Research ArticleImproved Generalized1198672 Filtering for Static NeuralNetworks with Time-Varying Delay via Free-Matrix-BasedIntegral Inequality

Hui-Jun Yu12 Yong He 34 andMinWu34

1School of Information Science and Engineering Central South University Changsha Hunan 410083 China2School of Electrical and Information Engineering Hunan University of Technology Zhuzhou 412007 China3School of Automation China University of Geosciences Wuhan 430074 China4Hubei Key Laboratory of Advanced Control and Intelligent Automation for Complex Systems Wuhan 430074 China

Correspondence should be addressed to Yong He heyong08cugeducn

Received 18 July 2017 Revised 16 December 2017 Accepted 2 January 2018 Published 30 January 2018

Academic Editor Renming Yang

Copyright copy 2018 Hui-Jun Yu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper focuses on the generalized 1198672 filtering of static neural networks with a time-varying delay The aim of this problem isto design a full-order filter such that the filtering error system is globally asymptotically stable with guaranteed 1198672 performanceindex By constructing an augmented Lyapunov-Krasovskii functional and applying the free-matrix-based integral inequality toestimate its derivative an improved delay-dependent condition for the generalized 1198672 filtering problem is established in terms ofLMIs Finally a numerical example is presented to show the effectiveness of the proposed method

1 Introduction

During the past few decades neural network (NN) whichmodels the way a biological brain solving problems hasbeen successfully applied to various engineering fields suchas pattern recognition and vehicle control [1] Because ofits powerful features in learning ability data processingfunction approximation and adaptiveness it has gainedincreasing attention and become a hot topic for scholars [2]while due to the finite information processing speed timedelays inevitably occur in many NNs and possibly lead topoor performances and complex dynamical behaviors [3]Thus extensive researches have been addressed for NNs withtime-varying delays

It is well known that by choosing the external states ofneurons or the internal states of neurons as basic variablesa neural network can be usually classified as a static neuralnetwork or a local field neural network [4] Although thesetwo types of neural network can be equivalent under someassumptions inmany applications these assumptions cannotalways be satisfied A unified model which combines thesetwo systems together has been constructed recently [5]

Since NNs usually are some highly interconnectednetworks with a great deal of neurons it may be very hardor even impossible to acquire all neurons state informationcompletely [6] However in some practical applications toachieve some desired objectives it is needed to know theneurons state information or estimate it in advance [7] Andthe design of filter which aims to estimate the states of asystem via its outputmeasurement provides amethod for theabove problems Consequently it is of great practical interestto study the filtering problems of NNs with time-varyingdelays

For NNs with time-varying delays the filtering problemwas firstly investigated in [8] both 119867infin and generalized 1198672filter were obtained by solving a set of LMIs Since thenmuch effort has been made and many results have beenreported on this issue In [9] the delay-dependent 1198672 filterof the Luenberger form was derived for delayed switchedHopfield NNs Considering that there is only one gain matrixin the Luenberger-type filter to be determined which mayintroduce some restrictions to certain extent [10] furtherstudied the 1198672 filtering problem of delayed static NNs basedon an Arcak-type filter Since it contains two gain matrices

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 5147565 9 pageshttpsdoiorg10115520185147565

2 Mathematical Problems in Engineering

the Arcak-type filter is regarded as an extension of theLuenberger-type filter In addition another generalized 1198672full-order filter was designed for a class of delayed stochasticNNs in [11] in which three gain matrices were involved

On the other hand with the rapid development ofthe Lyapunov-Krasovskii functional (LKF) theory plenty oftechniques developed for delay systems have been appliedto the filter design of delayed NNs in order to reduce theconservatism of the derived conditions The condition withless conservatismmeans it can provide larger feasible regionssuch that the filtering error system is globally asymptoticallystable with an optimal performance index It has been wellrecognized that constructing a suitable LKF and effectivelyestimating its derivative are two key points to reduce theconservatism In early years the simple LKFs combined withthe Jensen inequality is considered to be the most popularmethod for the filtering problems of delayedNNs [8] In orderto consider more information about time delays some tripleintegral terms were introduced into the LKF in [10] In [12]the delay-decomposition idea was used to derive the delay-dependent condition such that the filtering error system isglobally asymptotically stable with a guaranteed 119867infin perfor-mance Based on the works of [13] less conservative filteringdesign results were obtained via the free-weighting matrixapproach [14] And by introducing a additional zero equationinto the negative-definite conditions of LKFsrsquo derivative [15]derived a suitable 119867infin filter for the static NNs with mixedtime-varying delays Very recently [16] achieved the lessconservative generalized1198672 performance state estimation viaan augmented LKF Wirtinger integral inequality and theauxiliary function-based integral inequality

Although research on the filtering problem of delayedNNs is in progress due to its importance approaches orresults on this topic are lacking compared with the stabilityanalysis For instance some new proposed techniques whichcan effectively reduce the conservatismof the derived stabilitycriteria have not been applied to the filtering problem ofdelayedNNs tomention a few the free-matrix-based integralinequality [17] the relaxed integral inequality [18] and thegeneralized free-weighting-matrix approach [19]

Due to this consideration this paper further studies thefiltering problem of NNs with a time-varying delay Themain purpose of this paper is to develop a less conservativeapproach for the generalized 1198672 filter design To tacklethis problem a more general Arcak-type filter is adoptedfirstly Then by employing the LKF theory and using thefree-matrix-based integral inequality some other bound-ing inequalities and the free-weighting matrix approachnew delay-dependent condition is derived to guarantee theasymptotic stability of the filtering error system of delayedstatic NNs with guaranteed 1198672 performance index Finally anumerical example is provided to illustrate the effectivenessof our proposed method

Notations Throughout this paper the notations are standardR119899 is the set of all 119899 times 119899 real matrices 119875 gt 0 (ge0) means that119875 is a real symmetric and positive-definite (semi-positive-definite) matrix diagsdot sdot sdot denotes a block-diagonal matrix119868 and 0 represent the identity matrix and the zero-matrix

respectively the symmetric term in a symmetric matrix isdenoted by lowast and Sym119883 = 119883 + 1198831198792 Preliminaries

Consider the following delayed static NN with externaldisturbance

(119905) = minus119860119909 (119905) + 119891 (119882119909 (119905 minus 119889 (119905)) + 119869) + 1198611120596 (119905) 119910 (119905) = 119862119909 (119905) + 119863119909 (119905 minus 119889 (119905)) + 1198612120596 (119905) 119911 (119905) = 119867119909 (119905) 119909 (119905) = 120601 (119905) 119905 isin [minusℎ 0]

(1)

where 119909(119905) = [1199091(119905) 1199092(119905) 119909119899(119905)]119879 isin R119899 is the neuronstate vector 119891(119909(119905)) = [1198911(1199091(119905)) 1198912(1199092(119905)) 119891119899(119909119899(119905))]119879is the neural activation functions 120596(119905) is the external dis-turbance belonging to 1198712[0 infin) 119910(119905) isin R119901 is the networkoutput 119911(119905) is the linear combination of states to be estimated119869 = [1198691 1198692 119869119899]119879 is an exogenous input vector 119860 119882 11986111198612 119862 119863 and 119867 are known real matrices with appropriatedimensions 120601(119905) is the initial function and 119889(119905) is time-varying delay satisfying

0 le 119889 (119905) le ℎ119889 (119905) le 120583 (2)

where ℎ and 120583 are known scalars

Assumption 1 For each 119894 = 1 2 119899 there exist real scalars119897119894 gt 0 such that the continuous activation function 119891119894(sdot)satisfies

0 le 119891119894 (119886) minus 119891119894 (119887)119886 minus 119887 le 119897119894 119886 = 119887 isin R (3)

A full-order Arcak-type filter for the delayed static NN (1)is constructed as

119909 (119905) = minus119860119909 (119905)+ 119891 (119882119909 (119905 minus 119889 (119905)) + 119869 + 1198701 (119910 (119905) minus 119910 (119905)))+ 1198702 (119910 (119905) minus 119910 (119905))

119910 (119905) = 119862119909 (119905) + 119863119909 (119905 minus 119889 (119905)) (119905) = 119867119909 (119905) 119909 (119905) = 0 119905 isin [minusℎ 0]

(4)

where 119909(119905) = [1199091(119905) 1199092(119905) 119909119899(119905)]119879 isin R119899 andmatrices11987011198702 are the gains of the filter to be determinedDefine the error signals as 119890(119905) = 119909(119905) minus 119909(119905) and 119911119890(119905) =119911(119905) minus (119905) Then the filtering error system is expressed as

follows119890 (119905) = minus (119860 + 1198702119862) 119890 (119905) minus 1198702119863119890 (119905 minus 119889 (119905)) + 119892 (119905)

+ (1198611 minus 11987021198612) 120596 (119905) 119911119890 (119905) = 119867119890 (119905)

(5)

Mathematical Problems in Engineering 3

where 119892(119905) = 119891(119882119909(119905 minus 119889(119905)) + 119869) minus 119891(119882119909(119905 minus 119889(119905)) + 119869 +1198701(119910(119905) minus 119910(119905)))This paper aims to present a less conservative approach

for the 1198672 filtering problem of delayed static NN (1) Towardthis problem the following definition is indispensable

Definition 2 For given 120574 gt 0 the 1198672 filtering problem is saidto be solved if a suitable full-order filter (4) can be found suchthat

(1) the filtering error system (5) with 120596(119905) = 0 is globallyasymptotically stable

(2) the 1198672 performance 119911119890(119905)infin le 120574120596(119905)2 is guar-anteed under zero initial conditions for all nonzero120596(119905) isin L2[0 infin) where 119911119890(119905)infin = sup119905radic119911119879119890 (119905)119911119890(119905)and 120596(119905)2 = radicintinfin

0120596119879(119905)120596(119905)119889119905

Before proceeding we present the following lemmaswhich will be used in the sequel

Lemma 3 (free-matrix-based integral inequality [17]) Let 120596be a differentiable signal in [120572 120573] rarr R119899 for positive definitematrices 119877 isin R119899times119899 119883 119885 isin R3119899times3119899 and any matrices 119884 isinR3119899times3119899 119872 119873 isin R3119899times119899 satisfying

[[[

119883 119884 119872lowast 119885 119873lowast lowast 119877

]]]

ge 0 (6)

the following inequality holds

minus int120572120573

119879 (119904) 119877 (119904) 119889119904 le 120603119879Ω120603 (7)

where

Ω = (120572 minus 120573) (119883 + 13119885) + Sym 1198721198661 + 1198731198662 1198661 = [119868 minus119868 119874] 1198662 = [minus119868 minus119868 2119868] 120603 = [120596119879 (120572) 120596119879 (120573) 1120572 minus 120573 int120572

120573120596119879 (119904) 119889119904]119879

(8)

Lemma 4 (Jensenrsquos inequality [20]) Let 120596 be a differentiablesignal in [120572 120573] rarr R119899 for positive definite matric 119877 isin R119899 thefollowing inequalities hold

(120572 minus 120573)22 int120572

120573int120572119904

120596119879 (119904) 119877120596 (119904) 119889119904 119889120579ge (int120572120573

int120572119904

120596 (119904) 119889119904 119889120579)119879 119877 (int120572120573

int120572119904

120596 (119904) 119889119904 119889120579) (9)

3 Main Result

In this section by employing an augmented LKF and usingthe free-matrix-based integral inequality to estimate its

derivative an improved delay-dependent condition is derivedsuch that the filtering error system (5) is globally asymp-totically stable with guaranteed generalized 1198672 performanceindexes

Theorem 5 For given scalars ℎ and 120583 the 1198672 filteringproblem of NN (1) with time-varying delay satisfying (2) issolved with 120574min if there exist positive definite matrices 119875 =[ 1198751 1198752lowast 1198753 ] isin R2119899 1198761 1198762 119877 119885 isin R119899 positive definite diagonalmatrices Λ = diag1205821 1205822 120582119899 isin R119899 symmetric matrices1198831 1198833 1198841 1198843 1198851 1198853 isin R3119899 and any matrices 1198832 1198842 1198852 isinR3119899 1198721 1198722 1198731 1198732 1198661 1198662 isin R3119899times119899 119879 isin R119899 1198811 1198812 isinR119899times119901 satisfying the following LMIs

[[[

1198831 1198832 1198721lowast 1198833 1198722lowast lowast 119877]]]

ge 0 (10)

[[[

1198841 1198842 1198731lowast 1198843 1198732lowast lowast 119877]]]

ge 0 (11)

[[[

1198851 1198852 1198661lowast 1198853 1198662lowast lowast 119885]]]

ge 0 (12)

[[[[

1198751 1198752 119867119879lowast 1198753 119874lowast lowast 1205742119868

]]]]

ge 0 (13)

Γ (0) lt 0Γ (ℎ) lt 0 (14)

where

Γ (119889 (119905)) = Φ1 (119889 (119905)) + Φ2 + Φ3 (119889 (119905)) + Φ4 (119889 (119905))+ Ω

Φ1 (119889 (119905)) = Sym Π1198791119875Π2 Φ2 = 1198901198791 (1198761 + 1198762) 1198901 minus (1 minus 120583) 119890119879211987611198902 minus 119890119879311987621198903Φ3 (119889 (119905)) = ℎ11989011987971198771198907 + 119889 (119905) Π1198793 (1198831 + 131198833) Π3 + (ℎ

minus 119889 (119905)) Π1198794 (1198841 + 131198843) Π4 + Sym Π1198793119872Π5+ Π1198794119873Π6

Φ4 (119889 (119905)) = ℎ22 11989011987971198851198907 minus 2 (1198901 minus 1198904)119879119885 (1198901 minus 1198904)minus 2 (1198902 minus 1198905)119879119885 (1198902 minus 1198905) + ℎ (ℎ minus 119889 (119905)) Π1198793 (1198851+ 131198853) Π3 + (ℎ minus 119889 (119905)) Sym Π1198793119866Π5

4 Mathematical Problems in Engineering

Ω = Sym 1198901198796Λ (1198711198821198902 minus 1198906)minus 11989011987961198811 (1198621198901 + 1198631198902 + 11986121198908) + (1198901 + 1198907)119879sdot 119879 (1198907 + 1198601198901 minus 1198906 minus 11986111198908) + (1198901 + 1198907)119879sdot 1198812 (1198621198901 + 1198631198902 + 11986121198908)

Π1 = [1198901198791 119889 (119905) 1198901198794 + (ℎ minus 119889 (119905)) 1198901198795 ]119879 Π2 = [1198901198797 1198901198791 minus 1198901198793 ]119879 Π3 = [1198901198791 1198901198792 1198901198794 ]119879 Π4 = [1198901198792 1198901198793 1198901198795 ]119879 Π5 = [1198901198791 minus 1198901198792 minus1198901198791 minus 1198901198792 + 21198901198794 ]119879 Π6 = [1198901198792 minus 1198901198793 minus1198901198792 minus 1198901198793 + 21198901198795 ]119879 119872 = [1198721 1198722] 119873 = [1198731 1198732] 119866 = [1198661 1198662] 119890119894 = [0119899times(119894minus1)119899 119868119899 0119899times(7minus119894)119899 0119899times119901] (119894 = 1 2 7) 1198908 = [0119901times7119899 119868119901] 119871 = diag 1198971 1198972 119897119899 120585 (119905) = [119890119879 (119905) 119890119879 (119905 minus 119889 (119905)) 119890119879 (119905 minus ℎ) int119905

119905minus119889(119905)

119890119879 (119904)119889 (119905) 119889119904

int119905minus119889(119905)119905minusℎ

119890119879 (119904)ℎ minus 119889 (119905)119889119904 119892119879 (119905) 119890119879 (119905) 120596119879 (119905)]119879

(15)and the gain matrices 1198701 1198702 of the filter of (4) can be designedas

1198701 = (Λ119871)minus1 11988111198702 = 119879minus11198812 (16)

Proof Let us construct a new augmented LKF candidate as

119881 (119890119905) = 4sum119894=1

119881119894 (119890119905) (17)

where

1198811 (119890119905) = [[[

119909 (119905)int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

119875 [[[

119909 (119905)int119905119905minusℎ

119890 (119904) 119889119904]]]

1198812 (119890119905) = int119905119905minus119889(119905)

119890119879 (119904) 1198761119890 (119904) 119889119904+ int119905119905minusℎ

119890119879 (119904) 1198762119890 (119904) 119889119904

1198813 (119890119905) = int0minusℎ

int119905119905+120579

119890119879 (119904) 119877 119890 (119904) 119889119904 1198891205791198814 (119890119905) = int0

minusℎint119905120579

int119905119905+119906

119890119879 (119904) 119885 119890 (119904) 119889119904 119889119906 119889120579(18)

Taking the derivative of1198811(119890119905) along the filtering error system(5) yields

1 (119890119905) = 2 [[[

119909 (119905)int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

119875 [ (119905)119890 (119905) minus 119890 (119905 minus ℎ)]

= 120585119879 (119905) Φ1 (119889 (119905)) 120585 (119905) (19)

By the time derivative of 1198812(119890119905) we get2 (119890119905) = 119890119879 (119905) (1198761 + 1198762) 119890 (119905)

minus (1 minus 119889 (119905)) 119890119879 (119905 minus 119889 (119905)) 1198761119890 (119905 minus 119889 (119905))minus 119890119879 (119905 minus ℎ) 1198762119890 (119905 minus ℎ) = 120585119879 (119905) Φ2120585 (119905)

(20)

Calculating the derivative of 1198813(119890119905) we have3 (119890119905) = ℎ 119890119879 (119905) 119877 119890 (119905) minus int119905

119905minus119889(119905)119890119879 (119904) 119877 119890 (119904) 119889119904

minus int119905minus119889(119905)119905minusℎ

119890119879 (119904) 119877 119890 (119904) 119889119904(21)

Then by applying Lemma 3 to estimate the above 119877-dependent integral term if LMIs (10) and (11) hold one has

minus int119905119905minus119889(119905)

119890119879 (119904) 119877 119890 (119904) 119889119904le 119889 (119905) 1205941198791 (1198831 + 131198833) 1205941

+ Sym 120594119879111987211205942 + 120594119879111987221205943 (22)

minus int119905minus119889(119905)119905minusℎ

119890119879 (119904) 119877 119890 (119904) 119889119904le (ℎ minus 119889 (119905)) 1205941198794 (1198841 + 131198843) 1205944

+ Sym 120594119879411987311205945 + 120594119879411987321205946 (23)

where

1205941 = [119890119879 (119905) 119890119879 (119905 minus 119889 (119905)) 1119889 (119905) int119905119905minus119889(119905)

119890119879 (119904) 119889119904]119879 1205942 = 119890 (119905) minus 119890 (119905 minus 119889 (119905))

Mathematical Problems in Engineering 5

1205943 = minus119890 (119905) minus 119890 (119905 minus 119889 (119905)) + 2119889 (119905) int119905119905minus119889(119905)

119890 (119904) 1198891199041205944 = [119890119879 (119905 minus 119889 (119905)) 119890119879 (119905 minus ℎ) 1ℎ minus 119889 (119905)

sdot int119905minus119889(119905)119905minusℎ

119890119879 (119904) 119889119904]119879 1205945 = 119890 (119905 minus 119889 (119905)) minus 119890 (119905 minus ℎ) 1205946 = minus119890 (119905 minus 119889 (119905)) minus 119890 (119905 minus ℎ) + 2ℎ minus 119889 (119905)

sdot int119905minus119889(119905)119905minusℎ

119890 (119904) 119889119904(24)

Combining (21) with (23) it is clear that

3 (119890119905) le 120585119879 (119905) Φ3 (119889 (119905)) 120585 (119905) (25)

Finally the derivative of 1198814(119890119905) can be obtained as

4 (119890119905) = ℎ22 119890119879 (119905) 119885 119890 (119905) minus int119905119905minusℎ

int119905120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579= ℎ22 119890119879 (119905) 119885 119890 (119905)

minus int119905119905minus119889(119905)

int119905120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579minus int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579minus (ℎ minus 119889 (119905)) int119905

119905minus119889(119905)119890119879 (119904) 119885 119890 (119904) 119889119904

(26)

For the first double integral term in 4(119890119905) we can do thefollowing treatment based on Lemma 4

minus int119905119905minus119889(119905)

int119905120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579le minus 21198892 (119905) (int119905

119905minus119889(119905)int119905120579

119890 (119904) 119889119904 119889120579)119879

sdot 119885 (int119905119905minus119889(119905)

int119905120579

119890 (119904) 119889119904 119889120579)= minus 21198892 (119905) (119889 (119905) 119890 (119905) minus int119905

119905minus119889(119905)119890 (119904) 119889119904)119879

sdot 119885 (119889 (119905) 119890 (119905) minus int119905119905minus119889(119905)

119890 (119904) 119889119904)= minus2 (119890 (119905) minus int119905

119905minus119889(119905)

119890 (119904)119889 (119905)119889119904)119879

sdot 119885 (119890 (119905) minus int119905119905minus119889(119905)

119890 (119904)119889 (119905)119889119904)

(27)

Similarly the estimation of the second integral term can bedone as

minus int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579le minus 2

(ℎ minus 119889 (119905))2 (int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890 (119904) 119889119904 119889120579)119879

sdot 119885 (int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890 (119904) 119889119904 119889120579)

= minus2 (119890 (119905 minus 119889 (119905)) minus int119905minus119889(119905)119905minusℎ

119890 (119904)ℎ minus 119889 (119905)119889119904)119879

sdot 119885 (119890 (119905 minus 119889 (119905)) minus int119905minus119889(119905)119905minusℎ

119890 (119904)ℎ minus 119889 (119905)119889119904)

(28)

And by utilizing Lemma 3 if LMI (12) holds we obtain

minus (ℎ minus 119889 (119905)) int119905119905minus119889(119905)

119890119879 (119904) 119885 119890 (119904) 119889119904 le (ℎ minus 119889 (119905))sdot (ℎ1205941198791 (1198851 + 131198853) 1205941+ Sym 1205941198791 11986611205942 + 1205941198791 11986621205943)

(29)

From (26) to (29) we get

4 (119890119905) le 120585119879 (119905) Φ4 (119889 (119905)) 120585 (119905) (30)

Then taking assumption of the activation function (3) intoaccount yields

0 le 2119892119879 (119905) Λ [119871 (119882119890 (119905 minus 119889 (119905)) minus 1198701119862119890 (119905)minus 1198701119863119890 (119905 minus 119889 (119905)) minus 11987011198612120596 (119905)) minus 119892 (119905)] (31)

where

1198811 = Λ1198711198701997904rArr 0le 21198901198796Λ (1198711198821198902 minus 1198906) minus 211989011987961198811 (1198621198901 + 1198631198902 + 11986121198908)

(32)

Based on the filtering error system (5) for any appropriatelydimensioned matrix 119879 the following equation holds

0 = 2 (119890 (119905) + 119890 (119905))119879 119879 [ 119890 (119905) + (119860 + 1198702119862) 119890 (119905)+ 1198702119863119890 (119905 minus 119889 (119905)) minus 119892 (119905) minus (1198611 minus 11987021198612) 120596 (119905)] (33)

where

1198812 = 1198791198702997904rArr 0= 2 (1198901 + 1198907)119879 119879 (1198907 + 1198601198901 minus 1198906 minus 11986111198908)

+ 2 (1198901 + 1198907)1198791198812 (1198621198901 + 1198631198902 + 11986121198908) (34)

6 Mathematical Problems in Engineering

Combining (19) (20) (21) (22) (23) (25) (26) (27) (28)(29) (30) (32) and (34) one can easily yield

(119890119905) le 120585119879 (119905) Γ (119889 (119905)) 120585 (119905) (35)

If Γ(119889(119905)) lt 0 then (119890119905) lt 0 Thus if Γ(0) lt 0 and Γ(ℎ) lt0 are satisfied then (119890119905) le minus120598119890(119905)2 for a sufficient smallscalar 120598 gt 0 which implies the filtering error system (5) with120596(119905) = 0 is asymptotically stable

Considering the definition of 1198672 performance we define

119869 (119905) = 119881 (119890119905) minus int1199050

120596119879 (119904) 120596 (119904) 119889119904 (36)

Then under the zero-initial conditions 119881(119890119905)|119905=0 = 0 and119881(119890119905) gt 0 for 119905 gt 0 the following inequality is given119869 (119905) = 119881 (119890119905) minus 119881 (119890119905)1003816100381610038161003816119905=0 minus int119905

0120596119879 (119904) 120596 (119904) 119889119904

= int1199050

[ (119890 (119904)) minus 120596119879 (119904) 120596 (119904)] 119889119904 lt 0(37)

That is

119881 (119890119905) lt int1199050

120596119879 (119904) 120596 (119904) 119889119904 (38)

In addition it follows that LMI (13) and Schur complementthat

[119867119879119867 119874lowast 119874] le 1205742 [1198751 1198752lowast 1198753] (39)

Then the following holds

119911119879119890 (119905) 119911119890 (119905) = 119890119879 (119905) 119867119879119867119890 (119905)

= [[[

119909 (119905)int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

[119867119879119867 119874lowast 119874] [[

[119909 (119905)

int119905119905minusℎ

119890 (119904) 119889119904]]]le 1205742 [[

[119909 (119905)

int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

[1198751 1198752lowast 1198753][[[

119909 (119905)int119905119905minusℎ

119890 (s) 119889119904]]]= 12057421198811 (119890119905) le 1205742119881 (119890119905) le 1205742 int119905

0120596119879 (119904) 120596 (119904) 119889119904

le 1205742 intinfin0

120596119879 (119904) 120596 (119904) 119889119904

(40)

Taking the supremum over 119905 gt 0 one has 119911119890(119905)2infin le120574120596(119905)22 which means 119911119890(119905)infin le 120574120596(119905)2 Therefore when(10)ndash(14) hold the generalized1198672 filtering problem of NN (1)with time-varying delay satisfying (2) is solvedwith 120574minThiscompletes the proof

Remark 6 The generalized 1198672 design of a type of static NNwith a time-varying delay is solved in Theorem 5 Different

from [8ndash10 12ndash15] we solve the filtering problem based onan Arcak-type filter in which there are two gain matrices tobe determined Compared with the widely used Luenberger-type filter it contains an additional gainmatrix1198701 included inthe activation function So it can be expected to lead a bettergeneralized1198672 performance And if we set1198701 = 0 the Arcak-type filter is then reduced to the Luenberger-type filter

Remark 7 The triple integral term is introduced into theLKF in this paper When bounding its derivative the deriveddouble integral term minus int119905

119905minusℎint119905120579

119890119879(119904)119885 119890(119904)119889119904 119889120579 is divided intothree parts two double integral terms and a single integralterm By using the double Jensen inequality to estimatethe double integral terms and using the free-matrix-basedintegral inequality to estimate the single integral term thederivation of the triple term is accomplishedwithout ignoringany terms It should be pointed out that this technique ismore effective than the ones used in [10] in which the deriveddouble integral term minus int119905

119905minusℎint119905120579

119890119879(119904)119885 119890(119904)119889119904 119889120579 is estimated inits entirety

Remark 8 To reduce the conservatism of the derived con-ditions Wirtinger integral inequality was used to estimatesingle integral terms in [16] while in the presented studythe free-matrix-based integral inequality is employed Theuse of the free-matrix-based integral inequality brings severaladvantages for Theorem 5 On the one hand by introducingsome free matrices it gives more degrees of freedom tothe derived condition On the other hand it has beenproved in [17] that the free-matrix-based integral inequalitycontains Wirtinger integral inequality as a special case Thatmeans it can provide a tighter bound thanWirtinger integralinequality And if the same LKF is chosen our designed filterwill be better than the one designed in [16]

Remark 9 There is some room for further improvement ofour proposed method

(i) Notice that when the free-matrix-based integralinequality is employed to improve the proposed con-dition the number of decision variables is increasedwhich further increases the computation complexityof the condition It has been pointed out in [3]that the calculation complexity is also an importantconsideration during the application of the proposedLMI-based condition to physical systems Thus theresults considering both the conservatism and thecomputation complexity need more investigation

(ii) Recently some new approaches have been developedfor the stability analysis of delayed neural networkswhich can be applied into the generalized 1198672 filter-ing problem of neural networks such as the delay-decomposition approach [21] and the free-matrix-based double integral inequality [22]

Notice that in many applications the derivative bound ofthe time delay is unknown By setting 1198761 = 0 we can easilyderive the delay-dependent but delay derivative-independent

Mathematical Problems in Engineering 7

condition of the generalized 1198672 filtering problem of neuralnetwork (1) based onTheorem 5

Theorem 10 For given scalars ℎ the 1198672 filtering problem ofNN (1) with time-varying delay satisfying 0 le 119889(119905) le ℎ issolved with 120574min if there exist positive definite matrices 119875 =[ 1198751 1198752lowast 1198753 ] isin R2119899 1198762 119877 119885 isin R119899 positive definite diagonalmatrices Λ = diag1205821 1205822 120582119899 isin R119899 symmetric matrices1198831 1198833 1198841 1198843 1198851 1198853 isin R3119899 and any matrices 1198832 1198842 1198852 isinR3119899 1198721 1198722 1198731 1198732 1198661 1198662 isin R3119899times119899 119879 isin R119899 1198811 1198812 isinR119899times119901 satisfying LMIs (10)ndash(14)

4 Numerical Examples

In this section we provide a well-used numerical example todemonstrate the effectiveness of the proposed approach

Example 1 Consider the delayed static NN (1) with thefollowing parameters

119860 = [[[

096 0 00 122 00 0 078

]]]

119882 = [[[

minus032 075 minus142121 041 minus050042 082 minus106

]]]

1198611 = [[[

020305

]]]

119867 = [[[

08 0 051 0 10 minus1 1

]]]

119862 = [1 05 00 minus05 06]

119863 = [0 minus12 020 0 05]

1198612 = [ 04minus03]

(41)

This example has been studied in [10 16] and we takethese literatures for comparison study To verify the advan-tages of our proposed approach we calculate the optimal 1198672performance bounds index 120574min for various ℎ 120583 and 119871 suchthat filtering error system (5) is globally asymptotically stablewith generalized 1198672 performance

Firstly let 119871 = diag1198971 1198972 119897119899 = 119868 for given ℎ and 120583the optimal generalized 1198672 performance index 120574min derivedby Theorem 5 and the ones reported in [10 16] are givenin Table 1 where (120572 120573) represents ℎ = 120572 120583 = 120573 and

minus15

minus1

minus05

0

05

1

15

e(t)

5 10 150Time (s)

e1(t)

e2(t)

e3(t)

Figure 1 State response of the error 119890(119905)

ldquoinfeasiblerdquo means that no feasible solution can be found inthe corresponding criteria

It is clearly shown from Table 1 that compared with [1016] much better generalized 1198672 performance is achieved byTheorem 5 proposed in this paper It should be noticed thatin this paper we choose the same filter as [10 16] thus thebetter 1198672 performance effectively illustrates the superiorityof our proposed approaches which is mainly based on thefree-matrix-based integral inequality

Then let 119891(119909) = 157tanh(119909) 119889(119905) = 03 sin(15119905) + 12and 119908(119905) = sin(119905)119890minus5119905 for ℎ = 15 120583 = 03 by solving theLMIs in Theorem 5 the gain matrices of our designed filter(4) are obtained as

1198701 = [[[

10591 1299001519 0268724379 33466

]]]

1198702 = [[[

00944 minus0600503315 minus0510021756 12563

]]]

(42)

with the optimal generalized 1198672 performance index 120574 =00505 Figure 1 shows the state responses of error system (5)under initial condition 119890(0) = [minus1 1 minus05]119879 The resultingresponses obviously demonstrate the asymptotic stability ofthe simulated error system

In addition the optimal generalized 1198672 performanceindex 120574min for ℎ = 05 120583 = 07 and different 119871 is listed inTable 2 from which one can see thatTheorem 5 in this paperoutperforms the ones in [10 16]

Moreover the optimal generalized1198672 performance index120574min derived by Theorem 10 for ℎ = 08 unknown 120583 anddifferent 119871 is listed in Table 3

8 Mathematical Problems in Engineering

Table 1 Optimal generalized 1198672 performance index 120574min for 119871 = 119868 and different (ℎ 120583)(ℎ 120583) (1 04) (15 03) (3 08) (5 12) (6 15)Theorem 1 [10] 07981 Infeasible Infeasible Infeasible InfeasibleTheorem 2 [16] 134 times 10minus4 12763 Infeasible Infeasible InfeasibleTheorem 5 133 times 10minus4 00505 06639 09137 19523

Table 2 Optimal generalized 1198672 performance index 120574min for ℎ = 05 120583 = 07 and different 119871119871 16119868 18119868 25119868 34119868Theorem 1 [10] 33728 Infeasible Infeasible InfeasibleTheorem 2 [16] 05983 15563 Infeasible InfeasibleTheorem 5 02765 04015 06896 47676

Table 3 Optimal generalized 1198672 performance index 120574min for ℎ =05 unknown 120583 and different 119871119871 16119868 18119868 25119868 291119868Theorem 10 05156 05868 10891 51122

5 Conclusion

In this paper the generalized 1198672 filtering problem of staticNNs with a time-varying delay has been investigated Firstan Arcak-type filter has been constructed Then based onan augmented LKF the free-matrix-based integral inequalitythe free-weighting matrix approach and Jensen inequalitya suitable delay-dependent condition expressed in terms ofLMIs has been derived such that the filtering error systemof the considered static NNs is globally asymptotically stablewith guaranteed1198672 performance indexMoreover by solvingsome coupled LMIs the optimal performance index and thegain matrices of the designed 1198672 filter have been obtainedThe effectiveness and advantages of the proposed methodhave been demonstrated by an illustrative example finally

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding this paper

References

[1] L O Chua and L Yang ldquoCellular Neural Networks Applica-tionsrdquo IEEE Transactions on Circuits and Systems II ExpressBriefs vol 35 no 10 pp 1273ndash1290 1988

[2] X-M Zhang Q-L Han and X Yu ldquoSurvey on RecentAdvances inNetworkedControl Systemsrdquo IEEETransactions onIndustrial Informatics vol 12 no 5 pp 1740ndash1752 2016

[3] C-K Zhang Y He L Jiang and M Wu ldquoStability analysisfor delayed neural networks considering both conservativenessand complexityrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 7 pp 1486ndash1501 2016

[4] H-B Zeng Y He P Shi M Wu and S-P Xiao ldquoDissipativityanalysis of neural networks with time-varying delaysrdquo Neuro-computing vol 168 pp 741ndash746 2015

[5] C-K Zhang Y He L Jiang Q H Wu and M Wu ldquoDelay-dependent stability criteria for generalized neural networks

with two delay componentsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1263ndash12762014

[6] Y He Q GWang andC Lin ldquoAn improvedH8 filter design forsystems with time-varying interval delayrdquo IEEE Trans CircuitsSyst II vol 53 p 1235 2006

[7] YHeG P LiuD Rees andMWu ldquoImprovedHinfinfiltering forsystems with a time-varying delayrdquo Circuits Systems and SignalProcessing vol 29 no 3 pp 377ndash389 2010

[8] H Huang and G Feng ldquoDelay-dependent Hinfin and generalizedH2 filtering for delayed neural networksrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 4 pp 846ndash857 2009

[9] C K Ahn and M K Song ldquoL2-Linfin filtering for time-delayedswitched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[10] D Hu H Huang and T Huang ldquoDesign of Arcak-type gen-eralized H2 filter for delayed static neural networksrdquo CircuitsSystems and Signal Processing vol 33 no 11 pp 3635ndash36482014

[11] M Hua H Tan and J Chen ldquoDelay-dependent Hinfin andgeneralized H2 filtering for stochastic neural networks withtime-varying delay and noise disturbancerdquo Neural Computingand Applications vol 25 no 3-4 pp 613ndash624 2014

[12] Y J Li and Y N Liang ldquoDelay-dependent Hinfin filtering forneural networks with time delayrdquo Applied Mechanics and Mate-rials vol 511-512 pp 875ndash879 2014

[13] YHeG P LiuD Rees andMWu ldquoStability analysis for neuralnetworks with time-varying interval delayrdquo IEEE Transactionson Neural Networks and Learning Systems vol 18 no 6 pp1850ndash1854 2007

[14] Y Li J Li and M Hua ldquoNew results of H8 filtering for neuralnetwork with time-varying delayrdquo Int J Innov Comput InfControl vol 10 pp 2309ndash2323 2014

[15] G Liu S Zhou X Luo and K Zhang ldquoNew filter design forstatic neural networks with mixed time-varying delaysrdquo LectureNotes in Computer Science (including subseries Lecture Notesin Artificial Intelligence and Lecture Notes in Bioinformatics)Preface vol 9772 pp 117ndash129 2016

[16] Y Shu X-G Liu Y Liu and J H Park ldquoImproved Results onGuaranteed Generalized H2 Performance State Estimation forDelayed Static Neural Networksrdquo Circuits Systems and SignalProcessing vol 36 no 8 pp 3114ndash3142 2017

Mathematical Problems in Engineering 9

[17] H-B Zeng Y He M Wu and J She ldquoFree-matrix-basedintegral inequality for stability analysis of systems with time-varying delayrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 60 no 10 pp 2768ndash2772 2015

[18] C-K Zhang Y He L Jiang M Wu and H-B Zeng ldquoStabilityanalysis of systems with time-varying delay via relaxed integralinequalitiesrdquo Systems amp Control Letters vol 92 pp 52ndash61 2016

[19] C-K Zhang Y He L Jiang W-J Lin and M Wu ldquoDelay-dependent stability analysis of neural networks with time-varying delay a generalized free-weighting-matrix approachrdquoApplied Mathematics and Computation vol 294 pp 102ndash1202017

[20] J Sun G P Liu J Chen and D Rees ldquoImproved delay-range-dependent stability criteria for linear systemswith time-varyingdelaysrdquo Automatica vol 46 no 2 pp 466ndash470 2010

[21] C Ge C Hua and X Guan ldquoNew delay-dependent stabilitycriteria for neural networks with time-varying delay usingdelay-decomposition approachrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1378ndash13832014

[22] C Hua SWu X Yang and X Guan ldquoStability analysis of time-delay systems via free-matrix-based double integral inequalityrdquoInternational Journal of Systems Science vol 48 no 2 pp 257ndash263 2017

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Mathematical Problems in Engineering 3

where 119892(119905) = 119891(119882119909(119905 minus 119889(119905)) + 119869) minus 119891(119882119909(119905 minus 119889(119905)) + 119869 +1198701(119910(119905) minus 119910(119905)))This paper aims to present a less conservative approach

for the 1198672 filtering problem of delayed static NN (1) Towardthis problem the following definition is indispensable

Definition 2 For given 120574 gt 0 the 1198672 filtering problem is saidto be solved if a suitable full-order filter (4) can be found suchthat

(1) the filtering error system (5) with 120596(119905) = 0 is globallyasymptotically stable

(2) the 1198672 performance 119911119890(119905)infin le 120574120596(119905)2 is guar-anteed under zero initial conditions for all nonzero120596(119905) isin L2[0 infin) where 119911119890(119905)infin = sup119905radic119911119879119890 (119905)119911119890(119905)and 120596(119905)2 = radicintinfin

0120596119879(119905)120596(119905)119889119905

Before proceeding we present the following lemmaswhich will be used in the sequel

Lemma 3 (free-matrix-based integral inequality [17]) Let 120596be a differentiable signal in [120572 120573] rarr R119899 for positive definitematrices 119877 isin R119899times119899 119883 119885 isin R3119899times3119899 and any matrices 119884 isinR3119899times3119899 119872 119873 isin R3119899times119899 satisfying

[[[

119883 119884 119872lowast 119885 119873lowast lowast 119877

]]]

ge 0 (6)

the following inequality holds

minus int120572120573

119879 (119904) 119877 (119904) 119889119904 le 120603119879Ω120603 (7)

where

Ω = (120572 minus 120573) (119883 + 13119885) + Sym 1198721198661 + 1198731198662 1198661 = [119868 minus119868 119874] 1198662 = [minus119868 minus119868 2119868] 120603 = [120596119879 (120572) 120596119879 (120573) 1120572 minus 120573 int120572

120573120596119879 (119904) 119889119904]119879

(8)

Lemma 4 (Jensenrsquos inequality [20]) Let 120596 be a differentiablesignal in [120572 120573] rarr R119899 for positive definite matric 119877 isin R119899 thefollowing inequalities hold

(120572 minus 120573)22 int120572

120573int120572119904

120596119879 (119904) 119877120596 (119904) 119889119904 119889120579ge (int120572120573

int120572119904

120596 (119904) 119889119904 119889120579)119879 119877 (int120572120573

int120572119904

120596 (119904) 119889119904 119889120579) (9)

3 Main Result

In this section by employing an augmented LKF and usingthe free-matrix-based integral inequality to estimate its

derivative an improved delay-dependent condition is derivedsuch that the filtering error system (5) is globally asymp-totically stable with guaranteed generalized 1198672 performanceindexes

Theorem 5 For given scalars ℎ and 120583 the 1198672 filteringproblem of NN (1) with time-varying delay satisfying (2) issolved with 120574min if there exist positive definite matrices 119875 =[ 1198751 1198752lowast 1198753 ] isin R2119899 1198761 1198762 119877 119885 isin R119899 positive definite diagonalmatrices Λ = diag1205821 1205822 120582119899 isin R119899 symmetric matrices1198831 1198833 1198841 1198843 1198851 1198853 isin R3119899 and any matrices 1198832 1198842 1198852 isinR3119899 1198721 1198722 1198731 1198732 1198661 1198662 isin R3119899times119899 119879 isin R119899 1198811 1198812 isinR119899times119901 satisfying the following LMIs

[[[

1198831 1198832 1198721lowast 1198833 1198722lowast lowast 119877]]]

ge 0 (10)

[[[

1198841 1198842 1198731lowast 1198843 1198732lowast lowast 119877]]]

ge 0 (11)

[[[

1198851 1198852 1198661lowast 1198853 1198662lowast lowast 119885]]]

ge 0 (12)

[[[[

1198751 1198752 119867119879lowast 1198753 119874lowast lowast 1205742119868

]]]]

ge 0 (13)

Γ (0) lt 0Γ (ℎ) lt 0 (14)

where

Γ (119889 (119905)) = Φ1 (119889 (119905)) + Φ2 + Φ3 (119889 (119905)) + Φ4 (119889 (119905))+ Ω

Φ1 (119889 (119905)) = Sym Π1198791119875Π2 Φ2 = 1198901198791 (1198761 + 1198762) 1198901 minus (1 minus 120583) 119890119879211987611198902 minus 119890119879311987621198903Φ3 (119889 (119905)) = ℎ11989011987971198771198907 + 119889 (119905) Π1198793 (1198831 + 131198833) Π3 + (ℎ

minus 119889 (119905)) Π1198794 (1198841 + 131198843) Π4 + Sym Π1198793119872Π5+ Π1198794119873Π6

Φ4 (119889 (119905)) = ℎ22 11989011987971198851198907 minus 2 (1198901 minus 1198904)119879119885 (1198901 minus 1198904)minus 2 (1198902 minus 1198905)119879119885 (1198902 minus 1198905) + ℎ (ℎ minus 119889 (119905)) Π1198793 (1198851+ 131198853) Π3 + (ℎ minus 119889 (119905)) Sym Π1198793119866Π5

4 Mathematical Problems in Engineering

Ω = Sym 1198901198796Λ (1198711198821198902 minus 1198906)minus 11989011987961198811 (1198621198901 + 1198631198902 + 11986121198908) + (1198901 + 1198907)119879sdot 119879 (1198907 + 1198601198901 minus 1198906 minus 11986111198908) + (1198901 + 1198907)119879sdot 1198812 (1198621198901 + 1198631198902 + 11986121198908)

Π1 = [1198901198791 119889 (119905) 1198901198794 + (ℎ minus 119889 (119905)) 1198901198795 ]119879 Π2 = [1198901198797 1198901198791 minus 1198901198793 ]119879 Π3 = [1198901198791 1198901198792 1198901198794 ]119879 Π4 = [1198901198792 1198901198793 1198901198795 ]119879 Π5 = [1198901198791 minus 1198901198792 minus1198901198791 minus 1198901198792 + 21198901198794 ]119879 Π6 = [1198901198792 minus 1198901198793 minus1198901198792 minus 1198901198793 + 21198901198795 ]119879 119872 = [1198721 1198722] 119873 = [1198731 1198732] 119866 = [1198661 1198662] 119890119894 = [0119899times(119894minus1)119899 119868119899 0119899times(7minus119894)119899 0119899times119901] (119894 = 1 2 7) 1198908 = [0119901times7119899 119868119901] 119871 = diag 1198971 1198972 119897119899 120585 (119905) = [119890119879 (119905) 119890119879 (119905 minus 119889 (119905)) 119890119879 (119905 minus ℎ) int119905

119905minus119889(119905)

119890119879 (119904)119889 (119905) 119889119904

int119905minus119889(119905)119905minusℎ

119890119879 (119904)ℎ minus 119889 (119905)119889119904 119892119879 (119905) 119890119879 (119905) 120596119879 (119905)]119879

(15)and the gain matrices 1198701 1198702 of the filter of (4) can be designedas

1198701 = (Λ119871)minus1 11988111198702 = 119879minus11198812 (16)

Proof Let us construct a new augmented LKF candidate as

119881 (119890119905) = 4sum119894=1

119881119894 (119890119905) (17)

where

1198811 (119890119905) = [[[

119909 (119905)int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

119875 [[[

119909 (119905)int119905119905minusℎ

119890 (119904) 119889119904]]]

1198812 (119890119905) = int119905119905minus119889(119905)

119890119879 (119904) 1198761119890 (119904) 119889119904+ int119905119905minusℎ

119890119879 (119904) 1198762119890 (119904) 119889119904

1198813 (119890119905) = int0minusℎ

int119905119905+120579

119890119879 (119904) 119877 119890 (119904) 119889119904 1198891205791198814 (119890119905) = int0

minusℎint119905120579

int119905119905+119906

119890119879 (119904) 119885 119890 (119904) 119889119904 119889119906 119889120579(18)

Taking the derivative of1198811(119890119905) along the filtering error system(5) yields

1 (119890119905) = 2 [[[

119909 (119905)int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

119875 [ (119905)119890 (119905) minus 119890 (119905 minus ℎ)]

= 120585119879 (119905) Φ1 (119889 (119905)) 120585 (119905) (19)

By the time derivative of 1198812(119890119905) we get2 (119890119905) = 119890119879 (119905) (1198761 + 1198762) 119890 (119905)

minus (1 minus 119889 (119905)) 119890119879 (119905 minus 119889 (119905)) 1198761119890 (119905 minus 119889 (119905))minus 119890119879 (119905 minus ℎ) 1198762119890 (119905 minus ℎ) = 120585119879 (119905) Φ2120585 (119905)

(20)

Calculating the derivative of 1198813(119890119905) we have3 (119890119905) = ℎ 119890119879 (119905) 119877 119890 (119905) minus int119905

119905minus119889(119905)119890119879 (119904) 119877 119890 (119904) 119889119904

minus int119905minus119889(119905)119905minusℎ

119890119879 (119904) 119877 119890 (119904) 119889119904(21)

Then by applying Lemma 3 to estimate the above 119877-dependent integral term if LMIs (10) and (11) hold one has

minus int119905119905minus119889(119905)

119890119879 (119904) 119877 119890 (119904) 119889119904le 119889 (119905) 1205941198791 (1198831 + 131198833) 1205941

+ Sym 120594119879111987211205942 + 120594119879111987221205943 (22)

minus int119905minus119889(119905)119905minusℎ

119890119879 (119904) 119877 119890 (119904) 119889119904le (ℎ minus 119889 (119905)) 1205941198794 (1198841 + 131198843) 1205944

+ Sym 120594119879411987311205945 + 120594119879411987321205946 (23)

where

1205941 = [119890119879 (119905) 119890119879 (119905 minus 119889 (119905)) 1119889 (119905) int119905119905minus119889(119905)

119890119879 (119904) 119889119904]119879 1205942 = 119890 (119905) minus 119890 (119905 minus 119889 (119905))

Mathematical Problems in Engineering 5

1205943 = minus119890 (119905) minus 119890 (119905 minus 119889 (119905)) + 2119889 (119905) int119905119905minus119889(119905)

119890 (119904) 1198891199041205944 = [119890119879 (119905 minus 119889 (119905)) 119890119879 (119905 minus ℎ) 1ℎ minus 119889 (119905)

sdot int119905minus119889(119905)119905minusℎ

119890119879 (119904) 119889119904]119879 1205945 = 119890 (119905 minus 119889 (119905)) minus 119890 (119905 minus ℎ) 1205946 = minus119890 (119905 minus 119889 (119905)) minus 119890 (119905 minus ℎ) + 2ℎ minus 119889 (119905)

sdot int119905minus119889(119905)119905minusℎ

119890 (119904) 119889119904(24)

Combining (21) with (23) it is clear that

3 (119890119905) le 120585119879 (119905) Φ3 (119889 (119905)) 120585 (119905) (25)

Finally the derivative of 1198814(119890119905) can be obtained as

4 (119890119905) = ℎ22 119890119879 (119905) 119885 119890 (119905) minus int119905119905minusℎ

int119905120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579= ℎ22 119890119879 (119905) 119885 119890 (119905)

minus int119905119905minus119889(119905)

int119905120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579minus int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579minus (ℎ minus 119889 (119905)) int119905

119905minus119889(119905)119890119879 (119904) 119885 119890 (119904) 119889119904

(26)

For the first double integral term in 4(119890119905) we can do thefollowing treatment based on Lemma 4

minus int119905119905minus119889(119905)

int119905120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579le minus 21198892 (119905) (int119905

119905minus119889(119905)int119905120579

119890 (119904) 119889119904 119889120579)119879

sdot 119885 (int119905119905minus119889(119905)

int119905120579

119890 (119904) 119889119904 119889120579)= minus 21198892 (119905) (119889 (119905) 119890 (119905) minus int119905

119905minus119889(119905)119890 (119904) 119889119904)119879

sdot 119885 (119889 (119905) 119890 (119905) minus int119905119905minus119889(119905)

119890 (119904) 119889119904)= minus2 (119890 (119905) minus int119905

119905minus119889(119905)

119890 (119904)119889 (119905)119889119904)119879

sdot 119885 (119890 (119905) minus int119905119905minus119889(119905)

119890 (119904)119889 (119905)119889119904)

(27)

Similarly the estimation of the second integral term can bedone as

minus int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579le minus 2

(ℎ minus 119889 (119905))2 (int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890 (119904) 119889119904 119889120579)119879

sdot 119885 (int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890 (119904) 119889119904 119889120579)

= minus2 (119890 (119905 minus 119889 (119905)) minus int119905minus119889(119905)119905minusℎ

119890 (119904)ℎ minus 119889 (119905)119889119904)119879

sdot 119885 (119890 (119905 minus 119889 (119905)) minus int119905minus119889(119905)119905minusℎ

119890 (119904)ℎ minus 119889 (119905)119889119904)

(28)

And by utilizing Lemma 3 if LMI (12) holds we obtain

minus (ℎ minus 119889 (119905)) int119905119905minus119889(119905)

119890119879 (119904) 119885 119890 (119904) 119889119904 le (ℎ minus 119889 (119905))sdot (ℎ1205941198791 (1198851 + 131198853) 1205941+ Sym 1205941198791 11986611205942 + 1205941198791 11986621205943)

(29)

From (26) to (29) we get

4 (119890119905) le 120585119879 (119905) Φ4 (119889 (119905)) 120585 (119905) (30)

Then taking assumption of the activation function (3) intoaccount yields

0 le 2119892119879 (119905) Λ [119871 (119882119890 (119905 minus 119889 (119905)) minus 1198701119862119890 (119905)minus 1198701119863119890 (119905 minus 119889 (119905)) minus 11987011198612120596 (119905)) minus 119892 (119905)] (31)

where

1198811 = Λ1198711198701997904rArr 0le 21198901198796Λ (1198711198821198902 minus 1198906) minus 211989011987961198811 (1198621198901 + 1198631198902 + 11986121198908)

(32)

Based on the filtering error system (5) for any appropriatelydimensioned matrix 119879 the following equation holds

0 = 2 (119890 (119905) + 119890 (119905))119879 119879 [ 119890 (119905) + (119860 + 1198702119862) 119890 (119905)+ 1198702119863119890 (119905 minus 119889 (119905)) minus 119892 (119905) minus (1198611 minus 11987021198612) 120596 (119905)] (33)

where

1198812 = 1198791198702997904rArr 0= 2 (1198901 + 1198907)119879 119879 (1198907 + 1198601198901 minus 1198906 minus 11986111198908)

+ 2 (1198901 + 1198907)1198791198812 (1198621198901 + 1198631198902 + 11986121198908) (34)

6 Mathematical Problems in Engineering

Combining (19) (20) (21) (22) (23) (25) (26) (27) (28)(29) (30) (32) and (34) one can easily yield

(119890119905) le 120585119879 (119905) Γ (119889 (119905)) 120585 (119905) (35)

If Γ(119889(119905)) lt 0 then (119890119905) lt 0 Thus if Γ(0) lt 0 and Γ(ℎ) lt0 are satisfied then (119890119905) le minus120598119890(119905)2 for a sufficient smallscalar 120598 gt 0 which implies the filtering error system (5) with120596(119905) = 0 is asymptotically stable

Considering the definition of 1198672 performance we define

119869 (119905) = 119881 (119890119905) minus int1199050

120596119879 (119904) 120596 (119904) 119889119904 (36)

Then under the zero-initial conditions 119881(119890119905)|119905=0 = 0 and119881(119890119905) gt 0 for 119905 gt 0 the following inequality is given119869 (119905) = 119881 (119890119905) minus 119881 (119890119905)1003816100381610038161003816119905=0 minus int119905

0120596119879 (119904) 120596 (119904) 119889119904

= int1199050

[ (119890 (119904)) minus 120596119879 (119904) 120596 (119904)] 119889119904 lt 0(37)

That is

119881 (119890119905) lt int1199050

120596119879 (119904) 120596 (119904) 119889119904 (38)

In addition it follows that LMI (13) and Schur complementthat

[119867119879119867 119874lowast 119874] le 1205742 [1198751 1198752lowast 1198753] (39)

Then the following holds

119911119879119890 (119905) 119911119890 (119905) = 119890119879 (119905) 119867119879119867119890 (119905)

= [[[

119909 (119905)int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

[119867119879119867 119874lowast 119874] [[

[119909 (119905)

int119905119905minusℎ

119890 (119904) 119889119904]]]le 1205742 [[

[119909 (119905)

int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

[1198751 1198752lowast 1198753][[[

119909 (119905)int119905119905minusℎ

119890 (s) 119889119904]]]= 12057421198811 (119890119905) le 1205742119881 (119890119905) le 1205742 int119905

0120596119879 (119904) 120596 (119904) 119889119904

le 1205742 intinfin0

120596119879 (119904) 120596 (119904) 119889119904

(40)

Taking the supremum over 119905 gt 0 one has 119911119890(119905)2infin le120574120596(119905)22 which means 119911119890(119905)infin le 120574120596(119905)2 Therefore when(10)ndash(14) hold the generalized1198672 filtering problem of NN (1)with time-varying delay satisfying (2) is solvedwith 120574minThiscompletes the proof

Remark 6 The generalized 1198672 design of a type of static NNwith a time-varying delay is solved in Theorem 5 Different

from [8ndash10 12ndash15] we solve the filtering problem based onan Arcak-type filter in which there are two gain matrices tobe determined Compared with the widely used Luenberger-type filter it contains an additional gainmatrix1198701 included inthe activation function So it can be expected to lead a bettergeneralized1198672 performance And if we set1198701 = 0 the Arcak-type filter is then reduced to the Luenberger-type filter

Remark 7 The triple integral term is introduced into theLKF in this paper When bounding its derivative the deriveddouble integral term minus int119905

119905minusℎint119905120579

119890119879(119904)119885 119890(119904)119889119904 119889120579 is divided intothree parts two double integral terms and a single integralterm By using the double Jensen inequality to estimatethe double integral terms and using the free-matrix-basedintegral inequality to estimate the single integral term thederivation of the triple term is accomplishedwithout ignoringany terms It should be pointed out that this technique ismore effective than the ones used in [10] in which the deriveddouble integral term minus int119905

119905minusℎint119905120579

119890119879(119904)119885 119890(119904)119889119904 119889120579 is estimated inits entirety

Remark 8 To reduce the conservatism of the derived con-ditions Wirtinger integral inequality was used to estimatesingle integral terms in [16] while in the presented studythe free-matrix-based integral inequality is employed Theuse of the free-matrix-based integral inequality brings severaladvantages for Theorem 5 On the one hand by introducingsome free matrices it gives more degrees of freedom tothe derived condition On the other hand it has beenproved in [17] that the free-matrix-based integral inequalitycontains Wirtinger integral inequality as a special case Thatmeans it can provide a tighter bound thanWirtinger integralinequality And if the same LKF is chosen our designed filterwill be better than the one designed in [16]

Remark 9 There is some room for further improvement ofour proposed method

(i) Notice that when the free-matrix-based integralinequality is employed to improve the proposed con-dition the number of decision variables is increasedwhich further increases the computation complexityof the condition It has been pointed out in [3]that the calculation complexity is also an importantconsideration during the application of the proposedLMI-based condition to physical systems Thus theresults considering both the conservatism and thecomputation complexity need more investigation

(ii) Recently some new approaches have been developedfor the stability analysis of delayed neural networkswhich can be applied into the generalized 1198672 filter-ing problem of neural networks such as the delay-decomposition approach [21] and the free-matrix-based double integral inequality [22]

Notice that in many applications the derivative bound ofthe time delay is unknown By setting 1198761 = 0 we can easilyderive the delay-dependent but delay derivative-independent

Mathematical Problems in Engineering 7

condition of the generalized 1198672 filtering problem of neuralnetwork (1) based onTheorem 5

Theorem 10 For given scalars ℎ the 1198672 filtering problem ofNN (1) with time-varying delay satisfying 0 le 119889(119905) le ℎ issolved with 120574min if there exist positive definite matrices 119875 =[ 1198751 1198752lowast 1198753 ] isin R2119899 1198762 119877 119885 isin R119899 positive definite diagonalmatrices Λ = diag1205821 1205822 120582119899 isin R119899 symmetric matrices1198831 1198833 1198841 1198843 1198851 1198853 isin R3119899 and any matrices 1198832 1198842 1198852 isinR3119899 1198721 1198722 1198731 1198732 1198661 1198662 isin R3119899times119899 119879 isin R119899 1198811 1198812 isinR119899times119901 satisfying LMIs (10)ndash(14)

4 Numerical Examples

In this section we provide a well-used numerical example todemonstrate the effectiveness of the proposed approach

Example 1 Consider the delayed static NN (1) with thefollowing parameters

119860 = [[[

096 0 00 122 00 0 078

]]]

119882 = [[[

minus032 075 minus142121 041 minus050042 082 minus106

]]]

1198611 = [[[

020305

]]]

119867 = [[[

08 0 051 0 10 minus1 1

]]]

119862 = [1 05 00 minus05 06]

119863 = [0 minus12 020 0 05]

1198612 = [ 04minus03]

(41)

This example has been studied in [10 16] and we takethese literatures for comparison study To verify the advan-tages of our proposed approach we calculate the optimal 1198672performance bounds index 120574min for various ℎ 120583 and 119871 suchthat filtering error system (5) is globally asymptotically stablewith generalized 1198672 performance

Firstly let 119871 = diag1198971 1198972 119897119899 = 119868 for given ℎ and 120583the optimal generalized 1198672 performance index 120574min derivedby Theorem 5 and the ones reported in [10 16] are givenin Table 1 where (120572 120573) represents ℎ = 120572 120583 = 120573 and

minus15

minus1

minus05

0

05

1

15

e(t)

5 10 150Time (s)

e1(t)

e2(t)

e3(t)

Figure 1 State response of the error 119890(119905)

ldquoinfeasiblerdquo means that no feasible solution can be found inthe corresponding criteria

It is clearly shown from Table 1 that compared with [1016] much better generalized 1198672 performance is achieved byTheorem 5 proposed in this paper It should be noticed thatin this paper we choose the same filter as [10 16] thus thebetter 1198672 performance effectively illustrates the superiorityof our proposed approaches which is mainly based on thefree-matrix-based integral inequality

Then let 119891(119909) = 157tanh(119909) 119889(119905) = 03 sin(15119905) + 12and 119908(119905) = sin(119905)119890minus5119905 for ℎ = 15 120583 = 03 by solving theLMIs in Theorem 5 the gain matrices of our designed filter(4) are obtained as

1198701 = [[[

10591 1299001519 0268724379 33466

]]]

1198702 = [[[

00944 minus0600503315 minus0510021756 12563

]]]

(42)

with the optimal generalized 1198672 performance index 120574 =00505 Figure 1 shows the state responses of error system (5)under initial condition 119890(0) = [minus1 1 minus05]119879 The resultingresponses obviously demonstrate the asymptotic stability ofthe simulated error system

In addition the optimal generalized 1198672 performanceindex 120574min for ℎ = 05 120583 = 07 and different 119871 is listed inTable 2 from which one can see thatTheorem 5 in this paperoutperforms the ones in [10 16]

Moreover the optimal generalized1198672 performance index120574min derived by Theorem 10 for ℎ = 08 unknown 120583 anddifferent 119871 is listed in Table 3

8 Mathematical Problems in Engineering

Table 1 Optimal generalized 1198672 performance index 120574min for 119871 = 119868 and different (ℎ 120583)(ℎ 120583) (1 04) (15 03) (3 08) (5 12) (6 15)Theorem 1 [10] 07981 Infeasible Infeasible Infeasible InfeasibleTheorem 2 [16] 134 times 10minus4 12763 Infeasible Infeasible InfeasibleTheorem 5 133 times 10minus4 00505 06639 09137 19523

Table 2 Optimal generalized 1198672 performance index 120574min for ℎ = 05 120583 = 07 and different 119871119871 16119868 18119868 25119868 34119868Theorem 1 [10] 33728 Infeasible Infeasible InfeasibleTheorem 2 [16] 05983 15563 Infeasible InfeasibleTheorem 5 02765 04015 06896 47676

Table 3 Optimal generalized 1198672 performance index 120574min for ℎ =05 unknown 120583 and different 119871119871 16119868 18119868 25119868 291119868Theorem 10 05156 05868 10891 51122

5 Conclusion

In this paper the generalized 1198672 filtering problem of staticNNs with a time-varying delay has been investigated Firstan Arcak-type filter has been constructed Then based onan augmented LKF the free-matrix-based integral inequalitythe free-weighting matrix approach and Jensen inequalitya suitable delay-dependent condition expressed in terms ofLMIs has been derived such that the filtering error systemof the considered static NNs is globally asymptotically stablewith guaranteed1198672 performance indexMoreover by solvingsome coupled LMIs the optimal performance index and thegain matrices of the designed 1198672 filter have been obtainedThe effectiveness and advantages of the proposed methodhave been demonstrated by an illustrative example finally

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding this paper

References

[1] L O Chua and L Yang ldquoCellular Neural Networks Applica-tionsrdquo IEEE Transactions on Circuits and Systems II ExpressBriefs vol 35 no 10 pp 1273ndash1290 1988

[2] X-M Zhang Q-L Han and X Yu ldquoSurvey on RecentAdvances inNetworkedControl Systemsrdquo IEEETransactions onIndustrial Informatics vol 12 no 5 pp 1740ndash1752 2016

[3] C-K Zhang Y He L Jiang and M Wu ldquoStability analysisfor delayed neural networks considering both conservativenessand complexityrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 7 pp 1486ndash1501 2016

[4] H-B Zeng Y He P Shi M Wu and S-P Xiao ldquoDissipativityanalysis of neural networks with time-varying delaysrdquo Neuro-computing vol 168 pp 741ndash746 2015

[5] C-K Zhang Y He L Jiang Q H Wu and M Wu ldquoDelay-dependent stability criteria for generalized neural networks

with two delay componentsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1263ndash12762014

[6] Y He Q GWang andC Lin ldquoAn improvedH8 filter design forsystems with time-varying interval delayrdquo IEEE Trans CircuitsSyst II vol 53 p 1235 2006

[7] YHeG P LiuD Rees andMWu ldquoImprovedHinfinfiltering forsystems with a time-varying delayrdquo Circuits Systems and SignalProcessing vol 29 no 3 pp 377ndash389 2010

[8] H Huang and G Feng ldquoDelay-dependent Hinfin and generalizedH2 filtering for delayed neural networksrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 4 pp 846ndash857 2009

[9] C K Ahn and M K Song ldquoL2-Linfin filtering for time-delayedswitched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[10] D Hu H Huang and T Huang ldquoDesign of Arcak-type gen-eralized H2 filter for delayed static neural networksrdquo CircuitsSystems and Signal Processing vol 33 no 11 pp 3635ndash36482014

[11] M Hua H Tan and J Chen ldquoDelay-dependent Hinfin andgeneralized H2 filtering for stochastic neural networks withtime-varying delay and noise disturbancerdquo Neural Computingand Applications vol 25 no 3-4 pp 613ndash624 2014

[12] Y J Li and Y N Liang ldquoDelay-dependent Hinfin filtering forneural networks with time delayrdquo Applied Mechanics and Mate-rials vol 511-512 pp 875ndash879 2014

[13] YHeG P LiuD Rees andMWu ldquoStability analysis for neuralnetworks with time-varying interval delayrdquo IEEE Transactionson Neural Networks and Learning Systems vol 18 no 6 pp1850ndash1854 2007

[14] Y Li J Li and M Hua ldquoNew results of H8 filtering for neuralnetwork with time-varying delayrdquo Int J Innov Comput InfControl vol 10 pp 2309ndash2323 2014

[15] G Liu S Zhou X Luo and K Zhang ldquoNew filter design forstatic neural networks with mixed time-varying delaysrdquo LectureNotes in Computer Science (including subseries Lecture Notesin Artificial Intelligence and Lecture Notes in Bioinformatics)Preface vol 9772 pp 117ndash129 2016

[16] Y Shu X-G Liu Y Liu and J H Park ldquoImproved Results onGuaranteed Generalized H2 Performance State Estimation forDelayed Static Neural Networksrdquo Circuits Systems and SignalProcessing vol 36 no 8 pp 3114ndash3142 2017

Mathematical Problems in Engineering 9

[17] H-B Zeng Y He M Wu and J She ldquoFree-matrix-basedintegral inequality for stability analysis of systems with time-varying delayrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 60 no 10 pp 2768ndash2772 2015

[18] C-K Zhang Y He L Jiang M Wu and H-B Zeng ldquoStabilityanalysis of systems with time-varying delay via relaxed integralinequalitiesrdquo Systems amp Control Letters vol 92 pp 52ndash61 2016

[19] C-K Zhang Y He L Jiang W-J Lin and M Wu ldquoDelay-dependent stability analysis of neural networks with time-varying delay a generalized free-weighting-matrix approachrdquoApplied Mathematics and Computation vol 294 pp 102ndash1202017

[20] J Sun G P Liu J Chen and D Rees ldquoImproved delay-range-dependent stability criteria for linear systemswith time-varyingdelaysrdquo Automatica vol 46 no 2 pp 466ndash470 2010

[21] C Ge C Hua and X Guan ldquoNew delay-dependent stabilitycriteria for neural networks with time-varying delay usingdelay-decomposition approachrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1378ndash13832014

[22] C Hua SWu X Yang and X Guan ldquoStability analysis of time-delay systems via free-matrix-based double integral inequalityrdquoInternational Journal of Systems Science vol 48 no 2 pp 257ndash263 2017

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4 Mathematical Problems in Engineering

Ω = Sym 1198901198796Λ (1198711198821198902 minus 1198906)minus 11989011987961198811 (1198621198901 + 1198631198902 + 11986121198908) + (1198901 + 1198907)119879sdot 119879 (1198907 + 1198601198901 minus 1198906 minus 11986111198908) + (1198901 + 1198907)119879sdot 1198812 (1198621198901 + 1198631198902 + 11986121198908)

Π1 = [1198901198791 119889 (119905) 1198901198794 + (ℎ minus 119889 (119905)) 1198901198795 ]119879 Π2 = [1198901198797 1198901198791 minus 1198901198793 ]119879 Π3 = [1198901198791 1198901198792 1198901198794 ]119879 Π4 = [1198901198792 1198901198793 1198901198795 ]119879 Π5 = [1198901198791 minus 1198901198792 minus1198901198791 minus 1198901198792 + 21198901198794 ]119879 Π6 = [1198901198792 minus 1198901198793 minus1198901198792 minus 1198901198793 + 21198901198795 ]119879 119872 = [1198721 1198722] 119873 = [1198731 1198732] 119866 = [1198661 1198662] 119890119894 = [0119899times(119894minus1)119899 119868119899 0119899times(7minus119894)119899 0119899times119901] (119894 = 1 2 7) 1198908 = [0119901times7119899 119868119901] 119871 = diag 1198971 1198972 119897119899 120585 (119905) = [119890119879 (119905) 119890119879 (119905 minus 119889 (119905)) 119890119879 (119905 minus ℎ) int119905

119905minus119889(119905)

119890119879 (119904)119889 (119905) 119889119904

int119905minus119889(119905)119905minusℎ

119890119879 (119904)ℎ minus 119889 (119905)119889119904 119892119879 (119905) 119890119879 (119905) 120596119879 (119905)]119879

(15)and the gain matrices 1198701 1198702 of the filter of (4) can be designedas

1198701 = (Λ119871)minus1 11988111198702 = 119879minus11198812 (16)

Proof Let us construct a new augmented LKF candidate as

119881 (119890119905) = 4sum119894=1

119881119894 (119890119905) (17)

where

1198811 (119890119905) = [[[

119909 (119905)int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

119875 [[[

119909 (119905)int119905119905minusℎ

119890 (119904) 119889119904]]]

1198812 (119890119905) = int119905119905minus119889(119905)

119890119879 (119904) 1198761119890 (119904) 119889119904+ int119905119905minusℎ

119890119879 (119904) 1198762119890 (119904) 119889119904

1198813 (119890119905) = int0minusℎ

int119905119905+120579

119890119879 (119904) 119877 119890 (119904) 119889119904 1198891205791198814 (119890119905) = int0

minusℎint119905120579

int119905119905+119906

119890119879 (119904) 119885 119890 (119904) 119889119904 119889119906 119889120579(18)

Taking the derivative of1198811(119890119905) along the filtering error system(5) yields

1 (119890119905) = 2 [[[

119909 (119905)int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

119875 [ (119905)119890 (119905) minus 119890 (119905 minus ℎ)]

= 120585119879 (119905) Φ1 (119889 (119905)) 120585 (119905) (19)

By the time derivative of 1198812(119890119905) we get2 (119890119905) = 119890119879 (119905) (1198761 + 1198762) 119890 (119905)

minus (1 minus 119889 (119905)) 119890119879 (119905 minus 119889 (119905)) 1198761119890 (119905 minus 119889 (119905))minus 119890119879 (119905 minus ℎ) 1198762119890 (119905 minus ℎ) = 120585119879 (119905) Φ2120585 (119905)

(20)

Calculating the derivative of 1198813(119890119905) we have3 (119890119905) = ℎ 119890119879 (119905) 119877 119890 (119905) minus int119905

119905minus119889(119905)119890119879 (119904) 119877 119890 (119904) 119889119904

minus int119905minus119889(119905)119905minusℎ

119890119879 (119904) 119877 119890 (119904) 119889119904(21)

Then by applying Lemma 3 to estimate the above 119877-dependent integral term if LMIs (10) and (11) hold one has

minus int119905119905minus119889(119905)

119890119879 (119904) 119877 119890 (119904) 119889119904le 119889 (119905) 1205941198791 (1198831 + 131198833) 1205941

+ Sym 120594119879111987211205942 + 120594119879111987221205943 (22)

minus int119905minus119889(119905)119905minusℎ

119890119879 (119904) 119877 119890 (119904) 119889119904le (ℎ minus 119889 (119905)) 1205941198794 (1198841 + 131198843) 1205944

+ Sym 120594119879411987311205945 + 120594119879411987321205946 (23)

where

1205941 = [119890119879 (119905) 119890119879 (119905 minus 119889 (119905)) 1119889 (119905) int119905119905minus119889(119905)

119890119879 (119904) 119889119904]119879 1205942 = 119890 (119905) minus 119890 (119905 minus 119889 (119905))

Mathematical Problems in Engineering 5

1205943 = minus119890 (119905) minus 119890 (119905 minus 119889 (119905)) + 2119889 (119905) int119905119905minus119889(119905)

119890 (119904) 1198891199041205944 = [119890119879 (119905 minus 119889 (119905)) 119890119879 (119905 minus ℎ) 1ℎ minus 119889 (119905)

sdot int119905minus119889(119905)119905minusℎ

119890119879 (119904) 119889119904]119879 1205945 = 119890 (119905 minus 119889 (119905)) minus 119890 (119905 minus ℎ) 1205946 = minus119890 (119905 minus 119889 (119905)) minus 119890 (119905 minus ℎ) + 2ℎ minus 119889 (119905)

sdot int119905minus119889(119905)119905minusℎ

119890 (119904) 119889119904(24)

Combining (21) with (23) it is clear that

3 (119890119905) le 120585119879 (119905) Φ3 (119889 (119905)) 120585 (119905) (25)

Finally the derivative of 1198814(119890119905) can be obtained as

4 (119890119905) = ℎ22 119890119879 (119905) 119885 119890 (119905) minus int119905119905minusℎ

int119905120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579= ℎ22 119890119879 (119905) 119885 119890 (119905)

minus int119905119905minus119889(119905)

int119905120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579minus int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579minus (ℎ minus 119889 (119905)) int119905

119905minus119889(119905)119890119879 (119904) 119885 119890 (119904) 119889119904

(26)

For the first double integral term in 4(119890119905) we can do thefollowing treatment based on Lemma 4

minus int119905119905minus119889(119905)

int119905120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579le minus 21198892 (119905) (int119905

119905minus119889(119905)int119905120579

119890 (119904) 119889119904 119889120579)119879

sdot 119885 (int119905119905minus119889(119905)

int119905120579

119890 (119904) 119889119904 119889120579)= minus 21198892 (119905) (119889 (119905) 119890 (119905) minus int119905

119905minus119889(119905)119890 (119904) 119889119904)119879

sdot 119885 (119889 (119905) 119890 (119905) minus int119905119905minus119889(119905)

119890 (119904) 119889119904)= minus2 (119890 (119905) minus int119905

119905minus119889(119905)

119890 (119904)119889 (119905)119889119904)119879

sdot 119885 (119890 (119905) minus int119905119905minus119889(119905)

119890 (119904)119889 (119905)119889119904)

(27)

Similarly the estimation of the second integral term can bedone as

minus int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579le minus 2

(ℎ minus 119889 (119905))2 (int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890 (119904) 119889119904 119889120579)119879

sdot 119885 (int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890 (119904) 119889119904 119889120579)

= minus2 (119890 (119905 minus 119889 (119905)) minus int119905minus119889(119905)119905minusℎ

119890 (119904)ℎ minus 119889 (119905)119889119904)119879

sdot 119885 (119890 (119905 minus 119889 (119905)) minus int119905minus119889(119905)119905minusℎ

119890 (119904)ℎ minus 119889 (119905)119889119904)

(28)

And by utilizing Lemma 3 if LMI (12) holds we obtain

minus (ℎ minus 119889 (119905)) int119905119905minus119889(119905)

119890119879 (119904) 119885 119890 (119904) 119889119904 le (ℎ minus 119889 (119905))sdot (ℎ1205941198791 (1198851 + 131198853) 1205941+ Sym 1205941198791 11986611205942 + 1205941198791 11986621205943)

(29)

From (26) to (29) we get

4 (119890119905) le 120585119879 (119905) Φ4 (119889 (119905)) 120585 (119905) (30)

Then taking assumption of the activation function (3) intoaccount yields

0 le 2119892119879 (119905) Λ [119871 (119882119890 (119905 minus 119889 (119905)) minus 1198701119862119890 (119905)minus 1198701119863119890 (119905 minus 119889 (119905)) minus 11987011198612120596 (119905)) minus 119892 (119905)] (31)

where

1198811 = Λ1198711198701997904rArr 0le 21198901198796Λ (1198711198821198902 minus 1198906) minus 211989011987961198811 (1198621198901 + 1198631198902 + 11986121198908)

(32)

Based on the filtering error system (5) for any appropriatelydimensioned matrix 119879 the following equation holds

0 = 2 (119890 (119905) + 119890 (119905))119879 119879 [ 119890 (119905) + (119860 + 1198702119862) 119890 (119905)+ 1198702119863119890 (119905 minus 119889 (119905)) minus 119892 (119905) minus (1198611 minus 11987021198612) 120596 (119905)] (33)

where

1198812 = 1198791198702997904rArr 0= 2 (1198901 + 1198907)119879 119879 (1198907 + 1198601198901 minus 1198906 minus 11986111198908)

+ 2 (1198901 + 1198907)1198791198812 (1198621198901 + 1198631198902 + 11986121198908) (34)

6 Mathematical Problems in Engineering

Combining (19) (20) (21) (22) (23) (25) (26) (27) (28)(29) (30) (32) and (34) one can easily yield

(119890119905) le 120585119879 (119905) Γ (119889 (119905)) 120585 (119905) (35)

If Γ(119889(119905)) lt 0 then (119890119905) lt 0 Thus if Γ(0) lt 0 and Γ(ℎ) lt0 are satisfied then (119890119905) le minus120598119890(119905)2 for a sufficient smallscalar 120598 gt 0 which implies the filtering error system (5) with120596(119905) = 0 is asymptotically stable

Considering the definition of 1198672 performance we define

119869 (119905) = 119881 (119890119905) minus int1199050

120596119879 (119904) 120596 (119904) 119889119904 (36)

Then under the zero-initial conditions 119881(119890119905)|119905=0 = 0 and119881(119890119905) gt 0 for 119905 gt 0 the following inequality is given119869 (119905) = 119881 (119890119905) minus 119881 (119890119905)1003816100381610038161003816119905=0 minus int119905

0120596119879 (119904) 120596 (119904) 119889119904

= int1199050

[ (119890 (119904)) minus 120596119879 (119904) 120596 (119904)] 119889119904 lt 0(37)

That is

119881 (119890119905) lt int1199050

120596119879 (119904) 120596 (119904) 119889119904 (38)

In addition it follows that LMI (13) and Schur complementthat

[119867119879119867 119874lowast 119874] le 1205742 [1198751 1198752lowast 1198753] (39)

Then the following holds

119911119879119890 (119905) 119911119890 (119905) = 119890119879 (119905) 119867119879119867119890 (119905)

= [[[

119909 (119905)int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

[119867119879119867 119874lowast 119874] [[

[119909 (119905)

int119905119905minusℎ

119890 (119904) 119889119904]]]le 1205742 [[

[119909 (119905)

int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

[1198751 1198752lowast 1198753][[[

119909 (119905)int119905119905minusℎ

119890 (s) 119889119904]]]= 12057421198811 (119890119905) le 1205742119881 (119890119905) le 1205742 int119905

0120596119879 (119904) 120596 (119904) 119889119904

le 1205742 intinfin0

120596119879 (119904) 120596 (119904) 119889119904

(40)

Taking the supremum over 119905 gt 0 one has 119911119890(119905)2infin le120574120596(119905)22 which means 119911119890(119905)infin le 120574120596(119905)2 Therefore when(10)ndash(14) hold the generalized1198672 filtering problem of NN (1)with time-varying delay satisfying (2) is solvedwith 120574minThiscompletes the proof

Remark 6 The generalized 1198672 design of a type of static NNwith a time-varying delay is solved in Theorem 5 Different

from [8ndash10 12ndash15] we solve the filtering problem based onan Arcak-type filter in which there are two gain matrices tobe determined Compared with the widely used Luenberger-type filter it contains an additional gainmatrix1198701 included inthe activation function So it can be expected to lead a bettergeneralized1198672 performance And if we set1198701 = 0 the Arcak-type filter is then reduced to the Luenberger-type filter

Remark 7 The triple integral term is introduced into theLKF in this paper When bounding its derivative the deriveddouble integral term minus int119905

119905minusℎint119905120579

119890119879(119904)119885 119890(119904)119889119904 119889120579 is divided intothree parts two double integral terms and a single integralterm By using the double Jensen inequality to estimatethe double integral terms and using the free-matrix-basedintegral inequality to estimate the single integral term thederivation of the triple term is accomplishedwithout ignoringany terms It should be pointed out that this technique ismore effective than the ones used in [10] in which the deriveddouble integral term minus int119905

119905minusℎint119905120579

119890119879(119904)119885 119890(119904)119889119904 119889120579 is estimated inits entirety

Remark 8 To reduce the conservatism of the derived con-ditions Wirtinger integral inequality was used to estimatesingle integral terms in [16] while in the presented studythe free-matrix-based integral inequality is employed Theuse of the free-matrix-based integral inequality brings severaladvantages for Theorem 5 On the one hand by introducingsome free matrices it gives more degrees of freedom tothe derived condition On the other hand it has beenproved in [17] that the free-matrix-based integral inequalitycontains Wirtinger integral inequality as a special case Thatmeans it can provide a tighter bound thanWirtinger integralinequality And if the same LKF is chosen our designed filterwill be better than the one designed in [16]

Remark 9 There is some room for further improvement ofour proposed method

(i) Notice that when the free-matrix-based integralinequality is employed to improve the proposed con-dition the number of decision variables is increasedwhich further increases the computation complexityof the condition It has been pointed out in [3]that the calculation complexity is also an importantconsideration during the application of the proposedLMI-based condition to physical systems Thus theresults considering both the conservatism and thecomputation complexity need more investigation

(ii) Recently some new approaches have been developedfor the stability analysis of delayed neural networkswhich can be applied into the generalized 1198672 filter-ing problem of neural networks such as the delay-decomposition approach [21] and the free-matrix-based double integral inequality [22]

Notice that in many applications the derivative bound ofthe time delay is unknown By setting 1198761 = 0 we can easilyderive the delay-dependent but delay derivative-independent

Mathematical Problems in Engineering 7

condition of the generalized 1198672 filtering problem of neuralnetwork (1) based onTheorem 5

Theorem 10 For given scalars ℎ the 1198672 filtering problem ofNN (1) with time-varying delay satisfying 0 le 119889(119905) le ℎ issolved with 120574min if there exist positive definite matrices 119875 =[ 1198751 1198752lowast 1198753 ] isin R2119899 1198762 119877 119885 isin R119899 positive definite diagonalmatrices Λ = diag1205821 1205822 120582119899 isin R119899 symmetric matrices1198831 1198833 1198841 1198843 1198851 1198853 isin R3119899 and any matrices 1198832 1198842 1198852 isinR3119899 1198721 1198722 1198731 1198732 1198661 1198662 isin R3119899times119899 119879 isin R119899 1198811 1198812 isinR119899times119901 satisfying LMIs (10)ndash(14)

4 Numerical Examples

In this section we provide a well-used numerical example todemonstrate the effectiveness of the proposed approach

Example 1 Consider the delayed static NN (1) with thefollowing parameters

119860 = [[[

096 0 00 122 00 0 078

]]]

119882 = [[[

minus032 075 minus142121 041 minus050042 082 minus106

]]]

1198611 = [[[

020305

]]]

119867 = [[[

08 0 051 0 10 minus1 1

]]]

119862 = [1 05 00 minus05 06]

119863 = [0 minus12 020 0 05]

1198612 = [ 04minus03]

(41)

This example has been studied in [10 16] and we takethese literatures for comparison study To verify the advan-tages of our proposed approach we calculate the optimal 1198672performance bounds index 120574min for various ℎ 120583 and 119871 suchthat filtering error system (5) is globally asymptotically stablewith generalized 1198672 performance

Firstly let 119871 = diag1198971 1198972 119897119899 = 119868 for given ℎ and 120583the optimal generalized 1198672 performance index 120574min derivedby Theorem 5 and the ones reported in [10 16] are givenin Table 1 where (120572 120573) represents ℎ = 120572 120583 = 120573 and

minus15

minus1

minus05

0

05

1

15

e(t)

5 10 150Time (s)

e1(t)

e2(t)

e3(t)

Figure 1 State response of the error 119890(119905)

ldquoinfeasiblerdquo means that no feasible solution can be found inthe corresponding criteria

It is clearly shown from Table 1 that compared with [1016] much better generalized 1198672 performance is achieved byTheorem 5 proposed in this paper It should be noticed thatin this paper we choose the same filter as [10 16] thus thebetter 1198672 performance effectively illustrates the superiorityof our proposed approaches which is mainly based on thefree-matrix-based integral inequality

Then let 119891(119909) = 157tanh(119909) 119889(119905) = 03 sin(15119905) + 12and 119908(119905) = sin(119905)119890minus5119905 for ℎ = 15 120583 = 03 by solving theLMIs in Theorem 5 the gain matrices of our designed filter(4) are obtained as

1198701 = [[[

10591 1299001519 0268724379 33466

]]]

1198702 = [[[

00944 minus0600503315 minus0510021756 12563

]]]

(42)

with the optimal generalized 1198672 performance index 120574 =00505 Figure 1 shows the state responses of error system (5)under initial condition 119890(0) = [minus1 1 minus05]119879 The resultingresponses obviously demonstrate the asymptotic stability ofthe simulated error system

In addition the optimal generalized 1198672 performanceindex 120574min for ℎ = 05 120583 = 07 and different 119871 is listed inTable 2 from which one can see thatTheorem 5 in this paperoutperforms the ones in [10 16]

Moreover the optimal generalized1198672 performance index120574min derived by Theorem 10 for ℎ = 08 unknown 120583 anddifferent 119871 is listed in Table 3

8 Mathematical Problems in Engineering

Table 1 Optimal generalized 1198672 performance index 120574min for 119871 = 119868 and different (ℎ 120583)(ℎ 120583) (1 04) (15 03) (3 08) (5 12) (6 15)Theorem 1 [10] 07981 Infeasible Infeasible Infeasible InfeasibleTheorem 2 [16] 134 times 10minus4 12763 Infeasible Infeasible InfeasibleTheorem 5 133 times 10minus4 00505 06639 09137 19523

Table 2 Optimal generalized 1198672 performance index 120574min for ℎ = 05 120583 = 07 and different 119871119871 16119868 18119868 25119868 34119868Theorem 1 [10] 33728 Infeasible Infeasible InfeasibleTheorem 2 [16] 05983 15563 Infeasible InfeasibleTheorem 5 02765 04015 06896 47676

Table 3 Optimal generalized 1198672 performance index 120574min for ℎ =05 unknown 120583 and different 119871119871 16119868 18119868 25119868 291119868Theorem 10 05156 05868 10891 51122

5 Conclusion

In this paper the generalized 1198672 filtering problem of staticNNs with a time-varying delay has been investigated Firstan Arcak-type filter has been constructed Then based onan augmented LKF the free-matrix-based integral inequalitythe free-weighting matrix approach and Jensen inequalitya suitable delay-dependent condition expressed in terms ofLMIs has been derived such that the filtering error systemof the considered static NNs is globally asymptotically stablewith guaranteed1198672 performance indexMoreover by solvingsome coupled LMIs the optimal performance index and thegain matrices of the designed 1198672 filter have been obtainedThe effectiveness and advantages of the proposed methodhave been demonstrated by an illustrative example finally

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding this paper

References

[1] L O Chua and L Yang ldquoCellular Neural Networks Applica-tionsrdquo IEEE Transactions on Circuits and Systems II ExpressBriefs vol 35 no 10 pp 1273ndash1290 1988

[2] X-M Zhang Q-L Han and X Yu ldquoSurvey on RecentAdvances inNetworkedControl Systemsrdquo IEEETransactions onIndustrial Informatics vol 12 no 5 pp 1740ndash1752 2016

[3] C-K Zhang Y He L Jiang and M Wu ldquoStability analysisfor delayed neural networks considering both conservativenessand complexityrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 7 pp 1486ndash1501 2016

[4] H-B Zeng Y He P Shi M Wu and S-P Xiao ldquoDissipativityanalysis of neural networks with time-varying delaysrdquo Neuro-computing vol 168 pp 741ndash746 2015

[5] C-K Zhang Y He L Jiang Q H Wu and M Wu ldquoDelay-dependent stability criteria for generalized neural networks

with two delay componentsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1263ndash12762014

[6] Y He Q GWang andC Lin ldquoAn improvedH8 filter design forsystems with time-varying interval delayrdquo IEEE Trans CircuitsSyst II vol 53 p 1235 2006

[7] YHeG P LiuD Rees andMWu ldquoImprovedHinfinfiltering forsystems with a time-varying delayrdquo Circuits Systems and SignalProcessing vol 29 no 3 pp 377ndash389 2010

[8] H Huang and G Feng ldquoDelay-dependent Hinfin and generalizedH2 filtering for delayed neural networksrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 4 pp 846ndash857 2009

[9] C K Ahn and M K Song ldquoL2-Linfin filtering for time-delayedswitched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[10] D Hu H Huang and T Huang ldquoDesign of Arcak-type gen-eralized H2 filter for delayed static neural networksrdquo CircuitsSystems and Signal Processing vol 33 no 11 pp 3635ndash36482014

[11] M Hua H Tan and J Chen ldquoDelay-dependent Hinfin andgeneralized H2 filtering for stochastic neural networks withtime-varying delay and noise disturbancerdquo Neural Computingand Applications vol 25 no 3-4 pp 613ndash624 2014

[12] Y J Li and Y N Liang ldquoDelay-dependent Hinfin filtering forneural networks with time delayrdquo Applied Mechanics and Mate-rials vol 511-512 pp 875ndash879 2014

[13] YHeG P LiuD Rees andMWu ldquoStability analysis for neuralnetworks with time-varying interval delayrdquo IEEE Transactionson Neural Networks and Learning Systems vol 18 no 6 pp1850ndash1854 2007

[14] Y Li J Li and M Hua ldquoNew results of H8 filtering for neuralnetwork with time-varying delayrdquo Int J Innov Comput InfControl vol 10 pp 2309ndash2323 2014

[15] G Liu S Zhou X Luo and K Zhang ldquoNew filter design forstatic neural networks with mixed time-varying delaysrdquo LectureNotes in Computer Science (including subseries Lecture Notesin Artificial Intelligence and Lecture Notes in Bioinformatics)Preface vol 9772 pp 117ndash129 2016

[16] Y Shu X-G Liu Y Liu and J H Park ldquoImproved Results onGuaranteed Generalized H2 Performance State Estimation forDelayed Static Neural Networksrdquo Circuits Systems and SignalProcessing vol 36 no 8 pp 3114ndash3142 2017

Mathematical Problems in Engineering 9

[17] H-B Zeng Y He M Wu and J She ldquoFree-matrix-basedintegral inequality for stability analysis of systems with time-varying delayrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 60 no 10 pp 2768ndash2772 2015

[18] C-K Zhang Y He L Jiang M Wu and H-B Zeng ldquoStabilityanalysis of systems with time-varying delay via relaxed integralinequalitiesrdquo Systems amp Control Letters vol 92 pp 52ndash61 2016

[19] C-K Zhang Y He L Jiang W-J Lin and M Wu ldquoDelay-dependent stability analysis of neural networks with time-varying delay a generalized free-weighting-matrix approachrdquoApplied Mathematics and Computation vol 294 pp 102ndash1202017

[20] J Sun G P Liu J Chen and D Rees ldquoImproved delay-range-dependent stability criteria for linear systemswith time-varyingdelaysrdquo Automatica vol 46 no 2 pp 466ndash470 2010

[21] C Ge C Hua and X Guan ldquoNew delay-dependent stabilitycriteria for neural networks with time-varying delay usingdelay-decomposition approachrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1378ndash13832014

[22] C Hua SWu X Yang and X Guan ldquoStability analysis of time-delay systems via free-matrix-based double integral inequalityrdquoInternational Journal of Systems Science vol 48 no 2 pp 257ndash263 2017

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Mathematical Problems in Engineering 5

1205943 = minus119890 (119905) minus 119890 (119905 minus 119889 (119905)) + 2119889 (119905) int119905119905minus119889(119905)

119890 (119904) 1198891199041205944 = [119890119879 (119905 minus 119889 (119905)) 119890119879 (119905 minus ℎ) 1ℎ minus 119889 (119905)

sdot int119905minus119889(119905)119905minusℎ

119890119879 (119904) 119889119904]119879 1205945 = 119890 (119905 minus 119889 (119905)) minus 119890 (119905 minus ℎ) 1205946 = minus119890 (119905 minus 119889 (119905)) minus 119890 (119905 minus ℎ) + 2ℎ minus 119889 (119905)

sdot int119905minus119889(119905)119905minusℎ

119890 (119904) 119889119904(24)

Combining (21) with (23) it is clear that

3 (119890119905) le 120585119879 (119905) Φ3 (119889 (119905)) 120585 (119905) (25)

Finally the derivative of 1198814(119890119905) can be obtained as

4 (119890119905) = ℎ22 119890119879 (119905) 119885 119890 (119905) minus int119905119905minusℎ

int119905120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579= ℎ22 119890119879 (119905) 119885 119890 (119905)

minus int119905119905minus119889(119905)

int119905120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579minus int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579minus (ℎ minus 119889 (119905)) int119905

119905minus119889(119905)119890119879 (119904) 119885 119890 (119904) 119889119904

(26)

For the first double integral term in 4(119890119905) we can do thefollowing treatment based on Lemma 4

minus int119905119905minus119889(119905)

int119905120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579le minus 21198892 (119905) (int119905

119905minus119889(119905)int119905120579

119890 (119904) 119889119904 119889120579)119879

sdot 119885 (int119905119905minus119889(119905)

int119905120579

119890 (119904) 119889119904 119889120579)= minus 21198892 (119905) (119889 (119905) 119890 (119905) minus int119905

119905minus119889(119905)119890 (119904) 119889119904)119879

sdot 119885 (119889 (119905) 119890 (119905) minus int119905119905minus119889(119905)

119890 (119904) 119889119904)= minus2 (119890 (119905) minus int119905

119905minus119889(119905)

119890 (119904)119889 (119905)119889119904)119879

sdot 119885 (119890 (119905) minus int119905119905minus119889(119905)

119890 (119904)119889 (119905)119889119904)

(27)

Similarly the estimation of the second integral term can bedone as

minus int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890119879 (119904) 119885 119890 (119904) 119889119904 119889120579le minus 2

(ℎ minus 119889 (119905))2 (int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890 (119904) 119889119904 119889120579)119879

sdot 119885 (int119905minus119889(119905)119905minusℎ

int119905minus119889(119905)120579

119890 (119904) 119889119904 119889120579)

= minus2 (119890 (119905 minus 119889 (119905)) minus int119905minus119889(119905)119905minusℎ

119890 (119904)ℎ minus 119889 (119905)119889119904)119879

sdot 119885 (119890 (119905 minus 119889 (119905)) minus int119905minus119889(119905)119905minusℎ

119890 (119904)ℎ minus 119889 (119905)119889119904)

(28)

And by utilizing Lemma 3 if LMI (12) holds we obtain

minus (ℎ minus 119889 (119905)) int119905119905minus119889(119905)

119890119879 (119904) 119885 119890 (119904) 119889119904 le (ℎ minus 119889 (119905))sdot (ℎ1205941198791 (1198851 + 131198853) 1205941+ Sym 1205941198791 11986611205942 + 1205941198791 11986621205943)

(29)

From (26) to (29) we get

4 (119890119905) le 120585119879 (119905) Φ4 (119889 (119905)) 120585 (119905) (30)

Then taking assumption of the activation function (3) intoaccount yields

0 le 2119892119879 (119905) Λ [119871 (119882119890 (119905 minus 119889 (119905)) minus 1198701119862119890 (119905)minus 1198701119863119890 (119905 minus 119889 (119905)) minus 11987011198612120596 (119905)) minus 119892 (119905)] (31)

where

1198811 = Λ1198711198701997904rArr 0le 21198901198796Λ (1198711198821198902 minus 1198906) minus 211989011987961198811 (1198621198901 + 1198631198902 + 11986121198908)

(32)

Based on the filtering error system (5) for any appropriatelydimensioned matrix 119879 the following equation holds

0 = 2 (119890 (119905) + 119890 (119905))119879 119879 [ 119890 (119905) + (119860 + 1198702119862) 119890 (119905)+ 1198702119863119890 (119905 minus 119889 (119905)) minus 119892 (119905) minus (1198611 minus 11987021198612) 120596 (119905)] (33)

where

1198812 = 1198791198702997904rArr 0= 2 (1198901 + 1198907)119879 119879 (1198907 + 1198601198901 minus 1198906 minus 11986111198908)

+ 2 (1198901 + 1198907)1198791198812 (1198621198901 + 1198631198902 + 11986121198908) (34)

6 Mathematical Problems in Engineering

Combining (19) (20) (21) (22) (23) (25) (26) (27) (28)(29) (30) (32) and (34) one can easily yield

(119890119905) le 120585119879 (119905) Γ (119889 (119905)) 120585 (119905) (35)

If Γ(119889(119905)) lt 0 then (119890119905) lt 0 Thus if Γ(0) lt 0 and Γ(ℎ) lt0 are satisfied then (119890119905) le minus120598119890(119905)2 for a sufficient smallscalar 120598 gt 0 which implies the filtering error system (5) with120596(119905) = 0 is asymptotically stable

Considering the definition of 1198672 performance we define

119869 (119905) = 119881 (119890119905) minus int1199050

120596119879 (119904) 120596 (119904) 119889119904 (36)

Then under the zero-initial conditions 119881(119890119905)|119905=0 = 0 and119881(119890119905) gt 0 for 119905 gt 0 the following inequality is given119869 (119905) = 119881 (119890119905) minus 119881 (119890119905)1003816100381610038161003816119905=0 minus int119905

0120596119879 (119904) 120596 (119904) 119889119904

= int1199050

[ (119890 (119904)) minus 120596119879 (119904) 120596 (119904)] 119889119904 lt 0(37)

That is

119881 (119890119905) lt int1199050

120596119879 (119904) 120596 (119904) 119889119904 (38)

In addition it follows that LMI (13) and Schur complementthat

[119867119879119867 119874lowast 119874] le 1205742 [1198751 1198752lowast 1198753] (39)

Then the following holds

119911119879119890 (119905) 119911119890 (119905) = 119890119879 (119905) 119867119879119867119890 (119905)

= [[[

119909 (119905)int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

[119867119879119867 119874lowast 119874] [[

[119909 (119905)

int119905119905minusℎ

119890 (119904) 119889119904]]]le 1205742 [[

[119909 (119905)

int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

[1198751 1198752lowast 1198753][[[

119909 (119905)int119905119905minusℎ

119890 (s) 119889119904]]]= 12057421198811 (119890119905) le 1205742119881 (119890119905) le 1205742 int119905

0120596119879 (119904) 120596 (119904) 119889119904

le 1205742 intinfin0

120596119879 (119904) 120596 (119904) 119889119904

(40)

Taking the supremum over 119905 gt 0 one has 119911119890(119905)2infin le120574120596(119905)22 which means 119911119890(119905)infin le 120574120596(119905)2 Therefore when(10)ndash(14) hold the generalized1198672 filtering problem of NN (1)with time-varying delay satisfying (2) is solvedwith 120574minThiscompletes the proof

Remark 6 The generalized 1198672 design of a type of static NNwith a time-varying delay is solved in Theorem 5 Different

from [8ndash10 12ndash15] we solve the filtering problem based onan Arcak-type filter in which there are two gain matrices tobe determined Compared with the widely used Luenberger-type filter it contains an additional gainmatrix1198701 included inthe activation function So it can be expected to lead a bettergeneralized1198672 performance And if we set1198701 = 0 the Arcak-type filter is then reduced to the Luenberger-type filter

Remark 7 The triple integral term is introduced into theLKF in this paper When bounding its derivative the deriveddouble integral term minus int119905

119905minusℎint119905120579

119890119879(119904)119885 119890(119904)119889119904 119889120579 is divided intothree parts two double integral terms and a single integralterm By using the double Jensen inequality to estimatethe double integral terms and using the free-matrix-basedintegral inequality to estimate the single integral term thederivation of the triple term is accomplishedwithout ignoringany terms It should be pointed out that this technique ismore effective than the ones used in [10] in which the deriveddouble integral term minus int119905

119905minusℎint119905120579

119890119879(119904)119885 119890(119904)119889119904 119889120579 is estimated inits entirety

Remark 8 To reduce the conservatism of the derived con-ditions Wirtinger integral inequality was used to estimatesingle integral terms in [16] while in the presented studythe free-matrix-based integral inequality is employed Theuse of the free-matrix-based integral inequality brings severaladvantages for Theorem 5 On the one hand by introducingsome free matrices it gives more degrees of freedom tothe derived condition On the other hand it has beenproved in [17] that the free-matrix-based integral inequalitycontains Wirtinger integral inequality as a special case Thatmeans it can provide a tighter bound thanWirtinger integralinequality And if the same LKF is chosen our designed filterwill be better than the one designed in [16]

Remark 9 There is some room for further improvement ofour proposed method

(i) Notice that when the free-matrix-based integralinequality is employed to improve the proposed con-dition the number of decision variables is increasedwhich further increases the computation complexityof the condition It has been pointed out in [3]that the calculation complexity is also an importantconsideration during the application of the proposedLMI-based condition to physical systems Thus theresults considering both the conservatism and thecomputation complexity need more investigation

(ii) Recently some new approaches have been developedfor the stability analysis of delayed neural networkswhich can be applied into the generalized 1198672 filter-ing problem of neural networks such as the delay-decomposition approach [21] and the free-matrix-based double integral inequality [22]

Notice that in many applications the derivative bound ofthe time delay is unknown By setting 1198761 = 0 we can easilyderive the delay-dependent but delay derivative-independent

Mathematical Problems in Engineering 7

condition of the generalized 1198672 filtering problem of neuralnetwork (1) based onTheorem 5

Theorem 10 For given scalars ℎ the 1198672 filtering problem ofNN (1) with time-varying delay satisfying 0 le 119889(119905) le ℎ issolved with 120574min if there exist positive definite matrices 119875 =[ 1198751 1198752lowast 1198753 ] isin R2119899 1198762 119877 119885 isin R119899 positive definite diagonalmatrices Λ = diag1205821 1205822 120582119899 isin R119899 symmetric matrices1198831 1198833 1198841 1198843 1198851 1198853 isin R3119899 and any matrices 1198832 1198842 1198852 isinR3119899 1198721 1198722 1198731 1198732 1198661 1198662 isin R3119899times119899 119879 isin R119899 1198811 1198812 isinR119899times119901 satisfying LMIs (10)ndash(14)

4 Numerical Examples

In this section we provide a well-used numerical example todemonstrate the effectiveness of the proposed approach

Example 1 Consider the delayed static NN (1) with thefollowing parameters

119860 = [[[

096 0 00 122 00 0 078

]]]

119882 = [[[

minus032 075 minus142121 041 minus050042 082 minus106

]]]

1198611 = [[[

020305

]]]

119867 = [[[

08 0 051 0 10 minus1 1

]]]

119862 = [1 05 00 minus05 06]

119863 = [0 minus12 020 0 05]

1198612 = [ 04minus03]

(41)

This example has been studied in [10 16] and we takethese literatures for comparison study To verify the advan-tages of our proposed approach we calculate the optimal 1198672performance bounds index 120574min for various ℎ 120583 and 119871 suchthat filtering error system (5) is globally asymptotically stablewith generalized 1198672 performance

Firstly let 119871 = diag1198971 1198972 119897119899 = 119868 for given ℎ and 120583the optimal generalized 1198672 performance index 120574min derivedby Theorem 5 and the ones reported in [10 16] are givenin Table 1 where (120572 120573) represents ℎ = 120572 120583 = 120573 and

minus15

minus1

minus05

0

05

1

15

e(t)

5 10 150Time (s)

e1(t)

e2(t)

e3(t)

Figure 1 State response of the error 119890(119905)

ldquoinfeasiblerdquo means that no feasible solution can be found inthe corresponding criteria

It is clearly shown from Table 1 that compared with [1016] much better generalized 1198672 performance is achieved byTheorem 5 proposed in this paper It should be noticed thatin this paper we choose the same filter as [10 16] thus thebetter 1198672 performance effectively illustrates the superiorityof our proposed approaches which is mainly based on thefree-matrix-based integral inequality

Then let 119891(119909) = 157tanh(119909) 119889(119905) = 03 sin(15119905) + 12and 119908(119905) = sin(119905)119890minus5119905 for ℎ = 15 120583 = 03 by solving theLMIs in Theorem 5 the gain matrices of our designed filter(4) are obtained as

1198701 = [[[

10591 1299001519 0268724379 33466

]]]

1198702 = [[[

00944 minus0600503315 minus0510021756 12563

]]]

(42)

with the optimal generalized 1198672 performance index 120574 =00505 Figure 1 shows the state responses of error system (5)under initial condition 119890(0) = [minus1 1 minus05]119879 The resultingresponses obviously demonstrate the asymptotic stability ofthe simulated error system

In addition the optimal generalized 1198672 performanceindex 120574min for ℎ = 05 120583 = 07 and different 119871 is listed inTable 2 from which one can see thatTheorem 5 in this paperoutperforms the ones in [10 16]

Moreover the optimal generalized1198672 performance index120574min derived by Theorem 10 for ℎ = 08 unknown 120583 anddifferent 119871 is listed in Table 3

8 Mathematical Problems in Engineering

Table 1 Optimal generalized 1198672 performance index 120574min for 119871 = 119868 and different (ℎ 120583)(ℎ 120583) (1 04) (15 03) (3 08) (5 12) (6 15)Theorem 1 [10] 07981 Infeasible Infeasible Infeasible InfeasibleTheorem 2 [16] 134 times 10minus4 12763 Infeasible Infeasible InfeasibleTheorem 5 133 times 10minus4 00505 06639 09137 19523

Table 2 Optimal generalized 1198672 performance index 120574min for ℎ = 05 120583 = 07 and different 119871119871 16119868 18119868 25119868 34119868Theorem 1 [10] 33728 Infeasible Infeasible InfeasibleTheorem 2 [16] 05983 15563 Infeasible InfeasibleTheorem 5 02765 04015 06896 47676

Table 3 Optimal generalized 1198672 performance index 120574min for ℎ =05 unknown 120583 and different 119871119871 16119868 18119868 25119868 291119868Theorem 10 05156 05868 10891 51122

5 Conclusion

In this paper the generalized 1198672 filtering problem of staticNNs with a time-varying delay has been investigated Firstan Arcak-type filter has been constructed Then based onan augmented LKF the free-matrix-based integral inequalitythe free-weighting matrix approach and Jensen inequalitya suitable delay-dependent condition expressed in terms ofLMIs has been derived such that the filtering error systemof the considered static NNs is globally asymptotically stablewith guaranteed1198672 performance indexMoreover by solvingsome coupled LMIs the optimal performance index and thegain matrices of the designed 1198672 filter have been obtainedThe effectiveness and advantages of the proposed methodhave been demonstrated by an illustrative example finally

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding this paper

References

[1] L O Chua and L Yang ldquoCellular Neural Networks Applica-tionsrdquo IEEE Transactions on Circuits and Systems II ExpressBriefs vol 35 no 10 pp 1273ndash1290 1988

[2] X-M Zhang Q-L Han and X Yu ldquoSurvey on RecentAdvances inNetworkedControl Systemsrdquo IEEETransactions onIndustrial Informatics vol 12 no 5 pp 1740ndash1752 2016

[3] C-K Zhang Y He L Jiang and M Wu ldquoStability analysisfor delayed neural networks considering both conservativenessand complexityrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 7 pp 1486ndash1501 2016

[4] H-B Zeng Y He P Shi M Wu and S-P Xiao ldquoDissipativityanalysis of neural networks with time-varying delaysrdquo Neuro-computing vol 168 pp 741ndash746 2015

[5] C-K Zhang Y He L Jiang Q H Wu and M Wu ldquoDelay-dependent stability criteria for generalized neural networks

with two delay componentsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1263ndash12762014

[6] Y He Q GWang andC Lin ldquoAn improvedH8 filter design forsystems with time-varying interval delayrdquo IEEE Trans CircuitsSyst II vol 53 p 1235 2006

[7] YHeG P LiuD Rees andMWu ldquoImprovedHinfinfiltering forsystems with a time-varying delayrdquo Circuits Systems and SignalProcessing vol 29 no 3 pp 377ndash389 2010

[8] H Huang and G Feng ldquoDelay-dependent Hinfin and generalizedH2 filtering for delayed neural networksrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 4 pp 846ndash857 2009

[9] C K Ahn and M K Song ldquoL2-Linfin filtering for time-delayedswitched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[10] D Hu H Huang and T Huang ldquoDesign of Arcak-type gen-eralized H2 filter for delayed static neural networksrdquo CircuitsSystems and Signal Processing vol 33 no 11 pp 3635ndash36482014

[11] M Hua H Tan and J Chen ldquoDelay-dependent Hinfin andgeneralized H2 filtering for stochastic neural networks withtime-varying delay and noise disturbancerdquo Neural Computingand Applications vol 25 no 3-4 pp 613ndash624 2014

[12] Y J Li and Y N Liang ldquoDelay-dependent Hinfin filtering forneural networks with time delayrdquo Applied Mechanics and Mate-rials vol 511-512 pp 875ndash879 2014

[13] YHeG P LiuD Rees andMWu ldquoStability analysis for neuralnetworks with time-varying interval delayrdquo IEEE Transactionson Neural Networks and Learning Systems vol 18 no 6 pp1850ndash1854 2007

[14] Y Li J Li and M Hua ldquoNew results of H8 filtering for neuralnetwork with time-varying delayrdquo Int J Innov Comput InfControl vol 10 pp 2309ndash2323 2014

[15] G Liu S Zhou X Luo and K Zhang ldquoNew filter design forstatic neural networks with mixed time-varying delaysrdquo LectureNotes in Computer Science (including subseries Lecture Notesin Artificial Intelligence and Lecture Notes in Bioinformatics)Preface vol 9772 pp 117ndash129 2016

[16] Y Shu X-G Liu Y Liu and J H Park ldquoImproved Results onGuaranteed Generalized H2 Performance State Estimation forDelayed Static Neural Networksrdquo Circuits Systems and SignalProcessing vol 36 no 8 pp 3114ndash3142 2017

Mathematical Problems in Engineering 9

[17] H-B Zeng Y He M Wu and J She ldquoFree-matrix-basedintegral inequality for stability analysis of systems with time-varying delayrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 60 no 10 pp 2768ndash2772 2015

[18] C-K Zhang Y He L Jiang M Wu and H-B Zeng ldquoStabilityanalysis of systems with time-varying delay via relaxed integralinequalitiesrdquo Systems amp Control Letters vol 92 pp 52ndash61 2016

[19] C-K Zhang Y He L Jiang W-J Lin and M Wu ldquoDelay-dependent stability analysis of neural networks with time-varying delay a generalized free-weighting-matrix approachrdquoApplied Mathematics and Computation vol 294 pp 102ndash1202017

[20] J Sun G P Liu J Chen and D Rees ldquoImproved delay-range-dependent stability criteria for linear systemswith time-varyingdelaysrdquo Automatica vol 46 no 2 pp 466ndash470 2010

[21] C Ge C Hua and X Guan ldquoNew delay-dependent stabilitycriteria for neural networks with time-varying delay usingdelay-decomposition approachrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1378ndash13832014

[22] C Hua SWu X Yang and X Guan ldquoStability analysis of time-delay systems via free-matrix-based double integral inequalityrdquoInternational Journal of Systems Science vol 48 no 2 pp 257ndash263 2017

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Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

Combining (19) (20) (21) (22) (23) (25) (26) (27) (28)(29) (30) (32) and (34) one can easily yield

(119890119905) le 120585119879 (119905) Γ (119889 (119905)) 120585 (119905) (35)

If Γ(119889(119905)) lt 0 then (119890119905) lt 0 Thus if Γ(0) lt 0 and Γ(ℎ) lt0 are satisfied then (119890119905) le minus120598119890(119905)2 for a sufficient smallscalar 120598 gt 0 which implies the filtering error system (5) with120596(119905) = 0 is asymptotically stable

Considering the definition of 1198672 performance we define

119869 (119905) = 119881 (119890119905) minus int1199050

120596119879 (119904) 120596 (119904) 119889119904 (36)

Then under the zero-initial conditions 119881(119890119905)|119905=0 = 0 and119881(119890119905) gt 0 for 119905 gt 0 the following inequality is given119869 (119905) = 119881 (119890119905) minus 119881 (119890119905)1003816100381610038161003816119905=0 minus int119905

0120596119879 (119904) 120596 (119904) 119889119904

= int1199050

[ (119890 (119904)) minus 120596119879 (119904) 120596 (119904)] 119889119904 lt 0(37)

That is

119881 (119890119905) lt int1199050

120596119879 (119904) 120596 (119904) 119889119904 (38)

In addition it follows that LMI (13) and Schur complementthat

[119867119879119867 119874lowast 119874] le 1205742 [1198751 1198752lowast 1198753] (39)

Then the following holds

119911119879119890 (119905) 119911119890 (119905) = 119890119879 (119905) 119867119879119867119890 (119905)

= [[[

119909 (119905)int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

[119867119879119867 119874lowast 119874] [[

[119909 (119905)

int119905119905minusℎ

119890 (119904) 119889119904]]]le 1205742 [[

[119909 (119905)

int119905119905minusℎ

119890 (119904) 119889119904]]]

119879

[1198751 1198752lowast 1198753][[[

119909 (119905)int119905119905minusℎ

119890 (s) 119889119904]]]= 12057421198811 (119890119905) le 1205742119881 (119890119905) le 1205742 int119905

0120596119879 (119904) 120596 (119904) 119889119904

le 1205742 intinfin0

120596119879 (119904) 120596 (119904) 119889119904

(40)

Taking the supremum over 119905 gt 0 one has 119911119890(119905)2infin le120574120596(119905)22 which means 119911119890(119905)infin le 120574120596(119905)2 Therefore when(10)ndash(14) hold the generalized1198672 filtering problem of NN (1)with time-varying delay satisfying (2) is solvedwith 120574minThiscompletes the proof

Remark 6 The generalized 1198672 design of a type of static NNwith a time-varying delay is solved in Theorem 5 Different

from [8ndash10 12ndash15] we solve the filtering problem based onan Arcak-type filter in which there are two gain matrices tobe determined Compared with the widely used Luenberger-type filter it contains an additional gainmatrix1198701 included inthe activation function So it can be expected to lead a bettergeneralized1198672 performance And if we set1198701 = 0 the Arcak-type filter is then reduced to the Luenberger-type filter

Remark 7 The triple integral term is introduced into theLKF in this paper When bounding its derivative the deriveddouble integral term minus int119905

119905minusℎint119905120579

119890119879(119904)119885 119890(119904)119889119904 119889120579 is divided intothree parts two double integral terms and a single integralterm By using the double Jensen inequality to estimatethe double integral terms and using the free-matrix-basedintegral inequality to estimate the single integral term thederivation of the triple term is accomplishedwithout ignoringany terms It should be pointed out that this technique ismore effective than the ones used in [10] in which the deriveddouble integral term minus int119905

119905minusℎint119905120579

119890119879(119904)119885 119890(119904)119889119904 119889120579 is estimated inits entirety

Remark 8 To reduce the conservatism of the derived con-ditions Wirtinger integral inequality was used to estimatesingle integral terms in [16] while in the presented studythe free-matrix-based integral inequality is employed Theuse of the free-matrix-based integral inequality brings severaladvantages for Theorem 5 On the one hand by introducingsome free matrices it gives more degrees of freedom tothe derived condition On the other hand it has beenproved in [17] that the free-matrix-based integral inequalitycontains Wirtinger integral inequality as a special case Thatmeans it can provide a tighter bound thanWirtinger integralinequality And if the same LKF is chosen our designed filterwill be better than the one designed in [16]

Remark 9 There is some room for further improvement ofour proposed method

(i) Notice that when the free-matrix-based integralinequality is employed to improve the proposed con-dition the number of decision variables is increasedwhich further increases the computation complexityof the condition It has been pointed out in [3]that the calculation complexity is also an importantconsideration during the application of the proposedLMI-based condition to physical systems Thus theresults considering both the conservatism and thecomputation complexity need more investigation

(ii) Recently some new approaches have been developedfor the stability analysis of delayed neural networkswhich can be applied into the generalized 1198672 filter-ing problem of neural networks such as the delay-decomposition approach [21] and the free-matrix-based double integral inequality [22]

Notice that in many applications the derivative bound ofthe time delay is unknown By setting 1198761 = 0 we can easilyderive the delay-dependent but delay derivative-independent

Mathematical Problems in Engineering 7

condition of the generalized 1198672 filtering problem of neuralnetwork (1) based onTheorem 5

Theorem 10 For given scalars ℎ the 1198672 filtering problem ofNN (1) with time-varying delay satisfying 0 le 119889(119905) le ℎ issolved with 120574min if there exist positive definite matrices 119875 =[ 1198751 1198752lowast 1198753 ] isin R2119899 1198762 119877 119885 isin R119899 positive definite diagonalmatrices Λ = diag1205821 1205822 120582119899 isin R119899 symmetric matrices1198831 1198833 1198841 1198843 1198851 1198853 isin R3119899 and any matrices 1198832 1198842 1198852 isinR3119899 1198721 1198722 1198731 1198732 1198661 1198662 isin R3119899times119899 119879 isin R119899 1198811 1198812 isinR119899times119901 satisfying LMIs (10)ndash(14)

4 Numerical Examples

In this section we provide a well-used numerical example todemonstrate the effectiveness of the proposed approach

Example 1 Consider the delayed static NN (1) with thefollowing parameters

119860 = [[[

096 0 00 122 00 0 078

]]]

119882 = [[[

minus032 075 minus142121 041 minus050042 082 minus106

]]]

1198611 = [[[

020305

]]]

119867 = [[[

08 0 051 0 10 minus1 1

]]]

119862 = [1 05 00 minus05 06]

119863 = [0 minus12 020 0 05]

1198612 = [ 04minus03]

(41)

This example has been studied in [10 16] and we takethese literatures for comparison study To verify the advan-tages of our proposed approach we calculate the optimal 1198672performance bounds index 120574min for various ℎ 120583 and 119871 suchthat filtering error system (5) is globally asymptotically stablewith generalized 1198672 performance

Firstly let 119871 = diag1198971 1198972 119897119899 = 119868 for given ℎ and 120583the optimal generalized 1198672 performance index 120574min derivedby Theorem 5 and the ones reported in [10 16] are givenin Table 1 where (120572 120573) represents ℎ = 120572 120583 = 120573 and

minus15

minus1

minus05

0

05

1

15

e(t)

5 10 150Time (s)

e1(t)

e2(t)

e3(t)

Figure 1 State response of the error 119890(119905)

ldquoinfeasiblerdquo means that no feasible solution can be found inthe corresponding criteria

It is clearly shown from Table 1 that compared with [1016] much better generalized 1198672 performance is achieved byTheorem 5 proposed in this paper It should be noticed thatin this paper we choose the same filter as [10 16] thus thebetter 1198672 performance effectively illustrates the superiorityof our proposed approaches which is mainly based on thefree-matrix-based integral inequality

Then let 119891(119909) = 157tanh(119909) 119889(119905) = 03 sin(15119905) + 12and 119908(119905) = sin(119905)119890minus5119905 for ℎ = 15 120583 = 03 by solving theLMIs in Theorem 5 the gain matrices of our designed filter(4) are obtained as

1198701 = [[[

10591 1299001519 0268724379 33466

]]]

1198702 = [[[

00944 minus0600503315 minus0510021756 12563

]]]

(42)

with the optimal generalized 1198672 performance index 120574 =00505 Figure 1 shows the state responses of error system (5)under initial condition 119890(0) = [minus1 1 minus05]119879 The resultingresponses obviously demonstrate the asymptotic stability ofthe simulated error system

In addition the optimal generalized 1198672 performanceindex 120574min for ℎ = 05 120583 = 07 and different 119871 is listed inTable 2 from which one can see thatTheorem 5 in this paperoutperforms the ones in [10 16]

Moreover the optimal generalized1198672 performance index120574min derived by Theorem 10 for ℎ = 08 unknown 120583 anddifferent 119871 is listed in Table 3

8 Mathematical Problems in Engineering

Table 1 Optimal generalized 1198672 performance index 120574min for 119871 = 119868 and different (ℎ 120583)(ℎ 120583) (1 04) (15 03) (3 08) (5 12) (6 15)Theorem 1 [10] 07981 Infeasible Infeasible Infeasible InfeasibleTheorem 2 [16] 134 times 10minus4 12763 Infeasible Infeasible InfeasibleTheorem 5 133 times 10minus4 00505 06639 09137 19523

Table 2 Optimal generalized 1198672 performance index 120574min for ℎ = 05 120583 = 07 and different 119871119871 16119868 18119868 25119868 34119868Theorem 1 [10] 33728 Infeasible Infeasible InfeasibleTheorem 2 [16] 05983 15563 Infeasible InfeasibleTheorem 5 02765 04015 06896 47676

Table 3 Optimal generalized 1198672 performance index 120574min for ℎ =05 unknown 120583 and different 119871119871 16119868 18119868 25119868 291119868Theorem 10 05156 05868 10891 51122

5 Conclusion

In this paper the generalized 1198672 filtering problem of staticNNs with a time-varying delay has been investigated Firstan Arcak-type filter has been constructed Then based onan augmented LKF the free-matrix-based integral inequalitythe free-weighting matrix approach and Jensen inequalitya suitable delay-dependent condition expressed in terms ofLMIs has been derived such that the filtering error systemof the considered static NNs is globally asymptotically stablewith guaranteed1198672 performance indexMoreover by solvingsome coupled LMIs the optimal performance index and thegain matrices of the designed 1198672 filter have been obtainedThe effectiveness and advantages of the proposed methodhave been demonstrated by an illustrative example finally

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding this paper

References

[1] L O Chua and L Yang ldquoCellular Neural Networks Applica-tionsrdquo IEEE Transactions on Circuits and Systems II ExpressBriefs vol 35 no 10 pp 1273ndash1290 1988

[2] X-M Zhang Q-L Han and X Yu ldquoSurvey on RecentAdvances inNetworkedControl Systemsrdquo IEEETransactions onIndustrial Informatics vol 12 no 5 pp 1740ndash1752 2016

[3] C-K Zhang Y He L Jiang and M Wu ldquoStability analysisfor delayed neural networks considering both conservativenessand complexityrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 7 pp 1486ndash1501 2016

[4] H-B Zeng Y He P Shi M Wu and S-P Xiao ldquoDissipativityanalysis of neural networks with time-varying delaysrdquo Neuro-computing vol 168 pp 741ndash746 2015

[5] C-K Zhang Y He L Jiang Q H Wu and M Wu ldquoDelay-dependent stability criteria for generalized neural networks

with two delay componentsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1263ndash12762014

[6] Y He Q GWang andC Lin ldquoAn improvedH8 filter design forsystems with time-varying interval delayrdquo IEEE Trans CircuitsSyst II vol 53 p 1235 2006

[7] YHeG P LiuD Rees andMWu ldquoImprovedHinfinfiltering forsystems with a time-varying delayrdquo Circuits Systems and SignalProcessing vol 29 no 3 pp 377ndash389 2010

[8] H Huang and G Feng ldquoDelay-dependent Hinfin and generalizedH2 filtering for delayed neural networksrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 4 pp 846ndash857 2009

[9] C K Ahn and M K Song ldquoL2-Linfin filtering for time-delayedswitched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[10] D Hu H Huang and T Huang ldquoDesign of Arcak-type gen-eralized H2 filter for delayed static neural networksrdquo CircuitsSystems and Signal Processing vol 33 no 11 pp 3635ndash36482014

[11] M Hua H Tan and J Chen ldquoDelay-dependent Hinfin andgeneralized H2 filtering for stochastic neural networks withtime-varying delay and noise disturbancerdquo Neural Computingand Applications vol 25 no 3-4 pp 613ndash624 2014

[12] Y J Li and Y N Liang ldquoDelay-dependent Hinfin filtering forneural networks with time delayrdquo Applied Mechanics and Mate-rials vol 511-512 pp 875ndash879 2014

[13] YHeG P LiuD Rees andMWu ldquoStability analysis for neuralnetworks with time-varying interval delayrdquo IEEE Transactionson Neural Networks and Learning Systems vol 18 no 6 pp1850ndash1854 2007

[14] Y Li J Li and M Hua ldquoNew results of H8 filtering for neuralnetwork with time-varying delayrdquo Int J Innov Comput InfControl vol 10 pp 2309ndash2323 2014

[15] G Liu S Zhou X Luo and K Zhang ldquoNew filter design forstatic neural networks with mixed time-varying delaysrdquo LectureNotes in Computer Science (including subseries Lecture Notesin Artificial Intelligence and Lecture Notes in Bioinformatics)Preface vol 9772 pp 117ndash129 2016

[16] Y Shu X-G Liu Y Liu and J H Park ldquoImproved Results onGuaranteed Generalized H2 Performance State Estimation forDelayed Static Neural Networksrdquo Circuits Systems and SignalProcessing vol 36 no 8 pp 3114ndash3142 2017

Mathematical Problems in Engineering 9

[17] H-B Zeng Y He M Wu and J She ldquoFree-matrix-basedintegral inequality for stability analysis of systems with time-varying delayrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 60 no 10 pp 2768ndash2772 2015

[18] C-K Zhang Y He L Jiang M Wu and H-B Zeng ldquoStabilityanalysis of systems with time-varying delay via relaxed integralinequalitiesrdquo Systems amp Control Letters vol 92 pp 52ndash61 2016

[19] C-K Zhang Y He L Jiang W-J Lin and M Wu ldquoDelay-dependent stability analysis of neural networks with time-varying delay a generalized free-weighting-matrix approachrdquoApplied Mathematics and Computation vol 294 pp 102ndash1202017

[20] J Sun G P Liu J Chen and D Rees ldquoImproved delay-range-dependent stability criteria for linear systemswith time-varyingdelaysrdquo Automatica vol 46 no 2 pp 466ndash470 2010

[21] C Ge C Hua and X Guan ldquoNew delay-dependent stabilitycriteria for neural networks with time-varying delay usingdelay-decomposition approachrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1378ndash13832014

[22] C Hua SWu X Yang and X Guan ldquoStability analysis of time-delay systems via free-matrix-based double integral inequalityrdquoInternational Journal of Systems Science vol 48 no 2 pp 257ndash263 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

condition of the generalized 1198672 filtering problem of neuralnetwork (1) based onTheorem 5

Theorem 10 For given scalars ℎ the 1198672 filtering problem ofNN (1) with time-varying delay satisfying 0 le 119889(119905) le ℎ issolved with 120574min if there exist positive definite matrices 119875 =[ 1198751 1198752lowast 1198753 ] isin R2119899 1198762 119877 119885 isin R119899 positive definite diagonalmatrices Λ = diag1205821 1205822 120582119899 isin R119899 symmetric matrices1198831 1198833 1198841 1198843 1198851 1198853 isin R3119899 and any matrices 1198832 1198842 1198852 isinR3119899 1198721 1198722 1198731 1198732 1198661 1198662 isin R3119899times119899 119879 isin R119899 1198811 1198812 isinR119899times119901 satisfying LMIs (10)ndash(14)

4 Numerical Examples

In this section we provide a well-used numerical example todemonstrate the effectiveness of the proposed approach

Example 1 Consider the delayed static NN (1) with thefollowing parameters

119860 = [[[

096 0 00 122 00 0 078

]]]

119882 = [[[

minus032 075 minus142121 041 minus050042 082 minus106

]]]

1198611 = [[[

020305

]]]

119867 = [[[

08 0 051 0 10 minus1 1

]]]

119862 = [1 05 00 minus05 06]

119863 = [0 minus12 020 0 05]

1198612 = [ 04minus03]

(41)

This example has been studied in [10 16] and we takethese literatures for comparison study To verify the advan-tages of our proposed approach we calculate the optimal 1198672performance bounds index 120574min for various ℎ 120583 and 119871 suchthat filtering error system (5) is globally asymptotically stablewith generalized 1198672 performance

Firstly let 119871 = diag1198971 1198972 119897119899 = 119868 for given ℎ and 120583the optimal generalized 1198672 performance index 120574min derivedby Theorem 5 and the ones reported in [10 16] are givenin Table 1 where (120572 120573) represents ℎ = 120572 120583 = 120573 and

minus15

minus1

minus05

0

05

1

15

e(t)

5 10 150Time (s)

e1(t)

e2(t)

e3(t)

Figure 1 State response of the error 119890(119905)

ldquoinfeasiblerdquo means that no feasible solution can be found inthe corresponding criteria

It is clearly shown from Table 1 that compared with [1016] much better generalized 1198672 performance is achieved byTheorem 5 proposed in this paper It should be noticed thatin this paper we choose the same filter as [10 16] thus thebetter 1198672 performance effectively illustrates the superiorityof our proposed approaches which is mainly based on thefree-matrix-based integral inequality

Then let 119891(119909) = 157tanh(119909) 119889(119905) = 03 sin(15119905) + 12and 119908(119905) = sin(119905)119890minus5119905 for ℎ = 15 120583 = 03 by solving theLMIs in Theorem 5 the gain matrices of our designed filter(4) are obtained as

1198701 = [[[

10591 1299001519 0268724379 33466

]]]

1198702 = [[[

00944 minus0600503315 minus0510021756 12563

]]]

(42)

with the optimal generalized 1198672 performance index 120574 =00505 Figure 1 shows the state responses of error system (5)under initial condition 119890(0) = [minus1 1 minus05]119879 The resultingresponses obviously demonstrate the asymptotic stability ofthe simulated error system

In addition the optimal generalized 1198672 performanceindex 120574min for ℎ = 05 120583 = 07 and different 119871 is listed inTable 2 from which one can see thatTheorem 5 in this paperoutperforms the ones in [10 16]

Moreover the optimal generalized1198672 performance index120574min derived by Theorem 10 for ℎ = 08 unknown 120583 anddifferent 119871 is listed in Table 3

8 Mathematical Problems in Engineering

Table 1 Optimal generalized 1198672 performance index 120574min for 119871 = 119868 and different (ℎ 120583)(ℎ 120583) (1 04) (15 03) (3 08) (5 12) (6 15)Theorem 1 [10] 07981 Infeasible Infeasible Infeasible InfeasibleTheorem 2 [16] 134 times 10minus4 12763 Infeasible Infeasible InfeasibleTheorem 5 133 times 10minus4 00505 06639 09137 19523

Table 2 Optimal generalized 1198672 performance index 120574min for ℎ = 05 120583 = 07 and different 119871119871 16119868 18119868 25119868 34119868Theorem 1 [10] 33728 Infeasible Infeasible InfeasibleTheorem 2 [16] 05983 15563 Infeasible InfeasibleTheorem 5 02765 04015 06896 47676

Table 3 Optimal generalized 1198672 performance index 120574min for ℎ =05 unknown 120583 and different 119871119871 16119868 18119868 25119868 291119868Theorem 10 05156 05868 10891 51122

5 Conclusion

In this paper the generalized 1198672 filtering problem of staticNNs with a time-varying delay has been investigated Firstan Arcak-type filter has been constructed Then based onan augmented LKF the free-matrix-based integral inequalitythe free-weighting matrix approach and Jensen inequalitya suitable delay-dependent condition expressed in terms ofLMIs has been derived such that the filtering error systemof the considered static NNs is globally asymptotically stablewith guaranteed1198672 performance indexMoreover by solvingsome coupled LMIs the optimal performance index and thegain matrices of the designed 1198672 filter have been obtainedThe effectiveness and advantages of the proposed methodhave been demonstrated by an illustrative example finally

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding this paper

References

[1] L O Chua and L Yang ldquoCellular Neural Networks Applica-tionsrdquo IEEE Transactions on Circuits and Systems II ExpressBriefs vol 35 no 10 pp 1273ndash1290 1988

[2] X-M Zhang Q-L Han and X Yu ldquoSurvey on RecentAdvances inNetworkedControl Systemsrdquo IEEETransactions onIndustrial Informatics vol 12 no 5 pp 1740ndash1752 2016

[3] C-K Zhang Y He L Jiang and M Wu ldquoStability analysisfor delayed neural networks considering both conservativenessand complexityrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 7 pp 1486ndash1501 2016

[4] H-B Zeng Y He P Shi M Wu and S-P Xiao ldquoDissipativityanalysis of neural networks with time-varying delaysrdquo Neuro-computing vol 168 pp 741ndash746 2015

[5] C-K Zhang Y He L Jiang Q H Wu and M Wu ldquoDelay-dependent stability criteria for generalized neural networks

with two delay componentsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1263ndash12762014

[6] Y He Q GWang andC Lin ldquoAn improvedH8 filter design forsystems with time-varying interval delayrdquo IEEE Trans CircuitsSyst II vol 53 p 1235 2006

[7] YHeG P LiuD Rees andMWu ldquoImprovedHinfinfiltering forsystems with a time-varying delayrdquo Circuits Systems and SignalProcessing vol 29 no 3 pp 377ndash389 2010

[8] H Huang and G Feng ldquoDelay-dependent Hinfin and generalizedH2 filtering for delayed neural networksrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 4 pp 846ndash857 2009

[9] C K Ahn and M K Song ldquoL2-Linfin filtering for time-delayedswitched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[10] D Hu H Huang and T Huang ldquoDesign of Arcak-type gen-eralized H2 filter for delayed static neural networksrdquo CircuitsSystems and Signal Processing vol 33 no 11 pp 3635ndash36482014

[11] M Hua H Tan and J Chen ldquoDelay-dependent Hinfin andgeneralized H2 filtering for stochastic neural networks withtime-varying delay and noise disturbancerdquo Neural Computingand Applications vol 25 no 3-4 pp 613ndash624 2014

[12] Y J Li and Y N Liang ldquoDelay-dependent Hinfin filtering forneural networks with time delayrdquo Applied Mechanics and Mate-rials vol 511-512 pp 875ndash879 2014

[13] YHeG P LiuD Rees andMWu ldquoStability analysis for neuralnetworks with time-varying interval delayrdquo IEEE Transactionson Neural Networks and Learning Systems vol 18 no 6 pp1850ndash1854 2007

[14] Y Li J Li and M Hua ldquoNew results of H8 filtering for neuralnetwork with time-varying delayrdquo Int J Innov Comput InfControl vol 10 pp 2309ndash2323 2014

[15] G Liu S Zhou X Luo and K Zhang ldquoNew filter design forstatic neural networks with mixed time-varying delaysrdquo LectureNotes in Computer Science (including subseries Lecture Notesin Artificial Intelligence and Lecture Notes in Bioinformatics)Preface vol 9772 pp 117ndash129 2016

[16] Y Shu X-G Liu Y Liu and J H Park ldquoImproved Results onGuaranteed Generalized H2 Performance State Estimation forDelayed Static Neural Networksrdquo Circuits Systems and SignalProcessing vol 36 no 8 pp 3114ndash3142 2017

Mathematical Problems in Engineering 9

[17] H-B Zeng Y He M Wu and J She ldquoFree-matrix-basedintegral inequality for stability analysis of systems with time-varying delayrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 60 no 10 pp 2768ndash2772 2015

[18] C-K Zhang Y He L Jiang M Wu and H-B Zeng ldquoStabilityanalysis of systems with time-varying delay via relaxed integralinequalitiesrdquo Systems amp Control Letters vol 92 pp 52ndash61 2016

[19] C-K Zhang Y He L Jiang W-J Lin and M Wu ldquoDelay-dependent stability analysis of neural networks with time-varying delay a generalized free-weighting-matrix approachrdquoApplied Mathematics and Computation vol 294 pp 102ndash1202017

[20] J Sun G P Liu J Chen and D Rees ldquoImproved delay-range-dependent stability criteria for linear systemswith time-varyingdelaysrdquo Automatica vol 46 no 2 pp 466ndash470 2010

[21] C Ge C Hua and X Guan ldquoNew delay-dependent stabilitycriteria for neural networks with time-varying delay usingdelay-decomposition approachrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1378ndash13832014

[22] C Hua SWu X Yang and X Guan ldquoStability analysis of time-delay systems via free-matrix-based double integral inequalityrdquoInternational Journal of Systems Science vol 48 no 2 pp 257ndash263 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

Table 1 Optimal generalized 1198672 performance index 120574min for 119871 = 119868 and different (ℎ 120583)(ℎ 120583) (1 04) (15 03) (3 08) (5 12) (6 15)Theorem 1 [10] 07981 Infeasible Infeasible Infeasible InfeasibleTheorem 2 [16] 134 times 10minus4 12763 Infeasible Infeasible InfeasibleTheorem 5 133 times 10minus4 00505 06639 09137 19523

Table 2 Optimal generalized 1198672 performance index 120574min for ℎ = 05 120583 = 07 and different 119871119871 16119868 18119868 25119868 34119868Theorem 1 [10] 33728 Infeasible Infeasible InfeasibleTheorem 2 [16] 05983 15563 Infeasible InfeasibleTheorem 5 02765 04015 06896 47676

Table 3 Optimal generalized 1198672 performance index 120574min for ℎ =05 unknown 120583 and different 119871119871 16119868 18119868 25119868 291119868Theorem 10 05156 05868 10891 51122

5 Conclusion

In this paper the generalized 1198672 filtering problem of staticNNs with a time-varying delay has been investigated Firstan Arcak-type filter has been constructed Then based onan augmented LKF the free-matrix-based integral inequalitythe free-weighting matrix approach and Jensen inequalitya suitable delay-dependent condition expressed in terms ofLMIs has been derived such that the filtering error systemof the considered static NNs is globally asymptotically stablewith guaranteed1198672 performance indexMoreover by solvingsome coupled LMIs the optimal performance index and thegain matrices of the designed 1198672 filter have been obtainedThe effectiveness and advantages of the proposed methodhave been demonstrated by an illustrative example finally

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding this paper

References

[1] L O Chua and L Yang ldquoCellular Neural Networks Applica-tionsrdquo IEEE Transactions on Circuits and Systems II ExpressBriefs vol 35 no 10 pp 1273ndash1290 1988

[2] X-M Zhang Q-L Han and X Yu ldquoSurvey on RecentAdvances inNetworkedControl Systemsrdquo IEEETransactions onIndustrial Informatics vol 12 no 5 pp 1740ndash1752 2016

[3] C-K Zhang Y He L Jiang and M Wu ldquoStability analysisfor delayed neural networks considering both conservativenessand complexityrdquo IEEE Transactions on Neural Networks andLearning Systems vol 27 no 7 pp 1486ndash1501 2016

[4] H-B Zeng Y He P Shi M Wu and S-P Xiao ldquoDissipativityanalysis of neural networks with time-varying delaysrdquo Neuro-computing vol 168 pp 741ndash746 2015

[5] C-K Zhang Y He L Jiang Q H Wu and M Wu ldquoDelay-dependent stability criteria for generalized neural networks

with two delay componentsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1263ndash12762014

[6] Y He Q GWang andC Lin ldquoAn improvedH8 filter design forsystems with time-varying interval delayrdquo IEEE Trans CircuitsSyst II vol 53 p 1235 2006

[7] YHeG P LiuD Rees andMWu ldquoImprovedHinfinfiltering forsystems with a time-varying delayrdquo Circuits Systems and SignalProcessing vol 29 no 3 pp 377ndash389 2010

[8] H Huang and G Feng ldquoDelay-dependent Hinfin and generalizedH2 filtering for delayed neural networksrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 4 pp 846ndash857 2009

[9] C K Ahn and M K Song ldquoL2-Linfin filtering for time-delayedswitched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[10] D Hu H Huang and T Huang ldquoDesign of Arcak-type gen-eralized H2 filter for delayed static neural networksrdquo CircuitsSystems and Signal Processing vol 33 no 11 pp 3635ndash36482014

[11] M Hua H Tan and J Chen ldquoDelay-dependent Hinfin andgeneralized H2 filtering for stochastic neural networks withtime-varying delay and noise disturbancerdquo Neural Computingand Applications vol 25 no 3-4 pp 613ndash624 2014

[12] Y J Li and Y N Liang ldquoDelay-dependent Hinfin filtering forneural networks with time delayrdquo Applied Mechanics and Mate-rials vol 511-512 pp 875ndash879 2014

[13] YHeG P LiuD Rees andMWu ldquoStability analysis for neuralnetworks with time-varying interval delayrdquo IEEE Transactionson Neural Networks and Learning Systems vol 18 no 6 pp1850ndash1854 2007

[14] Y Li J Li and M Hua ldquoNew results of H8 filtering for neuralnetwork with time-varying delayrdquo Int J Innov Comput InfControl vol 10 pp 2309ndash2323 2014

[15] G Liu S Zhou X Luo and K Zhang ldquoNew filter design forstatic neural networks with mixed time-varying delaysrdquo LectureNotes in Computer Science (including subseries Lecture Notesin Artificial Intelligence and Lecture Notes in Bioinformatics)Preface vol 9772 pp 117ndash129 2016

[16] Y Shu X-G Liu Y Liu and J H Park ldquoImproved Results onGuaranteed Generalized H2 Performance State Estimation forDelayed Static Neural Networksrdquo Circuits Systems and SignalProcessing vol 36 no 8 pp 3114ndash3142 2017

Mathematical Problems in Engineering 9

[17] H-B Zeng Y He M Wu and J She ldquoFree-matrix-basedintegral inequality for stability analysis of systems with time-varying delayrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 60 no 10 pp 2768ndash2772 2015

[18] C-K Zhang Y He L Jiang M Wu and H-B Zeng ldquoStabilityanalysis of systems with time-varying delay via relaxed integralinequalitiesrdquo Systems amp Control Letters vol 92 pp 52ndash61 2016

[19] C-K Zhang Y He L Jiang W-J Lin and M Wu ldquoDelay-dependent stability analysis of neural networks with time-varying delay a generalized free-weighting-matrix approachrdquoApplied Mathematics and Computation vol 294 pp 102ndash1202017

[20] J Sun G P Liu J Chen and D Rees ldquoImproved delay-range-dependent stability criteria for linear systemswith time-varyingdelaysrdquo Automatica vol 46 no 2 pp 466ndash470 2010

[21] C Ge C Hua and X Guan ldquoNew delay-dependent stabilitycriteria for neural networks with time-varying delay usingdelay-decomposition approachrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1378ndash13832014

[22] C Hua SWu X Yang and X Guan ldquoStability analysis of time-delay systems via free-matrix-based double integral inequalityrdquoInternational Journal of Systems Science vol 48 no 2 pp 257ndash263 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

[17] H-B Zeng Y He M Wu and J She ldquoFree-matrix-basedintegral inequality for stability analysis of systems with time-varying delayrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 60 no 10 pp 2768ndash2772 2015

[18] C-K Zhang Y He L Jiang M Wu and H-B Zeng ldquoStabilityanalysis of systems with time-varying delay via relaxed integralinequalitiesrdquo Systems amp Control Letters vol 92 pp 52ndash61 2016

[19] C-K Zhang Y He L Jiang W-J Lin and M Wu ldquoDelay-dependent stability analysis of neural networks with time-varying delay a generalized free-weighting-matrix approachrdquoApplied Mathematics and Computation vol 294 pp 102ndash1202017

[20] J Sun G P Liu J Chen and D Rees ldquoImproved delay-range-dependent stability criteria for linear systemswith time-varyingdelaysrdquo Automatica vol 46 no 2 pp 466ndash470 2010

[21] C Ge C Hua and X Guan ldquoNew delay-dependent stabilitycriteria for neural networks with time-varying delay usingdelay-decomposition approachrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 7 pp 1378ndash13832014

[22] C Hua SWu X Yang and X Guan ldquoStability analysis of time-delay systems via free-matrix-based double integral inequalityrdquoInternational Journal of Systems Science vol 48 no 2 pp 257ndash263 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom