Research Article Further Stability Criterion on Delayed Recurrent Neural Networks Based on Reciprocal Convex Technique

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1 Mathematica Probems in Engineering Voume 2012, Artice ID , 14 pages doi: /2012/ Research Artice Further Stabiity Criterion on Deayed Recurrent Neura Networks Based on Reciproca Convex Technique Guobao Zhang, 1 Tao Li, 2 and Shumin Fei 1 1 Key Laboratory of Measurement and Contro of CSE, Schoo of Automation, Southeast University, Ministry of Education, Nanjing , China 2 Schoo of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing , China Correspondence shoud be addressed to Tao Li, itaotianren@yahoo.com.cn Received 15 Apri 2011; Revised 3 June 2011; Accepted 6 Juy 2011 Academic Editor: Zidong Wang Copyright q 2012 Guobao Zhang et a. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. Together with Lyapunov-Krasovskii functiona theory and reciproca convex technique, a new sufficient condition is derived to guarantee the goba stabiity for recurrent neura networks with both time-varying and continuousy distributed deays, in which one improved deay-partitioning technique is empoyed. The LMI-based criterion heaviy depends on both the upper and ower bounds on state deay and its derivative, which is different from the existent ones and has more appication areas as the ower bound of deay derivative is avaiabe. Finay, some numerica exampes can iustrate the reduced conservatism of the derived resuts by thinning the deay interva. 1. Introduction Recenty, various casses of neura networks have been increasingy studied in the past few decades, due to their practica importance and successfu appications in many areas such as optimization, image processing, and associative memory design. In those appications, the key feature of the designed neura network is to be convergent. Meanwhie, since there inevitaby exist communication deay which is the main source of osciation and instabiity in various dynamica systems, great efforts have been made to anayze the dynamica behaviors of time-deay systems incuding deayed neura networks DNNs, and many eegant resuts have been reported; see 1 31 and the references therein. In practica appication, though it is difficut to describe the form of deay precisey, the ranges of time deay and its variation rate can be measured. Since Lyapunov functiona approach imposed no restriction on deay

2 2 Mathematica Probems in Engineering and its derivative and presented the simpe stabiity resuts, Lyapunov-Krasovskii functiona LKF one has been widey utiized due to that it can fuy utiize the information on timedeay system. Thus recenty, the deay-dependent stabiity or deay-derivative-dependent one for DNNs has become an important topic of primary significance, in which its main purpose was to derive the maximum aowabe upper bound on time deay such that the system can be convergent 7 9, Meanwhie, since a neura network usuay has a spatia nature due to the presence of an amount of parae pathways of a variety of axon sizes and engths, it is desired to mode them by introducing a distributed deay over a certain duration of time such that the distant past has ess infuence compared to the recent behavior of the state. In other words, when studying stabiity for DNNs, the distributed deay shoud be taken into consideration simutaneousy Presenty, during tacking the effect of time deay, the deay-partitioning idea has been verified to be more effective in reducing the conservatism and widey empoyed In 11, one deay-partitioning idea has been used to tacke time-varying deay of DNNs based on the improved idea in 10 and in 12, 13, and some researchers aso put forward the other nove deay-partitioning idea to tacke constant deay, which can be more evident and concise than the idea based on the one in 10. Later, though the idea was extended to timevarying deay case 14, 15, 17, the improved idea cannot dea with the interva varying deay efficienty, especiay as the ower bound of deay is greater than 0. Meanwhie, as for timevarying deay, because convex combination technique can pay an important roe in reducing the conservatism, it has received much attention and achieved some great improvements in studying the stabiity for time-deay systems incuding DNNs 15 18, Its basic idea is to approximate the integra terms of quadratic quantities into a convex combination of quadratic terms of the LMIs. However, owing to the inversey weighted nature of coefficients in Jensen inequaity approach or the imitation of free-weighting matrix method, when estimating the derivative of Lyapunov-Krasovskii functiona, some important terms sti have been ignored based on the convex combination technique In 31, together with integra inequaity emma, the authors put forward the reciproca convex technique, which can consider these previousy ignorant terms. Yet, we have noticed that the reciproca convex technique coud not tacke the case that the ower and upper bound of time deay can be measured simutaneousy, which sti needs the convex technique. Now, together with the improved deay-partitioning idea in 17 and combining reciproca convex technique with convex combination one, no researcher has investigated the stabiity for DNNs as the ower bound of deay derivative is avaiabe, which motivates the focus of this presented work. In the paper, we make some great efforts to investigate asymptotica stabiity for recurrent neura networks with both time-varying and continuousy distributed deay, in which both the upper and ower bounds of time deay and its derivative are treated. Through appying an improved deay-partitioning idea, one LMI-based condition is derived based on combination of reciproca convex technique and convex one, which can present the pretty deay dependence and computationa efficiency. Finay, we give three numerica exampes to iustrate the reduced conservatism. Notations For symmetric matrices X, Y, X > Y resp., X Y means that X Y > 0 X Y 0 is a positive-definite resp., positive-semidefinite matrix; Tr A denotes the trace of the matrix A; X Y Y T Z XY Z with denotes the symmetric term in a symmetric matrix.

3 Mathematica Probems in Engineering 3 2. Probem Formuations Consider the DNNs with continuousy distributed deay of the foowing form: t ż t Cz t Ag z t Bg z t τ t D K t s g z s ds J, 2.1 where z t z 1 t z n t T R n is the neuron state vector; C diag{c 1,...,c n } > 0and A, B, D are n n known constant matrices; g z g 1 z 1 g n z n T R n stands for the neuron activation function, J R n is a constant input vector, and K t s k ij t s n n ; here the deay kerne k ij is a rea-vaued nonnegative continuous function defined on 0, and satisfies 0 k ij θ dθ 1fori, j 1, 2,...,n. The foowing assumptions on system 2.1 are utiized throughout this paper. H1 The deay τ t denotes one continuous function satisfying 0 τ 0 τ t τ m, μ 0 τ t μ m, 2.2 in which τ 0,τ m,μ 0,μ m are constants. Here we denote τ m τ m τ 0 and μ m μ m μ 0. H2 For the constants δ j,δ j, the functions g j in 2.1 are bounded and satisfy the foowing condition: δ j g j x g j ( y ) x y δ j, x, y R, x / y, j 1, 2,...,n. 2.3 In what foows, we denote Σ diag{δ 1,...,δ n}, Σ diag{δ 1,...,δ n} and set Σ 1 ΣΣ, Σ 2 Σ Σ /2, respectivey. Remark 2.1. As pointed out in 22, the constants δ i,δ i in H2 are aowed to be positive, negative, or zero. Thus, some previousy used Lipschitz conditions are just specia cases of H2, which means that the activation functions are of more genera descriptions than these previous ones. Suppose z z 1 z n T is the equiibrium point of the system 2.1. In order to prove the resut, we wi shift the equiibrium to the origin by changing the variabe x z z. Then the system 2.1 can be transformed into t ẋ t Cx t Af x t Bf x t τ t D K t s f x s ds, 2.4 where x t x 1 t,...,x n t T,andf x f 1 x 1,...,f n x n T with f j x j g j x j z j g j z j,j 1, 2,...,n. Note that the function f j satisfies f j 0 0, and δ j f j x x δ j, x R, x / In order to estabish the stabiity criterion, firsty, the foowing emmas are introduced.

4 4 Mathematica Probems in Engineering Lemma 2.2 see 27. For any constant matrix W R n n, W W T > 0, scaar functiona 0 r t r M, and a vector function ė : r M, 0 R n such that the foowing integration is we 0 defined, then r M r t ėt t s Wė t s ds e t e t r t T W e t e t r t. Lemma 2.3 see 31. Let the functions f 1 t,f 2 t,...,f N t : R m R have the positive vaues in an open subset D of R m and satisfy 1 α 1 f 1 t 1 α 2 f 2 t 1 α N f N t : D R n 2.6 with α i > 0 and i α i 1, then the reciproca convex technique of f i t over the set D satisfies 1 f i t f i t i,j t g i,j t : R α i i i i / jg m R n,g i,j t g j,i t, fi t g i,j t 0. g j,i t f j t 2.7 Then the probem to be addressed in next section can be formuated as deveoping a condition ensuring that the DNNs 2.4 are asymptoticay stabe. 3. Deay-Derivative-Dependent Stabiity In the section, through utiizing the reciproca convex technique idea in 31, we present one nove deay-derivative-dependent stabiity criterion for the system 2.4 in terms of LMIs. Theorem 3.1. Given a positive integer and setting δ τ m /, Σ 1 diag {Σ 1,...,Σ 1 }, Σ 2 diag {Σ 2,...,Σ 2 }, then the system 2.4 satisfying 2.2 and 2.5 is gobay asymptoticay stabe if there exist n n matrices P>0, P 1 > 0, H 1,Q 1 > 0, V>0, W j > 0 j 1,...,, S j j 1,...,,E i i 1, 2, n n diagona matrices K>0,L > 0,T > 0,U i > 0 i 1, 2, 3,V j > 0,R j > 0 j 1,...,, n n matrices X ij > 0,Y ij,z ij > 0 i 1, 2; j 1,...,, and n n constant matrices P >0, Q >0, H such that the LMIs in hod: P1 H 1 > 0, Q 1 P H > 0, Q Xij Y ij > 0, Z ij Wj S j 0, W j 3.1 i 1, 2; j 1,...,, Υ 1 ΘΥ T 1 Υ 2ΞΥ T 2 μ m I T 1i Xki Y ki I 1i < 0, k 1, 2, 3.2 Z ki

5 Mathematica Probems in Engineering 5 where I 1i 0n 1 i n I n 0 n 2 1 n I n 0 n i 4 n 0 n 1 i n I n 0 n 2 1 n I n 0 n i 4 n, and Θ 11 V 0 Θ Θ 17 0 E T 1 B ET 1 D Θ H U 2 Σ U 2 Σ Θ Θ Q Θ U , Θ 77 0 E T 2 B ET 2 D U 3 Σ 1 U 3 Σ 2 0 U n T I n 0 n n I n 0 n 1 n I n Ξ 11 S W 1 S Ξ n n I n 0 n 2 2 n I n Ξ 22 W S 0 0 H 0 n 2n I n 0 n 2 3 n I n Ξ Υ 1, Ξ Ξ 36 0 n 2 n I n 0 n 3 3 n I n, Υ 2 Ξ n 2 3 n I n 0 n 4 3 n I n Q 0 0 n 2 4 n I n 0 n 4 4 n I n Ξ 66 0 n 3 4 n I n 0 n 4 5 n I n 0 n 4 6 n I n 3.3 with representing the appropriatey dimensiona 0 matrix making Υ i i 1, 2 of 4 8 n coumns, X i diag{x i1,...,x i }, Ỹ i diag{y i1...y i }, Z i diag{z i1,...,z i },i 1, 2, W diag{w 1,...,W }, S diag{s 1,...,S }, Ṽ diag{v 1,...,V }, R diag{r 1,...,R }, and Θ 11 E T 1 C CT E 1 P 1 V U 1 Σ 1, Θ 14 E T 1 A H 1 U 1 Σ 2, Θ 17 P ΣK ΣL E T 1 CT E 2, Θ 22 P 1 V, Θ 44 U 1 Q 1 Tr T I n, Θ 47 K L A T E 2, Θ 77 E T 2 E 2 τ 2 0 V δ 2 W i, Ξ 11 P X 1 W Ṽ Σ 1,

6 6 Mathematica Probems in Engineering Ξ 14 H Ṽ Σ 2 Ỹ 1, Ξ 22 P W, Ξ 33 2 W S S T R Σ 1, ( Ξ 36 1 μ ( m )Ỹ2 1 μ 0 )Ỹ1 R Σ 2, Ξ 44 Q Z 1 Ṽ, ( Ξ 66 1 μ ) ( m Z 2 1 μ ) 0 Z 1 R. 3.4 Proof. Based on 2.5 and denoting ρ t τ t τ 0 /, we construct the Lyapunov-Krasovskii functiona candidate as V x t V 1 x t V 2 x t V 3 x t V 4 x t, 3.5 where n xi V 1 x t x T n xi t Px t 2 k i fi s δ i s ds 2 i δ i s f i s ds, V 2 x t t t τ 0 0 T x s P1 H 1 x s ds f x s Q 1 f x s t τ0 i 1 δ t τ 0 i 1 δ ρ t t τ0 i 1 δ ρ t t τ 0 iδ t τ0 t τ 0 δ T x s X1i Y 1i x s ds f x s f x s Z 1i 0 σ s T P h σ s T x s X2i Y 2i x s ds, f x s f x s Z 2i H σ s ds Q h σ s V 3 x t n n i j 1t 0 t k ij θ f 2 ( j xj s ) ds dθ, t θ V 4 x t 0 t τ 0 t θ τ 0 ẋ T s V ẋ s ds dθ τ0 i 1 δ t τ 0 iδ t θ δẋ T s W i ẋ s ds dθ 3.6 with K diag{k 1,...,k n } > 0, L diag{ 1,..., n } > 0, T diag{t 1,...,t n } > 0 waiting to be determined, and σ T s x T s x T s 1 δ, h T σ s f T x s f T x s 1 δ. 3.7

7 Mathematica Probems in Engineering 7 Through directy cacuating and using any n n constant matrices E i i 1, 2, thetime derivative of functiona 3.5 aong the trajectories of system 2.4 yieds ( ) V 1 x t 2x T t Pẋ t 2 f T x t K L x T t ΣL ΣK ẋ t 2 x T t E T 1 ẋt t E T 2 ẋ t Cx t Af x t Bf x t τ t t D K t s f x s ds, 3.8 V 2 x t x T t P 1 x t 2x T t H 1 f x t f T x t Q 1 f x t x T t τ 0 P 1 x t τ 0 2x T t τ 0 H 1 f x t τ 0 f T x t τ 0 Q 1 f x t τ 0 σ T t τ 0 Pσ t τ 0 2σ T t τ 0 Hh σ t τ 0 h T σ t τ 0 Qh σ t τ 0 σ T t τ 0 δ Pσ t τ 0 δ 2σ T t τ 0 δ Hh σ t τ 0 δ h T σ t τ 0 δ Qh σ t τ 0 δ σ T t τ 0 X 1 σ t τ 0 σ T( t τ 0 ρ t )( 1 τ t σ T t τ 0 δ X 2 σ t τ 0 δ ) ( X 2 X 1 ) σ ( t τ 0 ρ t ) 2σ T t τ 0 Ỹ 1 h σ t τ 0 2σ T( t τ 0 ρ t )( 1 τ t ) ) (Ỹ2 Ỹ 1 h ( σ ( t τ 0 ρ t )) 2σ T t τ 0 δ Ỹ 2 h σ t τ 0 δ h T σ t τ 0 Z 1 h σ t τ 0 h T( σ ( t τ 0 ρ t ))( 1 τ t ) ( ) Z 2 Z 1 h ( σ ( t τ 0 ρ t )) h T σ t τ 0 δ Z 2 h T σ t τ 0 δ, 3.9 V 3 x t n n i j 1t 0 k ij θ f 2 ( j xj t ) n n dθ j 1 0 k ij θ f 2 ( j xj t θ ) dθ n n n ( t i f T x t f x t j 1 0 ( k ij θ f j xj t θ ) ) 2 dθ

8 8 Mathematica Probems in Engineering ( t ) T( ) 1 Tr T f T x t f x t K t s f x s ds n T ( t ) K t s f x s ds, 3.10 ( ) V 4 x t ẋ T t τ 2 0 V δ 2 W i ẋ t t τ 0 iδ t τ0 i 1 δ t δẋ T s W i ẋ s ds. t τ 0 τ 0 ẋ T s V ẋ s ds 3.11 Moreover, together with Lemmas 2.2 and 2.3 and 3.1, we can estimate t τ0 i 1 δ t τ 0 iδ δẋ T s W i ẋ s ds as foows: t τ0 i 1 δ t τ 0 iδ δẋ T s W i ẋ s ds { τ m ( x t τ0 i 1 δ ρ t ) x t τ 0 iδ T τ m τ t ( W i x t τ0 i 1 δ ρ t ) x t τ 0 iδ τ t τ 0 x t τ 0 i 1 δ x ( t τ 0 i 1 δ ρ t ) T W i x t τ0 i 1 δ x ( t τ 0 i 1 δ ρ t )} τ m T x t τ 0 i 1 δ x ( t τ 0 i 1 δ ρ t ) W i S i W i S i 2W i S i S T i S T i W i x t τ 0 iδ W i x t τ 0 i 1 δ x ( t τ 0 i 1 δ ρ t ) x t τ 0 iδ T σ t τ 0 W S W S σ ( t τ 0 ρ t ) σ t τ 0 2 W S S T S T W σ ( t τ 0 ρ t ). σ t τ 0 δ W σ t τ 0 δ 3.12

9 Mathematica Probems in Engineering 9 From 2.5, the foowing inequaity hods for any diagona matrices U i > 0 i 1, 2, 3, V j > 0, R j > 0 j 1,..., with the compatibe dimensions and setting Ṽ diag{v 1,...,V }, R diag{r 1,...,R }, 0 x T t U 1 Σ 1 x t 2x T t U 1 Σ 2 f x t f T x t U 1 f x t x T t τ m U 2 Σ 1 x t τ m 2x T t τ m U 2 Σ 2 f x t τ m f T x t τ m U 2 f x t τ m x T t τ t U 3 Σ 1 x t τ t 2x T t τ t U 3 Σ 2 f x t τ t f T x t τ t U 3 f x t τ t σ T t τ 0 Ṽ Σ 1 σ t τ 0 2σ T t τ 0 Ṽ Σ 2 h σ t τ 0 h T σ t τ 0 Ṽh σ t τ 0 σ T( t τ 0 ρ t ) R Σ 1 σ ( t τ 0 ρ t ) 2σ T( t τ 0 ρ t ) R Σ 2 h ( σ ( t τ 0 ρ t )) h T( σ ( t τ 0 ρ t )) Rh ( σ ( t τ 0 ρ t )) Now, adding the terms on the right-hand side of and empoying inequaity 3.13, we can deduce { V x t ζ T t Υ 1 ΘΥ T 1 Υ 2ΞΥ T 2 τ t μ 0 X1i Y I T 1i 3i Z 1i } X2i Y 2i I 3i ζ t ζ T t Λ t ζ t, I T 3i Z 2i I 3i μ m τ t 3.14 in which Θ, Ξ, Υ i i 1, 2 are presented in 3.2, and ζ T t x T t σ T t τ 0 x T t τ 0 τ m σ T( t τ 0 ρ t ) f T x t h T σ t τ 0 f T x t τ m h T( σ ( t τ 0 ρ t )) ẋ T t ( t ) T x T t τ t f T x t τ t K t s f x s ds Then by empoying convex combination technique, the LMIs in 3.2 can guarantee Λ t < 0, which indicates that there must exist a positive scaar χ>0 such that V x t χ x t 2 < 0 for x t / 0. Then it foows from Lyapunov-Krasovskii stabiity theory that the system 2.4 is asymptoticay stabe, and the proof is competed.

10 10 Mathematica Probems in Engineering Remark 3.2. Presenty, the convex combination technique has been widey empoyed to tacke time-varying deay owing to the truth that it coud reduce the conservatism more effectivey than the previous ones, see 15 18, In 31, the authors put forward the reciproca convex approach, which can reduce the conservatism more effectivey than convex combination ones. Yet, it has come to our attention that no researchers have utiized both of them simutaneousy to tacke the stabiity for DNNs. Remark 3.3. One can easiy check that the theorem in this work achieves some great improvements over the one in 17, which can be iustrated in the foowing. Firsty, some ignored terms in 17 have been fuy considered in this paper when estimating the derivative of Lyapunov-Krasovskii functiona in 3.11, such as the ignored ones. Secondy, owing to the introduction of reciproca convex approach, Theorem 3.1 can be much ess compicated than the ones in 17, which wi resut in some computation simpicity in some degree. Thirdy, though reciproca convex approach pays an important roe in tacking the range of time deay, it cannot efficienty dea with the effect of ower and upper bound on deay derivative, which can be checked in Thus we empoy the convex combination technique to overcome this shortcoming. Remark 3.4. When τ t is not differentiabe or μ 0 resp., μ m is unknown, through setting Xij Y ij X2j Y Z ij 0 i 1, 2 or 2j X1j Y Z 2j 0 resp., 1j Z 1j 0 in 3.5, our theorem sti can be true. Remark 3.5. Owing to the introduction of deay-partitioning idea in this work, the difficuty and compexity in checking the theorem wi become more and more evident when the integer increases and the dimension of the LMIs in 3.2 wi become much higher. Yet based on the resuts in 12 16, the maximum aowabe upper bound of τ m woud become unapparent enarging as 5. Thus if we want to empoy the idea to rea cases, we do not necessariy partition the interva τ 0,τ m into more than 5 subintervas. 4. Numerica Exampes In the section, three numerica exampes wi be presented to iustrate that our resuts are superior over the ones by convex combination technique. Exampe 4.1. We revisit the system considered in 9, 11, 17 with the foowing parameters: C, A, B, D Σ, Σ, in which τ 0 0 is set. If we do not consider the existence of μ 0, then by utiizing Theorem 3.1 and Remark 3.2, the corresponding maximum aowabe upper bounds MAUBs τ max for different μ m derived by Theorem 3.1 in 17 and in the paper can be summarized in Tabe 1,

11 Mathematica Probems in Engineering 11 Tabe 1: Cacuated MAUBs τ max for various, unavaiabe μ 0 in Exampe 4.1. Methods\μ m Li et a Theorem Tabe 2: Cacuated MAUBs τ max for various and μ inexampe 4.1. Methods\μ m Zhang et a Li et a Theorem which demonstrates that Theorem 3.1 in this paper of 1 is ess conservative than the theorem in 17. Yet, if we set μ 0 0.5, it is sti easy to verify that our resuts can yied much ess conservative resuts than the one in 17, which can be shown in Tabe 2. Based on Tabes 1 and 2, it indicates that the conservatism of stabiity criterion can be greaty deduced if we take μ 0 into consideration. Moreover, though the deay-partitioning idea has been used in 17, the corresponding MAUBs τ max derived by 17 and Theorem 3.1 are summarized in Tabe 3, which shows that the idea in this work can be more efficient than the one in 17 even for 1, 2. Exampe 4.2. Consider the deayed neura networks 2.1 with C diag{1.2769, , , }, Σ 0 3 3, Σ diag{0.1137, , , }, A , which has been addressed extensivey, see 2, 15, 17 and the references therein. Together with the deay-partitioning idea and for different μ m, the works 15, 17 have cacuated the MAUBs τ max such that the origin of the system is gobay asymptoticay stabe for τ t satisfying 3 τ 0 τ t τ m τ max. By resorting to Theorem 3.1 and Remark 3.2, the corresponding resuts can be given in Tabe 4, which indicates that our deay-partitioning idea can be more effective than the reevant ones in 15, 17 for 1, 2andμ 0 0.

12 12 Mathematica Probems in Engineering Tabe 3: Cacuated MAUBs τ max for various, μ 0 0.5inExampe 4.1 Methods\μ m Unknown μ m Li et a Theorem Tabe 4: Cacuated MAUBs τ max for various, μ m in Exampe 4.2. Methods\μ m Unknown μ m Hu et a Li et a Theorem Tabe 5: Cacuated MAUBs τ max for various μ 0,μ m,and in Exampe 4.3. Methods\ μ 0,μ m 0.1, , , , 1.1 Li et a Theorem Exampe 4.3. Consider the deayed neura networks 2.4 with C, A, B, D, e 2 t s 3e 3 t s K t s, 4e 4 t s 2e 2 t s 4.3 and f x i tanh x i,i 1, 2. For τ 0 0.5, choosing various μ 0,μ m in Tabe 5 and appying Theorem 3.1 in our work and the one in 17, we can find the MAUBs on τ m for which the system remains asymptoticay stabe. Based on Tabe 5, it can indicate that the deay-partitioning idea in our work can be ess conservative than the ones in Concusion This paper has investigated the asymptotica stabiity for DNNs with continuousy distributed deay. Through empoying one improved deay-partitioning idea and combining

13 Mathematica Probems in Engineering 13 reciproca convex technique with convex combination one, one stabiity criterion with significanty reduced conservatism has been estabished in terms of LMIs. The proposed stabiity condition benefits from the partition of deay intervas and reciproca convex technique. Three numerica exampes have been given to demonstrate the effectiveness of the presented criteria and the improvements over some existent ones. Finay, it shoud be worth noting that the deay-partitioning idea presented in this work is widey appicabe in many cases. Acknowedgment This work is supported by the Nationa Natura Science Foundation of China no , no , no , no , and the Specia Foundation of China Postdoctora Science Foundation Projects no References 1 B. Shen, Z. Wang, H. Shu, and G. Wei, H fitering for noninear discrete-time stochastic systems with randomy varying sensor deays, Automatica, vo. 45, no. 4, pp , B. Shen, Z. Wang, Y. S. Hung, and G. Chesi, Distributed H fitering for poynomia noninear stochastic systems in sensor networks, IEEE Transactions on Industria Eectronics, vo.58,no.5,pp , H. Dong, Z. Wang, D. W. C. Ho, and H. Gao, Robust H fuzzy output-feedback contro with mutipe probabiistic deays and mutipe missing measurements, IEEE Transactions on Fuzzy Systems, vo. 18, no. 4, pp , B. Shen, Z. Wang, and Y. S. Hung, Distributed H -consensus fitering in sensor networks with mutipe missing measurements: the finite-horizon case, Automatica, vo. 46, no. 10, pp , H. Dong, Z. Wang, and H. Gao, Robust H fitering for a cass of noninear networked systems with mutipe stochastic communication deays and packet dropouts, IEEE Transactions on Signa Processing, vo. 58, no. 4, pp , G. Wei, Z. Wang, and B. Shen, Error-constrained fitering for a cass of noninear time-varying deay systems with non-gaussian noises, IEEE Transactions on Automatic Contro, vo. 55, no. 12, pp , Z. Wang, H. Zhang, and W. Yu, Robust stabiity criteria for interva Cohen-Grossberg neura networks with time varying deay, Neurocomputing, vo. 72, no. 4 6, pp , P. Baasubramaniam and S. Lakshmanan, Deay-range dependent stabiity criteria for neura networks with Markovian jumping parameters, Noninear Anaysis: Hybrid Systems, vo. 3, no. 4, pp , Y. Zhang, D. Yue, and E. Tian, New stabiity criteria of neura networks with interva time-varying deay: a piecewise deay method, Appied Mathematics and Computation, vo. 208, no. 1, pp , K. Gu, Refined discretized Lyapunov functiona method for systems with mutipe deays, Internationa Journa of Robust and Noninear Contro, vo. 13, no. 11, pp , W. H. Chen and W. X. Zheng, Improved deay-dependent asymptotic stabiity criteria for deayed neura networks, IEEE Transactions on Neura Networks, vo. 19, no. 12, pp , S. Mou, H. Gao, J. Lam, and W. Qiang, A new criterion of deay-dependent asymptotic stabiity for Hopfied neura networks with time deay, IEEE Transactions on Neura Networks, vo.19,no.3,pp , R. Yang, H. Gao, and P. Shi, Nove robust stabiity criteria for stochastic Hopfied neura networks with time deays, IEEE Transactions on Systems, Man, and Cybernetics, Part B, vo. 39, no. 2, pp , X. M. Zhang and Q. L. Han, New Lyapunov-Krasovskii functionas for goba asymptotic stabiity of deayed neura networks, IEEE Transactions on Neura Networks, vo. 20, no. 3, pp , 2009.

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