dαx1=f11x+ag1x+bg2x+d11+u1, where a and b are two uncertain parameters, then based on Theorem 4 we can take(28)u1=x2-f11x-a¯g1x-b¯g2x+β1, and the update laws of a¯,b¯ are designed as(29)Dαa¯=ξ1+α2ξ2+α1α2ξ3g1x,Dαb¯=ξ1+α2ξ2+α1α2ξ3g2x,where a¯ and b¯ are the estimated values of a and b, respectively.Remark 7.
In system (5), f1i(x) and g1i(x) can be any continuous functions which means that the technique proposed in this paper can be applied for any 3-D fractional-order chaotic systems. However, the schemes presented in papers [10–14] are only valid for a class of strict-feedback chaotic systems. In addition, the bounded external disturbances d11,d12,d13 are taken into consideration in system (5). However, the controllers presented in papers [10–14] are under the same assumption that systems are free from external perturbations.


## 4. The Synchronization Scheme

Based on the results obtained in Section3, in this section the drive-response synchronization scheme is investigated.Suppose system (5) is the drive system, then the response system is given as(30)Dαy1=f21y+a2g21y+d21+u1,Dαy2=f22y+b2g22y+d22+u2,Dαy3=f23y+c2g23y+d23+u3, where 0<α≤1, y=(y1,y2,y3)T∈R3×1 is the state vector of system (30), f2i(y),g2i(y) are all known functions, i=1,2,3. a2,b2,c2 are unknown parameters, d21,d22,d23 are bounded external disturbances, u1,u2,u3 are controllers.Assumption 8.
Suppose there exist known constantsd¯2i>0 such that d2i≤d¯2i,i=1,2,3.
The synchronization error is defined asei=yi-xi,i = 1,, 2, 3. Then the error dynamic system is given as(31)Dαe1=f21y+a2g21y+d21-f11x+a1g11x+d11+u1,Dαe2=f22y+b2g22y+d22-f12x+b1g12x+d12+u2,Dαe3=f23y+c2g23y+d23-f13x+c1g13x+d13+u3,
Now we introduce some new variables:η1=e1, e2∗=-α1e1, η2=e2-e2∗, e3∗=-α2η2, η3=e3-e3∗, where α1>0,α2>0.By using these new variables, the controllersu1,u2,u3 in system (31) are designed as(32)u1=e2+g11x-f21y+a¯1g11x-a¯2g21y+γ1,u2=e3-f22y+g12x+b¯1g12x-b¯2g22y+γ2,u3=-α6η3-f23y+g13x+c¯1g13x-c¯2g23y+γ3,where a¯1,b¯1,c¯1,a¯2,b¯2,c¯2 are the estimated values of a1,b1,c1,a2,b2,c2, respectively, γ1=-(d¯11+d¯21)sign(e1+α2η2+α1α2η3), γ2=-(d¯12+d¯22)sign(η2+α2η3), γ3=-(d¯13+d¯23)sign(η3), α6>0 is a constant.After plugging controllers (32) into system (31), one derives(33)Dαe1=e2-a1-a¯1g11x+a2-a¯2g21y-d11+d21+γ1,Dαe2=e3-b1-b¯1g12x+b2-b¯2g22y-d12+d22+γ2,Dαe3=-α6η3-c1-c¯1g13x+c2-c¯2g23y-d13+d23+γ3,Obviously, the synchronization problem between systems (5) and (30) is transformed into the stability problem of the equilibrium point (0,0,0) of system (33). To solve the stability problem of system (33), we propose Theorem 9 whose proof is omitted.Theorem 9.
Suppose the update laws ofa¯1,b¯1,c¯1,a¯2,b¯2,c¯2 are designed as(34)Dαa¯1=-η1+α2η2+α1α2η3g11x,Dαb¯1=-η2+α2η3g12x,Dαc¯1=-η3g13x,Dαa¯2=η1+α2η2+α1α2η3g21y,Dαb¯2=η2+α2η3g22y,Dαc¯2=η3g23y. If there exist constants αi>0(i=1,2,3,4,5,6), such that the inequalities (10) are satisfied, then the equilibrium point of system (33) is asymptotically stable which means that the synchronization between systems (5) and (30) is achieved.


## 5. Simulation Results

There are many new chaotic systems in the literature [18–20]. In this section the new fractional-order chaotic system which has been introduced in [20] is considered since it has many interesting properties.Example 10 (the control of a new fractional-order chaotic system).
The new fractional-order chaotic system is described by the following set of equations:(35)Dαx1=x2-ax1+bx2x3,Dαx2=cx2-x1x3+x3,Dαx3=dx1x2-hx3, where α is the fractional-order. x=(x1,x2,x3)T∈R3 is the state variable, parameters a,b,c,d and h are positive real constants. This system shows chaotic behaviors for the parameters a=3,b=2.7,c∈(4.45,4,60)∪(4.86,4.94)∪[5.17,7),d=2 and h=9. The chaotic attractor of system (35) with α=0.995,a=3,b=2.7,c=5,d=2,h=9 with initial conditions x1(0)=-3,x2(0)=1.5,x3(0)=2 is shown in Figure 1.Figure 1
The chaos attractor of system (35) with (x1(0),x2(0),x3(0))=(-3,1.5,2).For the control purpose, system (35) can be rewritten as(36)Dαx1=x2-ax1+bx2x3+sin⁡t+u1,Dαx2=cx2-x1x3+x3+2cos⁡t+u2,Dαx3=dx1x2-hx3+2sin⁡tcos⁡t+u3,where sin⁡t,2cos⁡t,2sin⁡tcos⁡t are the external disturbances, u1,u2,u3 are controllers.Supposea,b,c,d,h are unknown parameters, then f11(x)=x2,f12(x)=-x1x3+x3. Based on Theorem 4 the controllers u1,u2,u3 are chosen as(37)u1=a¯1x1-b¯1x2x3+β1,u2=x1x3-c¯1x2+β2,u3=-α6ξ3-d¯x1x2+h¯x3+β3, where a¯,b¯,c¯,d¯,h¯ are the estimated values of a,b,c,d,h, respectively, β1=-sign(x1+α2ξ2+α1α2ξ3), β2=-2sign(ξ2+α2ξ3),β3=-2sign(ξ3).The update laws ofa¯,b¯,c¯,d¯,h¯ are designed as(38)Dαa¯=-ξ1+α2ξ2+α1α2ξ3x1,Dαb¯=ξ1+α2ξ2+α1α2ξ3x2x3,Dαc¯=ξ2+α2ξ3x2,Dαd¯=ξ3x1x2,Dαh¯=-ξ3x3.If we takeα1=2,α2=8,α3=α4=α5=1/2,α6=586, then the conditions of Theorem 4 are satisfied. According to Theorem 4 the equilibrium point of system (36) is asymptotically stable. The simulation results with x1(0)=-3,x2(0)=1.5,x3(0)=2 and a^(0)=b^(0)=c^(0)=d^(0)=h^(0)=1 are shown in Figures 2–5. Figures 2–4 are the time response of states x1,x2,x3 of system (36) with controller (37). Figure 5 displays the time response of states a^,b^,c^,d^,h^ of system (36) with controller (37).Figure 2
The time response of statex1 of system (36) with controller (37).Figure 3
The time response of statex2 of system (36) with controller (37).Figure 4
The time response of statex3 of system (36) with controller (37).Figure 5
The time response of statesa¯,b¯,c¯,d¯,h¯ of system (36) with controller (37).Example 11 (the synchronization between two new fractional-order chaotic systems).
Another new chaotic system considered in this section is the fractional-order modified unified chaotic system [21] which is given as(39)Dαy1=35-25θy2-y1,Dαy2=-7+35θy1-y1y3+28-29θy2,Dαy3=y1y2-139-θy3, where α is the fractional-order. y=(y1,y2,y3)T∈R3 is the state variable, parameter θ∈[0,1]. The chaotic attractor of system (39) with θ=1 with initial conditions y1(0)=2,y2(0)=-1.5,⁡y3(0)=5 is shown in Figure 6.Figure 6
The chaos attractor of fractional-order modified unified chaotic system with(y1(0),y2(0),y3(0))=(2,-1.5,5).Suppose system (35) is the drive system and system (39) is the response system. The drive system with external disturbances is described as(40)Dαx1=x2-ax1+bx2x3+sin⁡t,Dαx2=cx2-x1x3+x3+2cos⁡t,Dαx3=dx1x2-hx3+2sin⁡tcos⁡t. The controlled response system with external disturbances is given as(41)Dαy1=35-25θy2-y1+2cos⁡t+u1,Dαy2=-7+35θy1-y1y3+28-29θy2+sin⁡t+u2,Dαy3=y1y2-μy3+cos⁡t+u3,where u1,u2,u3 are controllers and μ=(1/3)(9-θ).Now for simplicity, we assume parametersa and c in system (40) and parameter μ in system (41) are unknown.According to Theorem9, the controllers u1,u2,u3 in system (41) are designed as(42)u1=e2+x2+bx2x3-35-25θy2-y1-a¯1x1+γ1,u2=e3-x1x3+x3--7+35θy1-y1y3+28-29θy2+c¯1x2+γ2,u3=-α6η3+dx1x2-hx3-y1y2+μ¯y3+γ3, where a¯1,c¯1,μ¯ are the estimated values of a1,c1,μ, respectively, γ1=-3sign(e1+α2η2+α1α2η3), γ2=-3sign(η2+α2η3), γ3=-3sign(η3), α6>0 is a constant.The update laws ofa¯1,c¯1,μ¯ are designed as(43)Dαa¯=η1+α2η2+α1α2η3x1,Dαc¯=-η2+α2η3x2,Dαμ¯=-η3y3.Similarly, we takeα1=2,α2=8,α3=α4=α5=1/2,α6=586. Then the conditions of Theorem 9 are satisfied. According to Theorem 9 the synchronization between systems (40) and (41) will be achieved. The simulation results with x1(0)=-3,x2(0)=1.5,x3(0)=2,(y1(0),y2(0),y3(0))=(2,-1.5,5) and a¯(0)=c¯(0)=μ¯(0)=1 are shown in Figures 7–10. Figures 7–9 are the time response of states e1,e2,e3 of the error system. Figure 10 shows the time response of states a¯,c¯,μ¯ of the error system.Figure 7
The time response of the error statee1.Figure 8
The time response of the error statee2.Figure 9
The time response of the error statee3.Figure 10
The time response of statesa¯,c¯,μ¯.Remark 12.
Although the chaotic systems investigated in our numerical simulations are not in strict-feedback form and subject to external disturbances, from Figures2–5 and 7–10 one can find that the numerical results are completely consistent with the theoretical analysis presented in this paper which in turn demonstrates the validity, effectiveness, and good performance of the proposed schemes.


## 6. Conclusions

This paper deals with the control and synchronization of a class of 3-D uncertain fractional-order chaotic systems with external disturbances by using the adding one power integrator control method. Based on the systematic step by step control design procedure, some novel criteria are presented. These proposed control strategies can not only be applied to a class of strict-feedback systems, but also be applied to more general class of fractional-order chaotic systems. In addition, the presented controllers are robust against uncertain parameters and external disturbances. In the end, some illustrative examples and numerical simulation results are given to demonstrate the effectiveness of the proposed method.



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