# Adaptive Fuzzy Synchronization of Fractional-Order Chaotic Neural Networks with Backlash-Like Hysteresis

Authors: Wenqing Fu; Heng Liu
Journal: Advances in Mathematical Physics
 (2018)

---

## Abstract

An adaptive fuzzy synchronization controller is designed for a class of fractional-order neural networks (FONNs) subject to backlash-like hysteresis input. Fuzzy logic systems are used to approximate the system uncertainties as well as the unknown terms of the backlash-like hysteresis. An adaptive fuzzy controller, which can guarantee the synchronization errors tend to an arbitrary small region, is given. The stability of the closed-loop system is rigorously analyzed based on fractional Lyapunov stability criterion. Fractional adaptation laws are established to update the fuzzy parameters. Finally, some simulation examples are provided to indicate the effectiveness and the robust of the proposed control method.


---

## Body

## 1. Introduction

In the past two decades, study results of fractional calculus have received more and more attention because, compared with the classical integer-order calculus, the fractional-order one has many interesting and special properties. It has also been proven that a lot kinds of actual systems, ranging from life science and engineering to secret communication and system control, can be better modeled by using fractional-order differential equations (FDE) [1–10]. The nonlinear system, which is described by FDE, has memory. This advantage makes it possible to describe the hereditary as well as memory characters of many systems and processes. On this account, a lot of scholars employed the fractional-order derivative to replace the integer-order one in neural networks to get the FONNs [11–18]. It is known that the fractional model equips the neurons with more powerful computation ability, and these abilities could be used in information processing, frequency-independent phase shifts of oscillatory neuronal firing, and stimulus anticipation [13, 19]. By far, lots of methods have been given to synchronize FONNs [5, 12, 13, 20–22]. It should be mentioned that, in above works, the model of the master FONN should be known in advance. How to design synchronization controller when the master system’s model is unknown is a challenging but interesting work.It is well known that hysteresis can be found in a great mount of physical systems or devices, for instance, biology optics, mechanical actuators, electromagnetism, and electronic circuits [6, 23–26]. Hysteresis can damage the control performance or even lead to the instability of the controlled system. How to construct proper controller for these kinds of systems is an interesting work. With respect to integer-order systems subject to hysteresis, a lot of results have been given. In [27], a feedback controller was introduced to control nonlinear systems with hysteresis. The control of systems subject to Prandtl-Ishlinskii hysteresis was studied in [28]. To see more results on the control of integer-order systems with hysteresis please refer to [29–33]. However, with respect to fractional-order nonlinear systems with hysteresis, the related literatures are very few.Up to now, fuzzy control methods have been studied extensively [34–42]. Specially, this approach has been particularly used to synchronize or control integer-order neural networks (IONNs) [43–47]. In above literature, fuzzy logic systems were employed to approximate the uncertain functions. To enhance the approximation ability of the fuzzy system, some robust terms, for example, sliding mode control, H∞ control should be used together with the main fuzzy adaptive control term. It should be pointed out that the above results are limited to uncertain IONNs. It is advisable to discuss the synchronization problem for uncertain FONNs.In our paper, an adaptive fuzzy control approach is introduced for synchronizing two uncertain FONNs. Based on some fractional Lyapunov stability theorems, the stability analysis and the controller implement are given. To show the effectiveness of the proposed synchronization method, some illustrative examples are presented. Bearing the results of aforementioned works in mind, the main contributions of our study consist of the following:(1) by designing an adaptive fuzzy controller, a practical synchronization is proposed for a class of uncertain FONNs. To the best of our knowledge, how to construct fuzzy adaptive control for FONNs has not been previously investigated up to now, except some preliminaries works in [8, 46]. It should be pointed out that, in these works, the integer-order stability analysis method is used. However, in this paper, we will use the fractional stability analysis approach, and the stability of the closed-loop system is proved rigorously. (2) The models of the FONNs are assumed to be fully unknown (i.e., the controller designed is free of the models of both master and slave systems). (3) The control of fractional-order nonlinear systems with backlash-like hysteresis input is studied.


## 2. Preliminaries

### 2.1. Some Basic Results of Fractional Calculus

Theqth fractional integral is defined by(1)I-qft=1Γq∫0tfτt-τ1-qdτ,where Γ(q)=∫0+∞τq-1e-τdτ. The qth fractional-order derivative is given as(2)Dqft=1Γn-q∫0tfnτt-τq+1-ndτ,where n-1≤q<n (n∈N). The Laplace transform of the Caputo fractional derivative is(3)∫0∞e-stDqftdt=sqFs-∑k=0n-1sq-k-1fk0,where F(s)=L{ft}. For convenience, we always assume that 0<q≤1 in the rest of this paper.The following results on fractional calculus will help us to facilitate the synchronization controller design as well as the stability analysis.Definition 1 (see [1]).
The Mittag-Leffler function is defined by(4)Eq1,q2ζ=∑k=0∞ζkΓq1k+q2,where q1,q2>0, and ζ∈C.The Laplace transform of (4) is as follows [1]:(5)Ltβ-1Eq1,q2-atq1=sq1-q2sq1+a.Lemma 2 (see [1]).
Letx(t)∈C1[0,T] with T>0; then one has(6)I0t-qDqxt=xt-x0,Dq0It-qxt=xt.Lemma 3 (see [1]).
Letβ∈C and to constants q,μ satisfying 0<q<2 and(7)πq2<μ<min⁡π,πq,and then the following equality holds:(8)Eq,βz=-∑j=1n1Γβ-qjzj+o1zn+1,z⟶∞,μ≤arg⁡z≤π.Lemma 4 (see [1]).
Let0<q<2 and β∈R. If there exists a positive constant μ such that πq/2<μ≤min⁡π,πq, then one has(9)Eq,βz≤b01+z, where b0 is a positive real constant, μ≤arg⁡z≤π and |z|≥0.Lemma 5 (see [2]).
Letx(t)=0 be an equilibrium point of the following fractional-order nonlinear system:(10)Dqxt=ft,xt.If one can find a Lyapunov function V(t,x(t)) as well as three class-K functions gi,i=1,2,3 such that(11)g1xt≤Vt,xt≤g2xt,DqVt,xt≤-g3xt,then system (10) will be asymptotically stable.Lemma 6 (see [4]).
Letx(t)∈Rn be a continuous and derivable function. Then, for any t>0,(12)12DqxTtxt≤xTtDqxt.Lemma 7 (see [3]).
Letx(t)∈Rn be a continuous and derivable function. Then, for any t>0,(13)12DqxTtBxt≤xTtBDqxt,where B∈Rn×n is a positive definite constant matrix.


### 2.2. Description of a Fuzzy System

A fuzzy logic system consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on the fuzzy rules, and the defuzzifier [34–40, 48]. Usually, a fuzzy logic system is modeled by(14)f^xt=∑j∈Jϑjtμjxt∑j∈Jμjxt,where f^ (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, 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 ϑj (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 ϑ(t)=ϑ1t,…,ϑNtT and φ(x(t))=q1xt,q2xt,…,qNxtT, where qj (called the jth fuzzy basis function, j∈J) is a continuous mapping (and thus φ:Ω→RN is continuous) defined by(15)qjxt=ϑjt∑s∈Jμsxt.Then system (14) can be rewritten as(16)f^xt=ϑTtφxt.


## 2.1. Some Basic Results of Fractional Calculus

Theqth fractional integral is defined by(1)I-qft=1Γq∫0tfτt-τ1-qdτ,where Γ(q)=∫0+∞τq-1e-τdτ. The qth fractional-order derivative is given as(2)Dqft=1Γn-q∫0tfnτt-τq+1-ndτ,where n-1≤q<n (n∈N). The Laplace transform of the Caputo fractional derivative is(3)∫0∞e-stDqftdt=sqFs-∑k=0n-1sq-k-1fk0,where F(s)=L{ft}. For convenience, we always assume that 0<q≤1 in the rest of this paper.The following results on fractional calculus will help us to facilitate the synchronization controller design as well as the stability analysis.Definition 1 (see [1]).
The Mittag-Leffler function is defined by(4)Eq1,q2ζ=∑k=0∞ζkΓq1k+q2,where q1,q2>0, and ζ∈C.The Laplace transform of (4) is as follows [1]:(5)Ltβ-1Eq1,q2-atq1=sq1-q2sq1+a.Lemma 2 (see [1]).
Letx(t)∈C1[0,T] with T>0; then one has(6)I0t-qDqxt=xt-x0,Dq0It-qxt=xt.Lemma 3 (see [1]).
Letβ∈C and to constants q,μ satisfying 0<q<2 and(7)πq2<μ<min⁡π,πq,and then the following equality holds:(8)Eq,βz=-∑j=1n1Γβ-qjzj+o1zn+1,z⟶∞,μ≤arg⁡z≤π.Lemma 4 (see [1]).
Let0<q<2 and β∈R. If there exists a positive constant μ such that πq/2<μ≤min⁡π,πq, then one has(9)Eq,βz≤b01+z, where b0 is a positive real constant, μ≤arg⁡z≤π and |z|≥0.Lemma 5 (see [2]).
Letx(t)=0 be an equilibrium point of the following fractional-order nonlinear system:(10)Dqxt=ft,xt.If one can find a Lyapunov function V(t,x(t)) as well as three class-K functions gi,i=1,2,3 such that(11)g1xt≤Vt,xt≤g2xt,DqVt,xt≤-g3xt,then system (10) will be asymptotically stable.Lemma 6 (see [4]).
Letx(t)∈Rn be a continuous and derivable function. Then, for any t>0,(12)12DqxTtxt≤xTtDqxt.Lemma 7 (see [3]).
Letx(t)∈Rn be a continuous and derivable function. Then, for any t>0,(13)12DqxTtBxt≤xTtBDqxt,where B∈Rn×n is a positive definite constant matrix.


## 2.2. Description of a Fuzzy System

A fuzzy logic system consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on the fuzzy rules, and the defuzzifier [34–40, 48]. Usually, a fuzzy logic system is modeled by(14)f^xt=∑j∈Jϑjtμjxt∑j∈Jμjxt,where f^ (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, 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 ϑj (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 ϑ(t)=ϑ1t,…,ϑNtT and φ(x(t))=q1xt,q2xt,…,qNxtT, where qj (called the jth fuzzy basis function, j∈J) is a continuous mapping (and thus φ:Ω→RN is continuous) defined by(15)qjxt=ϑjt∑s∈Jμsxt.Then system (14) can be rewritten as(16)f^xt=ϑTtφxt.


## 3. Main Results

### 3.1. Problem Description

Consider a class of FONNs described as(17)Dqxit=-aixit+∑k=1nbikfkxkt+Ii,where i=1,…,n,xi(t) is the state variable, ai>0 and bik,k=1,2,…,m are constants, Ii represents the external input, and fk(·) is a smooth nonlinear function.Writex(t)=[x1(t),…,xn(t)]T∈Rn, f(·)=[f1(·),…,fn(·)]T∈Rn, I=[I1,…,In]T∈Rn, A=-diag⁡a1,…,an∈Rn×n, ut=u1t,…,umt∈Rm,(18)B=b11⋯b1n⋮⋱⋮b1n⋯bnn∈Rn×n,then (17) can be written into the following compact form:(19)Dqxt=Axt+Bfxt+I.To guarantee the existence and uniqueness of the solutions of the fractional-order neural network (19) (see, [3]), we assume that the functions f are Lipschitz-continuous, i.e., for all x(t),y(t)∈Rn,(20)fxt-fyt≤σxt-yt,where σ is a positive constant.The slave system is expressed by(21)Dαyt=Ayt+Bfyt+I+Gνt+dt,where y(t)∈Rn is the state vector of the slave system, G∈Rn×n is a positive definite control gain matrix, d(t)=d1t,d2t,…,dntT∈Rn is an unknown external disturbance, and ν(t)∈Rn represents hysteresis type of nonlinear control input which is described by(22)dνtdt=γ1dutdtγ3ut-νt+γ2dutdt,where u(t)=[u1(t),…,un(t)]T∈Rn is the control input, γ1,γ2 and γ3 are three constants satisfying γ3>0 and γ3>γ2. One can rewrite (22) as(23)νt=γ3ut+ħtħt=ω0-cu0e-aut-u0sign⁡u˙t+e-autsign⁡u˙t∫u0utb-ceaτsign⁡u˙tdτ.Whenγ1=1.1, γ3=3.22, γ2=0.44, u(t)=5.1sin⁡(2t) and ν(0)=0, the behavior of the backlash-like hysteresis is depicted in Figure 1.Figure 1
Hysteresis curve.The objective of this paper is to construct an adaptive fuzzy controller such that the slave system (21) synchronizes the master system (19). To proceed, the following assumption is needed.Assumption 8.
The external disturbance is bounded, i.e.,|di(t)|≤d¯i,i=1,2,…,n where d¯i is an unknown positive constant.


### 3.2. Synchronization Controller Implement

In this part, we will give the detailed procedure of the adaptive fuzzy controller design as well as the stability analysis. Let the synchronization error bee(t)=x(t)-y(t). It follows from (19), (21), and (23) that(24)Dqet=Aet+Afxt-fyt-γ3Gut-Għt-dt.DenoteΞ=(1/γ3)G-1; (24) implies that(25)ΞDqet=ϖet-ut,where ϖ(e(t))=Ξ(Ae(t)+A(f(x(t))-f(y(t)))-Għ(t)-d(t)) is an unknown nonlinear function.Remark 9.
In fact, it is easy to know thatϖ is a function of x(t), y(t), e(t), and t. In this paper, the synchronization error will be used as the input of the fuzzy logic systems. In addition, the synchronization error e(t) can be seen as a bridge between the signals x(t) and y(t). Consequently, we will use the denotation ϖ(e(t)) for convenience.Firstly, let us consider an ideal condition. Suppose thatϖ(e(t)) is known (i.e., d(t), f(x(t)), and f(y(t)) are all known in advance). Thus, the ideal controller can be given by(26)ueq=ϖet+Ket,where K=diag⁡(k1,…,kn) with ki>0, i=1,2,…,n. Then, we can give the following theorem.Theorem 10.
If the models of the master and slave systems are known, and the synchronization controller is given by (26), then we have that the synchronization error converges to zero asymptotically.Proof.
Substituting (26) into (25), one has(27)ΞDqet=-Ket.Define the following Lyapunov function:(28)Vt=12eTtΞet.
According to Lemma7, the qth derivative of V(t) is bounded by(29)DαVt=12DqeTtΞet≤eTtΞDqet≤-k_t,where k_=min{k1,k2,…,kn}. Thus, according to Lemma 5 and (29) that the synchronization error will converge to zero asymptotically.Sinceϖ(e(t)) is an unknown function, the above controller (26) may be inapplicable. In such a case, by using the fuzzy logic system, let us approximate ϖ(e(t)) (16) as(30)ϖ^iϑit,et=ϑiTtφiet,where φi(x(t)) is a fuzzy basis function, and ϑi(t) is an adjustable parameter of the fuzzy system which drive sup{ϖi(e(t))-ϖ^i(ϑi∗,e(t)) small enough. Let the parameter estimation error with respect to ϑi be(31)ϑ~it=ϑit-ϑi∗,and the fuzzy approximation error with respect to ϑi∗ be(32)εiet=ϖiet-ϖ^iϑi∗,et.According to the universal approximation theorem, one can know that the fuzzy systems do not violate the universal approximator property. Consequently, one can make the following assumption.Assumption 11.
There exists an unknown constantε∗>0 such that(33)sup⁡εiet≤ε∗.Based on the above analysis, one has(34)ϖ^iϑit,et-ϖiet=ϖ^iϑit,et-ϖ^iϑi∗,et+ϖ^iϑi∗,et-ϖiet=ϖ^iϑit,et-ϖ^iϑi∗,et-εiet=ϑ~iTtφiet-εiet.Denoteϑ(t)φ(e(t))=[ϑ1T(t)φ1(e(t)),ϑ2T(t)φ2(e(t)),…,ϑnT(t)φn(e(t))]T and ϑ~(t)φ(e(t)) = ϑ~1T(t)φ1(e(t)),ϑ~2T(t)φ2(e(t)),…,ϑ~nT(t)φn(e(t))T, then one can rewrite (34) as(35)ϖ^ϑt,et-ϖet=ϑ~Ttφet-εet,where ϑ(t)=[ϑ1T(t),ϑ2Tt,…,ϑnT(t)]T∈RN×n (N represents the amount of the fuzzy rules).To simplify the stability analysis, we give the following results first.Theorem 12.
Suppose thath(t)∈R is a positive definite smooth function. δ1>0,δ2∈R are two adjustable parameters. If it holds that(36)Dqht≤-δ1ht+δ2,then h(t) will be small enough eventually if proper parameters are chosen.Proof.
According to (36), one can find a function ζ(t) satisfying ζ(t)≥0 and(37)Dqht+ζt=-δ1ht+δ2.It follows from (37) that(38)Hs=sq-1sq+δ1h0+sq-1+qδ2sq+δ1-Zssq+δ1,where H(s)=L{h(t)} and Z(s)=L{ζ(t)}. One solves (38) according to (5) as(39)ht=h0Eq,1-δ1tq+δ2tqEq,1+q-δ1tq-ζt∗t-1Eq,0-δ1tq,where ∗ represents the convolution operator. Since that t-1Eq,0(-δ1tq) and ζ(t) are all nonnegative, one has ζ(t)∗t-1Eq,0(-δ1tq)≥0. Consequently, one has(40)ht≤h0Eq,1-δ1tq+δ2tqEq,1+q-δ1tq.
It is easy to know thatarg⁡(-δ1tq)=-π, -δ1tq≥0 and q∈(0,2); thus, by using Lemma 4, one obtains that(41)Eq,1-δ1tq≤κ1+δ1tq,where κ>0, i.e.,(42)limt→∞⁡h0Eq,1-δ1tq=0.
Thus, by using Lemma3, one has for every ε>0 and large enough time t that(43)tαEα,α+1-Atα≤BA+ε3.
That is to say, if the design parameters are chosen asδ2/δ1≤ε, then according to (40) and (43) one has(44)ht<ε.This completes the proof of Theorem 12.Then, one can obtain the following theorem.Theorem 13.
Under Assumptions8 and 11 and proper control parameters, if ε^∗(t) is the estimation of ε∗, u(t) is implemented as(45)ut=Ket+ϑTtφet+ε^∗tsign⁡et,and ϑi(t) and ε^∗(t) are, respectively, updated by(46)Dqϑit=ς1ieitφiet-ς1iς2iϑit,(47)Dqε^∗t=ς3∑i=1neit-ς3ς4ε^∗t,with ς1i, ς2i, ς3, ς4 being positive design parameters, then one has that the synchronization error eventually converges to an arbitrary small region of zero.Proof.
It follows from (25), (35), and (45) that(48)ΞDqet=ϖet-ut=ϖet-Ket-ϑTtφet-ε^∗tsign⁡et=-Ket-ϑ~Ttφet-ε^∗tsign⁡et+εet.
MultiplyingeT(t) to both sides of (48) gives(49)eTtΞDqet=-eTtKet-eTtϑ~Ttφet+eTtεet-eTtε^∗tsign⁡et≤-k_eTtet-∑i=1neitϑ~iTtφiet+ε∗∑i=1neit-ε^∗t∑i=1neit=-k_eTtet-∑i=1neitϑ~iTtφiet-ε~∗t∑i=1neit,where(50)ε~∗t=ε^∗t-ε∗is the estimation error of ε∗.
It is known that theDqC=0 where C is an arbitrary constant. Thus, it follows from (31) and (50) that Dqϑ~i(t)=Dqϑi(t) and Dqε~∗(t)=Dqε^∗(t).
Define the Lyapunov function as(51)Vt=12eTtΞet+∑i=1n12ς1iϑ~iTtϑ~it+12ς3ε~∗2t.
Using (46), (47), (49), and Lemma 6 one has(52)DqVt≤eTtΞDqet+∑i=1n1ς1iϑ~iTtDqϑ~it+1ς3ε~∗tDqε~∗t=-k_eTtet-∑i=1neitϑ~iTtφiet-ε~∗t∑i=1neit+∑i=1n1ς1iϑ~iTtDqϑit+1ς3ε~∗tDqε^∗t=-k_eTtet-∑i=1neitϑ~iTtφiet+1ς3ε~∗tς3∑i=1neit-ς3ς4ε^∗t-ε~∗t∑i=1neit+∑i=1n1ς1iϑ~iTtς1ieitφiet-ς1iς2iϑit=-k_eTtet-∑i=1nς2iϑ~iTtϑit-ς4ε~∗tε^∗t=-k_eTtet-∑i=1nς2iϑ~iTtϑ~it-ς4ε~∗tε~∗t-ς4ε~∗tε∗t-∑i=1nς2iϑ~iTtϑi∗≤-k_eTtet-∑i=1nς2i2ϑ~iTtϑ~it-ς42ε~∗2t+∑i=1nς2i2ϑi∗Tϑi∗+ς42ε∗2≤-δ1Vt+δ2with δ1=min⁡{2k_,ς1iς2i,ς2ς4} and δ2=∑i=1nς2i/2ϑi∗Tϑi∗+ς4/2ε∗2 being two positive constants.
Thus, based on (52) and Theorem 12, one knows that the synchronization error eventually converges to an arbitrary small region of zero if proper control parameters are chosen. This completes the proof of Theorem 13.Remark 14.
It should be pointed out that the fractional-order adaptation law was also introduced in [3, 5, 49, 50]. However, the above adaptation laws only contain a positive term (i.e., the adaptation law is designed as Dqϑ(t)=σei(t)φi(t)). Despite using this kind of adaptation law, the asymptotical stability of the system can be guaranteed. Yet, the boundedness of the control parameters cannot be ensured. The proposed adaptation law contains a negative term (for example, -ς1iς2iϑi(t) in (46)) which will drive the updated parameter tends to a small neighborhood of the origin eventually (see the proof of Theorem 13).Remark 15.
To enforce the synchronization error tending to a region as small as possible, one must makeδ2/δ1 small enough. To meet this objective, one should choose large ki, ς1i,ς3 and choose small ς2i,ς4.Remark 16.
It is worth mentioning that, in [51], to discuss the stability of the fractional-order nonlinear systems, a very complicated boundary condition(53)∑k=1∞Γ1+αΓ1+kΓ1-k+αDkxtDα-kxt≤axis assumed to be known. The above condition was proven in [51]. Yet, how to get the exact value of a is a challenging work. But in our paper, by using the quadratic Lyapunov functions, the aforementioned problem is solved.Remark 17.
It should be pointed out that the proposed control method does not need the prior knowledge of systems models. Therefore, the control method can be easily extended to the following domains: control of fractional-order nonlinear systems, synchronization of fractional-order chaotic system, and secret communication, and so on. And relating out control method to a potential application is one of our future research directions.


## 3.1. Problem Description

Consider a class of FONNs described as(17)Dqxit=-aixit+∑k=1nbikfkxkt+Ii,where i=1,…,n,xi(t) is the state variable, ai>0 and bik,k=1,2,…,m are constants, Ii represents the external input, and fk(·) is a smooth nonlinear function.Writex(t)=[x1(t),…,xn(t)]T∈Rn, f(·)=[f1(·),…,fn(·)]T∈Rn, I=[I1,…,In]T∈Rn, A=-diag⁡a1,…,an∈Rn×n, ut=u1t,…,umt∈Rm,(18)B=b11⋯b1n⋮⋱⋮b1n⋯bnn∈Rn×n,then (17) can be written into the following compact form:(19)Dqxt=Axt+Bfxt+I.To guarantee the existence and uniqueness of the solutions of the fractional-order neural network (19) (see, [3]), we assume that the functions f are Lipschitz-continuous, i.e., for all x(t),y(t)∈Rn,(20)fxt-fyt≤σxt-yt,where σ is a positive constant.The slave system is expressed by(21)Dαyt=Ayt+Bfyt+I+Gνt+dt,where y(t)∈Rn is the state vector of the slave system, G∈Rn×n is a positive definite control gain matrix, d(t)=d1t,d2t,…,dntT∈Rn is an unknown external disturbance, and ν(t)∈Rn represents hysteresis type of nonlinear control input which is described by(22)dνtdt=γ1dutdtγ3ut-νt+γ2dutdt,where u(t)=[u1(t),…,un(t)]T∈Rn is the control input, γ1,γ2 and γ3 are three constants satisfying γ3>0 and γ3>γ2. One can rewrite (22) as(23)νt=γ3ut+ħtħt=ω0-cu0e-aut-u0sign⁡u˙t+e-autsign⁡u˙t∫u0utb-ceaτsign⁡u˙tdτ.Whenγ1=1.1, γ3=3.22, γ2=0.44, u(t)=5.1sin⁡(2t) and ν(0)=0, the behavior of the backlash-like hysteresis is depicted in Figure 1.Figure 1
Hysteresis curve.The objective of this paper is to construct an adaptive fuzzy controller such that the slave system (21) synchronizes the master system (19). To proceed, the following assumption is needed.Assumption 8.
The external disturbance is bounded, i.e.,|di(t)|≤d¯i,i=1,2,…,n where d¯i is an unknown positive constant.


## 3.2. Synchronization Controller Implement

In this part, we will give the detailed procedure of the adaptive fuzzy controller design as well as the stability analysis. Let the synchronization error bee(t)=x(t)-y(t). It follows from (19), (21), and (23) that(24)Dqet=Aet+Afxt-fyt-γ3Gut-Għt-dt.DenoteΞ=(1/γ3)G-1; (24) implies that(25)ΞDqet=ϖet-ut,where ϖ(e(t))=Ξ(Ae(t)+A(f(x(t))-f(y(t)))-Għ(t)-d(t)) is an unknown nonlinear function.Remark 9.
In fact, it is easy to know thatϖ is a function of x(t), y(t), e(t), and t. In this paper, the synchronization error will be used as the input of the fuzzy logic systems. In addition, the synchronization error e(t) can be seen as a bridge between the signals x(t) and y(t). Consequently, we will use the denotation ϖ(e(t)) for convenience.Firstly, let us consider an ideal condition. Suppose thatϖ(e(t)) is known (i.e., d(t), f(x(t)), and f(y(t)) are all known in advance). Thus, the ideal controller can be given by(26)ueq=ϖet+Ket,where K=diag⁡(k1,…,kn) with ki>0, i=1,2,…,n. Then, we can give the following theorem.Theorem 10.
If the models of the master and slave systems are known, and the synchronization controller is given by (26), then we have that the synchronization error converges to zero asymptotically.Proof.
Substituting (26) into (25), one has(27)ΞDqet=-Ket.Define the following Lyapunov function:(28)Vt=12eTtΞet.
According to Lemma7, the qth derivative of V(t) is bounded by(29)DαVt=12DqeTtΞet≤eTtΞDqet≤-k_t,where k_=min{k1,k2,…,kn}. Thus, according to Lemma 5 and (29) that the synchronization error will converge to zero asymptotically.Sinceϖ(e(t)) is an unknown function, the above controller (26) may be inapplicable. In such a case, by using the fuzzy logic system, let us approximate ϖ(e(t)) (16) as(30)ϖ^iϑit,et=ϑiTtφiet,where φi(x(t)) is a fuzzy basis function, and ϑi(t) is an adjustable parameter of the fuzzy system which drive sup{ϖi(e(t))-ϖ^i(ϑi∗,e(t)) small enough. Let the parameter estimation error with respect to ϑi be(31)ϑ~it=ϑit-ϑi∗,and the fuzzy approximation error with respect to ϑi∗ be(32)εiet=ϖiet-ϖ^iϑi∗,et.According to the universal approximation theorem, one can know that the fuzzy systems do not violate the universal approximator property. Consequently, one can make the following assumption.Assumption 11.
There exists an unknown constantε∗>0 such that(33)sup⁡εiet≤ε∗.Based on the above analysis, one has(34)ϖ^iϑit,et-ϖiet=ϖ^iϑit,et-ϖ^iϑi∗,et+ϖ^iϑi∗,et-ϖiet=ϖ^iϑit,et-ϖ^iϑi∗,et-εiet=ϑ~iTtφiet-εiet.Denoteϑ(t)φ(e(t))=[ϑ1T(t)φ1(e(t)),ϑ2T(t)φ2(e(t)),…,ϑnT(t)φn(e(t))]T and ϑ~(t)φ(e(t)) = ϑ~1T(t)φ1(e(t)),ϑ~2T(t)φ2(e(t)),…,ϑ~nT(t)φn(e(t))T, then one can rewrite (34) as(35)ϖ^ϑt,et-ϖet=ϑ~Ttφet-εet,where ϑ(t)=[ϑ1T(t),ϑ2Tt,…,ϑnT(t)]T∈RN×n (N represents the amount of the fuzzy rules).To simplify the stability analysis, we give the following results first.Theorem 12.
Suppose thath(t)∈R is a positive definite smooth function. δ1>0,δ2∈R are two adjustable parameters. If it holds that(36)Dqht≤-δ1ht+δ2,then h(t) will be small enough eventually if proper parameters are chosen.Proof.
According to (36), one can find a function ζ(t) satisfying ζ(t)≥0 and(37)Dqht+ζt=-δ1ht+δ2.It follows from (37) that(38)Hs=sq-1sq+δ1h0+sq-1+qδ2sq+δ1-Zssq+δ1,where H(s)=L{h(t)} and Z(s)=L{ζ(t)}. One solves (38) according to (5) as(39)ht=h0Eq,1-δ1tq+δ2tqEq,1+q-δ1tq-ζt∗t-1Eq,0-δ1tq,where ∗ represents the convolution operator. Since that t-1Eq,0(-δ1tq) and ζ(t) are all nonnegative, one has ζ(t)∗t-1Eq,0(-δ1tq)≥0. Consequently, one has(40)ht≤h0Eq,1-δ1tq+δ2tqEq,1+q-δ1tq.
It is easy to know thatarg⁡(-δ1tq)=-π, -δ1tq≥0 and q∈(0,2); thus, by using Lemma 4, one obtains that(41)Eq,1-δ1tq≤κ1+δ1tq,where κ>0, i.e.,(42)limt→∞⁡h0Eq,1-δ1tq=0.
Thus, by using Lemma3, one has for every ε>0 and large enough time t that(43)tαEα,α+1-Atα≤BA+ε3.
That is to say, if the design parameters are chosen asδ2/δ1≤ε, then according to (40) and (43) one has(44)ht<ε.This completes the proof of Theorem 12.Then, one can obtain the following theorem.Theorem 13.
Under Assumptions8 and 11 and proper control parameters, if ε^∗(t) is the estimation of ε∗, u(t) is implemented as(45)ut=Ket+ϑTtφet+ε^∗tsign⁡et,and ϑi(t) and ε^∗(t) are, respectively, updated by(46)Dqϑit=ς1ieitφiet-ς1iς2iϑit,(47)Dqε^∗t=ς3∑i=1neit-ς3ς4ε^∗t,with ς1i, ς2i, ς3, ς4 being positive design parameters, then one has that the synchronization error eventually converges to an arbitrary small region of zero.Proof.
It follows from (25), (35), and (45) that(48)ΞDqet=ϖet-ut=ϖet-Ket-ϑTtφet-ε^∗tsign⁡et=-Ket-ϑ~Ttφet-ε^∗tsign⁡et+εet.
MultiplyingeT(t) to both sides of (48) gives(49)eTtΞDqet=-eTtKet-eTtϑ~Ttφet+eTtεet-eTtε^∗tsign⁡et≤-k_eTtet-∑i=1neitϑ~iTtφiet+ε∗∑i=1neit-ε^∗t∑i=1neit=-k_eTtet-∑i=1neitϑ~iTtφiet-ε~∗t∑i=1neit,where(50)ε~∗t=ε^∗t-ε∗is the estimation error of ε∗.
It is known that theDqC=0 where C is an arbitrary constant. Thus, it follows from (31) and (50) that Dqϑ~i(t)=Dqϑi(t) and Dqε~∗(t)=Dqε^∗(t).
Define the Lyapunov function as(51)Vt=12eTtΞet+∑i=1n12ς1iϑ~iTtϑ~it+12ς3ε~∗2t.
Using (46), (47), (49), and Lemma 6 one has(52)DqVt≤eTtΞDqet+∑i=1n1ς1iϑ~iTtDqϑ~it+1ς3ε~∗tDqε~∗t=-k_eTtet-∑i=1neitϑ~iTtφiet-ε~∗t∑i=1neit+∑i=1n1ς1iϑ~iTtDqϑit+1ς3ε~∗tDqε^∗t=-k_eTtet-∑i=1neitϑ~iTtφiet+1ς3ε~∗tς3∑i=1neit-ς3ς4ε^∗t-ε~∗t∑i=1neit+∑i=1n1ς1iϑ~iTtς1ieitφiet-ς1iς2iϑit=-k_eTtet-∑i=1nς2iϑ~iTtϑit-ς4ε~∗tε^∗t=-k_eTtet-∑i=1nς2iϑ~iTtϑ~it-ς4ε~∗tε~∗t-ς4ε~∗tε∗t-∑i=1nς2iϑ~iTtϑi∗≤-k_eTtet-∑i=1nς2i2ϑ~iTtϑ~it-ς42ε~∗2t+∑i=1nς2i2ϑi∗Tϑi∗+ς42ε∗2≤-δ1Vt+δ2with δ1=min⁡{2k_,ς1iς2i,ς2ς4} and δ2=∑i=1nς2i/2ϑi∗Tϑi∗+ς4/2ε∗2 being two positive constants.
Thus, based on (52) and Theorem 12, one knows that the synchronization error eventually converges to an arbitrary small region of zero if proper control parameters are chosen. This completes the proof of Theorem 13.Remark 14.
It should be pointed out that the fractional-order adaptation law was also introduced in [3, 5, 49, 50]. However, the above adaptation laws only contain a positive term (i.e., the adaptation law is designed as Dqϑ(t)=σei(t)φi(t)). Despite using this kind of adaptation law, the asymptotical stability of the system can be guaranteed. Yet, the boundedness of the control parameters cannot be ensured. The proposed adaptation law contains a negative term (for example, -ς1iς2iϑi(t) in (46)) which will drive the updated parameter tends to a small neighborhood of the origin eventually (see the proof of Theorem 13).Remark 15.
To enforce the synchronization error tending to a region as small as possible, one must makeδ2/δ1 small enough. To meet this objective, one should choose large ki, ς1i,ς3 and choose small ς2i,ς4.Remark 16.
It is worth mentioning that, in [51], to discuss the stability of the fractional-order nonlinear systems, a very complicated boundary condition(53)∑k=1∞Γ1+αΓ1+kΓ1-k+αDkxtDα-kxt≤axis assumed to be known. The above condition was proven in [51]. Yet, how to get the exact value of a is a challenging work. But in our paper, by using the quadratic Lyapunov functions, the aforementioned problem is solved.Remark 17.
It should be pointed out that the proposed control method does not need the prior knowledge of systems models. Therefore, the control method can be easily extended to the following domains: control of fractional-order nonlinear systems, synchronization of fractional-order chaotic system, and secret communication, and so on. And relating out control method to a potential application is one of our future research directions.


## 4. Simulation Studies

In (19), letting x(t)∈R3,x(0)=[-0.301,0.400,0.299]T,q=0.96, fi(xi(t))=tanh⁡(xi(t)),ai=1,Ii=0,u(t)≡0, and(54)B=2.001-1.20102.0001.7131.154-4.75101.101,the FONN (19) shows chaotic behavior, which is depicted in Figure 2.Figure 2
Chaotic dynamic of the FONN (19) in (a) 3D space; (b) x1-x2 plane; (c) x1-x3 plane; (d) x2-x3 plane.
(a) (b) (c) (d)


### 4.1. Synchronization of FONN with Constant System Parameters

Let the initial condition of the slave FONN (21) be y(0)=[4.2,-3.1,-1.9], and G=I3×3. The external disturbance is defined as d(t)=[sin⁡t,cos⁡t,sin⁡t+cos⁡t]T. The control design parameters are given as k1=k2=k3=0.5. ς11=ς12=ς13=1.1,ς21=ς22=ς23=0.05,ς3=1.55,ς4=0.02. In the simulation, the noncontinuous sign function in (45) is replaced with arctan (10·).With respect to the fuzzy logic systems, its input variable is chosen as the synchronization errorei(t). For each input, we give seven Gaussian membership functions on -55. The parameters of the proposed membership functions, which are defined as e-(ei(t)-a)2/2b2, are given in Table 1. These functions are depicted in Figure 3. The initial conditions of the fuzzy parameters are given as ϑ1(0)=ϑ2(0)=ϑ3(0)=0_∈R343.Table 1
Fuzzy parameters.
Membership functions Parameterb Parametera Membership function 1 1 - 5 Membership function 2 0.8 - 2 Membership function 3 0.2 - 0.3 Membership function 4 0.09 0 Membership function 5 0.2 0.3 Membership function 6 0.8 2 Membership function 7 1 5Figure 3
Membership functions.The simulation results are depicted in Figures4–6. The results that the signals x1(t), x2(t), and x3(t), respectively, track y1(t), y2(t), and y3(t), as well as the time response of the synchronization errors are presented in Figure 4. The smoothness and the boundedness of the control inputs are given in Figure 5. The fuzzy parameters, which can be concluded to be bounded according to the proposed adaptation law (46), are shown in Figure 6. From these figures one knows that the proposed controller works well and has good synchronization performance.Figure 4
Simulation results in (a)x1(t) and y1(t); (b) x2(t) and y2(t); (c) x3(t) and y3(t); (d) synchronization errors.
(a) (b) (c) (d)Figure 5
Control inputs.Figure 6
Fuzzy parameters.It is well known that the conventional systems usually suffer from discontented performance resulting from modeled errors, parametric uncertainties, input nonlinearities, and external disturbances, because it is impossible to provide accurate mathematical models of practical systems. These system uncertainties can damage the control performance or even lead to unstable of the controlled system if they are not well handled. To show the robust of the proposed method, let us consider the condition that the master system (19) suffers from time-varying system parameters and uncertainties. Suppose that the model of the master FONN (19) is replaced with(55)Dqxt=ΔAxt+Bfxt+ΔCξt+I,where ΔA=A+A¯∈Rn×n, ΔC∈Rn×m, and ξ∈Rm×n is an unknown external input signal. A¯ and C¯ are two unknown matrices. That is, A¯ represents unknown system parameter perturbations, and ΔCξ(t) denotes unknown input nonlinearities.For convenience, let us suppose that[A¯,C¯]=N1D(t)[N2,N3], where N1∈Rn×g,N2∈Rg×n and N3∈Rg×m are known matrices, and D(t)∈Rg×g is an unknown matrix.In the simulation, letN1=0.1000.10.10.1, N2=0.10.100.10.20.1, N3=0.100.10.10.10, and D(t)=0.2sin⁡t000.2cos⁡t. It is easy to know that A¯ and ΔCξ(t) are time-varying matrices.The simulations are presented in Figure7. It should be pointed out that, in the simulation, the slave system and the controller are chosen to be the same as those in Section 4.1. From the simulation results we can see that good synchronization performance has been achieved even the master system suffers from time-varying parameters and input nonlinearities. That is, the proposed method has good robustness.Figure 7
Simulation results whenu(t) = [5sin⁡20t,3cos⁡20t,4sin⁡(10t)]T in (a) x1(t) and x^1(t); (b) x2(t) and x^2(t); (c) x3(t) and x^3(t); (d) synchronization errors.
(a) (b) (c) (d)To indicate the effectiveness of our methods, the simulation results whenξ(t)=-[2sin(10t)+15rand⁡t,2cos⁡10t+20rand⁡t,2sin⁡(5t)+18rand⁡t]T where rand⁡(·) represents the random function produced in MATLAB software are shown in Figure 8.Figure 8
Simulation results whenξ(t)=-[2sin⁡10t+15rand⁡t;2cos⁡10t+20rand⁡t;2sin⁡(5t)+18rand⁡t]T in (a) x1(t) and y1t; (b) x2t and y2t; (c) x3(t) and y3(t); (d) synchronization errors.
(a) (b) (c) (d)


## 4.1. Synchronization of FONN with Constant System Parameters

Let the initial condition of the slave FONN (21) be y(0)=[4.2,-3.1,-1.9], and G=I3×3. The external disturbance is defined as d(t)=[sin⁡t,cos⁡t,sin⁡t+cos⁡t]T. The control design parameters are given as k1=k2=k3=0.5. ς11=ς12=ς13=1.1,ς21=ς22=ς23=0.05,ς3=1.55,ς4=0.02. In the simulation, the noncontinuous sign function in (45) is replaced with arctan (10·).With respect to the fuzzy logic systems, its input variable is chosen as the synchronization errorei(t). For each input, we give seven Gaussian membership functions on -55. The parameters of the proposed membership functions, which are defined as e-(ei(t)-a)2/2b2, are given in Table 1. These functions are depicted in Figure 3. The initial conditions of the fuzzy parameters are given as ϑ1(0)=ϑ2(0)=ϑ3(0)=0_∈R343.Table 1
Fuzzy parameters.
Membership functions Parameterb Parametera Membership function 1 1 - 5 Membership function 2 0.8 - 2 Membership function 3 0.2 - 0.3 Membership function 4 0.09 0 Membership function 5 0.2 0.3 Membership function 6 0.8 2 Membership function 7 1 5Figure 3
Membership functions.The simulation results are depicted in Figures4–6. The results that the signals x1(t), x2(t), and x3(t), respectively, track y1(t), y2(t), and y3(t), as well as the time response of the synchronization errors are presented in Figure 4. The smoothness and the boundedness of the control inputs are given in Figure 5. The fuzzy parameters, which can be concluded to be bounded according to the proposed adaptation law (46), are shown in Figure 6. From these figures one knows that the proposed controller works well and has good synchronization performance.Figure 4
Simulation results in (a)x1(t) and y1(t); (b) x2(t) and y2(t); (c) x3(t) and y3(t); (d) synchronization errors.
(a) (b) (c) (d)Figure 5
Control inputs.Figure 6
Fuzzy parameters.It is well known that the conventional systems usually suffer from discontented performance resulting from modeled errors, parametric uncertainties, input nonlinearities, and external disturbances, because it is impossible to provide accurate mathematical models of practical systems. These system uncertainties can damage the control performance or even lead to unstable of the controlled system if they are not well handled. To show the robust of the proposed method, let us consider the condition that the master system (19) suffers from time-varying system parameters and uncertainties. Suppose that the model of the master FONN (19) is replaced with(55)Dqxt=ΔAxt+Bfxt+ΔCξt+I,where ΔA=A+A¯∈Rn×n, ΔC∈Rn×m, and ξ∈Rm×n is an unknown external input signal. A¯ and C¯ are two unknown matrices. That is, A¯ represents unknown system parameter perturbations, and ΔCξ(t) denotes unknown input nonlinearities.For convenience, let us suppose that[A¯,C¯]=N1D(t)[N2,N3], where N1∈Rn×g,N2∈Rg×n and N3∈Rg×m are known matrices, and D(t)∈Rg×g is an unknown matrix.In the simulation, letN1=0.1000.10.10.1, N2=0.10.100.10.20.1, N3=0.100.10.10.10, and D(t)=0.2sin⁡t000.2cos⁡t. It is easy to know that A¯ and ΔCξ(t) are time-varying matrices.The simulations are presented in Figure7. It should be pointed out that, in the simulation, the slave system and the controller are chosen to be the same as those in Section 4.1. From the simulation results we can see that good synchronization performance has been achieved even the master system suffers from time-varying parameters and input nonlinearities. That is, the proposed method has good robustness.Figure 7
Simulation results whenu(t) = [5sin⁡20t,3cos⁡20t,4sin⁡(10t)]T in (a) x1(t) and x^1(t); (b) x2(t) and x^2(t); (c) x3(t) and x^3(t); (d) synchronization errors.
(a) (b) (c) (d)To indicate the effectiveness of our methods, the simulation results whenξ(t)=-[2sin(10t)+15rand⁡t,2cos⁡10t+20rand⁡t,2sin⁡(5t)+18rand⁡t]T where rand⁡(·) represents the random function produced in MATLAB software are shown in Figure 8.Figure 8
Simulation results whenξ(t)=-[2sin⁡10t+15rand⁡t;2cos⁡10t+20rand⁡t;2sin⁡(5t)+18rand⁡t]T in (a) x1(t) and y1t; (b) x2t and y2t; (c) x3(t) and y3(t); (d) synchronization errors.
(a) (b) (c) (d)


## 5. Conclusions

In this paper, an synchronization method was proposed for a class of FONNs subject to backlash-like hysteresis by means of adaptive fuzzy control. We showed that fuzzy logic systems can be employed to estimate nonlinear functions in fractional-order nonlinear systems. Based on the fractional stability theorems, an adaptive fuzzy synchronization controller, which can guarantee the synchronization error tends to an arbitrary small region of zero, was constructed. How to combine the proposed method with other control method, such as fractional-order sliding mode control, is one of our future research directions.



---

*Source: 7535628-2018-06-03.xml*