# Synchronization of Discrete-Time Stochastic Neural Networks with Random Delay

Authors: Haibo Bao; Jinde Cao
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

By using a Lyapunov-Krasovskii functional method and the stochastic
analysis technique, we investigate the problem of synchronization for discrete-time stochastic neural networks (DSNNs) with random delays. A control law is designed, and sufficient conditions
are established that guarantee the synchronization of two identical DSNNs with random delays.
Compared with the previous works, the time delay is assumed to be existent in a random fashion.
The stochastic disturbances are described in terms of a Brownian motion and the time-varying
delay is characterized by introducing a Bernoulli stochastic variable. Two examples are given to
illustrate the effectiveness of the proposed results. The main contribution of this paper is that the
obtained results are dependent on not only the bound but also the distribution probability of the
time delay. Moreover, our results provide a larger allowance variation range of the delay, and are
less conservative than the traditional delay-independent ones.


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## Body

## 1. Introduction

Synchronization is one of the most important dynamic behavior of complex networks, which means if two or more systems have something in common, they will adjust each other to give rise to a common dynamical behavior. It has been found applications in many fields such as synchronous information exchange in the Internet WWW, crickets chirping in synchrony, rhythmic applause, and synchronous transfer of digital or analog signals in the communication networks.Since the pioneering works of Pecora and Carroll [1, 2], the control and synchronization problems have become an active topic that attracts a lot of researchers' interest, including general complex dynamical networks [3–8], and the array of coupled neural networks with or without delays [6, 9–11]. Several different approaches have been proposed for the synchronization of chaotic systems, such as linear and nonlinear feedback control [12, 13], adaptive control [14], impulsive control [15], and intermittent linear state feedback [16].Itô-type stochastic systems are well known for their important impact on practical applications such as chemistry, biology, ecology, control, and information systems. In real complex networks, the signal transmission could be a noisy process brought by random fluctuations from the release of probabilistic causes such as neurotransmitters. Stochastic neural networks, as a special case of complex networks, have gained much more researchers' interests; see for example, [4, 5, 13, 15, 17, 18] and the references therein.Discrete-time neural networks play a more and more important role in engineering application. As pointed out in [19–21], the discretization cannot preserve the dynamics of continuous-time counterpart even for a small sampling period. Recently, the synchronization problem for discrete-time networks has received more attention [4, 5, 15, 22].Time delays occur frequently in practical situations, it can cause undesirable dynamic network behaviors such as oscillation and instability. Therefore, dynamical behavior [23, 24], especially synchronization problem for discrete-time neural networks with constant and time-varying delays has gained interesting research attention; see, for example, [4, 5, 15, 22]. It is worth mentioning that as a particular kind of time delays, random delays have also received much researchers' attention [25–31]. This is mainly because in many real systems, some values of the delay are very large, but the variation range of time delay taking such large values are very small. In this case, if only the variation range of time delay is employed to derive the criteria, the results may be somewhat more conservative.Inspired by the above discussion, the aim of this paper is to study the synchronization problem for a class of DSNNs with random delay. The effect of both variation range and distribution probability of the time delay are taken into account in the proposed approach. The stochastic disturbances are described in terms of a Brownian motion, and the time-varying delay is characterized by introducing a Bernoulli stochastic variable. By employing a Lyapunov-Krasovskii functional, sufficient delay-distribution-dependent conditions are established in terms of linear matrix inequalities (LMIs) that guarantee the exponentially mean square synchronization of two identical DSNNs with random delays, which can be checked readily by Matlab toolbox.This paper is organized as follows. In Section2, the model formulation and some preliminaries are given. The main results are stated in Section 3. Two illustrative examples are given to demonstrate the effectiveness of the proposed results in Section 4. Finally, the conclusions are made in Section 5.


### 1.1. Notation

Throughout this paper,ℝn and ℝn×m, respectively, denote the n-dimensional Euclidean and the set of all n×m matrices. I is the identity matrix of appropriate dimensions. The superscript “T” denotes matrix transposition. The notation X>0 (resp., X≥0), where X is a real symmetric matrix, means X is positive definite (resp., positive semidefinite). |·| is the Euclidean norm in ℝn. If A is a matrix, ∥A∥ denotes its operator norm. That is, ∥A∥=sup{|Ax|:|x|=1}=λmax(ATA), where λmax(A) (resp., λmin(A)) means the largest (resp., smallest) eigenvalue of A. ℤ≥0 denotes the set including zero and positive integers. The asterisk * in a matrix is used to denote term that is induced by symmetry. 𝔼{·} denotes the expectation. Moreover, let (Ω,ℱ,{ℱt}t≥0,𝒫) be a complete probability space with a filtration {ℱt}t≥0 satisfying the usual conditions. Denote by ℒℱ02([-τ,0],ℝn) the family of all ℱ0-measurable C([-τ,0]:ℝn)-valued random variables ϕ={ϕ(s),-τ≤s≤0} with the norm ∥ϕ∥=sup-τ≤s≤0|ϕ(s)|2<∞.


## 1.1. Notation

Throughout this paper,ℝn and ℝn×m, respectively, denote the n-dimensional Euclidean and the set of all n×m matrices. I is the identity matrix of appropriate dimensions. The superscript “T” denotes matrix transposition. The notation X>0 (resp., X≥0), where X is a real symmetric matrix, means X is positive definite (resp., positive semidefinite). |·| is the Euclidean norm in ℝn. If A is a matrix, ∥A∥ denotes its operator norm. That is, ∥A∥=sup{|Ax|:|x|=1}=λmax(ATA), where λmax(A) (resp., λmin(A)) means the largest (resp., smallest) eigenvalue of A. ℤ≥0 denotes the set including zero and positive integers. The asterisk * in a matrix is used to denote term that is induced by symmetry. 𝔼{·} denotes the expectation. Moreover, let (Ω,ℱ,{ℱt}t≥0,𝒫) be a complete probability space with a filtration {ℱt}t≥0 satisfying the usual conditions. Denote by ℒℱ02([-τ,0],ℝn) the family of all ℱ0-measurable C([-τ,0]:ℝn)-valued random variables ϕ={ϕ(s),-τ≤s≤0} with the norm ∥ϕ∥=sup-τ≤s≤0|ϕ(s)|2<∞.


## 2. Problem Formulation

Consider the followingn-neuron DSNNs with time delay(2.1)x(k+1)=Ax(k)+BF̃(x(k))+DG̃(x(k-τ(k))),
where x(k)=(x1(k),x2(k),…,xn(k))T∈ℝn is the state vector associated with the n neurons, A=diag(a1,a2,…,an) with |ai|<1; B=(bij)n×n, D=(dij)n×n denote the connection weights matrix and the delayed connection weights matrix, respectively, F̃(x(k))=[f̃1(x1(k)),f̃2(x2(k)),…,f̃n(xn(k))]T and G̃(x(k))=[g̃1(x1(k)),g̃2(x2(k)),…,g̃n(xn(k))]T denote the neuron activation functions; τ(k) denotes the time-varying delay.In this paper, we consider the model (2.1) as the drive system, and the noise-perturbed response system is given as(2.2)x̂(k+1)=Ax̂(k)+BF̃(x̂(k))+DG̃(x̂(k-τ(k)))+u(k)+σ(k,x̂(k)-x(k))ω(k),
where u(k)=(u1(k),u2(k),…,un(k))T∈ℝn is the state feedback controller given to achieve the global exponental synchronization between the drive-response system; σ:ℝ×ℝn→ℝn is a continuous function, ω(k) is a scalar Wiener process on a probability space (Ω,ℱ,𝒫) with (2.3)E{ω(k)}=0,E{ω2(k)}=1,E{ω(i)ω(j)}=0(i≠j).Throughout this paper, the following assumptions are made.Assumption 2.1.
Fori∈{1,2,…,n}, the neuron activation functions f̃i(·) and g̃i(·) are continuous and bounded and satisfy the following conditions:
(2.4)li≤f̃i(s1)-f̃i(s2)s1-s2≤Li,ωi≤g̃i(s1)-g̃i(s2)s1-s2≤Wi,∀s1,s2∈R(s1≠s2),i=1,2,…,n.
As first discussed in [32], for brevity of the following representation, we denote
(2.5)C1=diag⁡{l1L1,…,lnLn},C2=diag⁡{-l1+L12,…,-ln+Ln2},H1=diag⁡{ω1W1,…,ωnWn},H2=diag⁡{-ω1+W12,…,-ωn+Wn2}.Assumption 2.2.
σ(k,x(k)) is a continuous function satisfying σ(0,0)=0 and
(2.6)σ(k,x(k))Tσ(k,x(k))≤ρx(k)Tx(k),
where ρ>0 is a known constant scalar.Assumption 2.3.
The time delayτ(k) is bounded, 0≤τm<τ(k)≤τM, and its probability distribution can be observed; that is, suppose that τ(k) takes values in [τm:τ0] or (τ0:τM] and Prob{τ(k)∈[τm:τ0]}=δ0, where τm≤τ0<τM, and 0≤δ0≤1.Remark 2.4.
It is noted that the introduction of binary stochastic variable was first introduced in [25] and then successfully used in [26–31]. Under the Assumption 2.3, we know that δ0 is dependent on the values of τm, τ0, τM. In addition, Prob{τ(k)∈(τ0:τM]}=1-δ0≜δ¯0.In order to describe the probability distribution of the time delay, we define two sets(2.7)B1={k∣τ(k)∈[τm,τ0]},B2={k∣τ(k)∈(τ0,τM]},
where τ0 is an integer satisfying τm≤τ0<τM. Define two mapping functions (2.8)τ1(k)={τ(k),k∈B1,τm,else,τ2(k)={τ(k),k∈B2,τ0,else.
It follows from (2.7) that ℬ1∪ℬ2=ℤ≥0 and ℬ1∩ℬ2=∅. From (2.8), it can be seen that k∈ℬ1 implies the event τ(k)∈[τm:τ0] occurs and k∈ℬ2 implies the event τ(k)∈(τ0:τM] occurs. Defining a stochastic variable as (2.9)δ(k)={1,k∈B1,0,k∈B2,then system (2.1) and (2.2) can be equivalently rewritten as (2.10)x(k+1)=Ax(k)+BF̃(x(k))+δ(k)DG̃(x(k-τ1(k)))+(1-δ(k))DG̃(x(k-τ2(k))),(2.11)x̂(k+1)=Ax̂(k)+BF̃(x̂(k))+δ(k)DG̃(x̂(k-τ1(k)))+(1-δ(k))DG̃(x̂(k-τ2(k)))+u(k)+σ(k,x̂(k)-x(k))ω(k).It is further assumed that the variablesδ(k) and ω(k) are mutually independent.Remark 2.5.
Under the Assumption2.3 and the definition of δ(k), it can be seen that δ(k) is a Bernoulli distributed white sequence with Prob{δ(k)=1}=𝔼{δ(k)}=δ0 and Prob{δ(k)=0}=δ¯0. Furthermore, we can show that 𝔼{δ(k)-δ0}=0 and 𝔼{(δ(k)-δ0)2}=δ0δ¯0.Lettinge(k)=x̂(k)-x(k) be the synchronization error, we can derive error system as follows:(2.12)e(k+1)=Ae(k)+BF(e(k))+δ(k)DG(e(k-τ1(k)))+(1-δ(k))DG(e(k-τ2(k)))+σ(k,e(k))ω(k)+u(k),
where (2.13)F(e(k))=[f1(e1(k)),f2(e2(k)),…,fn(en(k))]T=F̃(x̂(k))-F̃(x(k)),G(e(k))=[g1(e1(k)),g2(e2(k)),…,gn(en(k))]T=G̃(x̂(k))-G̃(x(k)).The initial condition associated with (2.12) is given as (2.14)e(s)=ϕ(s),s∈N[-τM,0],whereϕ(s)∈ℒℱ02([-τM,0],ℝn), and ℕ[-τM,0]={-τM,-τM+1,…,0}.Lete(k,ϕ) be the state trajectory of system (2.12) under the initial condition. It is obvious that e(k,0)=0 is a trivial solution of DSNNs (2.12).Definition 2.6.
The drive system (2.10) and the response system (2.11) are said to be exponentially synchronized if, for a suitably designed feedback controller, the trivial solution of the error system (2.12) is globally exponentially stable in the mean square. That is, there exist constants ϑ>0, and ν∈(0,1) such that for sufficiently integer T>0, the inequality
(2.15)E|e(k)|2≤ϑνksup⁡s∈N[-τM,0]E|e(s)|2
holds for all k>T.For brevity of the following analysis, we denotee(k), e(k-τ1(k)), e(k-τ2(k)), δ(k), 1-δ(k), ω(k) by ek, eτ,1, eτ,2, δk, δ¯k, ωk, respectively. Then, (2.12) can be rearranged as(2.16)ek+1=Aek+BF(ek)+δkDG(eτ,1)+δ¯kDG(eτ,2)+σ(k,ek)ωk+u(k).


## 3. Main Results

In order to realize the synchronization between the drive system (2.10) and the noise-perturbed response system (2.11) with different initial conditions, we will design a suitable feedback controller and develop theoretical results of the synchronization scheme. In fact, if the trivial solution of the controlled error system (2.12) is exponentially stable in the mean square, (2.10) and (2.11) can achieve globally synchronization.For the noised-perturbed response system (2.11), the feedback controller u(k) is designed as(3.1)u(k)=Ke(k)+δ0K1e(k-τ1(k))+(1-δ0)K1e(k-τ2(k)),
where K and K1 are the feedback gains to be determined.Theorem 3.1.
Suppose that Assumptions2.1–2.3 hold, and let the state-feedback controller (3.1) be given. The noised-perturbed response system (2.11) is globally exponentially synchronized with the drive model (2.10) if there exist three positive definite matrices P, Q1, Q2, three diagonal matrices Λi=diag(λi1,λi2,…,λin)>0(i=1,2,3), two real matrices R, R1 and a scalar λ*>0 such that the following two LMIs hold:
(3.2)Ω=[Ξ100-Λ1C200ATP+RT*Ξ200-Λ2H20δ0R1T**Ξ300-Λ3H2δ¯0R1T***-Λ100BTP****-Λ20δ0DTP*****-Λ3δ¯0DTP******-P]<0,(3.3)P<λ*I,
whereΞ1=-P+λ*ρI+(τ0-τm+1)Q1+(τM-τ0+2)Q2-Λ1𝒞1, Ξ2=-Q1-Λ2ℋ1, Ξ3=-Q2-Λ3ℋ1.
And then the feedback gains can be designed as(3.4)K=P-1R,K1=P-1R1.Proof.
See the AppendixB.Remark 3.2.
We would like to point out that there is still enough room to improve the result. Because of (1) in real-time systems, time delays always exist in a stochastic fashion, so we also can consider the time delays satisfy other distributions. (2) And we can also extend this method to the dynamics of discrete-time stochastic complex networks. (3) The results can be improved by combining with delay-fractioning method to reduce conservatism.Remark 3.3.
Our results are less conservative than some other existed results because they are dependent on not only the bound but also the distribution probability of the time delays, and we obtain a larger allowance variation range of the delay, while the delay-fractioning or delay-partitioning approach [22] can reduce conservatism lie in the methods increase the number of fraction of the time delay. The delay-fractioning approach is a very effective approach to reduce conservatism. And we will combine with the free-weighting methods or delay partitioning approaches to reduce the conservativeness of the results in the future.Remark 3.4.
Whenδ(k)≡1(∀k∈ℤ≥0), which means τ1(k)≡τ(k), (2.10) reduces to (2.1). By setting δ0=1, Q2=0 in Theorem 3.1 and deleting the third and sixth rows and the corresponding third and sixth columns of (3.2), we can obtain the following results.Now, (2.11) reduces to(3.5)x̂(k+1)=Ax̂(k)+BF̃(x̂(k))+DG̃(x̂(k-τ(k)))+u(k)+σ(k,x̂(k)-x(k))ω(k).
Equation (2.16) reduces to(3.6)ek+1=Aek+BF(ek)+DG(eτ)+σ(k,ek)ωk+u(k).Corollary 3.5.
Suppose that Assumptions2.1-2.2 hold, and let the state-feedback controller (3.1) be given. The noised-perturbed response system (3.5) is globally exponentially synchronized with the drive model (2.1) if there exist two positive definite matrices P,Q1, two diagonal matrices Λi=diag(λi1,λi2,…,λin)>0(i=1,2), a real matrix R, and a scalar λ*>0, such that the following two LMIs hold:
(3.7)Ω2=[Γ10-ΛC20ATP+RT*Γ20-Λ2H2R1T**-Λ10BTP***-Λ2DTP****-P]<0,(3.8)P<λ*I,
where Γ1=-P+λ*ρI+(τ0-τm+1)Q1-Λ1𝒞1, Γ2=-Q1-Λ2ℋ1.
And then the feedback gains can be designed as(3.9)K=P-1R.Remark 3.6.
The model proposed in this paper takes some well-studied models as special cases such as the model given in [4].If we neglect the effect of the stochastic termωk in (2.11), then (2.11) reduces to(3.10)x̂(k+1)=Ax̂(k)+BF̃(x̂(k))+δ(k)DG̃(x̂(k-τ1(k)))+(1-δ(k))DG̃(x̂(k-τ2(k)))+u(k).
Equation (2.16) reduces to(3.11)ek+1=Aek+BF(ek)+δkDG(eτ,1)+δ¯kDG(eτ,2)+u(k).
In this case, we have the following results.Corollary 3.7.
Suppose that Assumptions2.1 and 2.3 hold, and let the state-feedback controller (3.1) be given. The response system (3.10) is globally exponentially synchronized with the drive model (2.10) if there exist three positive definite matrices P,Q1,Q2, three diagonal matrices Λi=diag(λi1,λi2,…,λin)>0(i=1,2,3), two real matrices R,R1, such that the following LMI holds:
(3.12)Ω3=[Π100-Λ1C200ATP+RT*Π200-Λ2H20δ0R1T**Π300-Λ3H2δ¯0R1T***-Λ100BTP****-Λ20δ0DTP*****-Λ3δ¯0DTP******-P]<0,
where Π1=-P+(τ0-τm+1)Q1+(τM-τ0+2)Q2-Λ1𝒞1, Π2=-Q1-Λ2ℋ1, Π3=-Q2-Λ3ℋ1.
And then the feedback gains can be designed as(3.13)K=P-1R,K1=P-1R1.


## 4. Two Numerical Examples

Two numerical examples are presented to demonstrate the effectiveness of our results.Example 4.1.
Consider the DSNNs (2.10) with the following parameters:
(4.1)A=[0.40000.50000.4],B=[0.3-0.10.20-0.30.2-0.1-0.1-0.2],D=[0.20.10.1-0.20.30.10.1-0.20.3],f1(s)=tanh(0.6s)-0.2sin(s),f2(s)=tanh(-0.4s),f3(s)=tanh(-0.2s),g1(s)=tanh(-0.4s)+0.2sin(s),g2(s)=tanh(0.2s),g3(s)=tanh(0.4s),
It is easy to verify that
(4.2)C1=diag⁡(-0.08,0,0),C2=diag⁡(0.1,0.2,0.1),H1=diag⁡(-0.06,0,0),H2=diag⁡(0.06,-0.1,-0.2).
Setting τm=1, τ0=2, τM=5, and δ0=0.89, ρ=0.1 in Theorem 3.1, we can verify by Matlab toolbox that LMIs (3.2) and (3.3) are solved and the solutions are given as follows:
(4.3)P=[15.7114-0.86100.1575-0.861011.95851.05100.15751.051014.8306],Q1=[1.9720-0.15970.0202-0.15972.06450.20550.02020.20552.2099],Q2=[0.6299-0.01910.0012-0.01910.34600.02510.00120.02510.5228],Λ1=[369.9501000369.9501000369.9501],Λ2=[369.9501000369.9501000369.9501],Λ3=[369.9501000369.9501000369.9501],R=[-5.74160.31310.0554-0.0464-6.5348-0.35670.0554-0.3567-6.4303],R1=[0.1675-0.2223-0.1017-0.2223-0.4297-0.2061-0.1017-0.2061-0.7492],λ*=16.0757,K=P-1R=[-0.3656-0.01030.0084-0.0008-0.54850.00890.00770.0419-0.4343],K1=P-1R1=[0.0098-0.0160-0.0067-0.0174-0.0361-0.0134-0.0057-0.0112-0.495].Therefore, it can be seen from Theorem3.1, the response system (2.11) is globally exponentially synchronized with the drive system (2.10). The result is further verified by the simulation results given by Figures 1 and 2. Figures 1(a), 1(b), and 1(c) represent the trajectories of x1(k), x̂1(k), x2(k), x̂2(k), and x3(k), x̂3(k), respectively, and the red line represents the drive system state, the blue line stands for the response system state, and the initial conditions are taken as x(k)=[1.0,2.5,0.5]T, x̂(k)=[-0.5,-0.5,-0.5]T. Figure 2 shows that the error state goes to zero after a short period time. Furthermore, if we increase δ0, the maximum allowance value of τM will increase subsequently. Specially, if δ0=0.99, we get the maximum allowance value of τM=13.State trajectories ofx(k), x̂(k).
(a)(b)(c)Figure 2
Error state trajectories ofe(k).Example 4.2.
Consider the DSNNs (2.10) with the following parameters:
(4.4)A=[-0.1000.4],B=[0.1-0.20-0.1],D=[00.20.2-0.1],ρ=0.02,f1(s)=sin(0.2s)-0.6cos⁡(s),f2(s)=tanh(-0.4s),g1(s)=tanh(0.83s)+0.6cos⁡(s),g2(s)=tanh(0.2s).
It is easy to verify that
(4.5)C1=diag⁡(-0.64,0),C2=diag⁡(0,0.2),H1=diag⁡(-0.6,0),H2=diag⁡(-0.2,-0.1).
Setting τm=1, τ0=2, τM=3, and δ0=0.85 in Theorem 3.1, we can verify by Matlab toolbox that LMIs (3.2) and (3.3) are solved and the solutions are given below:
(4.6)P=[48.7972-2.3255-2.325542.7033],Q1=[4.2255-0.4620-0.46208.3487],Q2=[4.0513-0.1026-0.10261.8888],Λ1=[880.773500880.7735],Λ2=[880.773500880.7735],Λ3=[880.773500880.7735],R=[4.8936-0.7368-0.7368-17.8357],R1=[0.0860-1.2452-1.24520.3222],λ*=49.8783,K=P-1R=[0.0997-0.0351-0.0118-0.4196],K1=P-1R1=[0.0004-0.0252-0.02910.0062].Therefore, it can be seen from Theorem3.1, the response system (2.11) is globally exponentially synchronized with the drive system (2.10). The result is further verified by the simulation results given by Figures 3 and 4.  Figures 3(a) and 3(b) represent the trajectories of x1(k), x̂1(k) and x2(k), x̂2(k), respectively, and the red line represents the drive system state, the blue line stands for the response system state, the initial conditions are taken as x(k)=[1.0,2.5]T, x̂(k)=[-0.5,-0.5]T. Figure 4 shows that the error state goes to zero after a short period time.State trajectories ofx(k), x̂(k).
(a)(b)Figure 4
Error state trajectories ofe(k).


## 5. Conclusions

In recent years, synchronization in networks has become a hot research subject. However, the corresponding results are none for DSNNs with random delay. This paper has addressed the problem of exponential synchronization for the drive-response systems, where the drive system describes a class of discrete-time stochastic neural networks (DSNNs) with random delays and the response system is disturbed by some stochastic motions. Sufficient conditions have been established in terms of LMIs, which can be checked readily by Matlab toolbox. Two examples have been given to illustrate the effectiveness of the proposed results.It is worth noting that the following two interesting and important issues should be addressed in our future work. Firstly, within the same LMI framework, it is not difficult to extend our main results to the synchronization problem for DSNNs with randomly mixed time-varying delays. Another future work may be how to extend the obtained results to DSNNs with Markovian jump and so on.



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*Source: 713502-2011-03-24.xml*