Criteria for Discrete-Time Neural Networks with Time-Varying Delay

Volume 2009, Article ID 874582,23pages doi:10.1155/2009/874582

Research Article

New Improved Exponential Stability

Criteria for Discrete-Time Neural Networks with Time-Varying Delay

Zixin Liu,

^{1, 2}

Shu Lv,

¹

Shouming Zhong,

¹

and Mao Ye

³

1School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, China

2School of Mathematics and Statistics, Guizhou College of Finance and Economics, Guiyang 550004, China

3School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China

Correspondence should be addressed to Zixin Liu,xinxin905@163.com Received 13 March 2009; Accepted 11 May 2009

Recommended by Manuel De La Sen

The robust stability of uncertain discrete-time recurrent neural networks with time-varying delay is investigated. By decomposing some connection weight matrices, new Lyapunov-Krasovskii functionals are constructed, and serial new improved stability criteria are derived. These criteria are formulated in the forms of linear matrix inequalitiesLMIs. Compared with some previous results, the new results are less conservative. Three numerical examples are provided to demonstrate the less conservatism and effectiveness of the proposed method.

Copyrightq2009 Zixin Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introductionn

In recent years, recurrent neural networkssee 1–7, such as Hopfield neural networks, cellular neural networks, and other networks have been widely investigated and successfully applied in all kinds of science areas such as pattern recognition, image processing, and fixed-point computation. However, because of the finite switching speed of neurons and amplifiers, time delay is unavoidable in nature and technology. It can make important effects on the stability of dynamic systems. Thus, the studies on stability are of great significance.

There has been a growing research interest on the stability analysis problems for delayed neural networks, and many excellent papers and monographs have been available. On the other hand, during the design of neural network and its hardware implementation, the convergence of a neural network may often be destroyed by its unavoidable uncertainty due

to the existence of modeling error, the deviation of vital data, and so on. Therefore, the studies on robust convergence of delayed neural network have been a hot research direction. Up to now, many sufficient conditions, either delay-dependent or delay-independent, have been proposed to guarantee the global robust asymptotic or exponential stability for different class of delayed neural networkssee8–13.

It is worth pointing out that most neural networks have been assumed to be in continuous time, but few in discrete time. In practice, the discrete-time neural networks are more applicable to problems that are inherently temporal in nature or related to biological realities. And they can ideally keep the dynamic characteristics, functional similarity, and even the physical or biological reality of the continuous-time networks under mild restriction.

Thus, the stability analysis problems for discrete-time neural networks have received more and more interest, and some stability criteria have been proposed in literature see 14–

25. In 14, Liu et al. researched a class of discrete-time RNNs with time-varying delay, and proposed a delay-dependent condition guaranteeing the global exponential stability. By using a similar technique to that in21, the result obtained in14has been improved by Song and Wang in15. The results in15are further improved by Zhang et al. in16by introducing some useful terms. In17, Yu et al. proposed a new less conservative result than that obtained in16via constructing a new augment Lyapunov-Krasovskii functional.

In this paper, the connection weight matrix C is decomposed, and some new Lyapunov-Krasovskii functionals are constructed. Combined with linear matrix inequality LMI technique, serial new improved stability criteria are derived. Numerical examples show that these new criteria are less conservative than those obtained in14–17.

Notation 1. The notations are used in our paper except where otherwise specified.·denotes a vector or a matrix norm;R,Rⁿare real and n-dimension real number sets, respectively;N is nonnegative integer set. I is identity matrix; ∗ represents the elements below the main diagonal of a symmetric block matrix; Real matrixP > 0 < 0denotes thatP is a positive definitenegative definitematrix;Na, b {a, a1, . . . , b};λ_minλmaxdenotes the minimum and maximum eigenvalue of a real matrix.

2. Preliminaries

Consider a discrete-time recurrent neural network with time-varying delays17described by

Σ:xk1 Ckxk Akfxk Bkfxk−τk J, k 1,2, . . . , 2.1

where xk x1k, x2k, . . . , xnk^T ∈ Rⁿ denotes the neural state vector; fxk f1x1k, f2 x2k, . . . , fnxnk^T, fxk − τk f1x1k − τk, f2x2k − τk, . . . , fnxnk−τk^T are the neuron activation functions;J J1, J₂, . . . , J_n^T is the external input vector; Positive integerτk represents the transmission delay that satisfies 0 < τm ≤ τk ≤ τM, where τm, τM are known positive integers representing the lower and upper bounds of the delay.Ck C ΔCk,Ak A ΔAk,Bk B ΔBk.

C diagc1, c₂, . . . , c_nwith|ci|<1 describes the rate with which theith neuron will reset its potential to the resting state in isolation when disconnected from the networks and external inputs; C, A, B ∈ R^n×n represent the weighting matrices; ΔCk,ΔAk,ΔBk denote the

time-varying structured uncertainties which are of the following form:

ΔCk,ΔAk,ΔBk KFk Ec, Ea, Eb, 2.2

whereK, Ec, Ea, Eb are known real constant matrices with appropriate dimensions,Fkis unknown time-varying matrix function satisfyingF^TkFk≤I, for allk∈N.

The nominalΣ0ofΣcan be defined as

Σ0:xk1 Cxk Afxk Bfxk−τk J, k 1,2, . . . , 2.3

To obtain our main results, we need to introduce the following assumption, definition and lemmas.

Assumption 1. For anyx, y∈R,x /y,

σ_i⁻ ≤fix−fi

y

x−y ≤σ_i, i 1,2, . . . , n, 2.4

whereσ_i⁻, σ_iare known constant scalars.

As pointed out in 16 under Assumption 1, system 2.3 has equilibrium points.

Assume thatx^∗ x^∗₁, x^∗₂, . . . , x^∗_n^Tis an equilibrium point of2.3and lety_ik x_ik−x^∗_i, g_iyik f_iyik x^∗_i−f_ix_i^∗. Then, system2.3, can be transformed into the following form:

yk1 Cyk Ag yk

Bg

yk−τk

, k 1,2, . . . , 2.5

where yk y1k, y2k, . . . , ynk^T, gyk g1y1k, g2y2k, . . . , gnynk^T, gyk − τk g1y1k−τk, g2y2k−τk, . . . , gnynk−τk^T. From Assumption 1, for anyx, y ∈ R,x /y, functionsgi·satisfyσ_i⁻ ≤ gix−giy/x−y ≤ σ_i, i 1,2, . . . , n,andg_i0 0.

Remark 2.1. Assumption 1is widely used for dealing with the stability problem for neural networks. As pointed out in13,14,16,17,26,27, constantsσ⁻_i, σ_i i 1,2, . . . , ncan be positive, negative, and zero. Thus, this assumption is less restrictive than traditional Lipschitz condition.

Definition 2.2. The delayed discrete-time recurrent neural network in 2.5 is said to be globally exponentially stable if there exist two positive scalarsα > 0 and 0 < β < 1 such that

yk ≤α·β^k sup

s∈N−τM,0ys, ∀k≥1. 2.6

Lemma 2.3Tchebychev Inequality 28. For any given vectorsvi ∈ Rⁿ, i 1,2, . . . , n, the following inequality holds:

_n

i 1

vi

_{T n}

i 1

vi

≤n n

i 1

v^T_ivi. 2.7

Lemma 2.4see29. For given matricesQ Q^T, H, EandR R^T >0 of appropriate dimensions, then

QHFEE^TF^TH^T <0, 2.8

for allFsatisfyingF^TF≤R, if and only if there is anε >0, such that

Qε⁻¹HH^TεE^TRE <0. 2.9

Lemma 2.5see16. IfAssumption 1holds, then for any positive-definite diagonal matrixD diagd1, d2, . . . , dn>0, the following inequality holds:

g ykT

Dg yk

−y^TkD

1

2

g yk

y^Tk

1

D

2

yk≤0, k∈N, 2.10

where

1 diagσ₁⁻, σ₂⁻, . . . , σ_n⁻,

2 diagσ₁, σ₂, . . . , σ_n.

Lemma 2.6see30. Given constant symmetric matricesΣ1,Σ2,Σ3whereΣ^T₁ Σ1and 0<Σ2

Σ^T₂, thenΣ1 Σ^T₃Σ⁻¹₂ Σ3<0 if and only if Σ1 Σ^T₃

Σ3 −Σ2

<0, or,

−Σ2 Σ3

Σ^T₃ Σ1

<0. 2.11

Lemma 2.7see13. LetNandEbe real constant matrices with appropriate dimensions, matrix FksatisfyingF^TkFk ≤I, then, for any > 0,EFkNN^TF^TkE^T ≤⁻¹EE^TN^TN, k∈N.

3. Main Results

To obtain our main results, we decompose the connection weight matrixCas follows:

C C₁C₂. 3.1

Then, we can get the following stability results.

Theorem 3.1. For any given positive scalars 0< τm < τM, then, underAssumption 1, system2.5 without uncertainty is globally exponentially stable for any time-varying delayτksatisfyingτ_m ≤ τk≤τ_M, if there exist positive-definite matricesP₁, Q₁, Q₂, Q₃, positive-definite diagonal matrices D1, D2, Q4, Q5, Q6, and arbitrary matricesPi, Hi, i 2,3, . . . ,21 with appropriate dimensions, such that the following LMI holds:

Ξ

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎝

Ξ11 Ξ12 Ξ13 Ξ14 Ξ15 Ξ16 Ξ17 Ξ18 Ξ19 Ξ1,10

∗ Ξ22 Ξ23 Ξ24 Ξ25 Ξ26 Ξ27 Ξ28 Ξ29 Ξ2,10

∗ ∗ Ξ33 Ξ34 Ξ35 Ξ36 Ξ37 Ξ38 Ξ39 Ξ3,10

∗ ∗ ∗ Ξ44 Ξ45 Ξ46 Ξ47 Ξ48 Ξ49 Ξ4,10

∗ ∗ ∗ ∗ Ξ55 Ξ56 Ξ57 Ξ58 Ξ59 Ξ5,10

∗ ∗ ∗ ∗ ∗ Ξ66 Ξ67 Ξ68 Ξ69 Ξ6,10

∗ ∗ ∗ ∗ ∗ ∗ Ξ77 Ξ78 Ξ79 Ξ7,10

∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ88 Ξ89 Ξ8,10

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ99 Ξ9,10

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ10,10

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎠

<0, 3.2

where

Ξ11 2C^T₁P1C1−C^T₁P12C2−C^T₂P₁₂^TC1−2P1H12C2C^T₂H₁₂^T Q2Q3

τ_M−τ_m1Q₁ 1τ_mQ₄ 1τ_MQ₅ τ_M−τ_mQ₆−2

1

D₁

2

,

Ξ12 C^T₂H13−P₁₃^T, Ξ13 C^T₁P1P₁₂−H₁₂C^T₂H14−P₁₄−P₁₂^TC₁^TP₁^T, Ξ14 C^T₁P2−H2C^T₂H15−P15^T, Ξ15 −C^T₁P2H2C₂^TH16−P16^T,

Ξ16 −C^T₁P₁₂AH₁₂AC^T₂H17−P₁₇^TD₁

1

2

,

Ξ17 −C^T₁P12BH12BC^T₂H18−P18^T, Ξ18 −C^T₁P2H2C^T₂H19−P19^T, Ξ19 C^T₁P₂−H₂C^T₂H20−P₂₀^T, Ξ1,10 C^T₁P₂−H₂C^T₂H21−P₂₁^T, Ξ22 −Q1−2

1

D2

2

, Ξ23 P13−H13, Ξ24 P3−H3,

Ξ25 −P3H₃, Ξ26 −P13AH₁₃A,

Ξ27 −P13BH13BD2

1

2

, Ξ28 −P3H3,

Ξ29 P3−H3, Ξ2,10 P3−H3,

Ξ33 P12P14−H14P₁₂^T P₁₄^T −H₁₄^T 2P1, Ξ34 P2P4−H4P₁₅^T −H₁₅^T Ξ35 H4−P2−P4P₁₆^T −H₁₆^T, Ξ36 H14−P12−P14AP₁₇^T −H₁₇^T, Ξ37 H14−P12−P14BP₁₈^T −H₁₈^T, Ξ38 H4−P2−P4P₁₉^T −H₁₉^T, Ξ39 P4P2−H4P₂₀^T −H₂₀^T, Ξ3,10 P4P2−H4P₂₁^T −H₂₁^T, Ξ44 P₅−H₅P₅^T−H₅^T−Q₃, Ξ45 H₅−P₅P₆^T−H₆^T, Ξ46 H₁₅A−P₁₅AP₇^T−H₇^T, Ξ47 H₁₅B−P₁₅BP₈^T−H₈^T, Ξ48 H₅−P₅P₉^T−H₉^T, Ξ49 P₅−H₅P₁₀^T −H₁₀^T,

Ξ49 P₅−H₅P₁₁^T −H₁₁^T,

Ξ55 H6−P6H₆^T−P₆^T−Q2, Ξ56 H16A−P16AH₇^T−P₇^T, Ξ57 H₁₆B−P₁₆BH₈^T−P₈^T, Ξ58 H₆−P₆H₉^T−P₉^T, Ξ59 P6−H6H₁₀^T −P₁₀^T, Ξ5,10 P6−H6H₁₁^T −P₁₁^T, Ξ66 H₁₇A−P₁₇AA^TH₁₇^T −A^TP₁₇^T −D₁−D₁^T,

Ξ67 H17B−P17BA^TH₁₈^T −A^TP₁₈^T,

Ξ68 H₇−P₇A^TH₁₉^T −A^TP₁₉^T, Ξ69 P₇−H₇A^TH₂₀^T −A^TP₂₀^T, Ξ6,10 P7−H7A^TH₂₁^T −A^TP₂₁^T,

Ξ77 H₁₈B−P₁₈BB^TH₁₈^T −B^TP₁₈^T −D₂−D₂^T,

Ξ78 H₈−P₈B^TH₁₉^T −B^TP₁₉^T, Ξ79 P₈−H₈B^TH₂₀^T −B^TP₂₀^T, Ξ7,10 P8−H8B^TH₂₁^T −B^TP₂₁^T,

Ξ88 H9−P9H₉^T−P₉^T−1τM⁻¹Q5, Ξ89 P9−H9H₁₀^T −P₁₀^T, Ξ8,10 P₉−H₉H₁₁^T −P₁₁^T,

Ξ9,9 P₁₀−H₁₀P₁₀^T −H₁₀^T −1τ_m⁻¹Q₄, Ξ9,10 P₁₀−H₁₀H₁₁^T −P₁₁^T, Ξ10,10 P₁₁−H₁₁P₁₁^T −H₁₁^T −τM−τ_m⁻¹Q₆.

3.3

Proof. Construct a new augmented Lyapunov-Krasovskii functional candidate as follows:

Vk V1k V2k V3k V4k V5k V6k, 3.4

where

V₁k 2Y^Tk

⎛

⎜⎜

⎜⎜

⎝

P₁ 0 · · · 0 0 0 · · · 0 ... ... ... ... 0 0 · · · 0

⎞

⎟⎟

⎟⎟

⎠

10n×10n

Yk, 3.5

Y^Tk y^Tk, y^Tk − τk, η^Tk, y^Tk − τM, y^Tk − τm, g^Tyk, g^Tyk − τM, _k

i k−τMy^Ti,_k

i k−τmy^Ti,_k−τm

i k−τM1y^Ti^T, ηk yk1−C1yk; 0 is zero matrix with appropriate dimensions:

V2k ^k−1

i k−τk

y^TiQ1yi,

V₃k ^k−1

i k−τm

y^TiQ₂yi ^k−1

i k−τM

y^TiQ₃yi,

V4k ^k−1

j k−τm

k−1

i j

y^TiQ4yi ^k−1

j k−τM

k−1

i j

y^TiQ5yi,

V₅k ^k−τm

j k−τM1

k−1 i j

y^TiQ₁yi,

V6k ^k−τm

j k−τM1

k−1 i j

y^TiQ6yi.

3.6

SetY^Tk1 y^Tk 1, y^Tk−τk, η^Tk, y^Tk−τM, y^Tk−τm, g^Tyk, g^Tyk − τ_M,_k

i k−τMy^Ti,_k

i k−τmy^Ti,_k−τm

i k−τM1y^Ti^T y^TkC₁^T η^Tk, η^Tk, y^Tk −τ_M, y^Tk − τm, g^Tyk, g^Tyk − τM,_k

i k−τMy^Ti, _k

i k−τmy^Ti,_k−τm

i k−τM1y^Ti^T. Define ΔVk Vk1−Vk. Then along the solution of system2.5we have

ΔV1k 2Y^Tk1

⎛

⎜⎜

⎜⎜

⎜⎝

P1 0 · · · 0 0 0 · · · 0 ... ... ... ... 0 0 · · · 0

⎞

⎟⎟

⎟⎟

⎟⎠Yk1−2Y^Tk

⎛

⎜⎜

⎜⎜

⎜⎝

P1 0 · · · 0 0 0 · · · 0 ... ... ... ... 0 0 · · · 0

⎞

⎟⎟

⎟⎟

⎟⎠Yk

2Y^Tk1

⎛

⎜⎜

⎜⎜

⎜⎝

P₁ 0 · · · 0 0 0 · · · 0 ... ... ... ... 0 0 · · · 0

⎞

⎟⎟

⎟⎟

⎟⎠

Yk1−2Y^Tk

⎛

⎜⎜

⎜⎜

⎜⎝

P₁ 0 · · · 0 0 0 · · · 0 ... ... ... ... 0 0 · · · 0

⎞

⎟⎟

⎟⎟

⎟⎠Yk

2I1−2I2,

3.7

I₁ Y^Tk

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

C^T₁ 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 I 0 I 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 I

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

P₁ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

Yk1

Y^Tk

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

C₁^T 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 I 0 I 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 I

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

P₁ P₂ P₁₂ 0 P₃ P₁₃ 0 P4 P14

0 P₅ P₁₅ 0 P6 P16

0 P7 P17

0 P₈ P₁₈ 0 P9 P19

0 P10 P20

0 P₁₁ P₂₁

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

⎛

⎜⎜

⎝

yk1 0 0

⎞

⎟⎟

⎠.

3.8

On the other hand, since ηk−C₂yk−Agyk−Bgyk −τk 0, _k

i k−τmyi− _k

i k−τMyi _k−τm

i k1−τMyi−y^Tk−τm y^Tk−τM 0, we have

⎛

⎜⎜

⎝

yk1 0 0

⎞

⎟⎟

⎠

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

C₁yk ηk k

i k−τm

yi− ^k

i k−τM

yi ^k−τm

i k1−τM

yi−y^Tk−τm y^Tk−τM

ηk−C2yk−Ag yk

−Bg

yk−τk

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

3.9

⎛

⎜⎜

⎜⎝

C1 0 I 0 0 0 0 0 0 0 0 0 0 I −I 0 0 −I I I

−C2 0 I 0 0 −A −B 0 0 0

⎞

⎟⎟

⎟⎠Yk, 3.10

I₂ Y^Tk

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

P₁ H₂ H₁₂ 0 H3 H13

0 H4 H14

0 H₅ H₁₅ 0 H6 H16

0 H₇ H₁₇ 0 H₈ H₁₈ 0 H9 H19

0 H₁₀ H₂₀ 0 H₁₁ H₂₁

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

⎛

⎜⎜

⎝ yk

0 0

⎞

⎟⎟

⎠

Y^Tk

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

P₁ H₂ H₁₂ 0 H3 H13

0 H₄ H₁₄ 0 H₅ H₁₅ 0 H6 H16

0 H₇ H₁₇ 0 H8 H18

0 H9 H19

0 H₁₀ H₂₀ 0 H11 H21

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

⎛

⎜⎜

⎝

I 0 0 0 0 0 0 0 0 0 0 0 0 I −I 0 0 −I I I

−C2 0 I 0 0 −A −B 0 0 0

⎞

⎟⎟

⎠Yk.

3.11

ΔV2k≤y^TkQ₁yk−y^Tk−τkQ₁yk−τk ^k−τm

i k1−τM

y^TiQ₁yi,

3.12 ΔV3k y^Tk Q2Q₃yk−y^Tk−τ_mQ₂yk−τ_m−y^Tk−τ_MQ₃yk−τ_M. 3.13

FromLemma 2.3we can obtain

ΔV4k ^k

j k1−τm

k i j

y^TiQ₄yi− ^k−1

j k−τm

k−1 i j

y^TiQ₄yi

^k

j k1−τM

k i j

y^TiQ₅yi− ^k−1

j k−τM

k−1

i j

y^TiQ₅yi

k−1

j k−τm

k i j1

y^TiQ₄yi− ^k−1

j k−τm

k−1

i j

y^TiQ₄yi

^k−1

j k−τM

k i j1

y^TiQ₅yi− ^k−1

j k−τM

k−1 i j

y^TiQ₅yi

k−1

j k−τm

y^TkQ₄yk−y^T j

Q₄y j

^k−1

j k−τM

y^TkQ₅yk−y^T j

Q₅y j

≤1τ_my^TkQ₄yk− ^k

j k−τm

y^T j

Q₄y j

1τ_My^TkQ₅yk− ^k

j k−τM

y^T j

Q₅y j

≤1τ_my^TkQ₄yk− 1 1τ_m

⎡

⎣ ^k

j k−τm

yj

⎤

⎦

T

Q₄

⎡

⎣ ^k

j k−τm

y j⎤

⎦

1τ_My^TkQ₅yk− 1 1τM

⎡

⎣ ^k

j k−τM

yj

⎤

⎦

T

Q₅

⎡

⎣ ^k

j k−τM

y j⎤

⎦,

3.14

ΔV5k ^k1−τm

j k2−τM

k i j

y^TiQ₁yi− ^k−τm

j k1−τM

k−1 i j

y^TiQ₁yi

k−τm

j k1−τM

k i j1

y^TiQ₁yi− ^k−τm

j k1−τM

k−1 i j

y^TiQ₁yi

k−τm

j k1−τM

y^TkQ4yk−y^T j

Q1y j

τM−τmy^TkQ1yk− ^k

j k1−τM

y^T j

Q1y j

.

3.15

Similarly,

ΔV6k τM−τ_my^TkQ₆yk− ^k−τm

j k1−τM

y^T j

Q₆y j

≤τM−τ_my^TkQ₆yk− 1 τM−τm

⎡

⎣ ^k−τm

j k1−τM

yj

⎤

⎦

T

Q₆

⎡

⎣ ^k−τm

j k1−τM

y j⎤

⎦.

3.16

FromLemma 2.5, for any positive diagonal matrixD₁, D₂, it follows that

2y^Tk−τkD2

1

2

g

yk−τk

−2g^T

yk−τk D2g

yk−τk

−2y^Tk−τk

1

D₂

2

yk−τk≥0

2y^TkD1

1

2

g yk

−2g^T yk

D1g yk

−2y^Tk

1

D1

2

yk≥0.

3.17

Combining3.7–3.17, we get

ΔVk≤Y^Tk ΞYk. 3.18

If the LMI3.2holds, it follows that there exists a sufficient small scalarε >0 such that ΔVk≤ −εyk². 3.19 On the other hand, it can easily to get that

Vk≤2λ_mP1yk²λ_maxQ1 ^k−1

i k−τk

yi²λ_maxQ2 ^k−1

i k−τm

yi²

λmaxQ3 ^k−1

i k−τM

yi²λmaxQ4 ^k

j k−τm

k−1 i j

yi²λmaxQ5 ^k

j k−τM

k−1

i j

yi²

λmaxQ1 ^k−τm

j k−τM

k−1 i j

yi²λmaxQ6 ^k−τm

j k1−τM

k−1

i j

yi²

≤2λmaxPyk²λ k−1 i k−τM

yi²,

3.20

whereλ λmaxQ1 λmaxQ2 λmaxQ3 1τmλmaxQ4 1τMλmaxQ5 1τM− τ_mλmaxQ1 λ_maxQ6. Choose a scalarθ >1 such that−εθ2θ−1λmaxP1 θ−1λ· τ_Mθ^τM 0. Then by3.19and3.20, we get

θ^k1Vk1−θ^kVk θ^k1ΔVk θ^kθ−1Vk

≤ε₁θ^kyk²ε₂θ^k

k−1

i k−τM

yi²,

3.21

where ε₁ −εθ2λ_maxPθ−1, ε₂ λθ−1. Therefore, for arbitrary positive integer N≥τ_M1, summing up both sides of3.21from 0 toN−1, we can obtain

θ^NVN−V0≤ε₁

N−1

k 0

θ^kyk²ε₂

N−1

k 0 k−1

i k−τM

θ^kyi²

≤ε2τMτM1θ^τM sup

i∈N−τM,0yi² ε1ε2τMθ^τMN−1

k 0

θ^kyk². 3.22

Noting that

VN≥λ_minP1yN², V0≤λτM2λ_maxP1 sup

i∈N−τM,0

yi². 3.23

It follows that yN ≤ α · β^Nsup_i∈N−τ

M,0yi, where β √

θ⁻¹, α λτM2λmaxP ε2τMτM1θ^τM/λminP. By Definition 2.2, system 2.5 is globally exponentially stable, which completes the proof ofTheorem 3.1.

Remark 3.2. By constructing the new augmented Lyapunov functional, free-weighting matricesP_i, H_i, i 2,3, . . . ,21 are introduced so as to reduce the conservatism of the delay- dependent result. Moreover, the decomposition of matrixC C₁C2makes the conservatism of the stability criterion reduce further, since the elements of matricesC1, C2are not restricted to−1,1any more.

Remark 3.3. SinceTheorem 3.1holds for arbitrary matricesC₁, C₂satisfyingC₁C₂ C, then, whenC1 0 orC2 0, respectively, we can easily obtains the following simplified useful corollaries.

Corollary 3.4. For any given positive scalars 0< τ_m< τ_M, then, underAssumption 1, system2.5 is globally exponentially stable for any time-varying delayτksatisfyingτm ≤ τk≤ τM, if there exist positive-definite matricesP₁, Q₁, Q₂, Q₃, positive-definite diagonal matricesD₁, D₂, Q₄, Q₅, Q₆,

and arbitrary matricesPi, Hi, i 2,3, . . . ,21 with appropriate dimensions, such that the following LMI holds:

Ξ

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎝

Ξ11 Ξ12 Ξ13 Ξ14 Ξ15 Ξ16 Ξ17 Ξ18 Ξ19 Ξ1,10

∗ Ξ22 Ξ23 Ξ24 Ξ25 Ξ26 Ξ27 Ξ28 Ξ29 Ξ2,10

∗ ∗ Ξ33 Ξ34 Ξ35 Ξ36 Ξ37 Ξ38 Ξ39 Ξ3,10

∗ ∗ ∗ Ξ44 Ξ45 Ξ46 Ξ47 Ξ48 Ξ49 Ξ4,10

∗ ∗ ∗ ∗ Ξ55 Ξ56 Ξ57 Ξ58 Ξ59 Ξ5,10

∗ ∗ ∗ ∗ ∗ Ξ66 Ξ67 Ξ68 Ξ69 Ξ6,10

∗ ∗ ∗ ∗ ∗ ∗ Ξ77 Ξ78 Ξ79 Ξ7,10

∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ88 Ξ89 Ξ8,10

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ99 Ξ9,10

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ10,10

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎠

<0, 3.24

where

Ξ11 −2P1H₁₂CC^TH₁₂^T Q₂Q₃ τ_M−τ_m1Q₁ 1τmQ4 1τMQ5 τM−τmQ6−2

1

D1

2

,

Ξ12 C^TH13−P₁₃^T, Ξ13 C^TH14−P₁₄−P₁₂^T−H₁₂, Ξ14 C^TH15−P15^T−H2, Ξ15 H2C^TH16−P16^T,

Ξ16 H₁₂AC^TH17−P₁₇^TD₁

1

2

, Ξ17 H₁₂BC^TH18−P₁₈^T,

Ξ18 H2C^TH19−P19^T, Ξ19 C^TH20−P20^T−H2, Ξ1,10 C^TH21−P21^T−H2. 3.25

Corollary 3.5. For any given positive scalars 0< τm< τM, then, underAssumption 1, system2.5 is globally exponentially stable for any time-varying delayτksatisfyingτ_m ≤ τk≤ τ_M, if there exist positive-definite matricesP₁, Q₁, Q₂, Q₃, positive-definite diagonal matricesD₁, D₂, Q₄, Q₅, Q₆, and arbitrary matricesPi, Hi, i 2,3, . . . ,21 with appropriate dimensions, such that the following LMI holds: