# Centralized Data-Sampling Approach for GlobalOt-α Synchronization of Fractional-Order Neural Networks with Time Delays

Authors: Jin-E Zhang
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

In this paper, the globalO(t-α) synchronization problem is investigated for a class of fractional-order neural networks with time delays. Taking into account both better control performance and energy saving, we make the first attempt to introduce centralized data-sampling approach to characterize the O(t-α) synchronization design strategy. A sufficient criterion is given under which the drive-response-based coupled neural networks can achieve global O(t-α) synchronization. It is worth noting that, by using centralized data-sampling principle, fractional-order Lyapunov-like technique, and fractional-order Leibniz rule, the designed controller performs very well. Two numerical examples are presented to illustrate the efficiency of the proposed centralized data-sampling scheme.


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## Body

## 1. Introduction

Fractional-order calculus has gained an increasing attention in physical systems and engineering systems. Fractional-order dynamic systems containing fractional derivatives and integrals have been investigated in the field of control systems [1–19]. Investigating analytical skills in fractional-order dynamic systems is one important theme. However, it is considered that many control schemes for fractional differential equation are only a mathematical concept. A lot of control methods in integer-order dynamic systems cannot be introduced to fractional-order dynamic systems [2, 3].As a kind of the important dynamic systems, the concept of neurodynamic systems can be traced back to the early 1940s. Neurodynamic systems behave like a synthesizer evaluating the performance of system itself via the topology structure. In recent years, modeling fractional phenomenon to neurodynamic systems has been developed in an effort to improve the neurodynamic processes [2, 3, 14, 16]. Appealing feature of fractional-order neurodynamic systems is that the infinite memory property can take the past inherited information into account, which well suits describing complex dynamic processes. For control system applications, fractional-order neurodynamic systems have greatly expanded systems of conventional difference equations, which provide an adaptive control system for standard application.Data-sampling control of systems has been studied in a number of publications. Actually, as stated in [20–30], for complex or multivariable control systems, it is unrealistic or even impossible to sample all real-time physical signals at one single rate. In such situation, one is forced to use multirate data-sampling control. Multirate data-sampling control conditions have been derived there which are less conservative [31–35]. Multirate data-sampling control can achieve what single-rate data-sampling control cannot, for instance, gain margin improvement, centralized control, and decentralized control.Many control schemes have been established for complex control systems, such as adaptive control [36, 37] and sliding mode control [38]. In view of the feature of fractional-order systems, compared with other control strategies, centralized data-sampling scheme is more applicable for implementation in fractional-order systems. For one thing, centralized data-sampling scheme itself is relatively cheaper and simpler to operate. Unlike a lot of data-sampling designs, these schemes are usually designed for continuous sampling, and the control cost is very high. For another thing, considering that the system structures of fractional-order systems are complex and ever changing, which may have unpredictable nonlinear effects, it is more reasonable and implementable for centralized data-sampling only carried out at part of timing nodes.In this paper, a centralized data-sampling architecture enabled by low-bandwidth communication is proposed. The centralized data-sampling approach is developed to globallyO(t-α) synchronize the drive-response-based coupled fractional-order neural networks with time delays. And then we present an intelligent control method for designing synchronization scheme based on centralized data-sampling principle, fractional-order Lyapunov-like technique, and fractional-order Leibniz rule. The obtained results provide novel and higher performance extension for the designed controller. The use of centralized data-sampling approach facilitates utilizing low-bandwidth communication to transmit harmonic signals. The operation principle and numerical examples based on computer simulations are also presented.


## 2. Preliminaries and Problem Formulation

### 2.1. Notation

Forn-dimensional vector υ=υ1,υ2,…,υnT, the norm of vector υ is recorded as υ=∑i=1nυi. Denote Cιt0,+∞,Rn as the space of ι-order continuous and differentiable functions from [t0,+∞) into Rn. Cτ≔C-τ,0,R is a Banach space of all continuous functions φ:[-τ,0]→R. For any φ∈Cτ, let φC=sups∈[-τ,0]⁡φ(s).


### 2.2. Preliminaries

In order to facilitate understanding, we first introduce some concepts of fractional calculation.Fractional integral with orderα>0 of function F(t) is characterized as(1)Dt0RLt-αFt=1Γα∫t0tt-sα-1Fsds, where t≥t0, Γ(·) is the Gamma function, which is defined as (2)Γα=∫t0+∞sα-1exp⁡-sds.Riemann-Liouville derivative with orderα>0 of function F(t) is characterized as(3)Dt0RLtαFt=1Γn-αdndtn∫t0tFst-sα-n+1ds, where t≥t0, n-1<α<n, and n is a positive integer.Caputo derivative with orderα>0 of function F(t)∈Cn+1t0,+∞,R is characterized as(4)t0CDtαFt=1Γn-α∫t0tFnst-sα-n+1ds, where t≥t0, n-1<α<n, and n is a positive integer.In this paper, consider a class of fractional-order neural networks with time delays governed by(5)t0CDtαxt=-Dxt+Agxt+Bfxt-τt+I,yt=Cxt, where 0<α<1, x(t)=x1t,x2t,…,xntT∈Rn is the state vector, D=diag(d1,d2,…,dn) is a positive diagonal matrix, A=aijn×n∈Rn×n and B=bijn×n∈Rn×n denote the weight matrix and the delayed weight matrix, respectively, τ(t) represents the transmission delay satisfying 0≤τ(t)≤τ, g(x(t))=g1x1t,g2x2t,…,gnxntT∈Rn and f(x(t-τ(t)))=(f1x1t-τt,f2x2(t-τ(t))),…,fnxnt-τtT∈Rn denote the activation functions at times t and t-τ(t), respectively, I=I1,I2,…,InT∈Rn is the bias, C=(Cij)m×n∈Rm×n is a constant matrix, and y(t)=y1t,y2t,…,ymtT∈Rm is the output vector.Obviously, system (5) is a more general model. In [2, 16], the model is a fractional-order system without time-delay. By comparing the system models, system (5) contains some existing fractional-order neural networks.The initial condition of system (5) is x(t)=χ(t)∈Cτ.In addition, in (5), the activation functions g(·) and f(·) are global Lipschitz; that is, for i=1,2,…,n, there exist positive constants Gi>0 and Fi>0 such that(6)giu-giv≤Giu-v,foranyu∈R,v∈R,fiu-fiv≤Fiu-v,foranyu∈R,v∈R.


### 2.3. Problem Formulation

In this paper, consider system (5) as the master/drive system, and then the slave/response system is described as(7)t0CDtαx^t=-Dx^t+Agx^t+Bfx^t-τt+I+ut,y^t=Cx^t, where u(t)=u1t,u2t,…,untT∈Rn is the controller to be designed.The initial condition of system (7) is x^(t)=χ^(t)∈Cτ.Definee(t)=x^(t)-x(t)∈Rn, from (5) and (7); then we can obtain the error dynamics system(8)t0CDtαet=-Det+Aget+Bfet-τt+ut, where(9)get=g1e1t,g2e2t,…,gnentT=gx^t-gxt,fet-τt=f1e1t-τt,f2e2t-τt,…,fnent-τtT=fx^t-τt-fxt-τt.The initial condition of system (8) is e(t)=ψ(t)≔χ^(t)-χ(t)∈Cτ.In our control design, the structure-dependent centralized data-sampling is used. Moreover, the measured output of error dynamics system is sampled and then the data-sampling information is sent to the controller of the response system as(10)zt=y^tk-ytk=Cetk,t∈tk,tk+1,k=0,1,2,…, where z(t)∈Rm represents the output of (8) and tk denotes the sampling instant satisfying limk→+∞⁡tk=+∞.To study globalO(t-α) synchronization, next, we will introduce some related definitions.Definition 1.
For dynamic system(11)t0CDtαXt=Ft,Xt, where Xt(θ)=X(t+θ) is defined on -τ≤θ≤0, the initial condition of system (11) is X(t)=φ(t)∈Cτ. System (11) is said to be globally O(t-α) stable if there exists a positive constant M>0 such that X(t,t0,φ)≤MφCO(t-α) for any φ∈Cτ and t≥t0.Remark 2.
Convergence of fractional-order systems is totally different from conventional exponential convergence or absolute convergence, which possesses abnormal convergence behavior. In addition, according to Definition1, global O(t-α) stability and global Mittag-Leffler stability are “essentially the same.” On Mittag-Leffler stability, please see some publications [2, 16].Definition 3.
For the master/drive system(12)t0CDtαXt=Ft,Xt and the slave/response system(13)t0CDtαYt=Ft,Yt,ut, where Xt(θ)=X(t+θ) and Yt(θ)=Y(t+θ) are defined on -τ≤θ≤0, the initial conditions for systems (12) and (13) are X(t)=ϕ(t)∈Cτ and Y(t)=ϕ^(t)∈Cτ, respectively. The coupled systems (12) and (13) are said to be globally O(t-α) synchronized if the zero solution of the error dynamics system(14)t0CDtαEt=Ft,Yt,ut-Ft,Xt is globally O(t-α) stable, where E(t)=Y(t)-X(t).


## 2.1. Notation

Forn-dimensional vector υ=υ1,υ2,…,υnT, the norm of vector υ is recorded as υ=∑i=1nυi. Denote Cιt0,+∞,Rn as the space of ι-order continuous and differentiable functions from [t0,+∞) into Rn. Cτ≔C-τ,0,R is a Banach space of all continuous functions φ:[-τ,0]→R. For any φ∈Cτ, let φC=sups∈[-τ,0]⁡φ(s).


## 2.2. Preliminaries

In order to facilitate understanding, we first introduce some concepts of fractional calculation.Fractional integral with orderα>0 of function F(t) is characterized as(1)Dt0RLt-αFt=1Γα∫t0tt-sα-1Fsds, where t≥t0, Γ(·) is the Gamma function, which is defined as (2)Γα=∫t0+∞sα-1exp⁡-sds.Riemann-Liouville derivative with orderα>0 of function F(t) is characterized as(3)Dt0RLtαFt=1Γn-αdndtn∫t0tFst-sα-n+1ds, where t≥t0, n-1<α<n, and n is a positive integer.Caputo derivative with orderα>0 of function F(t)∈Cn+1t0,+∞,R is characterized as(4)t0CDtαFt=1Γn-α∫t0tFnst-sα-n+1ds, where t≥t0, n-1<α<n, and n is a positive integer.In this paper, consider a class of fractional-order neural networks with time delays governed by(5)t0CDtαxt=-Dxt+Agxt+Bfxt-τt+I,yt=Cxt, where 0<α<1, x(t)=x1t,x2t,…,xntT∈Rn is the state vector, D=diag(d1,d2,…,dn) is a positive diagonal matrix, A=aijn×n∈Rn×n and B=bijn×n∈Rn×n denote the weight matrix and the delayed weight matrix, respectively, τ(t) represents the transmission delay satisfying 0≤τ(t)≤τ, g(x(t))=g1x1t,g2x2t,…,gnxntT∈Rn and f(x(t-τ(t)))=(f1x1t-τt,f2x2(t-τ(t))),…,fnxnt-τtT∈Rn denote the activation functions at times t and t-τ(t), respectively, I=I1,I2,…,InT∈Rn is the bias, C=(Cij)m×n∈Rm×n is a constant matrix, and y(t)=y1t,y2t,…,ymtT∈Rm is the output vector.Obviously, system (5) is a more general model. In [2, 16], the model is a fractional-order system without time-delay. By comparing the system models, system (5) contains some existing fractional-order neural networks.The initial condition of system (5) is x(t)=χ(t)∈Cτ.In addition, in (5), the activation functions g(·) and f(·) are global Lipschitz; that is, for i=1,2,…,n, there exist positive constants Gi>0 and Fi>0 such that(6)giu-giv≤Giu-v,foranyu∈R,v∈R,fiu-fiv≤Fiu-v,foranyu∈R,v∈R.


## 2.3. Problem Formulation

In this paper, consider system (5) as the master/drive system, and then the slave/response system is described as(7)t0CDtαx^t=-Dx^t+Agx^t+Bfx^t-τt+I+ut,y^t=Cx^t, where u(t)=u1t,u2t,…,untT∈Rn is the controller to be designed.The initial condition of system (7) is x^(t)=χ^(t)∈Cτ.Definee(t)=x^(t)-x(t)∈Rn, from (5) and (7); then we can obtain the error dynamics system(8)t0CDtαet=-Det+Aget+Bfet-τt+ut, where(9)get=g1e1t,g2e2t,…,gnentT=gx^t-gxt,fet-τt=f1e1t-τt,f2e2t-τt,…,fnent-τtT=fx^t-τt-fxt-τt.The initial condition of system (8) is e(t)=ψ(t)≔χ^(t)-χ(t)∈Cτ.In our control design, the structure-dependent centralized data-sampling is used. Moreover, the measured output of error dynamics system is sampled and then the data-sampling information is sent to the controller of the response system as(10)zt=y^tk-ytk=Cetk,t∈tk,tk+1,k=0,1,2,…, where z(t)∈Rm represents the output of (8) and tk denotes the sampling instant satisfying limk→+∞⁡tk=+∞.To study globalO(t-α) synchronization, next, we will introduce some related definitions.Definition 1.
For dynamic system(11)t0CDtαXt=Ft,Xt, where Xt(θ)=X(t+θ) is defined on -τ≤θ≤0, the initial condition of system (11) is X(t)=φ(t)∈Cτ. System (11) is said to be globally O(t-α) stable if there exists a positive constant M>0 such that X(t,t0,φ)≤MφCO(t-α) for any φ∈Cτ and t≥t0.Remark 2.
Convergence of fractional-order systems is totally different from conventional exponential convergence or absolute convergence, which possesses abnormal convergence behavior. In addition, according to Definition1, global O(t-α) stability and global Mittag-Leffler stability are “essentially the same.” On Mittag-Leffler stability, please see some publications [2, 16].Definition 3.
For the master/drive system(12)t0CDtαXt=Ft,Xt and the slave/response system(13)t0CDtαYt=Ft,Yt,ut, where Xt(θ)=X(t+θ) and Yt(θ)=Y(t+θ) are defined on -τ≤θ≤0, the initial conditions for systems (12) and (13) are X(t)=ϕ(t)∈Cτ and Y(t)=ϕ^(t)∈Cτ, respectively. The coupled systems (12) and (13) are said to be globally O(t-α) synchronized if the zero solution of the error dynamics system(14)t0CDtαEt=Ft,Yt,ut-Ft,Xt is globally O(t-α) stable, where E(t)=Y(t)-X(t).


## 3. Main Results

Based on the discussion in preceding section, then the data-sampling controller can be designed as(15)ut=Kzt=KCetk,t∈tk,tk+1,k=0,1,2,…, where K=Kijm×m∈Rm×m is the constant gain matrix to be determined. Therefore, the error dynamics system can be transformed into the following form:(16)t0CDtαet=-Det+Aget+Bfet-τt+KCetk,t∈tk,tk+1.For technical convenience, we also give mathematical expression for (16) represented by components: for i=1,2,…,n,(17)t0CDtαeit=-dieit+∑j=1naijgjejt+∑j=1nbijfjejt-τt+∑l=1m∑ħ=1mKlħCħieitk,t∈tk,tk+1.Before we began to develop theoretical criterion, we first state an important lemma, which will be used in the proving process.Lemma 4 (see [3]).
IfF∈C10,+∞,R, then the following properties hold:(1)
Dt0CtαF(t)=Dt0RLtαF(t)-Ft0/Γ1-αt-t0-α.(2)
IfF(t) and ϑ(t) and their all derivatives are continuous in [t0,t], then(18)Dt0RLtαϑtFt=∑κ=0ndκϑtdtκακDt0RLtα-κFt-Rnαt, where 0<α<1, n≥α,(19)Rnαt=-1nt-αn-α+1n!Γ-α∫01∫01Fαt,ξ,ηdξdη,Fαt,ξ,η=Ft0+ηt-t0ϑn+1t0+t-t0ξ+η-ξη,ακ=Γα+1κ!Γα-κ+1.Theorem 5.
If there exist positive constantsr>τ≥0 and βi>0i=1,2,…,n such that(20)di-1βi∑j=1nβjaijGj-∑l=1m∑ħ=1mKlħCħi-rr-τα∑j=1nβjβibijFj-1+αrαΓ2-α≥0, for i=1,2,…,n, set tk+1 as the sampling time point such that(21)tk+1=supδ≥tk⁡δ:eitk≤βimax1≤σ≤n⁡eσtβσ,∀t∈tk,δ,i=1,2,…,n for all k=0,1,2,…; then system (17) is globally O(t-α) stable. That is, the drive-response-based coupled systems (5) and (7) can reach global O(t-α) synchronization.Proof.
Consider the Lyapunov functions candidate as(22)Wt=max⁡eitβi,i=1,2,…,n,Vt=t-t0+rαmax⁡eitβi,i=1,2,…,nand set(23)W¯t=sup-τ≤θ≤t⁡Wθ,V¯t=sup-τ≤θ≤t⁡Vθ, for t≥t0.Using Lemma4, we can obtain(24)Dt0CtαVt=Dt0RLtαVt-Vt0Γ1-αt-t0-α=Dt0RLtαt-t0+rαmax⁡eitβi,i=1,2,…,n-rαmax⁡eit0/βi,i=1,2,…,nΓ1-αt-t0-α≤t-t0+rαDt0Ctαmax⁡eitβi,i=1,2,…,n+t-t0+rαmax⁡eit0/βi,i=1,2,…,nΓ1-αt-t0-α+α2t-t0+rα-1Dt0RLtα-1max⁡eitβi,i=1,2,…,n+α2α-122t-t0+rα-2×Dt0RLtα-2max⁡eitβi,i=1,2,…,n-rαmax⁡eit0/βi,i=1,2,…,nΓ1-αt-t0-α≤t-t0+rαDt0Ctαmax⁡eitβi,i=1,2,…,n+t-t0+rt-t0α-rt-t0αmax⁡eit0/βi,i=1,2,…,nΓ1-α+α2t-t0+rα-1Dt0RLtα-1max⁡eitβi,i=1,2,…,n+α2α-122t-t0+rα-2×Dt0RLtα-2max⁡eitβi,i=1,2,…,n≤t-t0+rαDt0CtαWt+1+2αV¯trαΓ1-α+α2t-t0+rα-1Dt0RLtα-1Wt+α2α-122t-t0+rα-2Dt0RLtα-2Wt,for t≥t0.On the other hand, it is not difficult to follow(25)α2t-t0+rα-1Dt0RLtα-1Wt=α2t-t0+r-1-αDt0RLt-1-αmax⁡eitβi,i=1,2,…,n≤α2t-t0+r-1-α1Γ1-α×sup-τ≤θ≤t⁡max⁡eiθβi,i=1,2,…,n∫t0tt-s-αds≤α2t-t0+r-1-αt-t01-α1-αΓ1-α×sup-τ≤θ≤t⁡max⁡eiθβi,i=1,2,…,n≤α2t-t0t-t0+r1-α1Γ2-α×sup-τ≤θ≤t⁡max⁡eiθβi,i=1,2,…,n≤α2Γ2-αWθ~≤α2rαΓ2-αV¯t,α2α-122t-t0+rα-2Dt0RLtα-2Wt≤α2α-122t-t0+rα-21Γ2-α×∫t0tt-s1-αmax⁡eisβi,i=1,2,…,nds≤α2α-122t-t0+rα-2×sup-τ≤θ≤t⁡max⁡eiθ/βi,i=1,2,…,nΓ2-α∫t0tt-s1-αds≤α2α-122t-t0+rα-2×t-t02-αsup-τ≤θ≤t⁡max⁡eiθ/βi,i=1,2,…,n2-αΓ2-α≤α2α-122t-t0t-t0+r2-α×sup-τ≤θ≤t⁡max⁡eiθ/βi,i=1,2,…,n2-αΓ2-α≤α2α-122sup-τ≤θ≤t⁡max⁡eiθ/βi,i=1,2,…,n2-αΓ2-α≤α2α-122rα2-αΓ2-αV¯t≤α2rαΓ2-αV¯t,for t≥t0, W¯(t)=W(θ~), where θ~∈-τ,t.Substituting (25) into (24), it is obvious to derive that(26)Dt0CtαVt≤t-t0+rαDt0Ctαmax⁡eitβi,i=1,2,…,n+ΔrαΓ2-α+ΘrαΓ1-α=t-t0+rαDt0CtαWt+1+αrαΓ2-αV¯t, where(27)Δ=2α2sup-τ≤θ≤t⁡θ-t0+rαmax⁡eiθβi,i=1,2,…,n,Θ=1+2αsup-τ≤θ≤t⁡θ-t0+rαmax⁡eiθβi,i=1,2,…,n.Utilizing (22), it is obvious that there exists a k^∈{1,2,…,n} such that(28)Wt=ek^tβk^, for t≥t0.From (17), together with the data-sampling principle (21), we compute(29)Dt0CtαWt=1βk^Dt0Ctαek^t≤1βk^sgn⁡ek^tDt0Ctαek^t≤1βk^sgn⁡ek^t-dk^ek^t+∑j=1nak^jgjejt+∑j=1nbk^jfjejt-τt+∑l=1m∑ħ=1mKlħCħk^ek^tk≤-dk^ek^tβk^+1βk^∑j=1nβjak^jGjejtβj+1βk^∑j=1nβjbk^jFjejt-τtβj+1βk^∑l=1m∑ħ=1mKlħCħk^ek^tk≤-dk^-1βk^∑j=1nβjak^jGj-∑l=1m∑ħ=1mKlħCħk^Wt+1βk^∑j=1nβjbk^jFjW¯t-τt. Then(30)t-t0+rαDt0CtαWt≤-t-t0+rαdk^-1βk^∑j=1nβjak^jGj-∑l=1m∑ħ=1mKlħCħk^Wt+t-t0+rα1βk^∑j=1nβjbk^jFjW¯t-τt≤-dk^-1βk^∑j=1nβjak^jGj-∑l=1m∑ħ=1mKlħCħk^Vt+t-t0+rαβk^t-t0+r+θ¯α∑j=1nβjbk^jFjV¯t≤-dk^-1βk^∑j=1nβjak^jGj-∑l=1m∑ħ=1mKlħCħk^-rr+θ¯α∑j=1nβjβk^bk^jFjVt≤-dk^-1βk^∑j=1nβjak^jGj-∑l=1m∑ħ=1mKlħCħk^-rr-τα∑j=1nβjβk^bk^jFjVt, where θ¯∈-τ,t such that V¯(t)=t-t0+θ¯+rαW¯(t), when V¯(t)=V(t).Substituting (30) into (26), we have(31)Dt0CtαVt≤-dk^-1βk^∑j=1nβjak^jGj-∑l=1m∑ħ=1mKlħCħk^-rr-τα∑j=1nβjβk^bk^jFjVt+1+αrαΓ2-αV¯t≤-dk^-1βk^∑j=1nβjak^jGj-∑l=1m∑ħ=1mKlħCħk^-rr-τα∑j=1nβjβk^bk^jFj-1+αrαΓ2-αVt, when V¯(t)=V(t).According to (20) and (31),(32)Dt0CtαV¯t≤0, for all t≥t0.By the definition of Caputo derivative, from (32), we can obtain(33)1Γ1-αddt∫t0tV¯st-sαds≤V¯t0Γ1-αt-t0-α; then(34)V¯t≤V¯t0, for t≥t0. Thus, for i=1,2,…,n,(35)eit≤βiWt=βiVtt-t0+rα≤βiV¯tt-t0+rα≤βiVt0t-t0+rα=βirαW¯t0t-t0+rα≤βirαψCβmint-t0+rα, for t≥t0, where βmin=min⁡{βi,i=1,2,…,n}; then(36)et≤ΛrαψCt-t0+rα, where Λ=1/βmin∑i=1nβi. Therefore, system (17) is globally O(t-α) stable, which implies that the drive-response-based coupled systems (5) and (7) can reach global O(t-α) synchronization.Remark 6.
In Theorem5, (20) is an algebraic condition, which relies on only system parameters, free-weighting parameters r and βi(i=1,2,…,n). Besides, (21) is used to characterize the sampling time point, which can be determined by trigger mechanism. Generally, the discrimination conditions in Theorem 5 can be easily rectified. In addition, although the results established are based on 0<α<1, clearly, such results can be also applied to α=1.Remark 7.
Aiming at the realistic environment under the limited bandwidth of communication channel, it is very critical to reduce the data transmission rate for networked systems. As (21), the centralized data-sampling mechanism for the sampling time point is an effective way, which does not waste the bandwidth of network to be with needless signals, and then reduces the data transmission and power consumption.


## 4. Two Illustrative Examples

In this section, two illustrative examples are given to demonstrate the effectiveness of theoretical criterion.Example 1.
Consider a two-dimensional neural network model (5) described by(37)α=0.5,D=6006,A=0.20.20.50.5,B=0.20.80.20.8,τt=sin⁡t,I=0.5,0.5T,C=1111,gΥ=fΥ=tanh⁡Υ.Figure1 shows simulation result of the above neural network model, which can exhibit chaotic behavior.Figure 1
Chaotic behavior.Obviously, we can obtainG1=G2=1, F1=F2=1, and τ=1; to apply Theorem 5, it requires(38)6-1β1∑j=12βja1j-∑l=12∑ħ=12Klħ-rr-10.5∑j=12βjβ1b1j-1.5r0.5Γ1.5≥0,6-1β2∑j=12βja2j-∑l=12∑ħ=12Klħ-rr-10.5∑j=12βjβ2b2j-1.5r0.5Γ1.5≥0.Selectr=4, β1=1, β2=2, and(39)K=K11K12K21K22=0.60.6-0.6-0.6;then (38) are satisfied, so the designed controller is(40)ut=KCetk=1.21.2-1.2-1.2etk,t∈tk,tk+1.Figure 2 is utilized to show the simulation result for the master/drive system state x1(t) and the slave/response system state x^1(t) in Example 1. Figure 3 is utilized to show the simulation result for the master/drive system state x2(t) and the slave/response system state x^2(t) in Example 1. Clearly, the drive-response-based coupled systems in Example 1 can reach global O(t-α) synchronization. Figure 4 shows the centralized data-sampling release instants and the corresponding release intervals. These results via computer simulations nicely demonstrate that the designed controller performs very well.Figure 2
Evolutive behavior ofx1(t) and x^1(t).Figure 3
Evolutive behavior ofx2(t) and x^2(t).Figure 4
Release instants and release intervals.Example 2.
Consider a one-dimensional neural network model (5) described by(41)α=0.5,D=6,A=1,B=-3.5,τt=sin⁡t,I=1,C=1,gΥ=fΥ=tanh⁡Υ.Figure5 shows simulation result of the above neural network model without any external control, which generates disorganized behavior.Figure 5
Disorganized behavior.Obviously, we can obtainG=1, F=1, and τ=1; to apply Theorem 5, it requires(42)D-A-K-rr-10.5B-1.5r0.5Γ1.5≥0.Selectr=4 and(43)K=K=0.1;then (42) is satisfied, so the designed controller is(44)ut=KCetk=0.1etk,t∈tk,tk+1.Figure 6 is utilized to show the simulation result for the master/drive system state x(t) and the slave/response system state x^(t) in Example 2. Figure 7 shows the centralized data-sampling release instants and the corresponding release intervals. Such results via computer simulations also indicate that the designed controller performs very well.Figure 6
Evolutive behavior ofx(t) and x^(t).Figure 7
Release instants and release intervals.


## 5. Conclusion

In this paper, the centralized data-sampling approach for globalO(t-α) synchronization of fractional-order neural networks with time delays is investigated. Some sufficient conditions for centralized data-sampling principle are derived and proved to guarantee the global O(t-α) synchronization for drive-response-based coupled neural networks. The proposed theoretical results are sufficient conditions for global O(t-α) synchronization and contain a lot of space for further improvement. Future research can be extended to (1) more economical and efficient event-triggered mechanism and (2) more realistic networked systems involving stochastic effect or incomplete measurement.



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*Source: 6157292-2017-02-09.xml*