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Gaoetal.
AdvancesinDierenceEquations(2019)2019:380https://doi.
org/10.
1186/s13662-019-2298-7RESEARCHOpenAccessVariance-constrainedresilientH∞stateestimationfortime-varyingneuralnetworkswithrandomlyvaryingnonlinearitiesandmissingmeasurementsYanGao1,2,JunHu1,2,3*,DongyanChen1,2andJunhuaDu4*Correspondence:hujun2013@gmail.
com1SchoolofScience,HarbinUniversityofScienceandTechnology,Harbin,ChinaFulllistofauthorinformationisavailableattheendofthearticleAbstractThispaperaddressestheresilientH∞stateestimationproblemundervarianceconstraintfordiscreteuncertaintime-varyingrecurrentneuralnetworkswithrandomlyvaryingnonlinearitiesandmissingmeasurements.
ThephenomenaofmissingmeasurementsandrandomlyvaryingnonlinearitiesaredescribedbyintroducingsomeBernoullidistributedrandomvariables,inwhichtheoccurrenceprobabilitiesareknownapriori.
Besides,themultiplicativenoiseisemployedtocharacterizetheestimatorgainperturbation.
Ourmainpurposeistodesignatime-varyingstateestimatorsuchthat,forallmissingmeasurements,randomlyvaryingnonlinearitiesandestimatorgainperturbation,boththeestimationerrorvarianceconstraintandtheprescribedH∞performancerequirementaremetsimultaneouslybyprovidingsomesucientcriteria.
Finally,thefeasibilityoftheproposedvariance-constrainedresilientH∞stateestimationmethodisveriedbysomesimulations.
Keywords:Time-varyingneuralnetworks;Resilientstateestimation;Randomlyvaryingnonlinearities;Missingmeasurements;H∞performance;Varianceconstraint1IntroductionInthepasttwodecades,thepopularizationoftheInternethasgreatlychangedourwayoflifethroughtherapidcommunicationways[1–3].
Accordingly,manyscholarshavewit-nessedsuccessfulapplicationsofrecurrentneuralnetworks(RNNs)inwideeldsinclud-ingpatternrecognition,imageprocessing,associatememoryandoptimizationdomains[4–6].
Nevertheless,itshouldbenotedthatthenonlinearitiesarecommonlyinherentcharacteristicsbetweenneurons,whichindeedaecttheunderstandingandanalysisoftheneuralnetworks(NNs).
Thus,someecientmethodshavebeengiventoanalyzedif-ferentNNs.
Forexample,aneectivenite-timesynchronizationcriterionhasbeenpro-posedin[7]forcoupledstochasticNNs,whereboththeMarkovianswitchingparametersandsaturationhavebeenaddressed.
Moreover,someusefulstateestimationalgorithmshavebeengivenin[8]fordelayedNNstoguaranteetheH∞aswellaspassivityandin[9]forbidirectionalassociativeNNssubjecttomixedtime-delays.
DuringtheanalysisandimplementationofthemethodsrelatedtoRNNs,itshouldbenoticedthattheneuronTheAuthor(s)2019.
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Gaoetal.
AdvancesinDierenceEquations(2019)2019:380Page2of23statesmaynotalwaysavailableinreality,sothereisaneedtoestimatethembyutilizingeectiveestimationmethods[10–12].
Untilnow,manyresultshavebeenpublishedwithrespecttothestateestimationproblemofdierenttypesofdynamicalnetworks[13–16].
Nevertheless,itisworthmentioningthatmuchpublishedliteratureisonlyapplicabletotime-invariantcase,whichmightleadtocertainapplicationlimitations.
Hence,moreandmoreresearchershavepaidattentionontheanalysisofthestateestimationproblemfortime-varyingsystemandproposedavarietyofmethodstoanalyzethebehaviorsofdy-namicssystems,seee.
g.
[17–20]andthereferencestherein.
Tobespecic,anevent-basedjointinputandstateestimationstrategyhavebeenpresentedin[19]toensurethattheco-varianceoftheestimationerrorhasanupperboundatanytimefortheaddressedlineardiscretetime-varyingsystems.
Accordingly,thereexistthedemandsonthedevelopmentofecientanalysismethodsfortime-varyingRNNs.
Duringthenetworkcommunicationsortransmissions,theperfectmeasurementscannotbealwaysavailable,thusincreasingresearcheorthasbeenmadeonthestateestima-tionproblemsagainstthemissingmeasurements[20–22].
Themissingmeasurementsareinevitableprimarilyduetothesignalinterferenceduringthetransmissionincludinglim-itedcommunicationchannels,noiseinterferencesandsoforth.
Forexample,thestudyofmissingmeasurementshasbeenconductedintheareasofprincipalcomponentanalysisandpartialleastsquaresmodels[23].
Actually,itiswellrecognizedthattheexistenceofthemissingmeasurementswouldleadtopoorsystemperformancefortheaddressedNNs[24,25].
Accordingly,alargenumberofresultshavebeengiventodealwiththestatees-timationproblemfordynamicsnetworkssubjecttomissingmeasurements[26–29].
Forexample,in[27,28],theH∞stateestimationdesignstrategieshavebeenproposedfordiscretedelayedneuralnetworks,wheretheimpactsfromthemultiplemissingmeasure-mentshavebeendiscussed.
Nevertheless,fewresultshavebeenreportedtohandletheproblemofstateestimationfortime-varyingdynamicsnetworkswithmissingmeasure-ments[20,30].
Recently,anewoptimalstateestimationalgorithmhasbeendevelopedin[20]fortime-varyingcomplexnetworks,wheretheimpactsfromboththestochasticcouplingandthemissingmeasurementswithintheframeworkofuncertainoccurrenceprobabilitieshavebeenaddressed.
However,itisworthwhiletonoticethatfewscholarshavestudiedthestateestimationproblemoftime-varyingneuralnetworkswithmissingmeasurements,whichconstitutesoneofresearchmotivations.
Recently,thephenomenaofrandomlyvaryingnonlinearitieshavebeenmodeledanddiscussedinvariouselds[31,32].
Theexistenceofrandomlyvaryingnonlinearitieshasbroughtmoreinuencesinphysics,engineering,informationscienceandotherelds,andhencethedynamicssystemanalysisproblemswithrandomlyvaryingnonlinearitieshavealreadyreceivedincreasinginterest[33–35].
Forexample,thestateestimationalgorithmindistributedwayhasbeenestablishedin[33]foraclassofdiscretesystemsoversensornetworkswithincompletemeasurementsandrandomlyvaryingnonlinearities,andtheprobability-dependentH∞synchronizationconditionhasbeenpresentedin[35]fordy-namicsnetworksbydesigningnewcontrolmethod.
Duringtheimplementationprocess,thepresentedestimationmethodwouldbefragile/non-resilientduetovariousreasons,suchasthenumericalroundingerror,nite-wordlengtheectsandanalog-to-digitalcon-versionaccuracy[36,37].
Accordingly,thenon-fragile/resilientestimationschemeshavewideapplicationeldsasinwaterqualityprocessing[38]andsynchronousgenerators[39].
In[40,41],thenon-fragileH∞stateestimationalgorithmshavebeengivenfordynamicalGaoetal.
AdvancesinDierenceEquations(2019)2019:380Page3of23systemswithMarkovswitchingeects,wherethepossibleestimatorparametervariationshavebeenmodeledanddiscussed.
TheresilientH∞lteringmethodshavebeengivenin[42]forswitcheddelayedneuralnetworkswithensurednite-timecriterionandin[43]fortime-varyingdynamicssystemsthroughsensornetworkswithcommunicationpro-tocols.
However,fewresilientH∞estimationmethodscanbeavailablefortime-varyingRNNs.
Ontheotherhand,fromtheengineering-orientedviewpointonthestateestima-tionproblems,itisofgreatpracticalsignicancetoensuretheestimationperformancehasacertainupperboundofrelatedtotheestimatederrorcovariance[44].
Unliketotheminimalestimationoferrorcovariance,thevariance-constrainedstateestimationstrat-egycouldprovidealooserevaluationbyintroducingaspecicupperboundconstraint,whichcanreecttheadmissibleaccuracyoftheproposedestimationmethods.
Veryre-cently,anewvariance-constrainedlteringalgorithmwithdistributedfeaturehasbeenestablishedin[45]fortime-varyingsensornetworkssubjecttodeceptionattacksandstateestimationscheme;see[46]fortime-varyingcomplexnetworkstoattenuatetheimpactsinducedbyrandomlyvaryingtopologies.
Assuch,wemaketherstattempttohandletheH∞resilientestimationproblemundervarianceconstraintfortime-varyingRNNswithrandomlyvaryingnonlinearitiesandmissingmeasurements.
Inspiredbytheabovediscussions,weaimtodesigntheresilientH∞stateestimationap-proachfortheaddresseddiscretetime-varyingRNNssubjecttomissingmeasurementsandrandomlyvaryingnonlinearitiesunderthevarianceconstraint.
Inparticular,theesti-matorgainperturbationisconsideredbyemployingtheGaussianwhitenoise.
Byresortingtothematrixtheoryandstochasticanalysistechnique,anewnonlineartime-varyingstateestimationmethodisproposed,whichcanensuretheerrorvarianceboundednessandprescribedH∞performancerequirementssimultaneously.
Fromanengineeringview-point,theproposedrecursivestateestimationschemehasatime-varyingfeatureappli-cableforhandlingtheestimationproblemsofneuralnetworks,whichissuitablefortheonlineestimationapplications.
Moreover,somesucientconditionscharacterizedbythematrixinequalitiesaregiventoensuretwomixedperformanceindies,whichcanbetterachievesatisfactorydisturbanceattenuationlevelandtheestimationcovarianceperfor-mance,thusperformingwideapplicationdomains.
Themajorfeaturesofthepapercanbesummarizedasfollows:(1)theH∞stateestimationproblemundervarianceconstraintis,forthersttime,investigatedforaclassofdiscretetime-varyingRNNssubjecttomiss-ingmeasurementsandrandomlyvaryingnonlinearities;(2)anewprobability-dependenttime-varyingstateestimationalgorithmisproposed,whichcanbeimplementedintermsofthesolutionstocertainmatrixinequalities;(3)theimpactscausedbythemissingmea-surementsandrandomlyvaryingnonlinearitiesontotwoestimationperformanceindicesarediscussedandexaminedsimultaneously;and(4)theproposedestimationalgorithmhastime-varyingcharacteristicapplicableforonlineapplications,whichperformsnewadvantagescomparedwiththeexistingestimationresultsfortime-invariantneuralnet-works.
Notations.
Thesymbolsusedthroughoutthepaperarefairlystandard.
RrdenotestherdimensionalEuclideanspace.
N+standsforthesetsofpositiveintegers.
ForthematrixAandthevectorx,ATandxTrepresentsthetransposeAandx,respectively.
TheidentitymatrixisdenotedbyIandthezeromatrixisdenotedby0.
E{x}meansthemathematicalexpectationofx.
X>0meansthatXisapositive-denitesymmetricmatrix.
Insymmetricblockmatrices,weuseanasterisktorepresentatermthatisinducedbysymmetry,andGaoetal.
AdvancesinDierenceEquations(2019)2019:380Page4of23diag{.
.
.
}standsforablock-diagonalmatrix.
Itisassumedthatthematriceshavecompat-ibledimensionsifitisnotexplicitlyspecied.
2ProblemformulationandpreliminariesInthispaper,weconsiderthen-neuronstime-varyingneuralnetworkgivenbyxk+1=(Ak+Ak)xk+αkB1kf(xk)+(1–αk)B2kg(xk)+Ckv1k,yk=λkDkxk+v2k,(1)zk=Mkxk,wherexk=[x1,kx2,k.
.
.
xn,k]T∈Rnrepresentsthestatevectorofneuralnetwork,yk∈Rmdenotesthemeasurementoutput,zk∈Rrstandsforthecontrolledoutput,Ak=diag{a1,k,a2,k,.
.
.
,an,k}isthestatecoecientmatrix,Ck,DkandMkareknownrealma-triceswithappropriatedimensions,v1k∈Rnwandv2k∈RmareGaussianwhitenoiseswithzeromeanvaluesandcovariancesV1>0andV2>0,respectively.
B1k=[b1ij(k)]n*nandB2k=[b2ij(k)]n*naretheconnectionweightmatrices.
f(xk)=[f1(x1,k).
.
.
fn(xn,k)]Tandg(xk)=[g1(x1,k).
.
.
gn(xn,k)]Taretheneuronactivationfunctions.
Akdescribesthepa-rameteruncertaintysatisfyingAk=HkFkNk,(2)whereHkandNkareappropriatelydimensionalknownmatrices,theunknownmatrixFksatisesFTkFk≤I,k∈N+.
(3)TheBernoullidistributedrandomvariablesαkandλkareutilizedtodescribethephe-nomenaofrandomlyvaryingnonlinearitiesandmissingmeasurements,respectively,andsatisfyProb{αk=1}=E{αk}=α,Prob{αk=0}=1–α,Prob{λk=1}=E{λk}=λ,Prob{λk=0}=1–λ,whereα∈[0,1]andλ∈[0,1]areknownconstants.
Assumption1Fortheactivationfunctionsf(·)andg(·)withf(0)=g(0)=0,thereexistfourscalarsλ–i,λ+i,σ–iandσ+isatisfyingthefollowingconditions:λ–i≤fi(s1)–fi(s2)s1–s2≤λ+i,σ–i≤gi(s1)–gi(s2)s1–s2≤σ+i,s1,s2∈R,s1=s2.
(4)Gaoetal.
AdvancesinDierenceEquations(2019)2019:380Page5of23Inordertoestimatethestatesofneurons,thefollowingtime-varyingnonlinearstateestimatorisconstructed:xk+1=Akxk+αB1kf(xk)+(1–α)B2kg(xk)+(Kk+δkKk)(yk–λDkxk),zk=Mkxk,(5)wherexkistheestimationofneuronstatexk,Kkisaknownrealmatrixwithappropriatedimension,δkiszeromeanGaussianwhitenoisewithunitycovariance.
Kkistheestimatorgainmatrixtobedeterminedlater.
Inthesequel,supposethatαk,λk,v1k,v2kandδkaremutuallyindependent.
Remark1Duringtheimplementationprocess,thestateestimationperformanceoftheneuralnetworksisusuallyaectedbythenumericalroundingerror,nite-wordlengtheectsandanalog-to-digitalconversionaccuracy,whichinevitablyleadtothepossibledeviationsoftheestimationvaluesprovidedbythestateestimator.
Thus,thepresentedestimationmethodmightbeafragile/non-resilientone.
Therefore,weaimtotakethees-timatorgainperturbationsintoaccountduringtheestimatordesignwithhopetoprovidearesilienttime-varyingstateestimatorwithadmissibleadjustmentability.
Accordingly,theestimatorgainperturbationsaremodeledbyazeromeanGaussianwhitenoiseδkandthenominalmatrixKk,wherethechangesofthewhitenoiseδkareutilizedtocharacter-izetheadmissibleerrorsoftheestimatorgain.
Assuch,aresilientstateestimationschemeunderprescribedperformanceindicesisexpectedtobegivenlaterfortheaddressedtime-varyingRNNs.
Lettheestimationerrorbeek=xk–xkandthecontrolledoutputestimationerrorbezk=zk–zk.
Thentheestimationerrordynamicscanbeobtainedfrom(1)and(5)asfollows:ek+1=Ak–(Kk+δkKk)λDkek+Akxk+αkB1kf(xk)+αB1kf(ek)–αkB2kg(xk)+(1–α)B2kg(ek)–(Kk+δkKk)(λkDkxk+v2k)+Ckv1k,zk=Mkek,(6)withf(ek)=f(xk)–f(xk),g(ek)=g(xk)–g(xk),αk=αk–αandλk=λk–λ.
Tosimplifythenotation,wecandeneηk=xTkeTkT,vk=vT1kvT2kT,f(ηk)=fT(xk)fT(ek)T,g(ηk)=gT(xk)gT(ek)T.
Considering(1),(6)andtheabovenotations,wecaneasilyderivethefollowingaugmentedsystem:ηk+1=(Ak+λkD1k+λkδkD2k+δkA2k)ηk+(B1k+αkBk)f(ηk)+(B2k–αkˇBk)g(ηk)+(C1k+δkC2k)vk,zk=Mkηk,(7)Gaoetal.
AdvancesinDierenceEquations(2019)2019:380Page6of23whereAk=Ak+Ak0AkAk–λKkDk,D1k=00–KkDk0,D2k=00–KkDk0,A2k=000–λKkDk,B1k=αB1k00αB1k,Bk=B1k0B1k0,B2k=(1–α)B2k00(1–α)B2k,ˇBk=B2k0B2k0,C1k=Ck0Ck–Kk,C2k=000–Kk,Mk=0Mk.
Next,weintroducethecovariancematrixXkdescribedbyXk=EηkηTk=ExkekxkekT.
(8)Themainpurposeofthispaperistodesignatime-varyingnonlinearstateestimator(5)suchthatthefollowingtworequirementsareachievedsimultaneously.
(R1)Letthescalarγ>0,thepositive-denitematricesWandWφbegiven.
Fortheinitialstateη0,theestimationerrorzksatisesthefollowingconstraint:J1:=EN–1k=0zk2–γ2vk2W–γ2EηT0Wφη00suchthatQ+RRT+–1HTH0,α∈[0,1]andλ∈[0,1],matricesW>0andWφ>0,stateestimatorgainmatrixKkin(5)aregiven.
IfQ0≤γ2Wφandthereexistsaseriesofpositive-denitereal-valuematrices{Qk}1≤k≤N+1satisfyingthefollowingrecursivematrixinequality:Φ=Φ11Λ21+ATkQk+1B1kΣ22+ATkQk+1B2k0Φ22Φ230Φ330Φ440andtheinitialconditionQ0≤γ2Wφ,itfollowsthatJ10,wecanobtainEAkηkfT(ηk)BT1k+B1kf(ηk)ηTkATk≤Eε1AkηkηTkATk+ε–11B1kf(ηk)fT(ηk)BT1k,Gaoetal.
AdvancesinDierenceEquations(2019)2019:380Page12of23EAkηkgT(ηk)BT2k+B2kg(ηk)ηTkATk≤Eε2AkηkηTkATk+ε–12B2kg(ηk)gT(ηk)BT2k,EB1kf(ηk)gT(ηk)BT2k+B2kg(ηk)fT(ηk)BT1k≤Eε3B1kf(ηk)fT(ηk)BT1k+ε–13B2kg(ηk)gT(ηk)BT2k,E–α(1–α)Bkf(ηk)gT(ηk)ˇBTk–α(1–α)ˇBkg(ηk)fT(ηk)BTk≤Eε4α(1–α)Bkf(ηk)fT(ηk)BTk+ε–14α(1–α)ˇBkg(ηk)gT(ηk)ˇBTk,whereεi(i=1,2,3,4)arepositivescalars.
ThenitisstraightforwardtoseethatXk+1≤E(1+ε1+ε2)AkηkηTkATk+λ(1–λ)D1kηkηTkDT1k+λ(1–λ)D2kηk*ηTkDT2k+A2kηkηTkAT2k+1+ε–11+ε3B1kf(ηk)fT(ηk)BT1k+α(1–α)1+ε–14ˇBkg(ηk)gT(ηk)ˇBTk+α(1–α)(1+ε4)Bkf(ηk)fT(ηk)BTk+1+ε–12+ε–13B2kg(ηk)gT(ηk)BT2k+C1kVCT1k+C2kVCT2k.
Furthermore,itfollowsfromLemma3thatEf(ηk)fT(ηk)≤EY1ηk2=EY1ηTkηk,Eg(ηk)gT(ηk)≤EY2ηk2=EY2ηTkηk,whereY1andY2aredenedin(22).
Thus,onehasXk+1≤E(1+ε1+ε2)AkηkηTkATk+λ(1–λ)D1kηkηTkDT1k+λ(1–λ)D2kηkηTkDT2k+A2kηkηTkAT2k+1+ε–11+ε3B1kY1ηTkηkBT1k+α(1–α)1+ε–14ˇBkY2ηTkηkˇBTk+α(1–α)(1+ε4)BkY1ηTkηk*BTk+1+ε–12+ε–13B2kY2ηTkηkBT2k+C1kVCT1k+C2kVCT2k.
(23)Accordingtothefeatureofthetrace,wecanobtainEηTkηk=EtrηkηTk=tr(Xk).
(24)Combining(23)with(24)resultsinXk+1≤(1+ε1+ε2)AkXkATk+λ(1–λ)D1kXkDT1k+λ(1–λ)D2kXkDT2k+A2kXkAT2k+1+ε–11+ε3tr(Xk)B1kY1BT1k+α(1–α)1+ε–14*tr(Xk)ˇBkY2ˇBTk+α(1–α)(1+ε4)tr(Xk)BkY1BTk+1+ε–12+ε–13tr(Xk)B2kY2BT2k+C1kVCT1k+C2kVCT2k=Ψ(Xk).
NotingthatG0≥X0andlettingGk≥Xk,wecanderivethefollowinginequality:Ψ(Gk)≥Ψ(Xk)≥Xk+1.
(25)Gaoetal.
AdvancesinDierenceEquations(2019)2019:380Page13of23Then,from(21)and(25),wearriveatGk+1≥Ψ(Gk)≥Ψ(Xk)≥Xk+1.
(26)Therefore,theproofiscomplete.
Basedontheabovetheorems,asucientconditioncanbepresentedtoguaranteethespeciedH∞performanceandestimationerrorvarianceconstraintbysolvingtherecur-sivematrixinequalities.
Theorem3Considerthetime-varyingRNNs(1)andsupposethattheestimatorgainma-trixKkisgiven.
Forgivenscalarsγ>0,α∈[0,1]andλ∈[0,1],positive-denitematricesW>0andWφ>0,undertheinitialconditionsQ0≤γ2WφandG0=X0,iftherearetwoseriesofpositive-denitereal-valuedmatrices{Qk}1≤k≤N+1and{Gk}1≤k≤N+1satisfyingthefollowingmatrixinequalities:Ξ11Ξ12Ξ13Ξ140Ξ22000Ξ330Ξ35Ξ440Ξ550,thescalarsα∈[0,1]andλ∈[0,1],thepositive-denitematricesW>0andWφ=Wφ1Wφ2WTφ2Wφ4>0andasetofpre-denedvarianceupperboundmatrices{Ψk}0≤k≤N+1,undertheinitialconditionsL0–γ2Wφ1Wφ2WTφ2Z0–γ2Wφ4≤0,Ee0eT0=G2,0≤Ψ0,(29)Gaoetal.
AdvancesinDierenceEquations(2019)2019:380Page15of23ifthereexistseriesofpositive-denitematrices{Lk}1≤k≤N+1,{Zk}1≤k≤N+1,{G1k}1≤k≤N+1and{G2k}1≤k≤N+1,positivescalars{1,k}0≤k≤N+1and{2,k}0≤k≤N+1,matrices{Kk}0≤k≤N+1and{G3k}1≤k≤N+1withappropriatedimensionssatisfyingthefollowingrecursivematrixinequalities:Θ11Θ12Θ13Θ1400Θ22000WTkΘ330Θ35YTkΘ4400Θ5501,kI<0,(30)Π11Π12Π13Π140Π2200XTkΠ3300Π4402,kI<0,(31)G2,k+1–Ψk+1≤0,(32)whereΘ11=–Γ1Λ1–Γ3Σ1+1,kNkNTk–Lk00–Γ2Λ1–Γ4Σ1–Zk,Θ12=αATkLk+1B1k+Γ1Λ20(1–α)ATkLk+1B2k+Γ3Σ200Ω50Ω6,Θ13=0ATkLk+100–2DTkKTkZk+100Ω700,Θ14=0–2DTkKTkZk+1000000–λDTkKTkZk+1MTk,Θ22=Ω80–α(1–α)BT1kZk+1B2k0–Γ2+α2BT1kZk+1B1k0Ω3Ω90Ω4,Θ33=diag–γ2W,–Lk+1,–Zk+1,–Lk+1,–Zk+1,Θ35=CTkLk+1CTkZk+1000–KTkZk+10–KTkZk+1000000000000,Θ44=diag{–Lk+1,–Zk+1,–Lk+1,–Zk+1,–I},Θ55=diag{–Lk+1,–Zk+1,–Lk+1,–Zk+1},Gaoetal.
AdvancesinDierenceEquations(2019)2019:380Page16of23Π11=Ω10Ω11Ω12,Π12=4AkG1k4AkGT3k4(Ak–λKkDk)G3k4(Ak–λKkDk)G2k,Π13=0000–2KkDkG1k–2KkDkGT3k–2KkDkG1k–2KkDkGT3k,Π14=00CkV1000–λKkDkG3k–λKkDkG2kCkV1–KkV20–KkV2,Π22=–G1k–GT3k–G2k,Π33=diag{Π22,Π22},Π44=diag{Π22,–V1,–V2,–V1,–V2},Ω3=α(1–α)BT1kZk+1B2k,Ω4=–Γ4+(1–α)2BT2kZk+1B2k,Ω5=αATk–λDTkKTkZk+1B1k+Γ2Λ2,Ω6=(1–α)ATk–λDTkKTkZk+1B2k+Γ4Σ2,Ω7=ATkZk+1–λDTkKTkZk+1,Ω8=–Γ1+αBT1kLk+1B1k+α(1–α)BT1kZk+1B1k,Ω9=–Γ3+(1–α)BT2kLk+1B2k+α(1–α)BT2kZk+1B2k,Ω10=–G1,k+1+α25tr(Gk)B1kY1BT1k+1tr(Gk)B1kY1BT1k+(1–α)23*tr(Gk)B2kY2BT2k+6tr(Gk)B2kY2BT2k+2,kHTkHk,Ω11=–GT3,k+1+1tr(Gk)B1kY1BT1k+6tr(Gk)B2kY2BT2k+2,kHTkHk,Ω12=–G2,k+1+α25tr(Gk)B1kY1BT1k+1tr(Gk)B1kY1BT1k+(1–α)23*tr(Gk)B2kY2BT2k+6tr(Gk)B2kY2BT2k+2,kHTkHk,Wk=αHTkLk+1B1kαHTkZk+1B1k(1–α)HTkLk+1B2k(1–α)HTkZk+1B2k,Yk=0HTkLk+1HTkZk+100,NTk=Nk0,Xk=4NkG1k4NkGT3k,FTk=HTkHTk,thenwecanconcludethattheestimatordesignproblemissolvable.
Gaoetal.
AdvancesinDierenceEquations(2019)2019:380Page17of23ProofFirstly,thematricesQkandGkaredecomposedasfollows:Qk=Lk0Zk,Gk=G1kGT3kG2k.
Secondly,inordertodealwithparameteruncertainty,werewrite(27)asfollows:Ξ11Ξ012Ξ013Ξ140Ξ22000Ξ330Ξ35Ξ440Ξ55+NkFkHk+HTkFTkNTk<0,whereΞ012=αATkLk+1B1k+Γ1Λ20(1–α)ATkLk+1B2k+Γ3Σ200Ω50Ω6,Ξ013=0ATkLk+100–2DTkKTkZk+100Ω700,NTk=NTk0000,Hk=0WkYk00.
Subsequently,itfollowsfromLemma2thatΞ11Ξ012Ξ013Ξ140Ξ22000Ξ330Ξ35Ξ440Ξ55+1,kNkNTk+–11,kHTkHk<0.
Similarly,(28)canberewrittenas–Gk+1Υ012Υ13Υ14–Gk00Υ330Υ44+NkFkHk+HTkFTkNTk<0,whereΥ012=4AkG1k4AkGT3k4(Ak–λKkDk)G3k4(Ak–λKkDk)G2k,NTk=FTk000,Hk=0Xk00.
Gaoetal.
AdvancesinDierenceEquations(2019)2019:380Page18of23ThenitfollowsfromLemma2that–Gk+1Υ012Υ13Υ14–Gk00Υ330Υ44+2,kNkNTk+–12,kHTkHk<0.
Now,itshouldbenotedthat(30)implies(27).
Similarly,wecanseethat(31)leadsto(28).
Assuch,boththeestimationerrorcovarianceconstraintandH∞performancerequire-mentofsystem(7)areensured.
TheproofofTheorem4iscomplete.
Remark5Uptonow,wehavediscussedthevariance-constrainedresilientH∞statees-timationproblemforaclassoftime-varyingRNNswithrandomlyvaryingnonlinearitiesandmissingmeasurements.
Byapplyingtherecursivematrixinequalitytechnique,somecriteriahavebeenestablishedtoguaranteetheprescribedH∞performanceandtheesti-mationerrorcovarianceconstraintsfortheaddressedestimationproblemoftime-varyingneuralnetworkswithinthenite-horizonframework.
Itshouldbenoticedthatthepro-posedestimationapproachhasthefollowingthreeadvantages:(i)thedisturbanceeectscanbeeectivelyattenuatedbytheH∞performanceindexovernitehorizon;(ii)theprescribedupperboundoftheestimationerrorcovariancecanbeguaranteedbyveri-fyingcertainmatrixinequalities;and(iii)thenewlydesignedstateestimationapproachcanbeappliedtotheonlinecalculationsandimplementationsforsolvingtheestimationproblemsoftime-varyingRNNs,whichconstitutesanotherappealingfeature.
Remark6Infact,almostallexistingestimationschemescanbeappliedtotime-invariantNNsonly,butwehavemadeoneoftherstattemptstodiscussthecharacteristicsofthetime-varyingRNNsandaddresstwocombinedperformanceindicestomeetthepracti-calrequirements,whicharetheessentialsuperiorityoftheproposedresult.
Forexample,comparedwiththenon-fragile/resilientstateestimationmethodin[10],ourestimationschemehastheadvantagetorevealthewholeimpactsfrommissingmeasurementsandrandomlyvaryingnonlinearitiesontotheestimationalgorithmperformance,whichcanpresentanewtreatmentway.
Incontrasttotheresultsin[11,12],thesuperioritydeal-ingwiththetime-varyingcharacteristicscanbeobservedfromournewstateestimationscheme.
5AnillustrativeexampleInthissection,wegiveasimulationtoillustratethefeasibilityofproposedestimationapproachundervarianceconstraint.
Theparametersoftime-varyingRNNs(1)aregivenasfollows:Ak=–0.
500–0.
1sin(2k),B1k=–sin(2k)0.
5–0.
20.
5,Γ1=0.
9000.
9,B2k=–0.
27sin(k)0.
2–0.
1–0.
14,Ck=–0.
1–0.
3sin(2k)T,Γ2=1001,Dk=–0.
55sin(k)1.
5,Kk=–0.
27sin(2k)0.
15T,Γ3=1.
1001.
1,Gaoetal.
AdvancesinDierenceEquations(2019)2019:380Page19of23Table1ThevaluesofestimatorgainKkkKk1K1=[–0.
17010.
1102]T2K2=[–0.
14810.
2838]T3K3=[–0.
24380.
1024]T.
.
.
.
.
.
Γ4=1.
3001.
3,Hk=0.
10.
15T,Nk=0.
20.
1,Fk=sin(0.
6k),Mk=–0.
01–0.
12sin(k),λ=0.
34,α=0.
1,ρ=0.
5,ε1=0.
5,ε2=0.
3,ε3=0.
2,ε4=0.
1.
Moreover,theactivationfunctionscanbetakenasf(xk)=g(xk)=tanh(x1,k)tanh(0.
8x2,k)withxk=[x1,kx2,k]Tbeingtheneuronstatevectorofneuralnetwork.
Itiseasytoob-tainΛ0=diag{0.
1,0.
1},Λ1=diag{0.
2,0.
2},Λ2=diag{0.
9,0.
9},Σ0=diag{0.
2,0.
2},Σ1=diag{0.
3,0.
3}andΣ2=diag{0.
5,0.
5}.
Letthedisturbanceattenuationlevelbeγ=0.
9andN=94,weightedmatricesasW(1)=W(2)=1,upperboundmatricesas{Ψk}1≤k≤N=diag{0.
3,0.
3},andcovariancesasV1=V2=1.
Choosetheparameters'initialmatricessat-isfying(29).
Thenthematrixinequalities(30)–(32)inTheorem4canbesolved,andKkisdesignedasinTable1.
Supposetheinitialstatesasx0=[–1.
50.
3]Tandx0=[–1.
20.
3]T.
BasedonthestateestimationmethodinTheorem4,thesimulationresultscanbeshowninFigs.
1–4.
Fig-ures1–2plottheoutputzkanditsestimationzk,respectively.
Figure3depictstheoutputestimationerrorzk.
TheerrorvarianceupperboundandactualerrorvarianceareplottedinFig.
4,whichindeedillustratesthattheactualerrorvariancebelowtheerrorvarianceupperbound.
Fromthesimulations,wecanconcludethatthenewlypresentedvariance-constrainedresilientH∞estimationalgorithmisecient.
6ConclusionsInthispaper,wehavediscussedthevariance-constrainedresilientH∞stateestimationproblemforaclassoftime-varyingneuralnetworkswithrandomlyvaryingnonlinearitiesandmissingmeasurements.
TworandomvariablesthatobeyBernoullidistributionhavebeenadoptedtodescribethephenomenaofrandomlyvaryingnonlinearitiesandmiss-ingmeasurements.
Anewvariance-constrainedH∞stateestimationmethodhasbeendesignedbasedontheavailableinformation.
Byapplyingtherecursivematrixinequal-itytechnique,somecriteriahavebeenestablishedtoguaranteetheprescribedH∞per-formanceandtheestimationerrorcovarianceconstraintsfortheaddressedestimationproblemofthetime-varyingneuralnetworks.
Inaddition,thegainmatrixofstateesti-matorhasbeenobtainedbytestingthefeasibilityoftheconcernedrecursivematrixin-equalities.
Finally,thevalidityoftheproposedestimationmethodhasbeenveriedbyasimulationexample.
OurfutureresearchtopicsincludethestateestimationproblemsforGaoetal.
AdvancesinDierenceEquations(2019)2019:380Page20of23Figure1Thecontrolledoutputz1,kanditsestimationFigure2Thecontrolledoutputz2,kanditsestimationtime-varyingRNNswiththenite-timecriterionasin[36]andtheinaccuracyoccurrenceprobabilityasmentionedin[49].
Gaoetal.
AdvancesinDierenceEquations(2019)2019:380Page21of23Figure3TheoutputestimationerrorsFigure4TheupperboundoferrorvarianceandactualerrorvarianceFundingThisworkwassupportedinpartbytheOutstandingYouthScienceFoundationofHeilongjiangProvinceofChinaundergrantJC2018001,theNationalNaturalScienceFoundationofChinaunderGrant61673141,theFokYingTungEducationFoundationofChinaunderGrant151004,theNaturalScienceFoundationofHeilongjiangProvinceofChinaundergrantA2018007,theFundamentalResearchFundsinHeilongjiangProvincialUniversitiesofChinaunderGrant135209250,theEducationalResearchProjectofQiqiharUniversityofChinaunderGrant2017028,andtheAlexandervonHumboldtFoundationofGermany.
Gaoetal.
AdvancesinDierenceEquations(2019)2019:380Page22of23AvailabilityofdataandmaterialsNotapplicable.
CompetinginterestsAllauthorsdeclaredthattheyhavenocompetinginterests.
ConsentforpublicationBothauthorsreadandapprovedthenalversionofthepaper.
Authors'contributionsTheauthorscontributedequallytothispaper.
Theauthorsreadandapprovedthenalversionofthesubmission.
Authordetails1SchoolofScience,HarbinUniversityofScienceandTechnology,Harbin,China.
2HeilongjiangProvincialKeyLaboratoryofOptimizationControlandIntelligentAnalysisforComplexSystems,HarbinUniversityofScienceandTechnology,Harbin,China.
3SchoolofEngineering,UniversityofSouthWales,Pontypridd,UK.
4QiqiharCollegeofScience,QiqiharUniversity,Qiqihar,China.
Publisher'sNoteSpringerNatureremainsneutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutionalaliations.
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