# Adaptive Feedback Control for Synchronization of Chaotic Neural Systems with Parameter Mismatches

Authors: Qian Ye; Zhengxian Jiang; Tiane Chen
Journal: Complexity
 (2018)

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## Abstract

This work pertains to the study of the synchronization problem of a class of coupled chaotic neural systems with parameter mismatches. By means of an invariance principle, a rigorous adaptive feedback method is explored for synchronization of a class of coupled chaotic delayed neural systems in the presence of parameter mismatches. Finally, the performance is illustrated with simulations in a two-order neural systems.


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## Body

## 1. Introduction

Pecora and Carroll firstly addressed the chaotic synchronization in systems and used the drive-response conception in [1]. The idea is to control the response signal by using the output signal of drive system such that the two kinds of signals synchronize. The problems of synchronization in chaotic dynamical systems have received increasing attention in the control areas [2–6]. Different approaches including adaptive design control [2], intermittent control [3], adaptive-impulsive control [4], and sliding mode control [6] have been proposed. In particular, Liu et al. [3] obtained novel synchronization criteria for exponential synchronization of chaotic systems with time delays via periodically intermittent control. By an adaptive feedback control technique, the synchronization of a class of chaotic systems with unknown parameters is achieved via mimicking model reference adaptive control-like structure in [2]. Tam et al. [5] addressed adaptive synchronization of complicated chaotic systems with unknown parameters via a set of fuzzy modeling-based adaptive strategy. Chen et al. [6] designed a sliding mode control scheme for adaptive synchronization of multiple response systems under the effects of external disturbances.Recently, there has sprung up hot research topics in the synchronization of chaotic neural systems (CNSs) due to possible chaotic behaviors in such systems [7–13]. For instance, synchronization of coupled delayed CNSs and applications to memristive CNSs in [12] have resulted in a theoretical condition under an irreducible assumption on coupling matrix. Cao and Lu [13] proposed a simple adaptive method for the synchronization of uncertain CNSs with or without variable delay via invariant principles. In particular, some efforts have been devoted to adaptive synchronization of CNSs [10–12, 14]. However, most existing works were applicable only for the CNSs with parameter matching. While in practical implementation of synchronized CNSs, it is well known that parameter mismatch in systems is generally inevitable [15–17], which will result in poor performance or loss of synchronization [17, 18]. For example, Zhang et al. [18] discussed asymptotic synchronization for delayed CNSs with fully unknown parameters based on the Lyapunov method and the inverse optimal method. Therefore, it is of importance to explore the effects of parameter mismatch in synchronization of CNSs.In this paper, we present theoretical analysis and numerical simulations of the parameter mismatch effect on synchronization for a class of coupled CNSs. By using adaptive control approaches [13, 19, 20] instead of traditional linear coupling scheme, and combining the invariance principle, we show that adaptive synchronization of such CNSs with parameter mismatches under loose conditions can be rapidly achieved. In addition, by adjusting the update gain of coupling strength introduced in this work, one can control the synchronization speed.The organization of this work is as follows. In Section2, we present needed formulation of synchronization of CNSs. Section 3 presents an adaptive control scheme in CNSs and provides two criteria for synchronization for CNSs. In Section 4, numerical simulations on a two-order CNS are provided to show the effectiveness of the proposed results. Section 5 concludes the paper.Notation 1.
Throughout the paper, we denoteAT and A−1 the transpose and the inverse of any square matrix A. A>0A<0 denotes a positive- (negative-) definite matrix A; and I is used to denote the n×n identity matrix. A denotes the spectral norm of matrix A. Let R denote the set of real numbers, Rn denotes the n-dimensional Euclidean space, and Rn×m denotes the set of all n×m real matrices. λmax· or λmin· denotes the largest or smallest eigenvalue of a matrix, respectively.


## 2. Formulation of Synchronization in Neural Networks

Consider the following CNS in a general form:(1)ẋt=−C1xt+A1fxt+B1fxt−τt+J,where xt=x1t,…,xntT∈Rn denotes the state vector; C1 represents a diagonal matrix with c1i>0,i=1,2,…,n; A1=a1ijn×n denotes the weight matrix; B1=b1ijn×n denotes the delayed weight matrix; J=J1,…,JnT∈Rn is the input vector function; τt represents the transmission variable delay; and fxt=f1x1t,…,fnxntT represents the activation function.Throughout the paper, we have the following two assumptions:A1 Each fj:R→R satisfies the Lipschitz condition, that is, there exist positive scalars kj>0 such that
(2)fjx−fjy≤kjx−y,j=1,2,…,n,for any x,y∈R.A2 τt≥0 is a function satisfying τ∗=maxtτt and 0≤τ̇t≤σ<1, for all t.xit=ϕit∈C−τ∗,0,R denotes initial conditions of (1), where C−τ∗,0,R represents the set of continuous functions from −τ∗,0 to R.To synchronize the drive (or master) system (1), the controlled response (or slave) system is given by
(3)ẏt=−C2yt+A2fyt+B2fyt−τt+J+ut,where ut is the driving signal, yt=y1t,…,yntT∈Rn. C2, A2, and B2 are generally different from C1, A1, and B1, respectively. Namely, parameter mismatches exist between the drive system and the response system. The initial conditions of system (3) denote yit=ψit∈C−τ∗,0,Ri=1,2,…,n. Denote the mismatch errors by ΔC=C2−C1, ΔA=A2−A1, and ΔB=B2−B1.We aim to design an appropriate controllerut in order to make the coupled CNSs remain synchronized in the presence of even large parameter mismatches. First, consider the feedback controller ut=ϵ∘yt−xt, where ϵ=ϵ1,…,ϵnT∈Rn is the coupling strength, and the symbol ∘ is defined as
(4)ϵ∘yt−xt≜ϵ1y1t−x1t,…,ϵnynt−xntT.Define the synchronization error aset=yt−xt, which leads to the following synchronization error system:
(5)ėt=−C2et+A2get+B2get−τt−C2−C1xt+A2−A1fxt+B2−B1fxt−τt+ϵ∘et,or
(6)ėt=−C2et+A2get+B2get−τt−ΔCxt+ΔAfxt+ΔBfxt−τt+ϵ∘et,where
(7)et=e1t,…,entT,get=ge1t,…,gentT,with gieit=fieit+xit−fixit,i=1,2,…,n.Obviously, using the assumptionA1, gi· has the following properties:
(8)giei≤kiei.


## 3. Adaptive Control Scheme

In this section, based on Lyapunov function and an invariance principle by combining an adaptive control approach, we consider the adaptive synchronization for two CNSs with time-varying delay and parameter mismatches.Theorem 1.
Suppose thatχ=x∈Rn∣x≤α1 and the parameter mismatches satisfy ΔC+ΔA+1/1−σΔBK2≤α2, where K=max1≤i≤nki. Under the assumptions A1 and A2, let α=α1·α2 and the controller ut=ϵ∘yt−xt=ϵ∘et with the following update law:
(9)ϵ̇i=−δiei2+αϵi+lxi2t,where δi>0 i=1,2,…,n are arbitrary constants, and l>0 is a constant to be determined. Then, the controlled uncertain response system (3) will globally synchronize with the drive system (1).Proof 1.
Consider the following Lyapunov function for the error dynamical system:(10)Vt=eTtet+∑i=1n1δiϵi+l2+11−σ∫t−τttgTesgesds+11−σ∫t−τttfTxsΔBTΔBfxsds.
Calculating the derivative of (10) along the trajectories of (6), we have
(11)V̇t=2eTtėt−2∑i=1nϵi+lei2+αϵi+lxi2t+11−σgTetget+11−σfTxtΔBTΔBfxt−1−τ̇t1−σgTet−τtget−τt−1−τ̇t1−σfTxt−τtΔBTΔBfxt−τt≤2eTt−C2et+A2get+B2get−τt−2eTtΔCxt+2eTtΔAfxt+2eTtΔBfxt−τt+2eTtϵet−2∑i=1nϵi+lei2+αϵi+lxi2t+11−σgTetget+11−σfTxtΔBTΔBfxt−gTet−τtget−τt−fTxt−τtΔBTΔBfxt−τt≤2eTt−C2et+A2get+B2get−τt−2eTtΔCxt+2eTtΔAfxt+2eTtΔBfxt−τt−2α∑i=1nxi2t−leTtet+11−σgTetget+11−σfTxtΔBTΔBfxt−gTet−τtget−τt−fTxt−τtΔBTΔBfxt−τt.
Using (8) and the Lemma 2 in [21], one can obtain
(12)2eTtA2get≤eTtA2A2Tet+gTetget,(13)2eTtB2get−τt≤eTtB2B2Tet+gTet−τtget−τt,(14)−2eTtΔCxt≤et+ΔCxt,(15)2eTtΔAfxt≤et+ΔAK2xt,(16)2eTtΔBfxt−τt≤et+fTxt−τtΔBTΔBfxt−τt,(17)11−σfTxtΔBTΔBfxt≤11−σΔBK2xt.
Substituting (12), (13), (14), (15), (16), and (17) into (11) and combining
(18)ΔC+ΔA+11−σΔBK2xt≤2αyield
(19)V̇t≤eTt−2C2+A2A2T+B2B2T+3−lIet+gTetget+11−σgTetget≤eTt2λmax−C2+λmaxA2A2T+λmaxB2B2T+K2+11−σK2+3−let.
We properly choose the constantl as
(20)l=2λmax−C2+λmaxA2A2T+λmaxB2B2T+K2+11−σK2+4,then we have V̇≤−eTtet.
It is obvious thatV̇=0 if and only if ei=0,i=1,2,…,n. It implies that the set E=et,ϵT∈R2n:et=0,ϵ=ϵ0∈Rn is the largest invariant set included in M=V̇=0 for system (6). Then, using the well-known Lyapunov-LaSalle-type theorem, the error converges asymptotically to E, that is, et→0 and ϵ→ϵ0 as t→∞. Therefore, the synchronization of the CNSs (1) and (3) is achieved under the coupling (9). The proof is completed.
For the coupled CNSs without time-varying delay (i.e.,B1=0 in (1) and B2 in (3)), one can easily derive the following corollary for two CNSs without delay (drive system and response system, resp.):
(21)ẋt=−C1xt+A1fxt+J,and
(22)ẏt=−C2yt+A2fyt+J+ϵ∘yt−xt.Corollary 1.
Suppose thatχ=x∈Rn∣x≤α1 and the parameter mismatches satisfy ΔC+ΔAK2≤α2, where K=max1≤i≤nki. Under the assumptions A1 and A2, set α=α1·α2 and the controller ut=ϵ∘yt−xt=ϵ∘et with the following update law:
(23)ϵ̇i=−δiei2+αϵi+lxi2t,where δi>0 i=1,2,…,n are arbitrary constants; l is chosen as
(24)l=2λmax−C2+λmaxA2A2T+K2+11−σK2+3.
Then, the controlled uncertain response system (3) will globally synchronize with the drive system (1).Remark 1.
It is noted from Theorem1 that one can choose the constant δi properly to adjust the synchronization speed. Large adaptive gain δi will lead to fast synchronization, while small adaptive gain δi will result in slow synchronization. In addition, such a way is robust against the effect of noise. An extension effort that extends the results for systems with hybrid characteristics as in [7, 22–24] is possible, which remains an open problem.Remark 2.
In [9, 11, 13], the derived results are applicable for CNSs with parameter matches. While the results here are suitable for parameter mismatches. Therefore, our results have more expansive application foreground. In addition, using adaptive feedback method, the criterion obtained here improves and extends the results reported in [9, 11, 13].


## 4. Numerical Simulations

In this section, a numerical example is employed to illustrate our results. Simulation results show that the proposed adaptive synchronization scheme is valid.Example 1.
Consider the following two-order CNSs with time-varying delay:(25)ẋt=−C1xt+A1fxt+B1fxt−τt+J,with
(26)C1=1001,A1=2.1−0.12−5.13.2,B1=−1.6−0.1−0.2−2.4,J=00,and
τt=et/1+et, where xt=x1t,x2tT, fxt=tanhx1t,tanhx2tT.
It is seen thatk1=k2=1, and thus K=1. Moreover,
(27)τ∗=1,τ̇t=et1+et2∈0,0.5,that is, σ=0.5. Therefore, A1 and A2 hold.Note that the neural system in this example is chaotic. Figure1 shows the phase plot of the CNS, and Figure 2 illustrates the power spectral with initial values ϕ1s=−0.5 and ϕ2s=0.3,∀s∈−1,0. It is found in Figure 1 that the double-scroll attractor is confined within the set.
(28)χ=x=x1,x2T∣−1≤x1≤1,−5≤x2≤5.Figure 1
Attractor of neural network model [21].Figure 2
Power spectral plot of neural network model [21].In this case, it is verified thatα1≃5.0990.To verify the effectiveness of the proposed method, consider the output signals of drive system in CNS (25). Then, the controlled response system is given by
(29)ẏt=−C~yt+A~fyt+B~fyt−τ~t+J+ut,where τ~t=1+1/2sint, yt=y1t,y2tT and
(30)C2=1.1000.8,A2=2.2−0.1−5.03.1,B2=−1.7−0.05−0.3−2.3,J=00.Therefore, we obtain(31)ΔC+ΔA+11−σΔBK2≤α2=0.7128,then α=α1·α2=3.6345.By Theorem1, since
(32)2λmax−C2+λmaxA2A2T+λmaxB2B2T+K2+11−σK2+4=48.9486,take l=50. Then, we can design the controller ut=ϵ∘yt−xt with the adaptive update laws
(33)ϵ̇1=−0.5y1t−x1t2+αϵ1+lx12t,ϵ̇2=−0.5y2t−x2t2+αϵ2+lx22t.Here, the adaptive gains are taken asδ1=δ2=0.5. Next, suppose the initial conditions are
(34)ϕ1s,ϕ2sT=0.2,0.5T,=ψ1s,ψ2sT=−1.3,2.1T,s∈−1,0,respectively, and ϵ10=ϵ20=0. The simulation results are depicted in Figures 3–6. Figure 3 shows the temporal evolution of states and errors for δ1=δ2=0.5. When δ1=δ2=0.3, namely, decreasing the update gain of coupling strength, Figure 4 shows the corresponding simulation results and it is revealed that it takes longer to achieve synchronization. From Figures 3 and 4, we found that less time is needed to achieve synchronization when larger δ1 and δ2 are taken. When δ1=0 and δ2=0.5, Figure 5 shows the results for the case that only x2 is chosen as the drive signal. When δ1=0.5 and δ2=0, Figure 6 shows the results for the case that only x1 is chosen as the drive signal. It can be seen that the coupling between x1 and y1 would drive the two CNSs (25) and (29) synchronized, while the coupling between x2 and y2 is invalid.Figure 3
The temporal evolution of each variable and the plot of synchronization errors when the adaptive gains areδ1=δ2=0.5.
(a)
System states
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Synchronization errorsFigure 4
The temporal evolution of each variable and the plot of synchronization errors when the adaptive adaptive gains areδ1=δ2=0.3.
(a)
System states
(b)
Synchronization errorsFigure 5
The temporal evolution of each variable and the plot of synchronization errors driven only by the signalx2, and the synchronization cannot be achieved in Example 1.
(a)
System states
(b)
Synchronization errorsFigure 6
The temporal evolution of each variable and the plot of synchronization errors driven only by the signalx1, and the synchronization is achieved in Example 1.
(a)
System states
(b)
Synchronization errors


## 5. Conclusions

This paper has analyzed the adaptive synchronization between two coupled CNSs with parameter mismatches by applying an invariance principle and a simple adaptive feedback approach. Practical and less restrictive conditions have been presented for adaptive synchronization of CNSs. Numerical simulations of two-order-coupled CNSs have also been provided to verify the usefulness and practicability of proposed theoretical results.



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