# Synchronization of Different Uncertain Fractional-Order Chaotic Systems with External Disturbances via T-S Fuzzy Model

Authors: Lin Wang; Jinbo Ni; Chunzhi Yang
Journal: Journal of Function Spaces
 (2018)

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## Abstract

This paper presents an adaptive fuzzy synchronization control strategy for a class of different uncertain fractional-order chaotic/hyperchaotic systems with unknown external disturbances via T-S fuzzy systems, where the parallel distributed compensation technology is provided to design adaptive controller with fractional adaptation laws. T-S fuzzy models are employed to approximate the unknown nonlinear systems and tracking error signals are used to update the parametric estimates. The asymptotic stability of the closed-loop system and the boundedness of the states and parameters are guaranteed by fractional Lyapunov theory. This approach is also valid for synchronization of fractional-order chaotic systems with the same system structure. One constructive example is given to verify the feasibility and superiority of the proposed method.


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## Body

## 1. Introduction

Fractional calculus is a mathematical topic being more than 300 years old, which can be traced back to the birth of integer-order calculus. The fundamentals results of fractional calculus were concluded in [1]. At present, researchers found that fractional differential equations not only improve the veracity in modeling physical systems but also generate a lot of applications in physics, electrical engineering, robotics, control systems, and chemical mixing [2–11]. In addition, the chaotic behavior has been discovered in many fractional-order systems, for instance, the fractional-order Chen’s system, the fractional-order Chua’s system, and the fractional-order Liu system. In view of chaotic potential value in control systems and secure communication [12], chaos synchronization was studied by more and more researches [13, 14].The conventional nonlinear systems control approaches suffer from discontented performance resulting from structure and parametric uncertainties, external disturbances. Usually, it is very hard to provide accurate mathematical models [15–25]. To control these uncertain systems, adaptive fuzzy/neural-network control was proposed [26, 27]. This method is effective and superior for handling parametric and structure uncertainties, external disturbances in integer-order nonlinear systems [28, 29], where tracking error is developed to update adjusted parameters and fuzzy logic systems or neural networks are introduced to model unknown physical systems as well as to approximate unknown nonlinear functions. There are two types of fuzzy logic systems: Mamdani type and T-S type. T-S fuzzy logic system is first proposed by Takagi and Sugeno [30]. Subsequently, many works found that T-S fuzzy systems can uniformly approximate any continuous functions on a compact set with random accuracy based on the Weierstrass approximation theorem [31]. Moreover, it was also shown that the approximation ability of T-S fuzzy systems was better than the Mamdani fuzzy systems [32]. Therefore, many studies focused on the chaos synchronization of fractional-order chaotic systems via T-S fuzzy models. For example, synchronization of fractional-order modified chaotic system via new linear control, backstepping control, and T-S fuzzy approaches was investigated in [33]. Impulsive control for fractional-order chaotic system was presented in [34]. Other results about the synchronization of a fractional-order chaotic system via T-S fuzzy approaches can be found in [35, 36]. However, only chaos synchronization of fractional-order nonlinear systems with same structure based on T-S fuzzy systems is considered in above previous works.This work investigates the chaos synchronization of fractional-order chaotic systems with different structures based on T-S fuzzy systems, where external disturbances in slaves system are considered. T-S fuzzy systems with random rule consequents are introduced to model controlled systems, whereas T-S fuzzy systems that have the same rule consequents with Mamdani fuzzy systems are used to approximate unknown nonlinear functions. The asymptotic stability of closed-loop system is proofed based on fractional Lyapunov stability theory. Compared to previous literature, the main contributions of this paper are as follows:( 1 ) This paper first considers the chaos synchronization of the master system and slave system with different structure based on T-S fuzzy systems, and the external disturbances are assumed to be unknown. The required knowledge of the disturbances is weaker than above previous works, for example, in [34–36]. In these works, the external disturbances are assumed to be bounded with known upper bounds. However, in our control method, we do not need to know the exact value of the upper bounds of external disturbances.( 2 ) T-S fuzzy logic systems are used to model the controlled system and the final outputs of system can be obtained. By combining the adaptive fuzzy control method and parallel distributed compensation technique, an adaptive controller with fractional-order laws is designed. The proposed method is superior to some works based on linear matrix inequality (LMI) and modified LMI [37].


## 2. Fundamentals of Fractional Calculus and Fuzzy Logic Systems

### 2.1. Fractional Calculus

There are two frequently used definitions for fractional integration and differentiation: Riemann-Liouville (denote R-L) and Caputo definitions. In this paper, we will consider Caputo’s definition, whose initial conditions are as the same form of the integer-order one [38–40]. The fractional integral is designed as [1](1)0CDt-μut=1Γμ∫0tuξt-ξ1-μdξ,where μ>0, n-1≤μ<n, and Γ(·) is Euler’s Gamma function, which is defined as Γ(s)=∫0∞ts-1e-tdt. The fractional derivative operator is given as(2)0CDtμut=1Γn-μ∫0tunξt-ξμ+1-ndξ.Some useful properties of fractional calculus that will be used in the controller design are listed as follows.Property 1 (see [1, 41–44]).
Caputo’s fractional derivative and integral are linear operations withλ1,λ2,μ∈R(3)0CDtμλ1u1t+λ2u2t=λ10CDtμu1t+λ20CDtμu2t.Property 2.
Letx(t)∈Cn[0,T]. Then we have(4)0CDt-μ0CDtμut=ut-∑k=0n-1uk0k!tk.Property 3 (see [1, 45–47]).
The Laplace transform of (2) is(5)L0CDtμft=sμFs-∑k=0n-1sμ-k-1fk0, with F(s)=L[f(t)].Definition 4.
The two-parameter Mittag-Leffler function was defined by [1](6)Eα,βz=∑k=0+∞zkΓkα+β, with α,β>0 and z∈C. The Laplace transform of the Mittag-Leffler function is given as(7)Ltβ-1Eα,β-atα=sα-βsα+a.In the subsequent paper, we only consider the case thatμ∈(0,1).


### 2.2. Takagi-Sugeno Fuzzy Logic Systems

Unlike the Mamdani fuzzy logic systems, the ith rule of a Multi-Input and Multioutput general fractional-order Takagi-Sugeno (T-S) fuzzy systems can be expressed as follows(i=1,2,⋯,N):Ri: If x1(t) is F1i and ⋯ and xn(t) is Fni, then 0CDtμx(t)=fi(t,x(t)),withFji∈R,j=1,2,⋯,n are fuzzy sets, x(t)=(x1(t),x2(t),⋯,xn(t))T∈Rn is a state vector, and fi(t,x(t)) is a random function. In this paper, singleton fuzzification, center average defuzzification, and product inference are adopted and a general fractional-order T-S fuzzy system can be rewritten in the form(8)0CDtμxt=∑i=1Nμixtfit,xt, where μi(x(t))=∏j=1nFji(xj(t))/∑i=1N(∏j=1nFji(xj(t))) satisfying ∑i=1Nμi(x(t))=1 and μi(x(t))≥0.Depending on the above statements, a main difference of Mamdani fuzzy logic systems and T-S fuzzy systems is that the rule consequents are functions for T-S fuzzy system whereas the rule consequents are fuzzy sets for Mamdani fuzzy logic systems. Moreover, the T-S fuzzy logic systems are also universal approximators [31].


## 2.1. Fractional Calculus

There are two frequently used definitions for fractional integration and differentiation: Riemann-Liouville (denote R-L) and Caputo definitions. In this paper, we will consider Caputo’s definition, whose initial conditions are as the same form of the integer-order one [38–40]. The fractional integral is designed as [1](1)0CDt-μut=1Γμ∫0tuξt-ξ1-μdξ,where μ>0, n-1≤μ<n, and Γ(·) is Euler’s Gamma function, which is defined as Γ(s)=∫0∞ts-1e-tdt. The fractional derivative operator is given as(2)0CDtμut=1Γn-μ∫0tunξt-ξμ+1-ndξ.Some useful properties of fractional calculus that will be used in the controller design are listed as follows.Property 1 (see [1, 41–44]).
Caputo’s fractional derivative and integral are linear operations withλ1,λ2,μ∈R(3)0CDtμλ1u1t+λ2u2t=λ10CDtμu1t+λ20CDtμu2t.Property 2.
Letx(t)∈Cn[0,T]. Then we have(4)0CDt-μ0CDtμut=ut-∑k=0n-1uk0k!tk.Property 3 (see [1, 45–47]).
The Laplace transform of (2) is(5)L0CDtμft=sμFs-∑k=0n-1sμ-k-1fk0, with F(s)=L[f(t)].Definition 4.
The two-parameter Mittag-Leffler function was defined by [1](6)Eα,βz=∑k=0+∞zkΓkα+β, with α,β>0 and z∈C. The Laplace transform of the Mittag-Leffler function is given as(7)Ltβ-1Eα,β-atα=sα-βsα+a.In the subsequent paper, we only consider the case thatμ∈(0,1).


## 2.2. Takagi-Sugeno Fuzzy Logic Systems

Unlike the Mamdani fuzzy logic systems, the ith rule of a Multi-Input and Multioutput general fractional-order Takagi-Sugeno (T-S) fuzzy systems can be expressed as follows(i=1,2,⋯,N):Ri: If x1(t) is F1i and ⋯ and xn(t) is Fni, then 0CDtμx(t)=fi(t,x(t)),withFji∈R,j=1,2,⋯,n are fuzzy sets, x(t)=(x1(t),x2(t),⋯,xn(t))T∈Rn is a state vector, and fi(t,x(t)) is a random function. In this paper, singleton fuzzification, center average defuzzification, and product inference are adopted and a general fractional-order T-S fuzzy system can be rewritten in the form(8)0CDtμxt=∑i=1Nμixtfit,xt, where μi(x(t))=∏j=1nFji(xj(t))/∑i=1N(∏j=1nFji(xj(t))) satisfying ∑i=1Nμi(x(t))=1 and μi(x(t))≥0.Depending on the above statements, a main difference of Mamdani fuzzy logic systems and T-S fuzzy systems is that the rule consequents are functions for T-S fuzzy system whereas the rule consequents are fuzzy sets for Mamdani fuzzy logic systems. Moreover, the T-S fuzzy logic systems are also universal approximators [31].


## 3. Adaptive Fuzzy Synchronization Control

### 3.1. Problem Statement

Consider the following fractional-order chaotic system as the master system via T-S type fuzzy systems. Theith rule can be expressed as (i=1,2,⋯,N)Ri: If x1(t) is F1i and ⋯ and xn(t) is Fni, then 0CDtμx(t)=Aix(t)+b1,whereAi is a constant matrix, x(t)∈D1⊆Rn is the state vector (D1 is a compact set), b1 is a constant vector, and Fji,j=1,2,⋯,n, are fuzzy sets. Hence, the final output of master system can be rewritten as(9)0CDtμxt=∑i=1NμixtAixt+b1,with μi(x(t))=∏j=1nFji(xj(t))/∑i=1N∏j=1nFji(xj(t)) satisfying μi(x(t))≥0 and ∑i=1Nμi(x(t))=1.Consider the following fractional-order chaotic system with external disturbances in the equation as the slave system based on T-S fuzzy models. Theith rule can be written in the following form (i=1,⋯,N):Ri: If y1(t) is F^1i and y2(t) is F^2i and ⋯ and yn(t) is F^ni, then 0CDtμy(t)=Biy(t)+b2+u(t)+di(t,y),whereBi is a constant matrix, y(t)∈D2⊆Rn is the state vector (D2 is a compact set), b2 is a constant vector, F^ji,j=1,2,⋯,n, are fuzzy sets, u(t)∈R is control input, and di(t,y)∈Rn are unknown external disturbances. Hence, the final output of slave system can be obtained as(10)0CDtμyt=∑i=1NμiytBiyt+b2+ut+dit,y,with μi(y(t))=∏j=1nF^ji(yj(t))/∑i=1N∏j=1nF^ji(yj(t)) satisfying μi(y(t))≥0 and ∑i=1Nμi(y(t))=1.The control objective of this work is to design a proper adaptive controlleru(t) to synchronize the above chaotic systems (9) and (10) with the tracking error signal(11)et=yt-xtasymptotically converging to zero with random accuracy, that is, limt→+∞e(t)=0. The norm adopts Euclid norm in this paper. In addition, all states and parameters in the closed-loop system are bounded. The following assumptions are necessary.Assumption 5.
The structure of master system (9) and slave system (10) is different. The parameters and the structure of the master system are complete unknown or partial unknown, but the parameters and structure of the slave system are known.Assumption 6.
The unknown disturbancesdi(t,y)=(di1(t,y),⋯,din(t,y))T, (i=1,2,⋯,N) satisfying dijt,y≤ρij(y) with ρij(y) being a continuous function, where ρij(y) is the estimated value of the observed value for dijt,y, for all y∈D2.Remark 7.
It is worth pointing out that Assumptions5 and 6 are rational. Due to the boundedness of chaos systems, we assume that D1 and D2 are compact sets. Since di(t,y) are unknown external disturbances and may be not continuous, they are assumed to be unknown measurable nonlinear functions. The slave systems and the controller lie on the receiving terminal; hence, the parameters and the structure of the master system may be complete unknown or partial unknown, but the parameters and structure of the slave system are known.


### 3.2. Control Design

The synchronization error dynamic equation can be obtained from (11) as(12)0CDtμet=0CDtμyt-0CDtμxt=∑i=1NμiytBiyt-∑i=1NμixtAixt+m+ut+∑i=1Nμiytdit,y,with m=b2-b1 being a constant vector.Based on T-S fuzzy logic system universal approximation theorem, T-S fuzzy systemsρ^ij(y,θij(t)) that have the same rule consequents with the Mamdani type fuzzy logic systems are used to approximate to ρij(y) in the Assumption 6, j∈1,2,⋯,n and i∈1,2,⋯,N, where θij(t) are adjusted parameters in fuzzy systems. Denote ρ^i(y,θi(t))=(ρ^i1(y,θi1(t)).⋯,ρ^in(y,θin(t)))T. Using [48–50], we obtain the ideal parameter as(13)θi∗j=argminθijt∈Rsupt≥0ρ^ijy,θijt-ρijy. Then we obtain the optimal parameter vector as θi∗=(θi∗1,⋯,θi∗n)T. Hence, ρ^ij(y,θi∗j) is the ideal approximator of ρij(y); that is, ρ^i(y,θi∗) is the ideal approximator of ρi(y). The minimum approximation errors and the ideal parameter errors of the fuzzy systems are defined as (j=1,2,⋯,n)(14)εijy=ρijy-ρ^ijy,θi∗j,(15)θ^ijt=θijt-θi∗j. According to [29, 51, 52], the approximation errors εij(y),j=1,2,⋯,n) are assumed to be bounded, that is, εijy≤εi∗j with the εi∗j being constants and ε^i∗j(y) being the estimate value of εi∗j. Thus, from the above analysis, we can obtain the equations ρ^i(y,θi(t))=θiT(t)φi(y) and ρ^i(y,θi∗)=θi∗Tφi(y), where φi(y) is fuzzy base functions. Denoting θ~i(t)=θi(t)-θi∗ and εi(y)=(εi1(y),⋯,εin(y))T, one has(16)ρ^iy,θit-ρiy=ρ^iy,θit-ρ^iy,θi∗+ρ^iy,θi∗-ρiy=ρ^iy,θit-ρ^iy,θi∗-εiy=θ~iTtφiy-εiy.Remark 8.
As shown in [53], if the rule consequences of T-S fuzzy systems have the same form with the rule consequences of Mamdani type logic systems, then T-S type is equivalent to Mamdani type fuzzy system.Based on above discussion, the controller is designed with the fuzzy systemρ^i(y,θi(t)) as well as the estimate value ε^i∗j(y) as(17)ut=udt+u~t+u1t=-∑i=1NμiytKet+Biyt+Hisignet+Tρ^iy,θit+m+∑i=1NμixtAixt,where the ith rule of ud(t) and u~(t) can be written as follows, respectively, (i=1,2,⋯,N):Ri: If y1(t) is F^1i and y2(t) is F^2i and ⋯ and yn(t) is F^ni, then ud(t)=-Ke(t)-Biy(t)-m, with K being an adjusted control gain matrix.Ri: If x1(t) is F1i and ⋯ and xn(t) is Fni, then u~(t)=Aix(t).Let us denoteu1(t)=-∑i=1Nμi(y(t))[Tρ^i(y,θi(t))+Hisign(e(t))], where T=diag[sign(e1(t)),⋯,sign(en(t))],Hi=diag[ε^i∗1(y),⋯,ε^i∗n(y)].In order to update parametric estimates, the fractional adaptation laws are designed as(18)0CDtμθ~ijt=αiμiytejtφijy,(19)0CDtμε~i∗jy=βiμiytejt,with αi,βi>0 being adaptation rates which are constant parameters. Taking the control law (17) into (12) and letting K=diag[k1,⋯,kn](kj>0), we have(20)0CDtμet=-Ket+∑i=1Nμiytdit,y-Tρ^iy,θit-Hisignet.Multiplying both sides of (20) by eT(t) and letting ε~i∗j(y)=ε^i∗j(y)-εi∗j, one gets(21)eTt0CDtμet≤-eTtKet-∑i=1Nμiyt∑j=1nejtε^i∗jy+ρ^ijy,θijt-ρijy≤-eTtKet+∑i=1Nμiyt∑j=1nejtεi∗j-ε^i∗jy-θ~ijtφijy=-eTtKet-∑i=1Nμiyt∑j=1nejtε~i∗jy+θ~ijtφijy.


### 3.3. Stability Analysis

Here, fractional Lyapunov’s theory is used to analyze the stability in closed-loop system. The following Lemmas are proposed to simplify the stability analysis.Lemma 9 (see [54]).
Ifμ∈(0,1) and x(t)∈C1, then 0CDtμx(t)Tx(t)≤2x(t)T0CDtμx(t).Lemma 10.
If0CDtμx(t)≤0, then one gets x(t)≤x(0); if 0CDtμx(t)≥0, then one gets x(t)≥x(0), with μ∈(0,1) and t∈[0,+∞).Proof.
We only consider the front part. If0CDtμx(t)≤0, let(22)ht=-0CDtμxt.Both sides of (22) take Laplace transform and one obtains(23)L0CDtμxt+Hs=0. Using Property 3, one obtains the following:(24)sμXs-sμ-1x0+Hs=0. Further, one gets(25)Xs=x0s-Hssμ,with X(s)=L[x(t)],H(s)=L[h(t)]. Both sides of (25) make Laplace inverse transform and using the fractional integral definition, one obtains(26)xt=x0-0CDt-μht=x0-1Γμ∫0tt-ξμ-1hξdξ. From the above equation, we get x(t)≤x(0),t∈[0,+∞).Lemma 11.
LetV(t)=(1/2)x(t)Tx(t)+(1/2)y(t)Ty(t) with x(t),y(t)∈Rn be continuous and derivable functions. If there exists a constant h>0 such that(27)0CDtμVt≤-hxtTxt,then x(t) and y(t) are bounded and limt→+∞x(t)=0.Proof.
According to Lemma10 and 0CDtμV(t)≤-hx(t)Tx(t)≤0, we obtain V(t)≤V(0). Further, we get the following:(28)xt≤2V0,yt≤2V0. This means that x(t) and y(t) are bounded.
We will proof thatx(t) tends to 0 asymptotically below. Both sides of (27) commute with μ-order integral; based on Property 2, one gets(29)Vt-V0≤-h0CDt-μxtTxt.Further, one obtains(30)xtTxt≤2V0-2h0CDt-μxtTxt.Hence, the following can be obtained from (30) with a nonnegative function z(t) as(31)xtTxt+zt=2V0-2h0CDt-μxtTxt.Applying the Laplace transform to formula (31) and according to the Definition 4, we have(32)LxtTxt=2V0sμ-1sμ+2h-Zssμsμ+2h=L2V0Eμ,1-2htμ-LztLt-1Eμ,0-2htμ.Hence, using the Laplace inverse transform to (32), we have(33)xtTxt=2V0Eμ,1-2htμ-zt∗t-1Eμ,0-2htμ, with ∗ being the convolution operator. Since t-1 and Eμ,0(-2htμ) are nonnegative functions, then x(t)Tx(t)≤2V(0)Eμ,1(-2htμ). According to the results in [55], one obtains that x(t) is M-L stability and x(t) tends to 0 asymptotically (namely, limt→∞x(t)=0).From above discussion, the boundedness of all signals in closed-loop system and the convergence of tracking error based on adaptive fuzzy control scheme via T-S fuzzy logic systems is presented in the following theorem.Theorem 12.
For the master system (9) and slave system (10) under the known initial conditions, if Assumptions 5 and 6 are satisfied and the adaptive controller is given as (17) with the fractional adaptation laws (18) and (19), then all signals in the closed-loop system are bounded and the tracking error signal tends to zero asymptotically.Proof.
Define the following Lyapunov function:(34)Vt=12eTtet+∑i=1N12αiθ~iTtθ~it+∑i=1N12βiε~i∗Tyε~i∗y, with θ~i(t)=θi(t)-θi∗ and ε~i∗(y)=ε^i∗(y)-εi∗. Hence, using the Lemma 9, the μ-order derivative of V(t) with respect to time t is obtained as(35)0CDtμVt≤eTt0CDtμet+∑i=1N1αiθ~iTt0CDtμθ~it+∑i=1N1βiε~i∗Ty0CDtμε~i∗ySubstituting (21) into (35), one gets(36)0CDtμVt≤-eTtKet+∑i=1N∑j=1n1αiθ~ijt0CDtμθ~ijt-μiytejtθ~ijtφijy+1βiε~i∗jy0CDtμε~i∗jy-μiytejtε~i∗jy,Taking (18) and (19) into (36), one gets the following inequality:(37)0CDtμV≤-eTtKet≤-λmineTtet, where λmin is the least eigenvalue of the positive definite matrix K. According to Lemma 11 and above discussion, we know that the tracking error signal e(t) tends to 0 asymptotically (that is, limt→∞e(t)=0) and θ~i(t) and ε~i∗(y) are bounded. Further, it means that θi(t) and ε^i∗(y) are bounded. Because of the boundedness of e(t) and x(t), we know that y(t) is bounded. Based on the control design, u(t) is bounded. Therefore, we know that all signals in the closed-loop system are bounded.


## 3.1. Problem Statement

Consider the following fractional-order chaotic system as the master system via T-S type fuzzy systems. Theith rule can be expressed as (i=1,2,⋯,N)Ri: If x1(t) is F1i and ⋯ and xn(t) is Fni, then 0CDtμx(t)=Aix(t)+b1,whereAi is a constant matrix, x(t)∈D1⊆Rn is the state vector (D1 is a compact set), b1 is a constant vector, and Fji,j=1,2,⋯,n, are fuzzy sets. Hence, the final output of master system can be rewritten as(9)0CDtμxt=∑i=1NμixtAixt+b1,with μi(x(t))=∏j=1nFji(xj(t))/∑i=1N∏j=1nFji(xj(t)) satisfying μi(x(t))≥0 and ∑i=1Nμi(x(t))=1.Consider the following fractional-order chaotic system with external disturbances in the equation as the slave system based on T-S fuzzy models. Theith rule can be written in the following form (i=1,⋯,N):Ri: If y1(t) is F^1i and y2(t) is F^2i and ⋯ and yn(t) is F^ni, then 0CDtμy(t)=Biy(t)+b2+u(t)+di(t,y),whereBi is a constant matrix, y(t)∈D2⊆Rn is the state vector (D2 is a compact set), b2 is a constant vector, F^ji,j=1,2,⋯,n, are fuzzy sets, u(t)∈R is control input, and di(t,y)∈Rn are unknown external disturbances. Hence, the final output of slave system can be obtained as(10)0CDtμyt=∑i=1NμiytBiyt+b2+ut+dit,y,with μi(y(t))=∏j=1nF^ji(yj(t))/∑i=1N∏j=1nF^ji(yj(t)) satisfying μi(y(t))≥0 and ∑i=1Nμi(y(t))=1.The control objective of this work is to design a proper adaptive controlleru(t) to synchronize the above chaotic systems (9) and (10) with the tracking error signal(11)et=yt-xtasymptotically converging to zero with random accuracy, that is, limt→+∞e(t)=0. The norm adopts Euclid norm in this paper. In addition, all states and parameters in the closed-loop system are bounded. The following assumptions are necessary.Assumption 5.
The structure of master system (9) and slave system (10) is different. The parameters and the structure of the master system are complete unknown or partial unknown, but the parameters and structure of the slave system are known.Assumption 6.
The unknown disturbancesdi(t,y)=(di1(t,y),⋯,din(t,y))T, (i=1,2,⋯,N) satisfying dijt,y≤ρij(y) with ρij(y) being a continuous function, where ρij(y) is the estimated value of the observed value for dijt,y, for all y∈D2.Remark 7.
It is worth pointing out that Assumptions5 and 6 are rational. Due to the boundedness of chaos systems, we assume that D1 and D2 are compact sets. Since di(t,y) are unknown external disturbances and may be not continuous, they are assumed to be unknown measurable nonlinear functions. The slave systems and the controller lie on the receiving terminal; hence, the parameters and the structure of the master system may be complete unknown or partial unknown, but the parameters and structure of the slave system are known.


## 3.2. Control Design

The synchronization error dynamic equation can be obtained from (11) as(12)0CDtμet=0CDtμyt-0CDtμxt=∑i=1NμiytBiyt-∑i=1NμixtAixt+m+ut+∑i=1Nμiytdit,y,with m=b2-b1 being a constant vector.Based on T-S fuzzy logic system universal approximation theorem, T-S fuzzy systemsρ^ij(y,θij(t)) that have the same rule consequents with the Mamdani type fuzzy logic systems are used to approximate to ρij(y) in the Assumption 6, j∈1,2,⋯,n and i∈1,2,⋯,N, where θij(t) are adjusted parameters in fuzzy systems. Denote ρ^i(y,θi(t))=(ρ^i1(y,θi1(t)).⋯,ρ^in(y,θin(t)))T. Using [48–50], we obtain the ideal parameter as(13)θi∗j=argminθijt∈Rsupt≥0ρ^ijy,θijt-ρijy. Then we obtain the optimal parameter vector as θi∗=(θi∗1,⋯,θi∗n)T. Hence, ρ^ij(y,θi∗j) is the ideal approximator of ρij(y); that is, ρ^i(y,θi∗) is the ideal approximator of ρi(y). The minimum approximation errors and the ideal parameter errors of the fuzzy systems are defined as (j=1,2,⋯,n)(14)εijy=ρijy-ρ^ijy,θi∗j,(15)θ^ijt=θijt-θi∗j. According to [29, 51, 52], the approximation errors εij(y),j=1,2,⋯,n) are assumed to be bounded, that is, εijy≤εi∗j with the εi∗j being constants and ε^i∗j(y) being the estimate value of εi∗j. Thus, from the above analysis, we can obtain the equations ρ^i(y,θi(t))=θiT(t)φi(y) and ρ^i(y,θi∗)=θi∗Tφi(y), where φi(y) is fuzzy base functions. Denoting θ~i(t)=θi(t)-θi∗ and εi(y)=(εi1(y),⋯,εin(y))T, one has(16)ρ^iy,θit-ρiy=ρ^iy,θit-ρ^iy,θi∗+ρ^iy,θi∗-ρiy=ρ^iy,θit-ρ^iy,θi∗-εiy=θ~iTtφiy-εiy.Remark 8.
As shown in [53], if the rule consequences of T-S fuzzy systems have the same form with the rule consequences of Mamdani type logic systems, then T-S type is equivalent to Mamdani type fuzzy system.Based on above discussion, the controller is designed with the fuzzy systemρ^i(y,θi(t)) as well as the estimate value ε^i∗j(y) as(17)ut=udt+u~t+u1t=-∑i=1NμiytKet+Biyt+Hisignet+Tρ^iy,θit+m+∑i=1NμixtAixt,where the ith rule of ud(t) and u~(t) can be written as follows, respectively, (i=1,2,⋯,N):Ri: If y1(t) is F^1i and y2(t) is F^2i and ⋯ and yn(t) is F^ni, then ud(t)=-Ke(t)-Biy(t)-m, with K being an adjusted control gain matrix.Ri: If x1(t) is F1i and ⋯ and xn(t) is Fni, then u~(t)=Aix(t).Let us denoteu1(t)=-∑i=1Nμi(y(t))[Tρ^i(y,θi(t))+Hisign(e(t))], where T=diag[sign(e1(t)),⋯,sign(en(t))],Hi=diag[ε^i∗1(y),⋯,ε^i∗n(y)].In order to update parametric estimates, the fractional adaptation laws are designed as(18)0CDtμθ~ijt=αiμiytejtφijy,(19)0CDtμε~i∗jy=βiμiytejt,with αi,βi>0 being adaptation rates which are constant parameters. Taking the control law (17) into (12) and letting K=diag[k1,⋯,kn](kj>0), we have(20)0CDtμet=-Ket+∑i=1Nμiytdit,y-Tρ^iy,θit-Hisignet.Multiplying both sides of (20) by eT(t) and letting ε~i∗j(y)=ε^i∗j(y)-εi∗j, one gets(21)eTt0CDtμet≤-eTtKet-∑i=1Nμiyt∑j=1nejtε^i∗jy+ρ^ijy,θijt-ρijy≤-eTtKet+∑i=1Nμiyt∑j=1nejtεi∗j-ε^i∗jy-θ~ijtφijy=-eTtKet-∑i=1Nμiyt∑j=1nejtε~i∗jy+θ~ijtφijy.


## 3.3. Stability Analysis

Here, fractional Lyapunov’s theory is used to analyze the stability in closed-loop system. The following Lemmas are proposed to simplify the stability analysis.Lemma 9 (see [54]).
Ifμ∈(0,1) and x(t)∈C1, then 0CDtμx(t)Tx(t)≤2x(t)T0CDtμx(t).Lemma 10.
If0CDtμx(t)≤0, then one gets x(t)≤x(0); if 0CDtμx(t)≥0, then one gets x(t)≥x(0), with μ∈(0,1) and t∈[0,+∞).Proof.
We only consider the front part. If0CDtμx(t)≤0, let(22)ht=-0CDtμxt.Both sides of (22) take Laplace transform and one obtains(23)L0CDtμxt+Hs=0. Using Property 3, one obtains the following:(24)sμXs-sμ-1x0+Hs=0. Further, one gets(25)Xs=x0s-Hssμ,with X(s)=L[x(t)],H(s)=L[h(t)]. Both sides of (25) make Laplace inverse transform and using the fractional integral definition, one obtains(26)xt=x0-0CDt-μht=x0-1Γμ∫0tt-ξμ-1hξdξ. From the above equation, we get x(t)≤x(0),t∈[0,+∞).Lemma 11.
LetV(t)=(1/2)x(t)Tx(t)+(1/2)y(t)Ty(t) with x(t),y(t)∈Rn be continuous and derivable functions. If there exists a constant h>0 such that(27)0CDtμVt≤-hxtTxt,then x(t) and y(t) are bounded and limt→+∞x(t)=0.Proof.
According to Lemma10 and 0CDtμV(t)≤-hx(t)Tx(t)≤0, we obtain V(t)≤V(0). Further, we get the following:(28)xt≤2V0,yt≤2V0. This means that x(t) and y(t) are bounded.
We will proof thatx(t) tends to 0 asymptotically below. Both sides of (27) commute with μ-order integral; based on Property 2, one gets(29)Vt-V0≤-h0CDt-μxtTxt.Further, one obtains(30)xtTxt≤2V0-2h0CDt-μxtTxt.Hence, the following can be obtained from (30) with a nonnegative function z(t) as(31)xtTxt+zt=2V0-2h0CDt-μxtTxt.Applying the Laplace transform to formula (31) and according to the Definition 4, we have(32)LxtTxt=2V0sμ-1sμ+2h-Zssμsμ+2h=L2V0Eμ,1-2htμ-LztLt-1Eμ,0-2htμ.Hence, using the Laplace inverse transform to (32), we have(33)xtTxt=2V0Eμ,1-2htμ-zt∗t-1Eμ,0-2htμ, with ∗ being the convolution operator. Since t-1 and Eμ,0(-2htμ) are nonnegative functions, then x(t)Tx(t)≤2V(0)Eμ,1(-2htμ). According to the results in [55], one obtains that x(t) is M-L stability and x(t) tends to 0 asymptotically (namely, limt→∞x(t)=0).From above discussion, the boundedness of all signals in closed-loop system and the convergence of tracking error based on adaptive fuzzy control scheme via T-S fuzzy logic systems is presented in the following theorem.Theorem 12.
For the master system (9) and slave system (10) under the known initial conditions, if Assumptions 5 and 6 are satisfied and the adaptive controller is given as (17) with the fractional adaptation laws (18) and (19), then all signals in the closed-loop system are bounded and the tracking error signal tends to zero asymptotically.Proof.
Define the following Lyapunov function:(34)Vt=12eTtet+∑i=1N12αiθ~iTtθ~it+∑i=1N12βiε~i∗Tyε~i∗y, with θ~i(t)=θi(t)-θi∗ and ε~i∗(y)=ε^i∗(y)-εi∗. Hence, using the Lemma 9, the μ-order derivative of V(t) with respect to time t is obtained as(35)0CDtμVt≤eTt0CDtμet+∑i=1N1αiθ~iTt0CDtμθ~it+∑i=1N1βiε~i∗Ty0CDtμε~i∗ySubstituting (21) into (35), one gets(36)0CDtμVt≤-eTtKet+∑i=1N∑j=1n1αiθ~ijt0CDtμθ~ijt-μiytejtθ~ijtφijy+1βiε~i∗jy0CDtμε~i∗jy-μiytejtε~i∗jy,Taking (18) and (19) into (36), one gets the following inequality:(37)0CDtμV≤-eTtKet≤-λmineTtet, where λmin is the least eigenvalue of the positive definite matrix K. According to Lemma 11 and above discussion, we know that the tracking error signal e(t) tends to 0 asymptotically (that is, limt→∞e(t)=0) and θ~i(t) and ε~i∗(y) are bounded. Further, it means that θi(t) and ε^i∗(y) are bounded. Because of the boundedness of e(t) and x(t), we know that y(t) is bounded. Based on the control design, u(t) is bounded. Therefore, we know that all signals in the closed-loop system are bounded.


## 4. Simulation Example

In this section, in order to further illustrate the effectiveness of the proposed control method designed in previous sections, one example about the synchronization for two different uncertain fractional-order chaotic system is given. The master system of a fractional-order chaotic system via T-S fuzzy model is given as(38)0CDtμxt=∑i=1NμixtAixt+b1,the ith rule of master system is given byR1: If x1 is F11 and x2 is F21 and x3 is F31, then 0CDt0.8x(t)=A1x(t)+b1,R2: If x1 is F12 and x2 is F22 and x3 is F32, then 0CDt0.8x(t)=A2x(t)+b1.The upper system is formulated to the alike form in (9) with(39)A1=-10100400-62.625.040-8,A2=-1010040060.7-24.280-8,b1t=0,0,0T.Figure 1 depicts the simulation results of the master system with the parameters N=2,μ=0.8 with time step h=0.005. Figure 1 shows x1(t), x2(t) and x3(t)∈[-6.07,6.26], [-10,10.6], and [0.2,9.05], respectively, that is, x(t)∈D1=[-6.07,6.26]×[-10,10.6]×[0.2,9.05]. Obviously, Chaos was found in system (38) with μ=0.8.Figure 1
Master system.
(a) (b) (c) (d)Two fuzzy sets are defined for the statex1 over the interval [-6.07,6.26] with the membership functions as(40)F11x1t=121-0.095-x1t6.165,F12x1t=121+0.095-x1t6.165.Two fuzzy sets are defined for the statex2 over the interval [-10,10.6] with the membership functions as(41)F21x2t=121-0.3-x2t10.3,F22x2t=121-0.3-x2t10.3.Two fuzzy sets are defined for the statex3 over the interval [0.2,9.05] with the membership functions as(42)F31x3t=121-4.625-x3t4.425,F32x3t=121-4.625-x3t4.425.The slave system of a fractional-order chaotic system with unknown disturbances via T-S fuzzy model is given as(43)0CDt0.8yt=∑i=12μiytBiyt+b2+ut+dit,y.The ith rule of slave system is given byR1: If y1 is F^11 and y2 is F^21 and y3 is F^31, then 0CDt0.8y(t)=B1y(t)+b2+u(t)+d1(t,y),R2: If y1 is F^12 and y2 is F^22 and y3 is F^32, then 0CDt0.8y(t)=B2y(t)+b2+u(t)+d2(t,y).The upper system is formulated to the alike form in (10) with(44)B1=-30300022.2-21.52021.52-2.94,B2=-30300022.229.210-29.21-2.94,b2t=0,0,0T.Figure2 with u(t)=0 and without the external disturbance is depicted the simulation results of the slave system with the parameters below: N=2,μ=0.8, for time step h=0.005. Moreover, Chaos was found in system (43) with μ=0.8. Figure 2 shows y1(t),y2(t) and y3t∈-29.21,21.52,-35.6,26.5 and [0,53.6], respectively, that is, y(t)∈D2=[-29.21,21.52]×[-35.6,26.5]×[0,53.6].Figure 2
Slave system.
(a) (b) (c) (d)Two fuzzy sets are defined for the statey1 over the interval [-29.21,21.52] with the membership functions as follows:(45)F^11y1t=121--3.845-y1t25.365,F^12y1t=121+-3.845-y1t25.365.Two fuzzy sets are defined for the statey2 over the interval [-35.6,26.5] with the membership functions as follows:(46)F^21y2t=121--4.55-y2t31.05,F^22y2t=121--4.55-y2t31.05.Two fuzzy sets are defined for the statey3 over the interval [0,53.6] with the membership functions as follows:(47)F^31y3t=121-26.8-y3t26.8,F^32y3t=121-26.8-y3t26.8.In the simulation, the initial conditions of master system and slave system are selected asx(0)=(1,1,-2)Tand y(0)=(-2,-3,3)T. The parameters relating the synchronization problem are set to K=E and ρ1(y)=ρ2(y)=(1.5y1,1.5y2,1.5y3)T. Let e(t)=(e1(t),e2(t),e3(t))T.The controller is designed as(48)ut=u1t,u2t,u3tT=-F^11y1tB1yt-Tρ^y,θt-F^12B2yt-Hsignet-et+F11A1xt+F12A2xt.Letρ^(y,θ(t))=(ρ^1,ρ^2,ρ^3)T,H=(ε^∗1(y), ε^∗2(y), and ε^∗3(y))T; then (49)u1t=-e1t-10x1+10x2+30y1-30y2-Tρ^1-ε^∗1y,u2t=-e2t+40x1-10x1x3-22.2y2+y1y3-Tρ^2-ε^∗2y,u3t=-e3t-8x3+4x12-y1y2+2.94y3-Tρ^3-ε^∗3y.The fractional adaptation laws ofθ~j(y) and ε~∗j(y),(j=1,2,3), with α=700,β=0.05 are designed to be (50)0CDtμθ~1t=αe1tφ1y,0CDtμε~∗1y=βe1t,0CDtμθ~2t=αe2tφ2y,0CDtμε~∗2y=βe2t,0CDtμθ~3t=αe3tφ3y,0CDtμε~∗3y=βe3t.The simulation results of the proposed adaptive control approach are shown in Figure3, where subgraph (a) denotes the tracking error trajectory and subgraph (b) denotes the control trajectory. Define the initial conditions of the approximation errors as ε^∗1(0)=0,ε^∗2(0)=0,ε^∗3(0)=0. In reducing the computation of the numerical simulation, x(t) and y(t) are replaced by e(t). Four fuzzy sets are defined for the tracking errors e1(t),e2(t),e3(t) over the interval [-3,3] with the Gaussian membership functions, where the first parameters are 1.1 and the second parameters are -3,-1,1,3, respectively. Comparing the conventional control method with the proposed method, we can see that the proposed approach can synchronize two chaotic plants to desired high accuracy and improve the performance as shown in Figure 3.Figure 3
(a) Synchronization error and (b) controller.
(a) (b)


## 5. Conclusions

In this paper, synchronization of different fractional-order chaotic or hyperchaotic systems with unknown disturbances and parametric uncertainties is addressed with adaptive fuzzy control algorithm based on T-S fuzzy models. The distinctive features of the proposed control approach are that T-S fuzzy logic systems are introduced to approximate the unknown disturbances and to model the unknown controlled systems; both adaptive fuzzy controller and fractional adaptation laws are developed based on combined fractional Lyapunov stability theory and parallel distributed compensation technique. It is shown that the proposed control method can guarantee that all the signals in the closed-loop system remain bounded and the synchronization error converges towards an arbitrary small neighbourhood of the origin asymptotically. A simulation example is used for verifying the effectiveness of the proposed control strategy. Further works would focus on chaos synchronization control of different uncertain fractional-order chaotic systems with time delay and input saturation.



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*Source: 2793673-2018-08-12.xml*