# Adaptive Fuzzy Synchronization of Fractional-Order Chaotic (Hyperchaotic) Systems with Input Saturation and Unknown Parameters

Authors: Heng Liu; Ye Chen; Guanjun Li; Wei Xiang; Guangkui Xu
Journal: Complexity
 (2017)

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## Abstract

We investigate the synchronization problem of fractional-order chaotic systems with input saturation and unknown external disturbance by means of adaptive fuzzy control. An adaptive controller, accompanied with fractional adaptation law, is established, fuzzy logic systems are used to approximate the unknown nonlinear functions, and the fractional Lyapunov stability theorem is used to analyze the stability. This control method can realize the synchronization of two fractional-order chaotic or hyperchaotic systems and the synchronization error tends to zero asymptotically. Finally, we show the effectiveness of the proposed method by two simulation examples.


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## Body

## 1. Introduction

Recent studies showed that a large number of physical phenomena of nature and chemical processes, such as viscosity systems, colored noise, electrolyte electrode polarization, electromagnetic waves, and many actual systems can be described by fractional-order differential equations, making the slowly developed fractional calculus be a resurgence of interest [1–10]. Today, fractional-order systems described by fractional operators play a very important role in control fields [11–13]. Using the traditional integer-order differential equations as a method of describing dynamic system models has great limitation in biological engineering, cell engineering, neural network engineering, and some other emerging fields. However, the models established by fractional calculus can often achieve more satisfying and unexpected results [9, 14, 15]. Actually, the physical models established by the theory of fractional-order calculus are more concise and accurate in presentation when describing the complex problems of physics. In addition, the fractional controller not only can expand the freedom of the controlled system but also is able to obtain better control performance. Furthermore, the feature that fractional calculus has the function of memory makes the system’s states in the future be related to the previous and current states. Thus the memory and genetic characteristics of certain processes and materials can be expressed more accurately, which is conducive to improving the control effect of the systems [16, 17].It is well known that chaos has potential application values and great prospect in secure communication and other areas [18–22]. Recently, fractional-order chaotic systems and hyperchaotic systems have been studied in a widespread way and have been payed close attention with the deepening of theoretical research of fractional-order systems [23–26]. Many scholars have studied the synchronization control problems for fractional-order chaotic systems. So far, there are many control methods for fractional-order nonlinear chaotic systems (such as drive-response method, finite-time synchronization, nonlinear feedback method, adaptive synchronization control method, nonlinear disturbance observer method, nonlinear coupling method, sliding method, PC method, Lyapunov function activated method, and synchronization control method [27–33]). It is worth noting that the above literatures which study the problem of fractional-order chaotic systems synchronization have a basic assumption that the controller does not have any restrictions. However, almost all actuators in actual control systems have full amplitude or amplitude constraint problem (the amplitude of the output of the controller is limited artificially for reliability [34]). In addition, the presence of the input saturation of the control systems tends to attenuate the good performance of the system and even leads to instability of the closed-loop system. So many scholars have conducted extensive research in integer-order systems with input saturation in the recent years (literatures [35–37], etc.). Little work has been done to study the synchronous control of fractional-order nonlinear chaotic systems with input saturation.In this paper, the adaptive fuzzy synchronization of uncertain fractional-order nonlinear systems with input saturation and external disturbance is investigated on the basis of the above discussions. Fuzzy logic systems are used to approximate the fully unknown nonlinear functions of the systems. A fractional adaptive fuzzy synchronization controller is designed, and we prove the stability of the chaotic systems according to the fractional Lyapunov stability criterion. The main work of this paper can be concluded as follows: (1) The synchronization of fractional-order chaotic systems with input saturation and external disturbance is discussed in this paper. (2) An adaptive fuzzy synchronization controller is designed and fractional adaptive laws are designed to update the values of the parameters online.


## 2. Preliminaries

### 2.1. Preliminaries of Fractional Calculus

With the history of more than 300 years, there are many definitions of fractional calculus. But the most commonly used definitions are Grünwald-Letnikov, Caputo, and Riemann-Liouville definitions [11]. We choose Caputo’s derivative in this paper as its Laplace transform requires the initial values of the classical integer-order systems.Theαth fractional integral operator is defined as(1)Iτ-αfτ=0Dτ-αfτ=1Γα∫0ττ-να-1fνdν,where Γ(z)=∫0∞τz-1e-τdτ represents Euler’s Gamma function. The αth fractional derivative operator is given as(2)Dtαfτ=1Γn-α∫0ττ-νn-α-1fnνdν,where n-1≤α<n and n is an integer. And the Laplace transform of the formed formula (2) is(3)∫0∞e-stDταfτdτ=sαFs-∑k=0n-1sα-k-1fk0.The following properties of fractional calculus hold.Property 1 (see [38]).
Suppose thatx(τ)∈C1[0,T], T>0; then(4)Dτα1Dτα2xτ=Dτα1+α2xτ,where α1,α2>0 and α1+α2⩽1.Property 2 (see [38]).
The linearity of the Caputo fractional operator is as follows:(5)Dταλxτ+ωyτ=λDταxτ+ωDταyτ,where λ and ω are two real constants.Remark 1.
Ifc is a constant, then its Caputo derivative is 0. Namely, Dταc=0. Particularly, we have Dτα0=0.Property 3 (see [11]).
Letx(τ)∈C1[0,T] and 0<α<1; then(6)Dτ-αDταxτ=xτ-x0,DταDτ-αxτ=xτ.Note that the above properties hold if and only if0<α<1. Consequently, only the case where 0<α<1 is involved in the controller design and stability analysis. For convenience, in the rest of this paper, we always assume that α∈(0,1).


### 2.2. Fuzzy Logic Systems

A fuzzy logic system (FLS) consists of four parts (cf. [8, 9, 39–44]): the knowledge base, the fuzzifier, the fuzzy inference engine working on the fuzzy rules, and the defuzzifier. Usually, a fuzzy logic system is modeled by(7)y^=∑j∈Jwjtμjxt∑j∈Jμjxt,where y^ (a Lipschitz continuous mapping from a compact subset Ω⊆Rn to the real line R) is called the output of the fuzzy logic system, x=[x1,…,xn]T∈C1[I,Ω] (the set of all continuous mappings from I=[0,+∞)⊆R to Ω which have continuous derivatives) is called the input vector which is defined by x(t)=[x1(t),…,xn(t)]T (∀t∈I), J=∏i=1nFi, Fi consists of Ni fuzzy sets (1≤i≤n), μj (a mapping from Rn to the closed unit interval [0,1]⊆R) is called the membership function of rule j(j∈J), and wj (a mapping from I to R) is called the centroid of the jth consequent set (j∈J); we may identify J with {1,2,…,N} for the sake of convenience. Write W(t)=[w1(t),…,wN(t)]T and φ(x(t))=[q1(x(t)),q2(x(t)),…,qN(x(t))]T, where qj (called the jth fuzzy basis function, j∈J) is a continuous mapping (and thus φΩ→RN is continuous) defined by (8)qjxt=wjt∑s∈Jμsxt.Then system (7) can be rewritten as(9)y=WTφX.In contrast to conventional control techniques, fuzzy logic systems are best utilized in complex ill-defined processes that can be controlled by a skilled human operator without much knowledge of their underlying dynamics. The basic idea behind fuzzy logic systems is to incorporate the “expert experience” of a human operator in the design of the controller in controlling a process whose input-output relationship is described by a collection of fuzzy control rules involving linguistic variables rather than a complicated dynamic model.The fuzzy logic system (9) is employed to approximate the unknown nonlinear function f(μ) in this paper. It can be expressed as(10)fμ=W∗Tφμ+ϵμ,where ϵ(μ) is the ideal vector of the approximation error. W∗ is the ideal weight matrix which can be expressed as(11)W∗=arg⁡minW⁡sup⁡f^μ-fμ,where f^(μ) is the estimation of f(μ).


## 2.1. Preliminaries of Fractional Calculus

With the history of more than 300 years, there are many definitions of fractional calculus. But the most commonly used definitions are Grünwald-Letnikov, Caputo, and Riemann-Liouville definitions [11]. We choose Caputo’s derivative in this paper as its Laplace transform requires the initial values of the classical integer-order systems.Theαth fractional integral operator is defined as(1)Iτ-αfτ=0Dτ-αfτ=1Γα∫0ττ-να-1fνdν,where Γ(z)=∫0∞τz-1e-τdτ represents Euler’s Gamma function. The αth fractional derivative operator is given as(2)Dtαfτ=1Γn-α∫0ττ-νn-α-1fnνdν,where n-1≤α<n and n is an integer. And the Laplace transform of the formed formula (2) is(3)∫0∞e-stDταfτdτ=sαFs-∑k=0n-1sα-k-1fk0.The following properties of fractional calculus hold.Property 1 (see [38]).
Suppose thatx(τ)∈C1[0,T], T>0; then(4)Dτα1Dτα2xτ=Dτα1+α2xτ,where α1,α2>0 and α1+α2⩽1.Property 2 (see [38]).
The linearity of the Caputo fractional operator is as follows:(5)Dταλxτ+ωyτ=λDταxτ+ωDταyτ,where λ and ω are two real constants.Remark 1.
Ifc is a constant, then its Caputo derivative is 0. Namely, Dταc=0. Particularly, we have Dτα0=0.Property 3 (see [11]).
Letx(τ)∈C1[0,T] and 0<α<1; then(6)Dτ-αDταxτ=xτ-x0,DταDτ-αxτ=xτ.Note that the above properties hold if and only if0<α<1. Consequently, only the case where 0<α<1 is involved in the controller design and stability analysis. For convenience, in the rest of this paper, we always assume that α∈(0,1).


## 2.2. Fuzzy Logic Systems

A fuzzy logic system (FLS) consists of four parts (cf. [8, 9, 39–44]): the knowledge base, the fuzzifier, the fuzzy inference engine working on the fuzzy rules, and the defuzzifier. Usually, a fuzzy logic system is modeled by(7)y^=∑j∈Jwjtμjxt∑j∈Jμjxt,where y^ (a Lipschitz continuous mapping from a compact subset Ω⊆Rn to the real line R) is called the output of the fuzzy logic system, x=[x1,…,xn]T∈C1[I,Ω] (the set of all continuous mappings from I=[0,+∞)⊆R to Ω which have continuous derivatives) is called the input vector which is defined by x(t)=[x1(t),…,xn(t)]T (∀t∈I), J=∏i=1nFi, Fi consists of Ni fuzzy sets (1≤i≤n), μj (a mapping from Rn to the closed unit interval [0,1]⊆R) is called the membership function of rule j(j∈J), and wj (a mapping from I to R) is called the centroid of the jth consequent set (j∈J); we may identify J with {1,2,…,N} for the sake of convenience. Write W(t)=[w1(t),…,wN(t)]T and φ(x(t))=[q1(x(t)),q2(x(t)),…,qN(x(t))]T, where qj (called the jth fuzzy basis function, j∈J) is a continuous mapping (and thus φΩ→RN is continuous) defined by (8)qjxt=wjt∑s∈Jμsxt.Then system (7) can be rewritten as(9)y=WTφX.In contrast to conventional control techniques, fuzzy logic systems are best utilized in complex ill-defined processes that can be controlled by a skilled human operator without much knowledge of their underlying dynamics. The basic idea behind fuzzy logic systems is to incorporate the “expert experience” of a human operator in the design of the controller in controlling a process whose input-output relationship is described by a collection of fuzzy control rules involving linguistic variables rather than a complicated dynamic model.The fuzzy logic system (9) is employed to approximate the unknown nonlinear function f(μ) in this paper. It can be expressed as(10)fμ=W∗Tφμ+ϵμ,where ϵ(μ) is the ideal vector of the approximation error. W∗ is the ideal weight matrix which can be expressed as(11)W∗=arg⁡minW⁡sup⁡f^μ-fμ,where f^(μ) is the estimation of f(μ).


## 3. Adaptive Fuzzy Synchronization Controller Design and Stability Analysis

### 3.1. Problem Statement

Consider the following fractional-order chaotic systems:(12)Dtαxt=Axt+fxt,(13)Dtαyt=Cyt+gyt+sat⁡u+dt,where x(t)=[x1(t),…,xn(t)]T∈Rn is the state vector of the drive system (12) and y(t)=[y1(t),…,yn(t)]T∈Rn is the state vector of the response system (13). fi(·) and gi(·), i=1,2,…,n, are two unknown nonlinear functions, A and C∈Rn×n are two constant matrices, d(t)∈Rn is the external disturbance, sat(u)∈Rn represents the input saturation, and u(t)∈Rn is the control input.Remark 2.
In theoretical analysis, one often hopes that the input value and the output value can keep proportionally synchronized change when the former is relatively small. However, when the input value increases to a certain extent due to the system limitation factor, the output value of the actual conditions is no longer increasing but tends to or stays at a certain value in practical systems. This is said to be “saturation” phenomenon which is shown in Figure1.Figure 1
The phenomenon of saturation.Definition 3 (see [45, 46]).
A mapping fromRn to Rnsat:ν→sat(ν) is called a saturator, where ν=[ν1,ν2,…,νn]T∈Rn and sat(ν)=[sat(ν1),sat(ν2),…,sat(νn)]T. The definition of sat(u) is as follows:(14)satut=ur,ut≥urut,ul<ut<urul,ut⩽ul.Suppose that the part that exceeds the saturation limiter is referred to as δ(t); then one has(15)δt=ur-ut,ut≥ur0,ul<ut<urul-ut,ut⩽ul,where ur and ul are called saturated amplitude satisfying ul<0 and ur>0.Assumption 4.
The external disturbancedi(·) is a bounded continuous function. Namely, there exists an unknown constant ρi>0 such that(16)di·⩽ρi,i=1,2,…,n.Remark 5.
It should be pointed out that Assumption4 is reasonable. We just need the boundaries of the external disturbances, and their exact values are not needed in the process of designing the controller.The objective of our work is to design an appropriate adaptive fuzzy controller such that the synchronization errore(t)=y(t)-x(t) tends to zero asymptotically (namely, limt→∞e(t)=0).


### 3.2. Controller Design

The dynamical equation of the synchronization error can be described as(17)Dtαet=Cet+C-Axt+gyt-fxt+satu+dt.Based on the definition of sat(u), we can obtain that(18)satu=ut+δt.Then (17) can be rewritten as(19)Dtαet=Cet+C-Axt+gyt-fxt+ut+δt+dt.Consider(20)hxt,yt=ht=C-Axt+gyt-fxt+δt.Nothing that the nonlinear functionh(t) is unknown, it can be approximated, through the fuzzy logic system (9), as(21)h^it=WiTtφix,where Wi∗=argminWi(t)[sup⁡hit-h^it,Wit]. Let the unknown constant estimation error of the fuzzy logic systems and the approximation error, respectively, be(22)W~it=Wit-Wi∗,ϵit=hit-h^it,Wi∗.The following assumption is needed in the controller design.Assumption 6.
Suppose that the estimation errorϵi(t) is bounded; namely, ϵi⩽ϵi∗, where ϵi∗>0 is an unknown constant (i=1,2,…,n).Then the estimated error of unknown nonlinear functionh(t) can be written as(23)h^t,W-ht=h^t,W-h^t,W∗+h^t,W∗-ht=h^t,W-h^t,W∗-ϵt=W~Ttφx-ϵt,where W∗=[W1∗,W2∗,…,Wm∗]T.Based on the above discussion, the synchronization controlleru(t) can be designed as(24)ut=-Ket-WTtφx-Hsign⁡et-ρ^sign⁡et,where K=diag⁡[k1,k2,…,kn]∈Rn×n and ki>0 is the designed parameter. H=diag⁡[ϵ1∗,…,ϵn∗], ρ^=diag⁡[ρ^1,ρ^2,…,ρ^n]∈Rn×n, and ρ^i is the estimation of ρi (i=1,2,…,n). The fuzzy parameters and ρ^i(t) are, respectively, updated by(25)DtαWit=μieitφix,(26)Dtαρ^it=γieit,where μi,γi>0 are positive design parameters.Remark 7.
The above fractional adaptive laws are used to update the adjustable parameters. Notice that (26) can also be written as the following equation:(27)ρ^iν=ρ^i0+1Γα∫0νγiν-τα-1eiτdτ.Definition 8 (see [11]).
Mittag-Leffler functions (M-L functions) with one parameter and two parameters are, respectively, defined as(28)Eαt=∑k=0∞tkΓαk+1,(29)Eα,βt=∑k=0∞tkΓαk+β,where α,β>0. The Laplace transform of (29) is expressed as(30)Ltβ-1Eα,β-λtα=sα-βsα+λ,Res>λ1/α,where t≥0 and λ∈R.Lemma 9 (fractional Lyapunov second method [11]).
Let the origin be the equilibrium point of the following system:(31)Dtαxt=ft,xt,where x(t)∈Rn is the system variable and f(t,x(t))∈Rn is nonlinear function that satisfies the local Lipschitz condition. If there exists Lyapunov function V(t,x(t)) and positive parameters h1, h2, and h3 such that(32)h1xt≤Vt,xt≤h2xt,DtαVt,xt≤-h3xt,then system (31) is asymptotically stable.Lemma 10 (see [47]).
Suppose thatx(t)∈Rn is a continuously differentiable function; then one has(33)12DtαxTtxt≤xTtDtαxt.Lemma 11 (fractional monotonic principle [47]).
IfDtαx(t)≥0, then x(t) is monotonically increasing in [0,+∞). If Dtαx(t)⩽0, then x(t) is monotonically decreasing in [0,+∞].Lemma 12 (see [9, 47]).
LetV1(t)=1/2x2(t)+1/2y2(t); x(t) and y(t)∈R are two continuous functions. If there exists a positive constant k satisfying(34)DtαV1t⩽-kx2t,then one has the following inequality:(35)x2t⩽2V10Eα-2ktα.Lemma 13.
Suppose thatV(t)=1/2xT(t)x(t)+1/2yT(t)y(t), where x(t),y(t)∈Rn have continuous derivative. If there exists a constant h>0 such that(36)DtαVt⩽-hxTtxt,then x(t) and y(t) are bounded for all t>0, and x(t) converges to zero asymptotically.Proof.
From (36), we have(37)DtαV2t⩽-hxTtxt⩽0.Based on Lemma 11, we know that V2(t) is monotonically decreasing in [0,+∞]. Then(38)V2t⩽V20.Thus, x(t)⩽2V2(0) and y(t)⩽2V2(0); namely, x(t) and y(t) are bounded. Next we will prove that x(t) tends to 0 asymptotically. Taking αth integral on both sides of (37), we have(39)V2t-V20⩽-hDt-αxTtxt.Noting that V2(t)=1/2xT(t)x(t)+1/2yT(t)y(t), we have xT(t)x(t)⩽2V2(t). Consequently,(40)xTtxt⩽2V20-2hDt-αxTtxt.Thus, we can find a nonnegative function Z(t) such that(41)xTtxt+Zt=2V20-2hDt-αxTtxt.TakingLaplace transform on both sides of (41), we have(42)XTsXs=2V20sα-1sα+2h-sαsα+2hZs.According to (30), the solution of (42) is(43)xTtxt=2V20Eα,0-2htα-Zt∗t-1Eα,0-2htα,where ∗ is convolution. Because t-1 and Eα,0(-2htα) are nonnegative functions, it follows from Lemma 12 that x(t) converges to zero asymptotically. This completes the proof of Lemma 13.Remark 14.
IfxT(t)x(t)⩽2V2(0)Eα(-2htα), we know that x(t) will tend to 0 asymptotically according to the results in [48]. Namely, limt→∞x(t)=0.


### 3.3. Stability Analysis

Theorem 15.
Ifd(t)=0, we can realize the synchronization of system (12) and (13) under the effect of the adaptive controller (24) and the fractional-order adaptive law (25). And all the variables in the closed-loop system remain bounded.Proof.
Substituting the synchronization controllers (24) and (23) into the error dynamical equation (19), we have(44)Dtαet=Cet+ht-Ket-WTφx-Hsign⁡et=-K-Cet+ϵt-W~Tφx-Hsign⁡et=-Pet+ϵt-W~Tφx-Hsign⁡et,where P=K-C. We can choose an appropriate gain matrix K such that P is a positive definite matrix. Multiplying eT(t) on both sides of (44) yields(45)eTtDtαet=-eTtPet+eTtϵt-eTtW~Tφx-eTtHsign⁡et⩽-eTtPet+∑i=1neitϵi∗t-∑i=1neitW~iTφix-∑i=1neitϵi∗t⩽-eTtPet-∑i=1neitW~iTφix.Consider the following Lyapunov function:(46)Vt=12eTtet+12∑i=1n1μiW~iTW~i.Because the αth Caputo derivative of a constant is 0, we have DtαW~i=DtαWi. Taking αth derivative of V(t) based on Lemma 10 gives(47)DtαVt⩽eTtDtαet+∑i=1n1μiW~iTDtαWi.Substituting (45) and (25) into (47), we have(48)DtαVt⩽-eTtPet-∑i=1neitW~iTφix+∑i=1n1μiW~TiDtαWi⩽-eTtPet-∑i=1neitW~iTφix+∑i=1neitW~iTφix⩽-eTtPet⩽-λmineTtet,where λmin is the minimum eigenvalue of matrix P. From (48) and Lemma 13, we know that the synchronization error e(t) tends to 0 asymptotically; namely, limt→∞e(t)=0.Theorem 16.
We can realize the synchronization of system (12) and (13) under the effect of the adaptive controller (24) and the fractional-order adaptive laws (see (25) and (26)). And all variables in the closed-loop system remain bounded when d(t)≠0.Proof.
Substituting (23) and (24) into the error dynamical equation (19), we obtain(49)Dtαet=Cet+ht+dt-Ket-WTφx-Hsign⁡et-ρ^sign⁡et=-K-Cet+dt+ϵt-W~Tφx-Hsign⁡et-ρ^sign⁡et⩽-Pet+ϵt-W~Tφx-Hsign⁡et-ρ~sign⁡et,where P=K-C is a positive matrix.(50)ρ~it=ρ^it-ρi.Multiplying eT(t) on both sides of (50) gives(51)eTtDtαet=-eTtPet+eTtϵt-eTtW~Tφx-eTtHsign⁡et-eTtρ~sign⁡et⩽-eTtPet+∑i=1neitϵi∗t-∑i=1neitW~iTφix-∑i=1nϵi∗teit-∑i=1nρ~iteit⩽-eTtPet-∑i=1neitW~iTφix-∑i=1nρ~iteit.Consider the following Lyapunov function:(52)Vt=12eTtet+12∑i=1n1μiW~iTW~i+12∑i=1n1γiρ~i2t.Taking αth derivative of V(t) based on Lemma 10, we have(53)DtαVt⩽eTtDtαet+∑i=1n1μiW~iTDtαWi+∑i=1n1γiρ~itDtαρit.Substituting (45), (25), and (26) into (53), we have(54)DtαVt⩽-eTtPet-∑i=1neitW~iTφix-∑i=1nρ~iteit+∑i=1n1μiW~iTDtαWi+∑i=1n1γiρ~itDtαρit⩽-eTtPet-∑i=1neitW~iTφix-∑i=1nρ~iteit+∑i=1neitW~iTφix+∑i=1nρ~iteit⩽-eTtPet⩽-λmineTtet,where λmin is the minimum eigenvalue of matrix P. From (54) and Lemma 13, we know that the synchronization error e(t) tends to 0 asymptotically; namely, limt→∞e(t)=0.


## 3.1. Problem Statement

Consider the following fractional-order chaotic systems:(12)Dtαxt=Axt+fxt,(13)Dtαyt=Cyt+gyt+sat⁡u+dt,where x(t)=[x1(t),…,xn(t)]T∈Rn is the state vector of the drive system (12) and y(t)=[y1(t),…,yn(t)]T∈Rn is the state vector of the response system (13). fi(·) and gi(·), i=1,2,…,n, are two unknown nonlinear functions, A and C∈Rn×n are two constant matrices, d(t)∈Rn is the external disturbance, sat(u)∈Rn represents the input saturation, and u(t)∈Rn is the control input.Remark 2.
In theoretical analysis, one often hopes that the input value and the output value can keep proportionally synchronized change when the former is relatively small. However, when the input value increases to a certain extent due to the system limitation factor, the output value of the actual conditions is no longer increasing but tends to or stays at a certain value in practical systems. This is said to be “saturation” phenomenon which is shown in Figure1.Figure 1
The phenomenon of saturation.Definition 3 (see [45, 46]).
A mapping fromRn to Rnsat:ν→sat(ν) is called a saturator, where ν=[ν1,ν2,…,νn]T∈Rn and sat(ν)=[sat(ν1),sat(ν2),…,sat(νn)]T. The definition of sat(u) is as follows:(14)satut=ur,ut≥urut,ul<ut<urul,ut⩽ul.Suppose that the part that exceeds the saturation limiter is referred to as δ(t); then one has(15)δt=ur-ut,ut≥ur0,ul<ut<urul-ut,ut⩽ul,where ur and ul are called saturated amplitude satisfying ul<0 and ur>0.Assumption 4.
The external disturbancedi(·) is a bounded continuous function. Namely, there exists an unknown constant ρi>0 such that(16)di·⩽ρi,i=1,2,…,n.Remark 5.
It should be pointed out that Assumption4 is reasonable. We just need the boundaries of the external disturbances, and their exact values are not needed in the process of designing the controller.The objective of our work is to design an appropriate adaptive fuzzy controller such that the synchronization errore(t)=y(t)-x(t) tends to zero asymptotically (namely, limt→∞e(t)=0).


## 3.2. Controller Design

The dynamical equation of the synchronization error can be described as(17)Dtαet=Cet+C-Axt+gyt-fxt+satu+dt.Based on the definition of sat(u), we can obtain that(18)satu=ut+δt.Then (17) can be rewritten as(19)Dtαet=Cet+C-Axt+gyt-fxt+ut+δt+dt.Consider(20)hxt,yt=ht=C-Axt+gyt-fxt+δt.Nothing that the nonlinear functionh(t) is unknown, it can be approximated, through the fuzzy logic system (9), as(21)h^it=WiTtφix,where Wi∗=argminWi(t)[sup⁡hit-h^it,Wit]. Let the unknown constant estimation error of the fuzzy logic systems and the approximation error, respectively, be(22)W~it=Wit-Wi∗,ϵit=hit-h^it,Wi∗.The following assumption is needed in the controller design.Assumption 6.
Suppose that the estimation errorϵi(t) is bounded; namely, ϵi⩽ϵi∗, where ϵi∗>0 is an unknown constant (i=1,2,…,n).Then the estimated error of unknown nonlinear functionh(t) can be written as(23)h^t,W-ht=h^t,W-h^t,W∗+h^t,W∗-ht=h^t,W-h^t,W∗-ϵt=W~Ttφx-ϵt,where W∗=[W1∗,W2∗,…,Wm∗]T.Based on the above discussion, the synchronization controlleru(t) can be designed as(24)ut=-Ket-WTtφx-Hsign⁡et-ρ^sign⁡et,where K=diag⁡[k1,k2,…,kn]∈Rn×n and ki>0 is the designed parameter. H=diag⁡[ϵ1∗,…,ϵn∗], ρ^=diag⁡[ρ^1,ρ^2,…,ρ^n]∈Rn×n, and ρ^i is the estimation of ρi (i=1,2,…,n). The fuzzy parameters and ρ^i(t) are, respectively, updated by(25)DtαWit=μieitφix,(26)Dtαρ^it=γieit,where μi,γi>0 are positive design parameters.Remark 7.
The above fractional adaptive laws are used to update the adjustable parameters. Notice that (26) can also be written as the following equation:(27)ρ^iν=ρ^i0+1Γα∫0νγiν-τα-1eiτdτ.Definition 8 (see [11]).
Mittag-Leffler functions (M-L functions) with one parameter and two parameters are, respectively, defined as(28)Eαt=∑k=0∞tkΓαk+1,(29)Eα,βt=∑k=0∞tkΓαk+β,where α,β>0. The Laplace transform of (29) is expressed as(30)Ltβ-1Eα,β-λtα=sα-βsα+λ,Res>λ1/α,where t≥0 and λ∈R.Lemma 9 (fractional Lyapunov second method [11]).
Let the origin be the equilibrium point of the following system:(31)Dtαxt=ft,xt,where x(t)∈Rn is the system variable and f(t,x(t))∈Rn is nonlinear function that satisfies the local Lipschitz condition. If there exists Lyapunov function V(t,x(t)) and positive parameters h1, h2, and h3 such that(32)h1xt≤Vt,xt≤h2xt,DtαVt,xt≤-h3xt,then system (31) is asymptotically stable.Lemma 10 (see [47]).
Suppose thatx(t)∈Rn is a continuously differentiable function; then one has(33)12DtαxTtxt≤xTtDtαxt.Lemma 11 (fractional monotonic principle [47]).
IfDtαx(t)≥0, then x(t) is monotonically increasing in [0,+∞). If Dtαx(t)⩽0, then x(t) is monotonically decreasing in [0,+∞].Lemma 12 (see [9, 47]).
LetV1(t)=1/2x2(t)+1/2y2(t); x(t) and y(t)∈R are two continuous functions. If there exists a positive constant k satisfying(34)DtαV1t⩽-kx2t,then one has the following inequality:(35)x2t⩽2V10Eα-2ktα.Lemma 13.
Suppose thatV(t)=1/2xT(t)x(t)+1/2yT(t)y(t), where x(t),y(t)∈Rn have continuous derivative. If there exists a constant h>0 such that(36)DtαVt⩽-hxTtxt,then x(t) and y(t) are bounded for all t>0, and x(t) converges to zero asymptotically.Proof.
From (36), we have(37)DtαV2t⩽-hxTtxt⩽0.Based on Lemma 11, we know that V2(t) is monotonically decreasing in [0,+∞]. Then(38)V2t⩽V20.Thus, x(t)⩽2V2(0) and y(t)⩽2V2(0); namely, x(t) and y(t) are bounded. Next we will prove that x(t) tends to 0 asymptotically. Taking αth integral on both sides of (37), we have(39)V2t-V20⩽-hDt-αxTtxt.Noting that V2(t)=1/2xT(t)x(t)+1/2yT(t)y(t), we have xT(t)x(t)⩽2V2(t). Consequently,(40)xTtxt⩽2V20-2hDt-αxTtxt.Thus, we can find a nonnegative function Z(t) such that(41)xTtxt+Zt=2V20-2hDt-αxTtxt.TakingLaplace transform on both sides of (41), we have(42)XTsXs=2V20sα-1sα+2h-sαsα+2hZs.According to (30), the solution of (42) is(43)xTtxt=2V20Eα,0-2htα-Zt∗t-1Eα,0-2htα,where ∗ is convolution. Because t-1 and Eα,0(-2htα) are nonnegative functions, it follows from Lemma 12 that x(t) converges to zero asymptotically. This completes the proof of Lemma 13.Remark 14.
IfxT(t)x(t)⩽2V2(0)Eα(-2htα), we know that x(t) will tend to 0 asymptotically according to the results in [48]. Namely, limt→∞x(t)=0.


## 3.3. Stability Analysis

Theorem 15.
Ifd(t)=0, we can realize the synchronization of system (12) and (13) under the effect of the adaptive controller (24) and the fractional-order adaptive law (25). And all the variables in the closed-loop system remain bounded.Proof.
Substituting the synchronization controllers (24) and (23) into the error dynamical equation (19), we have(44)Dtαet=Cet+ht-Ket-WTφx-Hsign⁡et=-K-Cet+ϵt-W~Tφx-Hsign⁡et=-Pet+ϵt-W~Tφx-Hsign⁡et,where P=K-C. We can choose an appropriate gain matrix K such that P is a positive definite matrix. Multiplying eT(t) on both sides of (44) yields(45)eTtDtαet=-eTtPet+eTtϵt-eTtW~Tφx-eTtHsign⁡et⩽-eTtPet+∑i=1neitϵi∗t-∑i=1neitW~iTφix-∑i=1neitϵi∗t⩽-eTtPet-∑i=1neitW~iTφix.Consider the following Lyapunov function:(46)Vt=12eTtet+12∑i=1n1μiW~iTW~i.Because the αth Caputo derivative of a constant is 0, we have DtαW~i=DtαWi. Taking αth derivative of V(t) based on Lemma 10 gives(47)DtαVt⩽eTtDtαet+∑i=1n1μiW~iTDtαWi.Substituting (45) and (25) into (47), we have(48)DtαVt⩽-eTtPet-∑i=1neitW~iTφix+∑i=1n1μiW~TiDtαWi⩽-eTtPet-∑i=1neitW~iTφix+∑i=1neitW~iTφix⩽-eTtPet⩽-λmineTtet,where λmin is the minimum eigenvalue of matrix P. From (48) and Lemma 13, we know that the synchronization error e(t) tends to 0 asymptotically; namely, limt→∞e(t)=0.Theorem 16.
We can realize the synchronization of system (12) and (13) under the effect of the adaptive controller (24) and the fractional-order adaptive laws (see (25) and (26)). And all variables in the closed-loop system remain bounded when d(t)≠0.Proof.
Substituting (23) and (24) into the error dynamical equation (19), we obtain(49)Dtαet=Cet+ht+dt-Ket-WTφx-Hsign⁡et-ρ^sign⁡et=-K-Cet+dt+ϵt-W~Tφx-Hsign⁡et-ρ^sign⁡et⩽-Pet+ϵt-W~Tφx-Hsign⁡et-ρ~sign⁡et,where P=K-C is a positive matrix.(50)ρ~it=ρ^it-ρi.Multiplying eT(t) on both sides of (50) gives(51)eTtDtαet=-eTtPet+eTtϵt-eTtW~Tφx-eTtHsign⁡et-eTtρ~sign⁡et⩽-eTtPet+∑i=1neitϵi∗t-∑i=1neitW~iTφix-∑i=1nϵi∗teit-∑i=1nρ~iteit⩽-eTtPet-∑i=1neitW~iTφix-∑i=1nρ~iteit.Consider the following Lyapunov function:(52)Vt=12eTtet+12∑i=1n1μiW~iTW~i+12∑i=1n1γiρ~i2t.Taking αth derivative of V(t) based on Lemma 10, we have(53)DtαVt⩽eTtDtαet+∑i=1n1μiW~iTDtαWi+∑i=1n1γiρ~itDtαρit.Substituting (45), (25), and (26) into (53), we have(54)DtαVt⩽-eTtPet-∑i=1neitW~iTφix-∑i=1nρ~iteit+∑i=1n1μiW~iTDtαWi+∑i=1n1γiρ~itDtαρit⩽-eTtPet-∑i=1neitW~iTφix-∑i=1nρ~iteit+∑i=1neitW~iTφix+∑i=1nρ~iteit⩽-eTtPet⩽-λmineTtet,where λmin is the minimum eigenvalue of matrix P. From (54) and Lemma 13, we know that the synchronization error e(t) tends to 0 asymptotically; namely, limt→∞e(t)=0.


## 4. Simulation Studies

In this section, two fractional-order chaotic systems and two fractional-order hyperchaotic systems will be synchronized to show the validity of the above method.


### 4.1. Example 1

Choose the frequently used fractional-order Arneodo system [49] in literature as the drive system:(55)Dtαx1t=x2tDtαx2t=x3tDtαx3t=ax1t-bx2t-cx3t+dx13t.When a=-5.5, b=3.5, c=0.4, d=-1, and α=0.9, the fractional-order Arneodo system shows chaotic phenomenon, which is indicated in Figures 2 and 3.Figure 2
Fractional-order Arneodo system.Figure 3
The chaotic behavior of fractional-order Arneodo system inx1-x2 plane.Let the response system be the following fractional-order Lorenz system [50]:(56)Dtαy1t=σy2t-y1tDtαy2t=y1tρ-y3t-y2tDtαy3t=y1ty2t-βy3t.When σ=10, ρ=28, β=8/3, and α=0.99, the chaotic behavior of system (56) is included in Figures 4 and 5.Figure 4
Fractional-order Lorenz system.Figure 5
The chaotic behavior of fractional-order Lorenz system iny1-y2 plane.It follows from (55) and (56) that (57)A=000001a-b-cC=-σσ0ρ-1000-β.Firstly, we consider the synchronization of system (55) and system (56) in the absence of external disturbance (namely, d(t)=0). In the simulation, we choose σ=0.8, α=0.96, and k1=k2=k3=3. The initial condition of system (55) is x(0)=[-3,3.4,-4]T, and the initial condition of system (56) is y(0)=[8,-6,9]T. With respect to the fuzzy logic system, we define 6 Gaussian membership functions uniformly distributed on [-2020]. The initial condition of the fuzzy parameter is chosen as zero vector. The input saturation limiter ur=-ul=5. The simulation results are depicted in Figures 6 and 7.Figure 6
Synchronization results: (a)x1(t) (solid line) and y1(t) (dashed-dotted line); (b) x2(t) (solid line) and y2(t) (dashed-dotted line); and (c) x3(t) (solid line) and y3(t) (dashed-dotted line). (d) Synchronization errors: e1(t) (dashed line), e2(t) (dashed-dotted line), and e3(t) (solid line).
(a)
(b)
(c)
(d)Figure 7
Control inputs and parameters: (a)u1(t); (b) u2(t); (c) u3(t); and (d) W1(t) (dashed line), W2(t) (dashed-dotted line), and W3(t) (solid line).
(a)
(b)
(c)
(d)Then we study the synchronization of system (55) and system (56) with the external disturbance. In simulation, we select σ=0.8,α=0.96,andk1=k2=k3=3. The initial conditions are x(0)=[-3,3.4,-4]T and y(0)=[8,-6,9]T. Let d(t)=[0.15sin(t),0.05cos(t),0.1cos(t)]T and let the input saturation limiter be ur=-ul=5. The simulation results are as shown in Figures 8 and 9.Figure 8
Synchronization results: (a)x1(t) (solid line) and y1(t) (dashed-dotted line); (b) x2(t) (solid line) and y2(t) (dashed-dotted line); and (c) x3(t) (solid line) and y3(t) (dashed-dotted line). (d) Synchronization errors: e1(t) (dashed line), e2(t) (dashed-dotted line), and e3(t) (solid line).
(a)
(b)
(c)
(d)Figure 9
Control inputs and parameters: (a)u1(t); (b) u2(t); (c) u3(t); and (d) W1(t) (dashed line), W2(t) (dotted dashed line), and W3(t) (solid line).
(a)
(b)
(c)
(d)


### 4.2. Example 2

Choose the fractional-order hyperchaotic Lorenz system [50] as the drive system:(58)Dtαx1t=ax2t-x1t+x4tDtαx2t=cx1t-x2t-x1tx3tDtαx3t=x1tx2t-bx3tDtαx4t=-x1tx3t+γx4t.When a=10, b=8/3, c=28, γ=-1, and α=0.93, the fractional-order hyperchaotic Lorenz system (58) has chaotic attractors, as shown in Figure 10.Figure 10
Fractional-order hyperchaotic Lorenz system (58).
(a)
(b)
(c)
(d)The response system is the following fractional-order hyperchaotic system:(59)Dtαy1t=ay2t-y1tDtαy2t=dy1t-y1ty3t+cy2t-y4tDtαy3t=y1ty2t-by3tDtαy4t=y1t+k.When a=36.01, b=3.00, c=28.00, d=-16.00, k=0.50, and α=0.95, the chaotic behavior of system (59) is given in Figure 11.Figure 11
Fractional-order hyperchaotic system (59).
(a)
(b)
(c)
(d)Firstly, we consider the synchronization of system (58) and system (59) when d(t)=0. In the simulation, we choose the control parameters as σ=0.8, α=0.95, and k1=k2=k3=k4=3. The initial conditions are x(0)=[-6,2.4,-4,-6]T and y(0)=[9.6,-6,9,5]T. The input saturation limiter ur1=-ul1=12 and uri=-uli=14, i=2,3,4. The simulation results are presented in Figures 12–15.Figure 12
Synchronization results: (a)x1(t) (solid line) and y1(t) (dashed line); (b) x2(t) (solid line) and y2(t) (dashed line); (c) x3(t) (solid line) and y3(t) (dashed line); and (d) x4(t) (solid line) and y4(t) (dashed line).
(a)
(b)
(c)
(d)Figure 13
Control inputs: (a)u1(t); (b) u2(t); (c) u3(t); and (d) u4(t).
(a)
(b)
(c)
(d)Figure 14
Synchronization errors:e1(t) (dotted line), e2(t) (dotted dashed line), e3(t) (solid line), and e4(t) (dashed line).Figure 15
Parameters:W1(t) (dotted line), W2(t) (dotted dashed line), W3(t) (solid line), and W4t (dashed line).Then, we consider the synchronization of system (58) and system (59) with external disturbance. In the simulation, let σ=0.8, α=0.95, k1=k2=k3=k4=3, and d(t)=[0.15sin(t),0.05cos(t),0.1cos(t),0.5sin(t)cos(t)]T. The initial conditions are x(0)=[-3,3.4,-4,-6]T and y(0)=[9.6,-6,9,5]T. The input saturation limiter is ur1=-ul1=12 and uri=-uli=14, i=2,3,4. The simulation results are depicted in Figures 16–19.Figure 16
Synchronization results: (a)x1(t) (solid line) and y1(t) (dashed line); (b) x2(t) (solid line) and y2(t) (dashed line); (c) x3(t) (solid line) and y3(t) (dashed line); and (d) x4(t) (solid line) and y4(t) (dashed line).
(a)
(b)
(c)
(d)Figure 17
Control inputs: (a)u1(t); (b) u2(t); (c) u3(t); and (d) u4(t).
(a)
(b)
(c)
(d)Figure 18
Synchronization errors:e1(t) (dotted line), e2(t) (dot dash line), e3(t) (solid line), and e4(t) (dashed line).Figure 19
Parameters:W1(t) (dotted line), W2(t) (dotted dashed line), W3(t) (solid line), and W4(t) (dashed line).


## 4.1. Example 1

Choose the frequently used fractional-order Arneodo system [49] in literature as the drive system:(55)Dtαx1t=x2tDtαx2t=x3tDtαx3t=ax1t-bx2t-cx3t+dx13t.When a=-5.5, b=3.5, c=0.4, d=-1, and α=0.9, the fractional-order Arneodo system shows chaotic phenomenon, which is indicated in Figures 2 and 3.Figure 2
Fractional-order Arneodo system.Figure 3
The chaotic behavior of fractional-order Arneodo system inx1-x2 plane.Let the response system be the following fractional-order Lorenz system [50]:(56)Dtαy1t=σy2t-y1tDtαy2t=y1tρ-y3t-y2tDtαy3t=y1ty2t-βy3t.When σ=10, ρ=28, β=8/3, and α=0.99, the chaotic behavior of system (56) is included in Figures 4 and 5.Figure 4
Fractional-order Lorenz system.Figure 5
The chaotic behavior of fractional-order Lorenz system iny1-y2 plane.It follows from (55) and (56) that (57)A=000001a-b-cC=-σσ0ρ-1000-β.Firstly, we consider the synchronization of system (55) and system (56) in the absence of external disturbance (namely, d(t)=0). In the simulation, we choose σ=0.8, α=0.96, and k1=k2=k3=3. The initial condition of system (55) is x(0)=[-3,3.4,-4]T, and the initial condition of system (56) is y(0)=[8,-6,9]T. With respect to the fuzzy logic system, we define 6 Gaussian membership functions uniformly distributed on [-2020]. The initial condition of the fuzzy parameter is chosen as zero vector. The input saturation limiter ur=-ul=5. The simulation results are depicted in Figures 6 and 7.Figure 6
Synchronization results: (a)x1(t) (solid line) and y1(t) (dashed-dotted line); (b) x2(t) (solid line) and y2(t) (dashed-dotted line); and (c) x3(t) (solid line) and y3(t) (dashed-dotted line). (d) Synchronization errors: e1(t) (dashed line), e2(t) (dashed-dotted line), and e3(t) (solid line).
(a)
(b)
(c)
(d)Figure 7
Control inputs and parameters: (a)u1(t); (b) u2(t); (c) u3(t); and (d) W1(t) (dashed line), W2(t) (dashed-dotted line), and W3(t) (solid line).
(a)
(b)
(c)
(d)Then we study the synchronization of system (55) and system (56) with the external disturbance. In simulation, we select σ=0.8,α=0.96,andk1=k2=k3=3. The initial conditions are x(0)=[-3,3.4,-4]T and y(0)=[8,-6,9]T. Let d(t)=[0.15sin(t),0.05cos(t),0.1cos(t)]T and let the input saturation limiter be ur=-ul=5. The simulation results are as shown in Figures 8 and 9.Figure 8
Synchronization results: (a)x1(t) (solid line) and y1(t) (dashed-dotted line); (b) x2(t) (solid line) and y2(t) (dashed-dotted line); and (c) x3(t) (solid line) and y3(t) (dashed-dotted line). (d) Synchronization errors: e1(t) (dashed line), e2(t) (dashed-dotted line), and e3(t) (solid line).
(a)
(b)
(c)
(d)Figure 9
Control inputs and parameters: (a)u1(t); (b) u2(t); (c) u3(t); and (d) W1(t) (dashed line), W2(t) (dotted dashed line), and W3(t) (solid line).
(a)
(b)
(c)
(d)


## 4.2. Example 2

Choose the fractional-order hyperchaotic Lorenz system [50] as the drive system:(58)Dtαx1t=ax2t-x1t+x4tDtαx2t=cx1t-x2t-x1tx3tDtαx3t=x1tx2t-bx3tDtαx4t=-x1tx3t+γx4t.When a=10, b=8/3, c=28, γ=-1, and α=0.93, the fractional-order hyperchaotic Lorenz system (58) has chaotic attractors, as shown in Figure 10.Figure 10
Fractional-order hyperchaotic Lorenz system (58).
(a)
(b)
(c)
(d)The response system is the following fractional-order hyperchaotic system:(59)Dtαy1t=ay2t-y1tDtαy2t=dy1t-y1ty3t+cy2t-y4tDtαy3t=y1ty2t-by3tDtαy4t=y1t+k.When a=36.01, b=3.00, c=28.00, d=-16.00, k=0.50, and α=0.95, the chaotic behavior of system (59) is given in Figure 11.Figure 11
Fractional-order hyperchaotic system (59).
(a)
(b)
(c)
(d)Firstly, we consider the synchronization of system (58) and system (59) when d(t)=0. In the simulation, we choose the control parameters as σ=0.8, α=0.95, and k1=k2=k3=k4=3. The initial conditions are x(0)=[-6,2.4,-4,-6]T and y(0)=[9.6,-6,9,5]T. The input saturation limiter ur1=-ul1=12 and uri=-uli=14, i=2,3,4. The simulation results are presented in Figures 12–15.Figure 12
Synchronization results: (a)x1(t) (solid line) and y1(t) (dashed line); (b) x2(t) (solid line) and y2(t) (dashed line); (c) x3(t) (solid line) and y3(t) (dashed line); and (d) x4(t) (solid line) and y4(t) (dashed line).
(a)
(b)
(c)
(d)Figure 13
Control inputs: (a)u1(t); (b) u2(t); (c) u3(t); and (d) u4(t).
(a)
(b)
(c)
(d)Figure 14
Synchronization errors:e1(t) (dotted line), e2(t) (dotted dashed line), e3(t) (solid line), and e4(t) (dashed line).Figure 15
Parameters:W1(t) (dotted line), W2(t) (dotted dashed line), W3(t) (solid line), and W4t (dashed line).Then, we consider the synchronization of system (58) and system (59) with external disturbance. In the simulation, let σ=0.8, α=0.95, k1=k2=k3=k4=3, and d(t)=[0.15sin(t),0.05cos(t),0.1cos(t),0.5sin(t)cos(t)]T. The initial conditions are x(0)=[-3,3.4,-4,-6]T and y(0)=[9.6,-6,9,5]T. The input saturation limiter is ur1=-ul1=12 and uri=-uli=14, i=2,3,4. The simulation results are depicted in Figures 16–19.Figure 16
Synchronization results: (a)x1(t) (solid line) and y1(t) (dashed line); (b) x2(t) (solid line) and y2(t) (dashed line); (c) x3(t) (solid line) and y3(t) (dashed line); and (d) x4(t) (solid line) and y4(t) (dashed line).
(a)
(b)
(c)
(d)Figure 17
Control inputs: (a)u1(t); (b) u2(t); (c) u3(t); and (d) u4(t).
(a)
(b)
(c)
(d)Figure 18
Synchronization errors:e1(t) (dotted line), e2(t) (dot dash line), e3(t) (solid line), and e4(t) (dashed line).Figure 19
Parameters:W1(t) (dotted line), W2(t) (dotted dashed line), W3(t) (solid line), and W4(t) (dashed line).


## 5. Conclusions

In this paper, we investigate the synchronization for two uncertain fractional-order nonlinear chaotic systems with saturated input and external disturbances in accordance with the fractional Lyapunov stability theory and adaptive fuzzy control method. A fractional-order adaptive controller that can guarantee that the synchronization error tends to a small region of origin and fractional parameters adaptive laws are designed. Based on the proposed method, we can achieve synchronization of many fractional-order chaotic systems and hyperchaotic systems. It should be mentioned that the proposed controller can guarantee that the synchronization errors converge to a small region of the origin eventually. How to design an adaptive fuzzy controller such that the dynamical system is asymptotic stable is one of our future research directions.



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