# Adaptive Synchronization for Uncertain Delayed Fractional-Order Hopfield Neural Networks via Fractional-Order Sliding Mode Control

Authors: Bo Meng; Xiaohong Wang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Adaptive synchronization for a class of uncertain delayed fractional-order Hopfield neural networks (FOHNNs) with external disturbances is addressed in this paper. For the unknown parameters and external disturbances of the delayed FOHNNs, some adaptive estimations are designed. Firstly, a fractional-order switched sliding surface is proposed for the delayed FOHNNs. Then, according to the fractional-order extension of the Lyapunov stability criterion, a fractional-order sliding mode controller is constructed to guarantee that the synchronization error of the two uncertain delayed FOHNNs converges to an arbitrary small region of the origin. Finally, a numerical example of two-dimensional uncertain delayed FOHNNs is given to verify the effectiveness of the proposed method.


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## Body

## 1. Introduction

The research of neural networks (NNs) is quite extensive, reflecting the characteristics of multidisciplinary technology. NNs have many successful applications in the fields of associative memories and image processing. Recently, the discussion on NNs has become a hot topic [1–3]. Guo et al. [4] studied the exponential stability analysis for complex-valued memristor-based bidirectional associative memory (BAM) NNs with time delays. Lv et al. [5] used NNs to discuss the adaptive tracking control for a class of uncertain nonlinear systems. Li et al. [6] studied Hopf bifurcation analysis of complex-valued neural networks model.Fractional calculus (FC) has a long history. As early as 1695, the concept of fractional differential was mentioned in Leibnitz’s letter to L’Hospital. For a long time, FC continues to grow. Podlubny’s book [7] systematically introduced the concepts and properties of FC. Bai et al. (see [8–13], and the references therein) studied the existence and uniqueness of solutions for fractional differential equations (FDE). Wang et al. [14–16] studied the numerical analysis of FDE. In recent years, fractional-order systems (FOS) have attracted wide attentions. The control problems of all kinds of FOS were studied recently [17–21]. Many researchers focused on fractional-order neural networks (FONNs) [22–27]. Cao et al. [28] investigated the existence and uniqueness of the nontrivial solution of NNs and the uniform stability of the FONNs.The researches on the stability of NNs, FOS, and stochastic systems have attracted the attention of a large number of researchers, and many achievements have been made [29–41]. The sliding mode control (SMC) is a very popular strategy for a general class of nonlinear uncertain systems, with a very large frame of applications fields. Due to the use of the discontinuous function, its main features are the robustness of closed-loop system and the finite-time convergence. Utkin et al. [42] studied the minimum possible value of control based on adaptation SMC methodology. Efe. et al. [43] discussed the fractional fuzzy adaptive SMC. Aghababa [44] designed a chatter-free terminal sliding mode controller for nonlinear fractional-order dynamical systems. The synchronization problems of FOHNNs have captured more and more researchers’ attention [45–48]. Xi et al. [22] have discussed SMC for uncertain FOHNNs. Liu et al. [24] have researched adaptive synchronization of a class of FONNs. It is well known that time delay is unavoidable due to finite switching speeds of the amplifiers, and it may cause oscillations or instability of dynamic systems. Wang et al. [26] have discussed the stability analysis of FOHNNs with time delay.However, to the best of our knowledge, there are few attentions to adaptive synchronization for a class of uncertain delayed FOHNNs subject to external disturbances. The SMC technology was used to solve the above problems in the paper. The rest of this paper is organized as follows: some necessary definitions and lemmas are given in Section2. The main works including the introduction of fractional-order network model, the fractional-order switched sliding mode surface (SMS), the design of adaptive synchronization controller, and stability analysis are included in Section 3. Section 4 presents a simulation example. Finally, the paper is concluded in Section 5.


## 2. Preliminaries

There are several kinds of definitions for fractional-order derivatives. The definitions of more frequency of use in literatures are Grünwald-Letnikov, Riemann-Liouville, and Caputo definitions [7]. These definitions are generally not equivalent with each other. The Caputo’s derivative’s Laplace transform requires integer-order derivatives for the initial conditions, which was used in engineering applications frequently. But, the Riemann-Liouville definition’s Laplace transform involved fractional-order derivatives for the initial conditions. It was hard to use physically. In the following parts, the Caputo’s derivative will be used [24]. Firstly, we give some definitions and lemmas.Definition 1 (Riemann-Liouville fractional-order integral [7]).
The Riemann-Liouville fractional integral of orderα for a function f(t) is defined as(1)Dt0t-αft=1Γα∫t0tfτt-τ1-αdτ.where α>0, t≥t0. Γ(α) is Euler’s gamma function.The gamma functionΓ(α) is defined for all complex numbers except the nonpositive integers. For complex numbers with a positive real part, it is defined via a convergent infinite integral: (2)Γα=∫0+∞τα-1e-τdτ.Definition 2 (Caputo fractional-order derivative [7]).
The Caputo fractional derivative of orderα for a function f(t) is defined as(3)Dt0Ctαft=1Γn-α∫t0tt-τn-α-1fnτdτ.where α>0, n is an integer satisfying n-1≤α<n. Particularly, for 0<α<1 case, one can get(4)Dt0Ctαft=1Γ1-α∫t0tt-τ-αf′τdτ.According to Definition2, for any constants L1∈R and L2∈R, the linearity of Caputo’s fractional derivative is described by (5)Dt0CtαL1ft+L2gt=L1Dt0Ctαft+L2Dt0Ctαgt.In nonlinear control systems, Lyapunov second method gives a way to analyze the stability of the system without explicitly solving the differential equations. The Lyapunov stability theory for FOS has been developed by Li et al. [33]. One of the main contributions of [33] is the following lemma.Consider the fractional-order nonlinear system:(6)Dt0Ctαxt=fx,t,xt0=xt0,where x(t)∈Rn is the state vector and f(x,t)∈Rn is a Lipschitz continuous nonlinear function.Lemma 3 (see [49]).
LetG(t) be a continuous function on [0,+∞), if there exist constants κ1>0 and κ2>0, such that(7)Dt0CtαGt≤-κ1Gt+κ2,t≥0.Then, (8)Gt≤G0Eα-κ1tα+κ2tαEα,α+1-κ1tα,t≥0. where 0<α<1, Eα(·) and Eα,α+1(·) are one-parameter Mittag-Leffler function and two-parameter Mittag-Leffler function, respectively.Remark 4.
Mittag-Leffler stability means asymptotical stability [32].Lemma 5 (see [7]).
Ifx(t)∈C1[0,T], for α>0 and T>0, then the following equations hold:(9)D0CtαD0t-αxt=xt. and(10)D0t-αD0Ctαxt=xt-∑k=0n-1xktk!tk. In particular, for 0<α<1,(11)D0t-αD0Ctαxt=xt-x0.Lemma 6 (see [21]).
Letx(t)⊂Rn be a continuous and derivable function. Then for any t>0 the following inequality holds:(12)12D0CtαxTtxt≤xTt0CDtαxt


## 3. Main Results

In this section, considering a system of the uncertain FOHNNs with delay (as a master system)(13)D0Ctαxit=-aitxit+∑j=1nbijfjxjt+∑j=1ncijgjxjt-τ+Ii.where i=1,2,…,n, 0<α<1, n is the number of units in a neural network, xi(t) is the state of the ith unit at time t, fj, gj denotes the activation function of the jth neuron, bij,cij denotes the constant connection weight of the jth neuron on the ith neuron, ai>0 represents the rate with which the ith neuron resets its potential to the resting state when disconnected from the network and ai>0 is unknown, Ii denotes the constant external inputs, and τ is the transmission constant delay.Let us discuss the synchronization results, assuming that (13) is a master system and the slave system is defined by the following equation:(14)D0Ctαyit=-aityit+∑j=1nbijfjyjt+∑j=1ncijgjyjt-τ+Ii+dit+uit.where yi(t) is the state of the ith unit at time t, di(t) is the unknown external disturbance, and ui(t) is the control input which will be given later.Defining the synchronization errorei(t) as(15)eit=yit-xitthen the error dynamics between the master system (13) and the slave system (14) can be written as(16)D0Ctαeit=-aiteit+∑j=1nbijfjyjt-fjxjt+∑j=1ncijgjyjt-τ-gjxjt-τ+dit+uit.Assumption 7.
Assuming that the nonlinear functionsfj and gj(j=1,2,…,n) satisfy local Lipschitz conditions, and existing positive constants Lj1 and Lj2 such that(17)fjyjt-fjxjt≤Lj1yjt-xjt,gjyjt-τ-gjxjt-τ≤Lj2yjt-τ-xjt-τ.Assumption 8.
Let the external disturbancedi(t)(i=1,2,…,n) be a bounded continuous function, so there exists an unknown positive constant ρi such that(18)dit≤ρiFor the sake of simplicity, this article only discusses the constant unknown disturbance.


### 3.1. Controller Design

Generally, designing the process of SMC has two steps. Firstly, an appropriate SMS is designed, which represents the required system dynamic characteristics. In this paper, a switching fractional-order SMS is given as(19)sit=D0t-1D0Ctαeit+pieit+qisigneit, where i=1,2,…,n, ei(t) is the state of the error system (16), and pi and qi are positive constants. sign(·) is the symbolic function.(20)signeit=1eit>0,-1eit<0,∈-1,1eit=0.According to the SMC theory, when the system operates in SMS, the SMS and its derivative must satisfy(21)sit=0,s˙it=0.As a result, considering ((19)-(21)), one obtains(22)s˙it=D0Ctαeit+pieit+qisigneit=0. Then, we have the sliding mode equation (SME)(23)0CDtαeit=-pieit+qisigneit. In the next parts, we construct the SMC law ui(t) to make sure the state trajectories of system (16) reach the SMS si(t)=0 and keep on it forever by the SMC method. The fractional-order SMC law is presented as(24)uit=--a^it+pieit-d^it-sit∑j=1nbijLjfejt+∑j=1ncijLjgejt-τ-qisigneit-ζiasit-ζibsignsit, where ζia>0 and ζib>0 are constant gains, a^i(t) is the estimation of ai(t), d^i(t) is the estimation of di(t), and the unknown parameters ai(t) and di(t) are estimated as(25)a^˙it=-ηiasiteit,d^˙it=ηidsit.where i=1,2,…,n, ηia>0 and ηid>0 are adaptation gains.In order to realize SMC, two steps are required. Firstly, the system trajectories are controlled to reach the SMSsi(t)=0, which is shown in Theorem 9. Secondly, once the system operates in SMS, we should get the stability of the error system (16) and make sure SMS converge to zero in finite time, which is shown in Theorem 12.Theorem 9.
For the uncertain delayed FOHNNs (16), if the system is controlled by the SMC law (24) and (25), then the system trajectories will converge to the SMS si(t)=0 in finite time.Proof.
Choose the positive definite Lyapunov function candidate(26)Vit=12sit2+12ηiaa^it-ait2+12ηidd^it-di2. Taking the integer-order derivative of Vi(t), we have(27)V˙it=sits˙it+1ηiaa^it-aia^˙it+1ηidd^it-did^˙it. Inserting s˙i(t) from (22) into the above equation, one has(28)V˙it=sitD0Ctαei+piei+qisignei+1ηiaa^it-aia^˙it+1ηidd^it-did^˙it. Based on (16), we get(29)V˙it=sit-aieit+∑j=1nbijfjyjt-fjxjt+∑j=1ncijgjyjt-τ-gjxjt-τ+di+uit+pieit+qisigneit+1ηiaa^it-aia^˙it+1ηidd^it-ditd^˙it. According to Assumption 7, we will obtain(30)sit∑j=1nbijfjyjt-fjxjt≤sit∑j=1nbijfjyjt-fjxjt≤sit∑j=1nbijLjfejt Correspondingly, we have(31)sit∑j=1ncijgjyjt-τ-gjxjt-τ≤sit∑j=1ncijLjgejt-τ Combining ((29)-(31)), we can get the following conclusion:(32)V˙it≤sit-ai+pieit+qisigneit+di+uit+sit∑j=1nbijLjfejt+sit∑j=1ncijLjgejt-τ+1ηiaa^it-aia^˙it+1ηidd^it-did^˙it. Substituting ui(t) from (24) into (32), it yields(33)V˙it≤sit-ai+pieit+qisigneit+di--a^i+pieit-d^it-sit∑j=1nbijLjfejt+∑j=1ncijLjgejt-τ-qisigneit-ζiasit-ζibsignsit+sit∑j=1nbijLjfejt+sit∑j=1ncijLjgejt-τ+1ηiaa^it-aia^˙it+1ηidd^it-did^˙it. Through operation, we get(34)V˙it≤sita^it-aieit-d^it-di-ζiasit-ζibsignsit+1ηiaa^it-aia^˙it+1ηidd^it-did^˙it. Insert (25)(35)V˙it≤sita^it-aieit-d^it-di-ζiasit-ζibsignsit-1ηiaa^it-aiηiasiteit+1ηidd^it-diηidsit. Then, one obtains(36)V˙it≤-ζiasi2t-ζibsignsitsit. Using sign(si(t))si(t)=sit and property of inequality, we get(37)V˙it≤-ζibsit<0,where ζib>0. Therefore, according to Lyapunov theory, the system states will converge to SMS si(t)=0. Hence, the proof is achieved completely.Remark 10.
Theorem9 gets the error systems trajectories to reach the sliding surface si(t)=0 in finite time.Remark 11.
Time delay and external disturbance have little influence on the error system (16).


### 3.2. Stability of Sliding Mode

For the SME (23), we choose the positive definite Lyapunov function(38)Vit=12ei2t. Taking the fractional-order derivative of Vi(t) and using Lemma 6, we get(39)D0CtαVit≤eitD0Ctαeit=eit-piei-qisigneit=-piei2t-qieitsigneit=-piei2t-qieit≤-qieit=-2qiVi1/2t,where qi>0. As a result, according to Lemma 3 and Remark 4, ei will converge to 0 asymptotically.Therefore, the state trajectories of system (23) will converge to 0, so one has the following conclusion.Theorem 12.
The sliding mode dynamics system (23) is asymptotically stable, and its states ei(t) converge to 0.Corollary 13.
By Theorems9 and 12, system (16) is asymptotically stable, which means that system (14) can synchronize system (13).


## 3.1. Controller Design

Generally, designing the process of SMC has two steps. Firstly, an appropriate SMS is designed, which represents the required system dynamic characteristics. In this paper, a switching fractional-order SMS is given as(19)sit=D0t-1D0Ctαeit+pieit+qisigneit, where i=1,2,…,n, ei(t) is the state of the error system (16), and pi and qi are positive constants. sign(·) is the symbolic function.(20)signeit=1eit>0,-1eit<0,∈-1,1eit=0.According to the SMC theory, when the system operates in SMS, the SMS and its derivative must satisfy(21)sit=0,s˙it=0.As a result, considering ((19)-(21)), one obtains(22)s˙it=D0Ctαeit+pieit+qisigneit=0. Then, we have the sliding mode equation (SME)(23)0CDtαeit=-pieit+qisigneit. In the next parts, we construct the SMC law ui(t) to make sure the state trajectories of system (16) reach the SMS si(t)=0 and keep on it forever by the SMC method. The fractional-order SMC law is presented as(24)uit=--a^it+pieit-d^it-sit∑j=1nbijLjfejt+∑j=1ncijLjgejt-τ-qisigneit-ζiasit-ζibsignsit, where ζia>0 and ζib>0 are constant gains, a^i(t) is the estimation of ai(t), d^i(t) is the estimation of di(t), and the unknown parameters ai(t) and di(t) are estimated as(25)a^˙it=-ηiasiteit,d^˙it=ηidsit.where i=1,2,…,n, ηia>0 and ηid>0 are adaptation gains.In order to realize SMC, two steps are required. Firstly, the system trajectories are controlled to reach the SMSsi(t)=0, which is shown in Theorem 9. Secondly, once the system operates in SMS, we should get the stability of the error system (16) and make sure SMS converge to zero in finite time, which is shown in Theorem 12.Theorem 9.
For the uncertain delayed FOHNNs (16), if the system is controlled by the SMC law (24) and (25), then the system trajectories will converge to the SMS si(t)=0 in finite time.Proof.
Choose the positive definite Lyapunov function candidate(26)Vit=12sit2+12ηiaa^it-ait2+12ηidd^it-di2. Taking the integer-order derivative of Vi(t), we have(27)V˙it=sits˙it+1ηiaa^it-aia^˙it+1ηidd^it-did^˙it. Inserting s˙i(t) from (22) into the above equation, one has(28)V˙it=sitD0Ctαei+piei+qisignei+1ηiaa^it-aia^˙it+1ηidd^it-did^˙it. Based on (16), we get(29)V˙it=sit-aieit+∑j=1nbijfjyjt-fjxjt+∑j=1ncijgjyjt-τ-gjxjt-τ+di+uit+pieit+qisigneit+1ηiaa^it-aia^˙it+1ηidd^it-ditd^˙it. According to Assumption 7, we will obtain(30)sit∑j=1nbijfjyjt-fjxjt≤sit∑j=1nbijfjyjt-fjxjt≤sit∑j=1nbijLjfejt Correspondingly, we have(31)sit∑j=1ncijgjyjt-τ-gjxjt-τ≤sit∑j=1ncijLjgejt-τ Combining ((29)-(31)), we can get the following conclusion:(32)V˙it≤sit-ai+pieit+qisigneit+di+uit+sit∑j=1nbijLjfejt+sit∑j=1ncijLjgejt-τ+1ηiaa^it-aia^˙it+1ηidd^it-did^˙it. Substituting ui(t) from (24) into (32), it yields(33)V˙it≤sit-ai+pieit+qisigneit+di--a^i+pieit-d^it-sit∑j=1nbijLjfejt+∑j=1ncijLjgejt-τ-qisigneit-ζiasit-ζibsignsit+sit∑j=1nbijLjfejt+sit∑j=1ncijLjgejt-τ+1ηiaa^it-aia^˙it+1ηidd^it-did^˙it. Through operation, we get(34)V˙it≤sita^it-aieit-d^it-di-ζiasit-ζibsignsit+1ηiaa^it-aia^˙it+1ηidd^it-did^˙it. Insert (25)(35)V˙it≤sita^it-aieit-d^it-di-ζiasit-ζibsignsit-1ηiaa^it-aiηiasiteit+1ηidd^it-diηidsit. Then, one obtains(36)V˙it≤-ζiasi2t-ζibsignsitsit. Using sign(si(t))si(t)=sit and property of inequality, we get(37)V˙it≤-ζibsit<0,where ζib>0. Therefore, according to Lyapunov theory, the system states will converge to SMS si(t)=0. Hence, the proof is achieved completely.Remark 10.
Theorem9 gets the error systems trajectories to reach the sliding surface si(t)=0 in finite time.Remark 11.
Time delay and external disturbance have little influence on the error system (16).


## 3.2. Stability of Sliding Mode

For the SME (23), we choose the positive definite Lyapunov function(38)Vit=12ei2t. Taking the fractional-order derivative of Vi(t) and using Lemma 6, we get(39)D0CtαVit≤eitD0Ctαeit=eit-piei-qisigneit=-piei2t-qieitsigneit=-piei2t-qieit≤-qieit=-2qiVi1/2t,where qi>0. As a result, according to Lemma 3 and Remark 4, ei will converge to 0 asymptotically.Therefore, the state trajectories of system (23) will converge to 0, so one has the following conclusion.Theorem 12.
The sliding mode dynamics system (23) is asymptotically stable, and its states ei(t) converge to 0.Corollary 13.
By Theorems9 and 12, system (16) is asymptotically stable, which means that system (14) can synchronize system (13).


## 4. Numerical Simulations

The effectiveness of the obtained theoretical results is shown by the example. Considering the two-dimensional uncertain delayed FOHNNs (as the Master system)(40)D0Ctαx1t=-x1+0.5sin⁡x1t+sin⁡x2t+0.5tanh⁡x1t-0.8+tanh⁡x2t-0.8+0.2,D0Ctαx2t=-0.5x2+sinx1t-0.5sinx2t-0.5tanhx1t-0.8-tanhx2t-0.8+0.3where α=0.9, the initial conditions are x1(0)=-5,x2(0)=5.The form of the slave system is given by(41)D0Ctαy1t=-y1+0.5siny1t+siny2t0.5tanhy1t-0.8+tanhy2t-0.8+0.2+0.1+u1t,D0Ctαy2t=-0.5y2-siny1t-0.5siny2t-0.5tanhy1t-0.8-tanhy2t-0.8+0.3+0.15+u2tAssume that the initial conditions are y1(0)=-3,y2(0)=3, and α=0.9.Choosingp1=p2=0.5,q1=q2=1.5,L1f=L2f=L1g=L1g=1, η1a=η2a=1.1,η1d=η2d=1.2,ζ1a=ζ2a=1.1,ζ1b=ζ2b=1.2, one gets(42)a^˙1t=-1.1e1ts1ta^˙2t=-1.1e2ts2tand(43)d^˙1t=1.2s1td^˙2t=1.2s2tWe use (19) and design the SMS(44)s1t=D0t-1D0Ct0.9e1t+0.5e1t+1.5signe1ts2t=D0t-1D0Ct0.9e2t+0.5e2t+1.5signe2t.Thus, according to (24), the control inputs are obtained as(45)u1t=a^1t-0.5e1t-d^1t-1.5signe1t-1.1s1t-1.2signs1t-s1t0.5e1t+e2t+0.5e1t-0.8+e2t-0.8,u2t=a^2t-0.5e2t-d^2t-1.5signe2t-1.1s2t-1.2signs2t-s2t-e1t-0.5e2t-0.5e1t-0.8-1e2t-0.8.The simulation results are depicted in Figures1–6. Figures 1–3 show the synchronization between two fractional-order neural networks and the time response of the synchronization errors. The time response of the updated parameters and the sliding surfaces are included in Figures 4, 5, and 6, respectively. From the results, we can see that the synchronization errors converge to origin rapidly, and favorable synchronization performance has been achieved.Figure 1
Synchronization betweenx1(t) (blue line) and y1(t) (green line).Figure 2
Synchronization betweenx2(t) (blue line) and y2(t) (green line).Figure 3
Synchronization errorse1(t) (blue line), e2(t) (green line).Figure 4
The time response of control parameters:a^1 (blue line) and a^2(green line).Figure 5
The time response of control parameters:d^1 (blue line) and d^2 (green line).Figure 6
The sliding surfaces:s1(t) (blue line) and s2(t) (green line).


## 5. Conclusion

The adaptive synchronization problem for FOHNNs with system uncertainties, time delay, and external disturbances has been studied by SMC. Some estimations for system uncertainties and external disturbances are made. Firstly, establishing a switched SMS, the finite-time stability of the SMS to origin is proved according to the fractional-order Lyapunov theory. Secondly, an adaptive synchronization fractional-order sliding mode controller is designed to force the error systems trajectories to reach the switching SMS and remain on it forever. The effectiveness and feasibility of theoretical results are verified by the numerical simulations. Furthermore, taking advantage of the SMC theory, the stability problems of FOHNNs with both multiple time delays and impulses will be discussed in future works.



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