Complex-Valued Neural Networks Systems with Time Delay. Stability Analysis and (Anti-)Synchronization Control

236 Pages·2022·5.688 MB·English

by  Ziye Zhang, Zhen Wang, Jian Chen, Chong Lin

Intelligent Control and Learning Systems 4
Ziye Zhang
Zhen Wang
Jian Chen
Chong Lin
Complex-Valued
Neural Networks
Systems with
Time Delay
Stability Analysis and
(Anti-)Synchronization Control
Intelligent Control and Learning Systems
Volume 4
SeriesEditor
DongShen ,SchoolofMathematics,RenminUniversityofChina,Beijing,
Beijing,China
The Springer book series Intelligent Control and Learning Systems addresses the
emergingadvancesinintelligentcontrolandlearningsystemsfrombothmathemat-
ical theory and engineering application perspectives. It is a series of monographs
andcontributedvolumesfocusingonthein-depthexplorationoflearningtheoryin
controlsuchasiterativelearning,machinelearning,deeplearning,andotherssharing
thelearningconcept,andtheircorrespondingintelligentsystemframeworksinengi-
neeringapplications.Thisseriesisfeaturedbythecomprehensiveunderstandingand
practicalapplicationoflearningmechanisms.Thisbookseriesinvolvesapplications
inindustrialengineering,controlengineering,andmaterialengineering,etc.
TheIntelligentControlandLearningSystembookseriespromotestheexchangeof
emergingtheoryandtechnologyofintelligentcontrolandlearningsystemsbetween
academia and industry. It aims to provide a timely reflection of the advances in
intelligent control and learning systems. This book series is distinguished by the
combination of the system theory and emerging topics such as machine learning,
artificialintelligence,andbigdata.Asacollection,thisbookseriesprovidesvaluable
resources to a wide audience in academia, the engineering research community,
industry and anyone else looking to expand their knowledge in intelligent control
andlearningsystems.
· · ·
Ziye Zhang Zhen Wang Jian Chen Chong Lin
Complex-Valued Neural
Networks Systems with Time
Delay
Stability Analysis and (Anti-)Synchronization
Control
ZiyeZhang ZhenWang
CollegeofMathematicsandSystems CollegeofMathematicsandSystems
Science Science
ShandongUniversityofScience ShandongUniversityofScience
andTechnology andTechnology
Qingdao,Shandong,China Qingdao,Shandong,China
JianChen ChongLin
SchoolofInformationandControl InstituteofComplexityScience,College
Engineering ofAutomation
QingdaoUniversityofTechnology QingdaoUniversity
Qingdao,Shandong,China Qingdao,Shandong,China
ISSN 2662-5458 ISSN 2662-5466 (electronic)
IntelligentControlandLearningSystems
ISBN 978-981-19-5449-8 ISBN 978-981-19-5450-4 (eBook)
https://doi.org/10.1007/978-981-19-5450-4
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature
SingaporePteLtd.2022
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse
ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and
transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd.
The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721,
Singapore
Preface
In recent years, with the emergence of new scientific and technological methods,
complexsignalshaveappearedinradar,imagingandotherfields,whichmakesthe
state variables of the controlled system extended from the real number domain to
thecomplexnumberone.Asaresult,thecomplex-valuedneuralnetworks(CVNNs)
model has entered scholars’ vision. It is a kind of network that processes infor-
mationwithcomplexparametersandvariablesinacomplexfield.Comparedwith
real-valuedneuralnetworks,ononehand,theneuronstate,activationfunctionand
weightofcomplex-valuedonesarecomplex-valued,whichcandirectlyprocesstwo-
dimensionaldata.Ontheotherhand,CVNNsmodelismoreefficient,fasterandhas
bettergeneralizationability.Inparticular,itcandealwithmanyproblemsthatcannot
besolvedbyreal-valuedneuralnetworks.Sofar,CVNNshavebeenwidelyusedin
manybiologicalandengineeringfields,suchasimageprocessing,optoelectronics,
patternrecognition,signalprocessingandsoon.Moreover,timedelayisinevitable
intheactualsystem,whichhasaverysignificantimpactonthesystem.Itoftenleads
tothedelayininformationtransmission,affectstheperformanceofthesystemand
evencausesinstabilityinthesystem.Therefore,itisofgreatsignificancetomakea
profoundstudyofCVNNsmodelwithtimedelay.
From the point of view of dynamic system, CVNNs with time delay model,
as a complex nonlinear dynamic system, its wide applications in various fields
largely depend on its dynamic behaviors. In the past ten or more years, a large
numberofoutstandingachievementsondynamicbehaviorsforthismodelhavebeen
published,involvingglobalstability,exponentialstability,Lagrangestability,input-
statestability,synchronization,stabilization,dissipativity,passivity,stateestimation
andsoon.However,inthisdirection,thereisalackofamonographthatprovides
comprehensivereportsonthisnewgrowtharea.
Alongthisline,thisbookaimstoprovidethelatestdevelopmentsandadvancesin
thedynamicsanalysisforcomplex-valuedneuralnetworksystemswithtimedelay.
It is not only a collection of results in the existing references, but also covers a
completeinterpretationandestablishesasystematicstatementoftheexistinganal-
ysisandmethods.Specifically,thebookismainlyconcentratedontherecentresearch
developmentcarriedonbytheauthorsandtheirco-workersinthisdomain,which
v
vi Preface
includesasymptoticstability,Hopfbifurcation,finite-timestability,Lagrangeexpo-
nential stability, synchronization control, anti-synchronization control, fixed-time
synchronization,fixed-timepinningsynchronizationandadaptivesynchronization.
It gives an update on recent developments for analysis techniques and synthesis
methods,putforwardasystematicstatement,bringsoutfurtherinsightsandcoversa
completetopicondynamicalbehaviorsforcomplex-valuedneuralnetworkssystems
withtimedelay.
Thebookissuitableforpostgraduates,seniorundergraduates,graduatestudents
andseniorresearchersinterestedinorworkingwiththesystemtheoryofcomplex-
valued neural networks, or for those relevant scientific and technical workers with
researchinstabilityandcontroltheoryinsomeareascontainingengineeringdisci-
plines, applied mathematics, control engineering, system analysis and integration,
automation,nonlinearscienceandcomputerscience.Theprerequisitesforthisbook
aremodest,anditisassumedthatthereadersofthisbookhavesomebackgroundin
relatedissuesinvolvingneuralnetworks,systemcontroltheoryandmatrixtheory.
We are deeply grateful to Dr. Mengchu Huang, Senior Editor at Springer, for
his constant guidance and support during the preparation and production of this
book.WewouldliketothankDr.DharaneeswaranSundaramurthyinSpringer,for
his help in the project coordination and supervision during the production process
ofthis.Bothauthorsappreciatetheloveandsupportfromtheirfamiliesduringthe
writingofthisbook.Finally,wealsoacknowledgevaluablesupportfromtheNational
NaturalScienceFoundationofChina(GrantNos.62173214,61803220,61673227,
61503222, 61573008, 61174033), the Natural Science Foundation of Shandong
ProvinceofChina(GrantNos.ZR2021MF100,ZR2011FM006),theResearchFund
fortheTaishanScholarProjectofShandongProvinceofChina,theScienceandTech-
nologySupportPlanforYouthInnovationofCollegesandUniversitiesofShandong
ProvinceofChina(GrantNos.2019KJI005),andtheSDUSTResearchFund(Grant
Nos.2019TDJH102).
Qingdao,China ZiyeZhang
June2022 ZhenWang
JianChen
ChongLin
Contents
1 Introduction .................................................. 1
1.1 HistoryofNeuralNetworks ................................ 1
1.2 HistoryofComplex-ValuedNeuralNetworks ................ 3
1.3 RecentProgressofComplex-ValuedNeuralNetworks ......... 5
1.3.1 SurveyofDynamicsofCVNNs ..................... 5
1.3.2 SurveyofDynamicsofCVBAMNNs ................ 12
1.3.3 SurveyofDynamicsofCVINNs .................... 13
1.4 ThisBook ............................................... 14
References .................................................... 16
2 StabilityCriterionforCVNNswithConstantDelay .............. 27
2.1 Introduction ............................................. 27
2.2 ProblemFormulation ..................................... 28
2.3 MainResult ............................................. 31
2.4 IllustrativeExamples ..................................... 34
2.5 Conclusion .............................................. 35
References .................................................... 35
3 FurtherStabilityAnalysisforCVNNswithConstantDelay ....... 37
3.1 Introduction ............................................. 37
3.2 Preliminaries ............................................ 38
3.3 FurtherStabilityAnalysisBasedonSeparableMethod ......... 41
3.4 StabilityAnalysisBasedonNonseparableMethod ............ 42
3.5 IllustrativeExamples ..................................... 46
3.6 ConclusionandNotes ..................................... 50
References .................................................... 51
4 Hopf Bifurcation Analysis for CVNNs with Discrete
andDistributedDelays ........................................ 53
4.1 Introduction ............................................. 53
4.2 ProblemFormulation ..................................... 54
4.3 HopfBifurcationResult ................................... 57
vii
viii Contents
4.4 DirectionoftheHopfBifurcation ........................... 62
4.5 IllustrativeExample ...................................... 73
4.6 Conclusion .............................................. 74
References .................................................... 76
5 Finite-TimeStabilityAnalysisforCVBAMNNswithConstant
Delay ........................................................ 79
5.1 Introduction ............................................. 79
5.2 ProblemFormulationandPreliminaries ...................... 81
5.3 SufficientCriterionfortheExistenceandUniqueness .......... 85
5.4 Finite-TimeStabilityCriterion ............................. 86
5.5 IllustrativeExamples ..................................... 89
5.6 Conclusion .............................................. 93
References .................................................... 93
6 Lagrange Exponential Stability for CVBAMNNs
withTime-VaryingDelays ...................................... 95
6.1 Introduction ............................................. 95
6.2 ProblemFormulationandPreliminaries ...................... 97
6.3 StabilityCriteriaBasedonAlgebraicStructure ............... 99
6.3.1 Stability Criterion Dependent on Separable
Method .......................................... 99
6.3.2 StabilityCriterionDependentonNonseparable
Method .......................................... 107
6.4 StabilityCriterioninTermsofLMI ......................... 110
6.5 IllustrativeExamples ..................................... 114
6.6 Conclusion .............................................. 121
References .................................................... 121
7 Anti-synchronization Control for CVBAMNNs
withTime-VaryingDelays ...................................... 125
7.1 Introduction ............................................. 125
7.2 ProblemFormulationandPreliminaries ...................... 126
7.3 Anti-SynchronizationControlCriterion ...................... 130
7.4 IllustrativeExample ...................................... 140
7.5 Conclusion .............................................. 143
References .................................................... 143
8 Anti-synchronizationControlforCVNNswithMixedDelays ...... 145
8.1 Introduction ............................................. 145
8.2 ProblemFormulationandPreliminaries ...................... 146
8.3 Anti-synchronizationCriterionBasedonSeparableMethod .... 148
8.4 Anti-synchronizationCriterionBasedonNonseparable
Method ................................................. 152
8.5 IllustrativeExamples ..................................... 155
8.6 Conclusion .............................................. 157
References .................................................... 159
Contents ix
9 (Anti)-SynchronizationforCVINNswithTime-VaryingDelays .... 161
9.1 Introduction ............................................. 161
9.2 ProblemFormulationandPreliminaries ...................... 162
9.3 SynchronizationControlCriterion .......................... 164
9.4 Anti-SynchronizationControlCriterion ...................... 167
9.5 IllustrativeExamples ..................................... 170
9.6 Conclusion .............................................. 178
References .................................................... 178
10 Fixed-Time Synchronization for CVBAMNNs with Time
Delays ........................................................ 181
10.1 Introduction ............................................. 181
10.2 ProblemFormulationandPreliminaries ...................... 183
10.3 Fixed-TimeSynchronizationCriterion ....................... 186
10.4 IllustrativeExamples ..................................... 194
10.5 Conclusion .............................................. 203
References .................................................... 203
11 Fixed-Time Pinning Synchronization for CVINNs
withTime-VaryingDelays ...................................... 205
11.1 Introduction ............................................. 205
11.2 ProblemFormulationandPreliminaries ...................... 207
11.3 MainResults ............................................ 210
11.3.1 Fixed-TimeSynchronizationCriterion ............... 210
11.3.2 Fixed-TimeAdaptiveSynchronizationCriterion ....... 217
11.4 IllustrativeExample ...................................... 221
11.5 Conclusion .............................................. 224
References .................................................... 224
Index ............................................................. 227