delay-dependent stability criteria of uncertain periodic...

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2011, Article ID 325371, 14 pagesdoi:10.1155/2011/325371

Research ArticleDelay-Dependent Stability Criteria ofUncertain Periodic Switched Recurrent NeuralNetworks with Time-Varying Delays

Xing Yin,1 Jun Li,2 Weigen Wu,1 and Qiranrong Tan1

1 College of Computer Science, Panzhihua University, Panzhihua 617000, China2 School of Computer Science and Technology, Nanjing University of Science and Technology,Nanjing 219004, China

Correspondence should be addressed to Xing Yin, [email protected]

Received 20 June 2011; Revised 29 August 2011; Accepted 17 September 2011

Academic Editor: Baodong Zheng

Copyright q 2011 Xing Yin et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

This paper deals with the problem of delay-dependent stability criterion of uncertain periodicswitched recurrent neural networks with time-varying delays.When uncertain discrete-time recur-rent neural network is a periodic system, it is expressed as switched neural network for the finiteswitching state. Based on the switched quadratic Lyapunov functional approach (SQLF) and free-weighting matrix approach (FWM), some linear matrix inequality criteria are found to guaranteethe delay-dependent asymptotical stability of these systems. Two examples illustrate the exactnessof the proposed criteria.

1. Introduction

Recurrent neural networks (RNNs) are a very important tool for many application areas suchas associative memory, pattern recognition, signal processing, model identification, and com-binatorial optimization. With the development of research on RNNs in theory and applica-tion, the model is more and more complex. When the continuous-time RNNs are simulatedusing computer, they should be discretized into discrete-time RNNs [1–3]. Simultaneously,in implementations of artificial neural networks, time-varying delay may occur due to finiteswitching speeds of the amplifiers and communication time [4, 5]. Therefore, researchershave considered that discrete-time RNNs with time-varying delay are incorporated in theprocessing and/or transmission parts of the network architectures [6–9]. Parameter uncer-tainties and nonautonomous phenomena often exist in real systems due to modeling inac-curacies [4]. Particularly when we consider a long-term dynamical behaviors of the systemand consider seasonality of the changing environment, the parameters of the system usually

2 Discrete Dynamics in Nature and Society

will change with time [10–14]. In order to model those systems with neural networks, the un-certain (or switched or jumping) neural networks with time-varying delay appear in manypapers [6, 15–24]. So in this paper we consider the stability of the following discrete-timerecurrent neural networks with time-varying delay:

u(k + 1) = (A + ΔA(k))u(k) + (W1 + ΔW1(k))g(u(k))

+ (W2 + ΔW2(k))f(u(k − d(k))) + (I + ΔI(k)),(1.1)

where u(k) = {u1(k), u2(k), . . . , un(k)} ∈ Rn is the state vector associated with n neurons,A =diag{a1, a2, . . . , an} is a diagonal matrix with positive entries, W1 and W2 are, respectively,the connection weight matrix and the delayed connection weight matrix, I is input vector,g(u) = {g1(u), g2(u), . . . , gn(u)} and f(u) = {f1(u), f2(u), . . . , fn(u)} are the neuron activationfunction vectors, and d(k) is nonnegative differential time-varying functions which denotethe time delays and satisfy

0 ≤ d1 ≤ d(k) ≤ d2. (1.2)

In most literatures it is required that parameter uncertainty matrices, such as ΔA(k),ΔW1(k), ΔW2(k), and ΔI(k), should be in the form

[ΔA(k) ΔW1(k) ΔW2(k) ΔI(k)] = DF(k)[Ea Ew1 Ew2 Ei], (1.3)

where Ea, Ew1, Ew2, and Ei are given constant matrices of appropriate dimensions and F(k)is an uncertain matrix such that

FT (k)F(k) ≤ I. (1.4)

In practice, however, ΔA(k), ΔW1(k), ΔW2(k), and ΔI(k) are generally difficult to have thedecomposition of matrices for D, Ea, Ew1, Ew2, and Ei. In addition, periodic oscillation inrecurrent neural networks is an interesting dynamic behavior as many biological and cogni-tive activities require repetition [7, 10, 11, 25]. Simultaneously, periodic oscillations in recur-rent neural networks have been found in many applications such as associative memories,pattern recognition, machine learning, and robot motion control [25]. So, if (1.1) is an un-certain periodic neural network in which the period is less than a constant b, then (1.1) canbe expressed as switched neural network for the finite switching state, that is, if (1.1) is aneural network with period a (0 < a ≤ b), then ΔA(k) = ΔA(k + a), ΔW1(k) = ΔW1(k + a),ΔW2(k) = ΔW2(k + a), and ΔI(k) = ΔI(k + a), which is corresponding to a switched neuralnetwork set: Ω =

⋃ba=1 Ua, where Ua = {Sai = (Aai,W1ai,W2ai, Iai) | 0 < i ≤ a}, Aai =

A+ΔA(k+ i),W1ai = W1+ΔW1(k+ i),W2ai = W2+ΔW2(k+ i), and Iai = I +ΔI(k+ i). SupposeN is the number of elements of Ω; then (1.1) is actually modified by

u(k + 1) = Ar(k)u(k) +W1r(k)g(u(k)) +W2r(k)f(u(k − d(k))) + Ir(k), (1.5)

Discrete Dynamics in Nature and Society 3

where r(k) is a switching rule defined by r(k) : N → ΩwithΩ = {S1, S2, . . . , SN}. Moreover,r(k) = j means the sub-recurrent neural network (sub-RNN) Sj , which is corresponding to(Aj,W1j ,W2j , Ij), is active.

The dynamic behaviors of those models are foundations for applications. Under (1.5)most papers discuss the stability of uncertain neural networks with the common Lyapunovfunction approach [5, 6, 15–23, 25]. To the best of the authors’ knowledge, up to now, there isscarcely any paper that studies the uncertain periodic neural networks using the SQLF. Thissituation motivates this research.

Motivated by the above discussions, the authors intend to study a problem of thedelay-dependent stability criterion of uncertain discrete-time recurrent neural networks withtime-varying delays that the uncertain recurrent neural networks have a finite number of sub-RNNs, and the sub-RNNs may change from one to another according to arbitrary switchingand restricted switching. The contributions of this paper are the following. (1) Using aswitching graph, uncertain periodic recurrent neural networks with time-varying delays aretransformed into switched recurrent neural networks; (2) the derivative of the SQLF (3.7) ofthe literature [8] is improved in (3.11), please see Remark 4.3 and Table 3; (3) based on theswitching graph, the delay-dependent stability criteria of switched recurrent neural networksare studied by FWM and SQLF. Then an effective LMI approach is developed to solve theproblem.

This paper is organized as follows. In Section 2, we give some basic definitions. Weanalyze the stability of the system (2.2)with the SQLF and FWM in Section 3. Some examplesare given in Section 4. Section 5 offers the conclusions of this paper.

2. Preliminaries

In many electronic circuits, nonmonotonic functions can be more appropriate to describe theneuron activation in designing and implementing an artificial neural network [7]; hence, wehave the following assumption.

For any j ∈ {1, 2, . . . , n}, there exist constants g−j , g

+j , f

−j , and f+

j such that

g−j ≤ gj(θ1) − gj(θ2)

θ1 − θ2≤ g+

j , ∀θ1, θ2 ∈ R,

f−j ≤ fj(θ1) − fj(θ2)

θ1 − θ2≤ f+

j , ∀θ1, θ2 ∈ R.

(2.1)

Under the assumption, the equilibrium points of UDNN (1.1) exist by the fixed pointtheorem [1]. In the following, let u◦ = {u◦

1, u◦2, . . . , u

◦n} be the equilibrium point of (1.1); then

x(·) = u(·) − u◦. The systems (1.1) and (1.5) are, respectively, shifted to the following form:

x(k + 1) = (A + ΔA(k))x(k) + (W1 + ΔW1(k))g(x(k)) + (W2 + ΔW2(k))f(x(k − τ(k))),

x(k + 1) = Ar(k)x(k) +W1r(k)g(x(k)) +W2r(k)f(x(k − d(k))), r(k) ∈ Ω.

(2.2)

For convenience, the switching graph is defined.


Definition 2.1. Let Γ = (Ω,W) be a switching graph, whereΩ is the set of sub-RNNs Si andWis the set of weighted arcs wij ∈ {0, 1}. wij = 1 (or 0) represents the sub-RNN Si switches (ordoes not switch) to the sub-RNN Sj .

Remark 2.2. When wij = 1, if∑N

l=1wjl = 0, that is sub-RNN Sj cannot switch to any othersub-RNN, we suppose that the uncertain neural networks will always stay in the sub-RNNSi that means wii = 1 and wij = 0.

Throughout this paper, the superscript T stands for the transpose of a matrix, P > 0means that the matrix P is positive definite, and the symmetric terms in a symmetric matrixare denoted by ∗, for example,

[M O

∗ N

]

=

[M O

OT N

]

,

G1 = diag{g−1 g

+1 , g

−2 g

+2 , . . . , g

−ng

+n

},

G2 = diag

{

−g−1 + g+

1

2,−g

−2 + g+

2

2, . . . ,−g

−n + g+

n

2

}

,

F1 = diag{f−1 f

+1 , f

−2 f

+2 , . . . , f

−n f

+n

},

F2 = diag

{

−f−1 + f+

1

2,−f

−2 + f+

2

2, . . . ,−f

−n + f+

n

2

}

.

(2.3)

3. Asymptotical Stability of Uncertain Periodic SwitchedRecurrent Neural Networks

Theorem 3.1. Let d1 and d2 be positive integers such that 0 ≤ d1 ≤ d2. Based on a switching graphΓ, the system (2.2) is asymptotical stable if, when wij = 1 (Si, Sj ∈ Ω), there exist the correspondingsymmetric matrices Pi = PT

i > 0, Q1 = QT1 > 0, Q2 = QT

2 > 0, R = RT > 0, Z1 = ZT1 > 0,

Z2 = ZT2 > 0, Xij = (Xij)T ≥ 0, Uij = (Uij)T ≥ 0, Hij = diag{hij

1 , hij

2 , . . . , hijn} ≥ 0, Oij =

diag{oij1 , oij

2 , . . . , oijn} ≥ 0, and any appropriate dimensional matrices Nij , Mij , and Tij such that the

following LMIs hold:

Φij =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Φij

11 Φij

12 Φij

13 Φij

14 Φij

15 Φij

16

∗ Φij

22 Φij

23 Φij

24 0 0

∗ ∗ Φij

33 Φij

34 0 0

∗ ∗ ∗ Φij

44 0 −F2Oij

∗ ∗ ∗ ∗ Φij

55 Φij

56

∗ ∗ ∗ ∗ ∗ Φij

66

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (3.1)

Λij

1 =

[Xij Nij

∗ Z1

]

> 0, (3.2)


Λij

2 =

[Uij Tij

∗ Z2

]

> 0, (3.3)

Λij

3 =

[Xij +Uij Mij

∗ Z1 + Z2

]

> 0, (3.4)

where

Φij

11 = ATi PjAi − Pi +Q1 +Q2 + (d2 − d1 + 1)R + (Ai − E)T

(d2Z1j + (d2 − d1)Z2j

)(Ai − E)

+(N

ij

1

)T+N

ij

1 + d2Xij

11 + (d2 − d1)Uij

11 −G1Hij ,

Φij

12 =(N

ij

2

)T+ T

ij

1 + d2Xij

12 + (d2 − d1)Uij

12,

Φij

13 =(N

ij

3

)T −Mij

1 + d2Xij

13 + (d2 − d1)Uij

13,

Φij

14 = −Nij

1 +(N

ij

4

)T+M

ij

1 − Tij

1 + d2Xij

14 + (d2 − d1)Uij

14,

Φij

15 = ATi PjW1i + (Ai − E)T (d2Z1 + (d2 − d1)Z2)W1i −G2H

ij ,

Φij

16 = ATi PjW2i + (Ai − E)T (d2Z1 + (d2 − d1)Z2)W2i,

Φij

22 = −Q1 − R +(Tij

2

)T+ T

ij

2 + d2Xij

22 + (d2 − d1)Uij

22,

Φij

23 = −Mij

2 +(Tij

3

)T+ d2X

ij

23 + (d2 − d1)Uij

23,

Φij

24 = −Nij

2 +(Tij

4

)T+M

ij

2 − Tij

2 + d2Xij

24 + (d2 − d1)Uij

24,

Φij

33 = −Q2 − R −Mij

3 −(M

ij

3

)T+ d2X

ij

33 + (d2 − d1)Uij

33,

Φij

34 = −Nij

3 −(M

ij

4

)T+M

ij

3 − Tij

3 + d2Xij

34 + (d2 − d1)Uij

34,

Φij

44 = −Nij

4 −(N

ij

4

)T+M

ij

4 +(M

ij

4

)T − Tij

4 −(Tij

4

)T+ d2X

ij

44 + (d2 − d1)Uij

44 − F1Oij ,

Φij

55 = WT1iPjW1i +WT

1i(d2Z1 + (d2 − d1)Z2)W1i −Hij ,

Φij

56 = WT1iPjW2i +WT

1i(d2Z1 + (d2 − d1)Z2)W2i,

Φij

66 = WT2iPjW2i +WT

2i(d2Z1 + (d2 − d1)Z2)W2i −Oij .

(3.5)

Proof. Suppose that y(l) = x(l + 1) − x(l); then we have x(k + 1) = x(k) + y(k) and x(k) =x(k − d(k)) +

∑k−1i=k−d(k)y(l).


We consider the following SQLF:

Vr(k)(k) = V1r(k)(k) + V2r(k)(k) + V3r(k)(k) + V4r(k)(k),

V1r(k)(k) = xT (k)Pr(k)x(k),

V2r(k)(k) =k−1∑

l=k−d1

xT (l)Q1r(k)x(l) +k−1∑

l=k−d2

xT (l)Q2r(k)x(l),

(3.6)

V3r(k)(k) =k−d1∑

θ=k−d2

k−1∑

l=θ

xT (l)Rr(k)x(l), (3.7)

V4r(k)(k) =−1∑

θ=−d2

k−1∑

l=k+θ

yT (l)Z1r(k)y(l) +−d1−1∑

θ=−d2

k−1∑

l=k+θ

yT (l)Z2r(k)y(l). (3.8)

It is clear that the following equations are true:

k−1∑

l=k−d2

yT (l)Z1r(k)y(l) =k−d(k)−1∑

l=k−d2

yT (l)Z1r(k)y(l) +k−1∑

l=k−d(k)yT (l)Z1r(k)y(l),

k−d1−1∑

l=k−d2

yT (l)Z2r(k)y(l) =k−d(k)−1∑

l=k−d2

yT (l)Z2r(k)y(l) +k−d1−1∑

l=k−d(k)yT (l)Z2r(k)y(l).

(3.9)

Firstly, we prove that under wij = 1 the SQLF is less than 0. Suppose that r(k) = i andr(k + 1) = j, that means the sub-RNN Si switches to the sub-RNN Sj ; we obtain

ΔV1i(k) = V1j(k + 1) − V1i(k) =(Aix(k) +W1ig(x(k)) +W2if(x(k − d(k)))

)T

× Pj

(Aix(k) +W1ig(x(k)) +W2if(x(k − d(k)))

) − xT (k)Pix(k),

ΔV2i(k) = V2j(k + 1) − V2i(k) = xT (k)(Q1j +Q2j

)x(k) − xT (k − d1)Q1ix(k − d1)

− xT (k − d2)Q2ix(k − d2) +k−1∑

l=k+1−d1

xT (l)(Q1j −Q1i

)x(l)

+k−1∑

l=k+1−d2

xT (l)(Q2j −Q2i

)x(l),

(3.10)

ΔV3i(k) = V3j(k + 1) − V3i(k) ≤ (d2 − d1 + 1)xT (k)Rjx(k)

− xT (k − d1)Rix(k − d1) − xT (k − d2)Rix(k − d2) − xT (k − d(k))Rix(k − d(k))

+k−1∑

l=k+1−d1

xT (l)(Rj − Ri

)x(l) +

k−d1∑

θ=k+1−d2

k−1∑

l=θ

xT (l)(Rj − Ri

)x(l),

(3.11)


ΔV4i(k) = V4j(k + 1) − V4i(k) = d2yT (k)Z1jy(k) + (d2 − d1)yT (k)Z2jy(k)

−k−1∑

l=k−d(k)yT (l)Z1iy(l) −

k−d1−1∑

l=k−d(k)yT (l)Z2iy(l)

−k−d(k)−1∑

l=k−d2

yT (l)(Z1i + Z2i)y(l) +−1∑

θ=−d2

k−1∑

l=k+1+θ

yT (l)(Z1j − Z1i

)y(l)

+−d1−1∑

θ=−d2

k−1∑

l=k+1+θ

yT (l)(Z2j − Z2i

)y(l).

(3.12)

In order to strictly guarantee ΔV2i(k) < 0,∑k−1

l=k+1−d1xT (l)(Q1j −Q1i)x(l) should be less

than 0. In the switching graph Γ if there exists sub-RNN Si (Si ∈ Ω), which satisfied∑N

j=1wij =0, which means the Si cannot switch to any other sub-RNN, the equation Q1i = Q1j can begrounded, otherwise the switching sequence must be β = {Sα1 → Sα2 → , . . . , → Sαj →· · · , → SαL → SαL+1} (Sαj ∈ Ω), and there exist l such that SαL+1 = Sαl (1 ≤ l < L). Becausethe affection of the sub-RNNs S1, S2 . . . , Sl−1 and Sl on the whole system is before time αl,after αl the β changes to a periodic sequence β′ = {Sαl → · · · , → SαL → SαL+1}. Suppose thati = αl = αL+1; then in switching sequence β′ the following LMIs all hold:

Q1αl −Q1αl+1 ≥ 0(Q1i −Q1(l+1) ≥ 0

),

Q1αl+1 −Q1αl+2 ≥ 0,

...

Q1αL −Q1αL+1 ≥ 0(Q1(L) −Q1i ≥ 0

);

(3.13)

then the solution of (3.13) is Q1i = Q1αl = Q1αl+1 = · · · = Q1αL . Thus, we suppose that

Q1 = Q1i = Q1j . (3.14)

Similar to∑k−1

l=k+1−d1xT (l)(Q1j −Q1i)x(l), together with

∑k−1l=k+1−d2

xT (l)(Q2j −Q2i)x(l),∑k−1

l=k+1−d1

xT (l)(Rj − Ri)x(l),∑k−d1

θ=k+1−d2

∑k−1l=θ x

T (l)(Rj − Ri)x(l),∑−1

θ=−d2

∑k−1l=k+1+θy

T (l)(Z1j − Z1i)y(l) and∑−d1−1

θ=−d2

∑k−1l=k+1+θy

T (l)(Z2j − Z2i)y(l), we suppose that

Q2 = Q2i = Q2j ,

R = Ri = Rj,(3.15)

Z1 = Z1i = Z1j , (3.16)

Z2 = Z2i = Z2j . (3.17)


On the other hand, for any appropriately dimensioned matrices Nij , Mij , and Tij the follow-ing equations are true:

2ξT (k)Nij

⎡

⎣x(k) − x(k − d(k)) −k−1∑

l=k−d(k)y(l)

⎤

⎦ = 0, (3.18)

2ξT (k)Mij

[

x(k − d(k)) − x(k − d2) −k−d(k)−1∑

l=k−d2

y(l)

]

= 0,

2ξT (k)Tij

⎡

⎣x(k − d1) − x(k − d(k)) −k−d1−1∑

l=k−d(k)y(l)

⎤

⎦ = 0,

(3.19)

where ξ(k) = [xT (k), xT (k − d1), xT (k − d2), xT (k − d(k))]T .In addition, for any semipositive definite matrix Xij = (Xij)T and Uij = (Uij)T , the fol-

lowing equations hold:

d2ξT (k)Xijξ(k) −

k−d(k)−1∑

l=k−d2

ξT (k)Xijξ(k) −k−1∑

l=k−d(k)ξT (k)Xijξ(k) = 0,

(d2 − d1)ξT (k)Uijξ(k) −k−d(k)−1∑

l=k−d2

ξT (k)Uijξ(k) −k−d1−1∑

l=k−d(k)ξT (k)Uijξ(k) = 0.

(3.20)

From the assumption, we have

(gl(x(k)) − g−

l xl(k)) × (

gl(x(k)) − g+l xl(k)

) ≤ 0, l = 1, 2, . . . , n,

(fl(x(k − dk)) − f−

l xl(k − dk)) × (

fl(x(k − dk)) − f+l xl(k − dk)

) ≤ 0, l = 1, 2, . . . , n.(3.21)

Similar to the conclusion in [8], for Hij = diag{h1, h2, . . . , hn} ≥ 0, Oij = diag{o1,o2, . . . , on} ≥ 0 the following inequalities are also true:

−[

x(k)

g(x(k))

]T[G1H

ij G2Hij

� Hij

][x(k)

g(x(k))

]

≥ 0, (3.22)

−[

x(k − d(k))

f(x(k − d(k)))

]T[F1O

ij F2Oij

� Oij

][x(k − d(k))

f(x(k − d(k)))

]

≥ 0. (3.23)


Then we add the terms on the right side of (3.16)–(3.23) to yield

ΔVi(k) = ΔV1i(k) + ΔV2i(k) + ΔV3i(k) + ΔV4i(k)

≤ ηT1 (k)Φ

ijη1(k) −k−1∑

l=k−d(k)ηT2 (k, l)Λ

ij

1 η2(k, l)

−k−d1−1∑

l=k−d(k)ηT2 (k, l)Λ

ij

2 η2(k, l) −k−d(k)−1∑

l=k−d2

ηT2 (k, l)Λ

ij

3 η2(k, l),

(3.24)

where ηT1 (k) = [ξT (k), gT (x(k)), fT (x(k − d(k)))]T , ηT

2 (k, l) = [ξT (k), yT (l)]T.

And ξ(k) is defined in (3.18). Φij , Λij

1 , Λij

2 , and Λij

3 are defined in (3.1)–(3.4). Therefore,when the corresponding LMIs satisfy Φij < 0, Λij

1 ≥ 0, Λij

2 ≥ 0, and Λij

3 ≥ 0, Si, Sj ∈ Ω,ΔVi(k) ≤ 0.

Secondly, based on the switching graph Γ, whenwij = 1 (Si, Sj ∈ Ω), all correspondingΔVi(k) are less than 0 that means the system (2.2) is asymptotical stable. This completes theproof of Theorem 3.1.

Remark 3.2. Using the method in [8], it is easily to know that the system is the globally ex-ponentially stable.

Remark 3.3. In V3i(k) of the literature [8], −∑k−d(k)l=k−d2

xT (l)Rix(l) ≤ −xT (k − d(k))Rix(k − d(k))

may lead to considerable conservativeness. Then, it is improved as −∑k−d(k)l=k−d2

xT (l)Rix(l) ≤−xT (k −d2)Rix(k −d2)−xT (k −d(k))Rix(k −d(k))−xT (k −d1)Rix(k −d1). Please see Table 3.

Combined with Theorem 3.1, we consider the common Lyapunov function approach;then we have the following.

Corollary 3.4. Let d1 and d2 be positive integers such that 0 ≤ d1 ≤ d2. The system (2.2) is asym-ptotical stable if there exist symmetric matrices P = PT > 0, Q1 = QT

1 > 0, Q2 = QT2 > 0, R =

RT > 0, Z1 = ZT1 > 0, Z2 = ZT

2 > 0, X = XT ≥ 0, U = UT ≥ 0, H = diag{h1, h2, . . . , hn} ≥ 0,O = diag{o1, o2, . . . , on} ≥ 0, and any appropriate dimensional matrices N, M, and T such that thefollowing LMIs hold:

Φi =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

Φ11 Φ12 Φ13 Φ14 Φ15 Φ16

∗ Φ22 Φ23 Φ24 0 0∗ ∗ Φ33 Φ34 0 0∗ ∗ ∗ Φ44 0 −F2O

∗ ∗ ∗ ∗ Φ55 Φ56

∗ ∗ ∗ ∗ ∗ Φ66

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, Si ∈ Ω,

Λ1 =

[X N

∗ Z1

]

> 0,

Λ2 =

[U T

∗ Z2

]

> 0,

Λ3 =

[X +U M

∗ Z1 + Z2

]

> 0,

(3.25)


where

Φ11 = ATi PAi − P +Q1 +Q2 + (d2 − d1 + 1)R + (Ai − E)T (d2Z1 + (d2 − d1)Z2)(Ai − E)

+NT1 +N1 + d2X11 + (d2 − d1)U11 −G1H,

Φ12 = NT2 + T1 + d2X12 + (d2 − d1)U12,

Φ13 = NT3 −M1 + d2X13 + (d2 − d1)U13,

Φ14 = −N1 +NT4 +M1 − T1 + d2X14 + (d2 − d1)U14,

Φ15 = ATi PW1i + (Ai − E)T (d2Z1 + (d2 − d1)Z2)W1i −G2H,

Φ16 = ATi PW2i + (Ai − E)T (d2Z1 + (d2 − d1)Z2)W2i,

Φ22 = −Q1 − R + TT2 + T2 + d2X22 + (d2 − d1)U22,

Φ23 = −M2 + TT3 + d2X23 + (d2 − d1)U23,

Φ24 = −N2 + TT4 +M2 − T2 + d2X24 + (d2 − d1)U24,

Φ33 = −Q2 − R −M3 −MT3 + d2X33 + (d2 − d1)U33,

Φ34 = −N3 −MT4 +M3 − T3 + d2X34 + (d2 − d1)U34,

Φ44 = −N4 −NT4 +M4 +MT

4 − T4 − TT4 − F1O + d2X44 + (d2 − d1)U44 − R,

Φ55 = WT1iPW1i +WT

1i(d2Z1 + (d2 − d1)Z2)W1i −H,

Φ56 = WT1iPW2i +WT

1i(d2Z1 + (d2 − d1)Z2)W2i,

Φ66 = WT2iPW2i +WT

2i(d2Z1 + (d2 − d1)Z2)W2i −O.

(3.26)

4. Examples

Example 4.1. Consider the discrete-time recurrent neural network (2.2)with

A1 =

⎡

⎢⎢⎣

0.7 0 0

0 0.2 0

0 0 0.5

⎤

⎥⎥⎦, W11 =

⎡

⎢⎢⎣

0.1 −0.2 0.1

0.3 −0.1 0

0 −0.1 −0.4

⎤

⎥⎥⎦, W21 =

⎡

⎢⎢⎣

−0.3 0.1 −0.20.2 −0.1 0.1

0 −0.02 0.07

⎤

⎥⎥⎦,

A2 =

⎡

⎢⎢⎣

0.1 0 0

0 0.2 0

0 0 0.4

⎤

⎥⎥⎦, W12 =

⎡

⎢⎢⎣

0.4 −0.2 0

0.2 −0.2 0

−0.3 −0.1 −0.5

⎤

⎥⎥⎦, W22 =

⎡

⎢⎢⎣

−0.3 0.1 −0.20 −0.03 0

0.5 −0.2 0.5

⎤

⎥⎥⎦,

A3 =

⎡

⎢⎢⎣

0.1 0 0

0 0.4 0

0 0 0.4

⎤

⎥⎥⎦, W13 =

⎡

⎢⎢⎣

0.2 0 0.3

0.04 −0.3 0

−0.1 −0.6 −0.1

⎤

⎥⎥⎦, W23 =

⎡

⎢⎢⎣

−0.08 0.1 −0.20.2 −0.4 −0.050 −0.2 −0.1

⎤

⎥⎥⎦,

Γ =

⎡

⎢⎢⎣

0 1 0

0 0 1

1 0 0

⎤

⎥⎥⎦,


g1(x) = tanh(0.4x), g2(x) = tanh(0.2x), g3(x) = tanh(−0.8x),f1(x) = tanh(−0.6x), f2(x) = tanh(0.4x), f3(x) = tanh(0.2x).

(4.1)

Then we have

G1 = F1 =

⎡

⎢⎢⎣

0 0 0

0 0 0

0 0 0

⎤

⎥⎥⎦, G2 =

⎡

⎢⎢⎣

0.2 0 0

0 0.1 0

0 0 −0.4

⎤

⎥⎥⎦, F2 =

⎡

⎢⎢⎣

0.3 0 0

0 −0.2 0

0 0 −0.1

⎤

⎥⎥⎦. (4.2)

Employing the LMIs in Theorem 3.1 yields upper bounds on d2 that guarantee thestability of system (1.1) for various lower bounds d1, which are listed in Table 1. When d1 = 1and d2 = 3, it can be seen from Figure 1 that all the state solutions corresponding to the 10random initial points are convergent asymptotically to the unique equilibrium x∗ = {0, 0, 0},and according to Theorem 3.1, LMIs (3.1)–(3.4) are solvable in Matlab 7.0.1.

Example 4.2. Consider the discrete-time recurrent neural network (2.2)with

A =

⎡

⎢⎢⎣

0.1 0 0

0 0.2 0

0 0 0.3

⎤

⎥⎥⎦, W1 =

⎡

⎢⎢⎣

0.5 −0.2 0.2

0.1 −0.1 0

−0.4 −0.1 −0.3

⎤

⎥⎥⎦, W2 =

⎡

⎢⎢⎣

−0.2 0.1 0.2

0 −0.3 0.1

0 −0.3 0.03

⎤

⎥⎥⎦,

g1(x) = f1(x) = tanh(0.6x), g2(x) = f2(x) = tanh(0.4x), g3(x) = f3(x) = tanh(0.2x).(4.3)

Then we have

G1 = F1 =

⎡

⎢⎢⎣

0 0 0

0 0 0

0 0 0

⎤

⎥⎥⎦, G2 = F2 =

⎡

⎢⎢⎣

0.3 0 0

0 0.2 0

0 0 0.1

⎤

⎥⎥⎦. (4.4)

Employing the LMIs in [8] and those in Corollary 3.4 yields upper bounds on d2 thatguarantee the stability of system for various lower bounds d1, which are listed in Table 2. It isclear that the obtained upper bounds of this paper are better than those of [8]. It can be seenfrom Figure 2 that, when d1 = 1 and d2 = 3, all the state solutions corresponding to the 10random initial points are convergent asymptotically to the unique equilibrium x∗ = {0, 0, 0}.

Remark 4.3. Employing the LMIs in [8] and those in Corollary 3.4 yields upper bounds on d2

that guarantee the stability of system (1.1) of the Example 1 of [8] for various lower boundsd1, which are listed in Table 3.


10

0

−10

10

0

−10

10

0

−10

0 5 10 15 20 25

0 5 10 15 20 25

0 5 10 15 20 25

x1

x2

x3

Figure 1: Global convergence of states x1, x2, and x3 in Example 4.1, when d1 = 1 and d2 = 3.

10

0

−10

10

0

−10

10

0

−10

0 5 10 15 20 25

0 5 10 15 20 25

0 5 10 15 20 25

x1

x2

x3

Figure 2: Global convergence of states x1, x2, and x3 in Example 4.2, when d1 = 1 and d2 = 3.


Table 1: Allowable upper bound of d2 with given d1.

d1 0 2 4 6 8 20Theorem 3.1 9 11 13 15 17 29


d1 0 2 4 6 8 10 20Reference [8] 20 22 24 26 28 30 40Corollary 3.4 38 40 42 44 46 48 58


d1 0 2 4 6 10Reference [8] 11 13 15 17 21Corollary 3.4 12 14 16 18 22

5. Conclusions

This paper was dedicated to the delay-dependent stability of uncertain periodic switchedrecurrent neural networks with time-varying delay. A less conservative LMI-based globallystability criterion is obtained with the switched quadratic Lyapunov functional approachand free-weighting matrix approach for periodic uncertain discrete-time recurrent neuralnetworks with a time-varying delay. One example illustrates the exactness of the proposedcriterion. Another example demonstrates that the proposed method is an improvement overthe existing one.

Acknowledgments

This work was supported by the Sichuan Science and Technology Department under Grant2011JY0114. The authors would like to thank the Associate Editor and the anonymous re-viewers for their detailed comments and valuable suggestions which greatly contributed tothis paper.

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