A Lyapunov approach to stability analysis of partial synchronization in delay-coupled networks

Libo Su; Yanling Wei; Wim Michiels; Erik Steur; Henk Nijmeijer

doi:10.1016/j.ifacol.2018.12.094

A Lyapunov approach to stability analysis of partial synchronization in delay-coupled networks

Libo Su
, Yanling Wei
, Wim Michiels
, Erik Steur
, Henk Nijmeijer

Dynamics and Control

Research output: Contribution to journal › Conference article › peer-review

2 Citations (Scopus)

Abstract

Networks of interconnected dynamical systems may exhibit a so-called partial synchronization phenomenon, which refers to synchronous behaviors of some but not all of the systems. The patterns of partial synchronization are often characterized by partial synchronization manifolds, which are linear invariant subspace of the state space of the network dynamics. Here, we propose a Lyapunov-Krasovskii approach to analyze the stability of partial synchronization manifolds in delay-coupled networks. First, the synchronization error dynamics are isolated from the network dynamics in a systematic way. Second, we use a parameter-dependent Lyapunov-Krasovskii functional to assess the local stability of the manifold, by employing techniques originally developed for linear parameter-varying (LPV) time-delay systems. The stability conditions are formulated in the form of linear matrix inequalities (LMIs) which can be solved by several available tools.

Original language	English
Pages (from-to)	198-204
Number of pages	7
Journal	IFAC-PapersOnLine
Volume	51
Issue number	33
DOIs	https://doi.org/10.1016/j.ifacol.2018.12.094
Publication status	Published - 2018

Funding

LiboSu∗∗YanlingWneei∗∗tWwoimrksMichiels∗∗ErikSteur∗∗∗∗ Libo Su∗ Yanling Wei∗ Wim Michiels∗ Erik Steur∗∗ Libo Su∗ YanlingHWeneki∗NWijmimeiMjerichiels∗ Erik Steur∗∗ ∗∗ Henk∗∗Nijmeijer∗∗∗∗∗∗∗∗∗∗∗∗ Libo Su YanlingHWeneki NWijmimeiMjerichiels Erik Steur ∗∗∗ Department of Computer Science, KU Le∗∗∗ufien, Celestijnenlaan 200A ∗∗Department of Computer Science, KU Leufien, Celestijnenlaan 200A Debpoaxr2tm40e2n,t 3o0f0C1oLmepuufietenr, SBceilegniucem, K(eUmaLielsu:fliiebno,[email protected], 0A ∗ box 2402, 3001 Leufien, Belgium (emails:[email protected], Debpoaxyra2tmn4l0ei2nn,[email protected],fiSBecnei.legbneiuc,emw, iK(meUm.maLiicelshu:ifilieebnso@,[email protected])n.b2e0, 0A ∗∗ [email protected], [email protected]) ∗∗∗∗ Dboexlyfta2nC4l0ei2nn,[email protected],sfiBeanen.lgbdeiu,Cmwoni(mterm.omla,iicMlsh:[email protected]@.k2cu,sl.e2ku6uf2ilee8nuC.fibeDen).bDee,lft, ∗∗Delft Center for Systems and Control, Mekelweg 2, 2628 CD Delft, [email protected].(bdem,Cwaoinilm:ter.o.mslt,eicMuhrei@[email protected],lle2)u6f2ie8nC.bDe) Delft, ∗∗∗∗∗ the Netherlands (email:[email protected]) ∗∗∗∗∗D∗ DeleftpaCretmnteentrhtefooNrf eMStyhesectrhelamannsidcasanl(deEmCngaoiinnl:teereo.srlti,enMugr,[email protected],xl2)561238, 5C6D00DMelBft, ∗∗∗Department of Mechanical Engineering, P.O. Box 513, 5600 MB DepEairntmdheontfhiteenoN,f teMhtheeecNrhlaeantnhidcesarl(aeEnmndsgaii(nl:eeeme.sratienilug:hr,[email protected]@l)5t1u3e,.n5l6)00 MB ∗∗∗ Eindhofien, the Netherlands (email:[email protected]) DepEairntmdheonfiteno,f tMhecNhaenthicearllaEnndsgi(neemerainilg:h,.Pni.jOm.eBijeorx@5t1u3e,.n5l6)00 MB Eindhofien, the Netherlands (email:[email protected]) Abstract: Networks of interconnected dynamical systems may effiflibit a so-called partial Abstract: Networks of interconnected dynamical systems may effiflibit a so-called partial sAybncsftlrroancitz:atNioentwpofrlkensoomfeninotneΦrcwonflnicefclterdefedrysnatomiscyanlcfslyrsotneomuss mbeafylaveifofifrlsibiotf asosmo-ecablluetd npoatrtiaalll syncflronization pflenomenonΦ wflicfl refers to syncflronous beflaviors of some but not all oAsyfbntcsfftllerroasncyitzs:taetNmioesntΠwpTofrfllkeensopomaftetnienortnneΦsrcwonfflnipceafclrtterideafledrsysynnatcomflrisocyannlizcafslytrisootneomuassrembeoaffyltaevneifofifcrlfsilbaiortafcatseosrmoiz-eecdabllubetdy npoatrtiaalll of tfle systemsΠ Tfle patterns of partial syncflronization are often cflaracterized by partial soyfntcffllerosnyizstaetmiosnΠ mpTafflnleenifoopmladtestnΦeowrnnflΦsicwfolffalirpcefalrlitnrieaeflaerrssyinnvtcoaflrrisoaynnntizcsaflutriboonsnpoauacsreebofeofftltfaelvenisotcrafsltaeorasfpctsaeocrmeizoeefdbtfulbetynnepotawtrtoiaralkl syncflronization manifoldsΦ wflicfl are linear invariant subspace of tfle state space of tfle network doyfynntcafflmleroicsnysiΠzstaΛetmeiorsenΠΦ mwTaeflnepifrooplpadtostsΦewrnaflsiLcfoylfaaprpueanrloitnviae-lKarsryainsnvocavflrsrikoainni tiazspautpibrosonpaacafclreetoofoafttnfelaenlystzcaefltaetrflasepctasetcraeibzoielfdittyfbleoyfnepptaawrrttoiiraakll dynamicsΠ ΛereΦ we propose a Lyapunov-Krasovskii approacfl to analyze tfle stability of partial sdyynncaffmllrrooicnnsiiΠzzaΛtteiioorenΦmwaennpiiffrooollpddosssΦiewnafdliLeclfyalayap-rcueonuloipnvle-eKadrrnainesvotwavrsoikariknistaΠspFupibrrssoptaΦactfclfelteosfyatnnflcaefllysrtzoaentietzfalsetpiasotcnaebeoirlfriottyrfledoyfnneptaawmrtoiircaksl syncflronization manifolds in delay-coupled networksΠ FirstΦ tfle syncflronization error dynamics asyrencflronization manifolds in deldayy-ncaomupiclsedinneatwsoyrsktseΠmFaitrisctΦ wtfaleyΠsySnecfolrnodnΦizwaetiouns eraropr adryanmaemteicrs-are isolated from tfle network dynamics in a systematic wayΠ SecondΦ we use a parameter-ayrencifslroolantiezdaLtiyforanopmunatonfvlief-oKnldreastswionovrsdkkeilidayyfu-ncnaocmutpiioclsendailnnetaotwsaoysrssktesesΠmsFatitrflisectΦlwtofacleyaΠlsySsnteacfbolrnilodintΦyizwaoetfioutnfsleerarmoaprnadiryfaonmlademΦteibcrys-dependent Lyapunov-Krasovskii functional to assess tfle local stability of tfle manifoldΦ by edmreeppeilnsooydlieanntgtedLteyfcrafolpmnuiqntuofveles-Knoreratiswgoiovnrsakklliyidyfdunenavcmetliioocpsneaidln tfaoorsayslssitneesemsarattflipecarlwoaacmyaeΠltSesrte-acvboanirldiytΦiynwgoef(uLtfsPleeVam) aptnaimirfaoeml-ddeΦetlebary-employing tecflniques originally developed for linear parameter-varying (LPV) time-delay employing Lteycaflpnuiqnuoves-Korraisgoivnsaklliyi fduenvcetlioopneadl tfooraslsinesesartflpearloacmaeltesrt-avbairliytiyngof(LtfPleVm) atnimifoel-ddΦelbay systemsΠ Tfle stability conditions are formulated in tfle form of linear matriffi inequalities (LMIs) weymfsliptcelfomlycsiaΠnnTg fblteecssftolanlbviqeilduiteybsycosnreidvgeiitnriaaollnlaysvaadrieleavfboelrleomptuoeldoaltsefΠodrinlitnfelearforpmaroafmlienteear-rvmaraytirnigffi i(nLePquVa)littieims (eL-dMelIasy) wflicfl can be solved by several available toolsΠ wyfslitcefml csaΠ nTfblee sstoalbveilditybycosnedveitriaolnasvarielafbolremtuoloaltseΠd in tfle form of linear matriffi inequalities (LMIs) © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Partial syncflronizationΦ linear parameter-varying systemsΦ time-delay systemsΦ Keywords: Partial syncflronizationΦ linear parameter-varying systemsΦ time-delay systemsΦ Kineeyawromrdast:rPiffai rintieaqlusaylnitcifelsronizationΦ linear parameter-varying systemsΦ time-delay systemsΦ linear matriffi inequalities lKineeyawromrdast:rPiffai rintieaqlusaylnitcifelsronizationΦ linear parameter-varying systemsΦ time-delay systemsΦ linear matriffi inequalities 1Π INTRODUCTION 1Π INTRODUCTION In recent decadesΦ1ΠsyINncTflRroOnDizUatCioTnIOofNnetworks witfl in-In recent decadesΦ syncflronization of networks witfl in-ItnercroencennetctdedecaddyensaΦmsyicnaclflsryosntiezmatsiofnlaosfrenceetiwvoedrksinwcrietaflsiing-terconnected dynamical systems flas received increasing taInetrtceronenctienonentcΠtdSedeycnadcdyflenrsoaΦnmiszyicanatciloflnsryoosntfieznmaettsiwofnolarkosefrdencseyetiwsvtoeedrmksinfwclarisetafblseiinenng-attentionΠ Syncflronization of networked systems flas been aotetbtrtsceeeonnrntvineoodenncΠtwSediydnedclyffyllnroainnmizzivcaaattrilooionnsuysoostffefnimeeltsdwsfooΦlarrrksaeenrddegcsiyneissgvtteeefdmmrossimnfclaarnsseaabtseuinernge observed widely in various fieldsΦ ranging from nature o(aBbttssuecnrrktvvieaodnndΠ wSBiiydduneecllkfyylr(o1inn97iz6vva)aatΦrrioiiLooneuuwssoifsffniieeeetlltddwassloΦΦΠ r(rrk2aae0nd1g4sii)ynn)sggteoffmrreonsmgfliannseaebttreuuinerrngee (Buck and Buck (1976)Φ Lewis et alΠ (2014)) to engineering (BNuijcmkeaijnedr BanudckR(o1d9r7i6vg)auΦreiLzo-euAwsnisgeetleaslΠ(2(020031)4Φ))PteotftreeonrmgseinnneaeetruianrlgeΠ (Nijmeijer and Rodriguez-Angeles (2003)Φ Pettersen et alΠ ((N2B0uii0jjcmm6k)eeΦaiijnPeedrrloBaaennugdckeRt(o1ad9lr7Πii6gg()uu2Φee0Lzz1--e4AAw))nniΠsggeeeIntlleeassnlΠ((a22(t002u000r331e))4ΦΦ)s)PyteenottcttefeelnrrrogsseeninnnizeeaetrtiianolngΠΠ (2006)Φ Ploeg et alΠ (2014))Π In natureΦ syncflronization (o(2Nft0ie0jnm6))efΦΦliajPPeprllpooaenngds eRstpooadnllrΠΠtiag((nu22ee0oz1-u4As))l))nyΠΠgΦ eIwnlefslnil(ae2tt0uui0nrr3ee)ΦΦeΦnssPgyyiennntcetffeellrrrroosinennngiizzΦeaaittttiiaooilnsΠ often flappens spontaneouslyΦ wflile in engineeringΦ it is ogo2feftt0nee0en6ra)fΦlllayPplappoednngesssiesgtpnoeandlΠtap(nf2lee0on1uo4sm)l)yΠeΦnIwonnffllΠniillaeetuiinnreΦensgyinncefelrroinngizΦaittioiinss generally a designed pflenomenonΠ gSgeeotnnmeeenerrtaafilllmllaypeaspΦednnesseitsgwpnoerndktsapnflmeeeonnauooysmmlyseeΦfnnlowoownnflΠΠilae ifnormengoinf eeinricnogmΦ pilteties SometimesΦ networks may sflow a form of incomplete SsgyeonnmceefrltarilomlnyeizasaΦ dtineosenitgΦwncoeardlklsepdflmepnaaoyrmtiaseflnl oswnynΠ cahrfoonrimzatoiof ninocromclupsleteter syncflronizationΦ called partial synchronization or cluster sSyonmcefltriomneizsaΦtionΦ calledmpaayrtiasflloswyncahrfoonrimzatoiofninocromclupsleteter synchronizationΦ wflicfl refers to tfle situation wflere only soyynmnncccefhhlrrrbooounntiiznaaottttiiioooannlΦl twcfalffellliilccesffdyllsrtpeeaffmreetrsisailntostyfttnlffellceehnrsseoiittnwuuioaazrattkiitooisonsnywnoffcllreeflrrrceeolunosinntzelleyyrΦ some but not all tfle systems in tfle networks syncflronizeΦ sPsoyomngcrehorbmounstkiznyaoettitoaanllΦlΠ t(wf2lf0eli0cs2fyl)sΦrtBeefmeelrsyskintfol tefttlfelaelnΠse(it2twu0oa0rt8ki)osΦnDsywanfflclmeflrrseoenotinzalelyΠΦ Pogromsky et alΠ (2002)Φ Belykfl et alΠ (2008)Φ Daflms et alΠ P(s2oo0mg1re2o)bmΠusPtkanyrotetitaalallslΠyt(nf2lc0ef0lsr2yo)snΦtiBezmaeltsyiokinnfl tiesftleoaflntΠee(tn2w0o0br8ks)esΦrDsvyeandflcmfilnrsoecntoimzael-ΠΦ (2012)Π Partial syncflronization is often observed in com-(pP22loe00gf11fir22os))ymΠstsPekaamyrrtsei;taflaolrsΠye(n2ffic0aff0mllr2op)nΦleiBzzΦ aaesylttyinookcnnffllriesotnoaofltΠue(sn2f0iro0ib8ns)geΦroDvfeandfelmuinrsoecntosmailn-Π pleffi systems; for effiampleΦ syncflronous firing of neurons in ppp2lalee0rff1tffiis2ss)yyoΠssfttPteefamlretssif;;aluffloomrrsyaeenffffiicaafbmlrroappnillneeizΦΦΦ assGyytirnnoaccnyffllrri(soo1n9oo9ftuu4ess)nΠffiiBrroiibennssggiedrooevffsenΦdeituuinrrsooflcnnoossumilindn-parts of tfle fluman brainΦ Gray (1994)Π BesidesΦ it sflould pbpaaeerrfttfpisssoyooinsfftttteeffmdlleesff;olluufotmr aetnfffliaabbmtrraapciilonnemΦΦΦ sGpylrneatcyeflr(((o11fn99uo99llu44)s))ΠΠsfiyBBrniencssgfiilddroeeofsnnΦΦieziittuatrssioffollnoonsuulliiddns be pointed out tflat complete (full) syncflronization is bnpoaetrtpasolwoinfattyefsdledfoeluusitmraatbnflleab:trsaycinonmcΦflGprlorenatyieza((t1fi9uo9lnl4))oΠfsyBenfeficcsfieldsresoisnvΦiezitaatsmifoloonuunlidts not always desirable: syncflronization of efficessive amount nobfooettnpaaeolluwwirnaaotynesds dcoeeaussniitrraactbbaflueea::stessyycbnnormccaffllpirnloentdiiezzisaao(ttfriiudoolnnel)rosfsyleeinkffffiiecccfeelssressopiinivliezpaaastmmyiooonauunidtts of neurons can cause brain disorders like epilepsy and onfotnaeluwraoynss dceasniracbalue:sesybnrcaflirnondizisaotridonerosf leikffiecesespiivleepasmy oaundt ★★ This work was supported by the project C14/17/072 of the ★ofThisneurowonsrkcwaans suppcausortede brabiny thedisoprordejrsectliC14/17/072ke epilepsyofathend KUThLiseuwveonrkRwesaesarscuhppCoorutendcil,bybythteheprporjoejcetctCG140/A1573/1077N2 of the KU Leuven Research Council, by the project G0A5317N of the KeUseaLrecuhveFnouRnedsaetairocnh-FClaonudnecrisl,(bFyWtOhe-pVrolajeacntdeGre0nA)5,3a1n7dNboyf the Research Foundation-Flanders (FWO - Vlaanderen), and by the RreosjeeacrtchUCFoouCnodSa,tifounn-dFeldandbeyrsth(eFWEuOro-peVanlaaUnndieornens ),Hoarnidzonby20th20e project UCoCoS, funded by the European Unions Horizon 2020 rersoejaerccthUaCndoCionSn,ovfautniodnedprboygrtahmemEeuuronpdeearnthUenMionasrieHSokrilzoodnow2s0k2a0-research and innovation programme under the Marie Sklodowska-reuseriaercGhraanntdAingrneoevmateinotnNporo6g7r5a0m8m0.e under the Marie Sklodowska-Curie Grant Agreement No 675080. CuriereuseriaercGGhrraaannnttdAAinggrrneeoeevmmateeinnottnNNpooro67506g7r5a0m80.8m0.e under the Marie Sklodowska- C24o0p5y-r8i9g6h3t ©© 22001188, IIFFAC (International Federation of Automatic Contr2o1l)4 Hosting by Elsevier Ltd. All rights reserved. Copyright © 2018 IFAC 214 Peer review under responsibility of International Federation of Automatic Control. Copyright © 2018 IFAC 214 10.1016/j.ifacol.2018.12.094 Copyright © 2018 IFAC 214 Πhis work was supporteff by the project C14/17/072 of the KU Leuven Research Council, by the project G0A5317N of the Research Founffation-Flanffers flFWO-Vlaanfferen), anff by the project UCoCoS, funffeff by the European Unions Horizon 2020 research anff innovation prograΦΦe unffer the Marie Skloffowska-Curie Grant AgreeΦent No 675080.

Keywords

Partial synchronization
linear parameter-varying systems
time-delay systems
linear matrix inequalities

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10.1016/j.ifacol.2018.12.094

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@article{62337b6631f34e2ebf43e09cd7502037,

title = "A Lyapunov approach to stability analysis of partial synchronization in delay-coupled networks",

abstract = "Networks of interconnected dynamical systems may exhibit a so-called partial synchronization phenomenon, which refers to synchronous behaviors of some but not all of the systems. The patterns of partial synchronization are often characterized by partial synchronization manifolds, which are linear invariant subspace of the state space of the network dynamics. Here, we propose a Lyapunov-Krasovskii approach to analyze the stability of partial synchronization manifolds in delay-coupled networks. First, the synchronization error dynamics are isolated from the network dynamics in a systematic way. Second, we use a parameter-dependent Lyapunov-Krasovskii functional to assess the local stability of the manifold, by employing techniques originally developed for linear parameter-varying (LPV) time-delay systems. The stability conditions are formulated in the form of linear matrix inequalities (LMIs) which can be solved by several available tools.",

keywords = "Partial synchronization, linear parameter-varying systems, time-delay systems, linear matrix inequalities",

author = "Libo Su and Yanling Wei and Wim Michiels and Erik Steur and Henk Nijmeijer",

year = "2018",

doi = "10.1016/j.ifacol.2018.12.094",

language = "English",

volume = "51",

pages = "198--204",

journal = "IFAC-PapersOnLine",

issn = "2405-8963",

publisher = "Elsevier",

number = "33",

}

TY - JOUR

T1 - A Lyapunov approach to stability analysis of partial synchronization in delay-coupled networks

AU - Su, Libo

AU - Wei, Yanling

AU - Michiels, Wim

AU - Steur, Erik

AU - Nijmeijer, Henk

PY - 2018

Y1 - 2018

N2 - Networks of interconnected dynamical systems may exhibit a so-called partial synchronization phenomenon, which refers to synchronous behaviors of some but not all of the systems. The patterns of partial synchronization are often characterized by partial synchronization manifolds, which are linear invariant subspace of the state space of the network dynamics. Here, we propose a Lyapunov-Krasovskii approach to analyze the stability of partial synchronization manifolds in delay-coupled networks. First, the synchronization error dynamics are isolated from the network dynamics in a systematic way. Second, we use a parameter-dependent Lyapunov-Krasovskii functional to assess the local stability of the manifold, by employing techniques originally developed for linear parameter-varying (LPV) time-delay systems. The stability conditions are formulated in the form of linear matrix inequalities (LMIs) which can be solved by several available tools.

AB - Networks of interconnected dynamical systems may exhibit a so-called partial synchronization phenomenon, which refers to synchronous behaviors of some but not all of the systems. The patterns of partial synchronization are often characterized by partial synchronization manifolds, which are linear invariant subspace of the state space of the network dynamics. Here, we propose a Lyapunov-Krasovskii approach to analyze the stability of partial synchronization manifolds in delay-coupled networks. First, the synchronization error dynamics are isolated from the network dynamics in a systematic way. Second, we use a parameter-dependent Lyapunov-Krasovskii functional to assess the local stability of the manifold, by employing techniques originally developed for linear parameter-varying (LPV) time-delay systems. The stability conditions are formulated in the form of linear matrix inequalities (LMIs) which can be solved by several available tools.

KW - Partial synchronization

KW - linear parameter-varying systems

KW - time-delay systems

KW - linear matrix inequalities

UR - https://www.scopus.com/pages/publications/85059150541

U2 - 10.1016/j.ifacol.2018.12.094

DO - 10.1016/j.ifacol.2018.12.094

M3 - Conference article

SN - 2405-8963

VL - 51

SP - 198

EP - 204

JO - IFAC-PapersOnLine

JF - IFAC-PapersOnLine

IS - 33

ER -

A Lyapunov approach to stability analysis of partial synchronization in delay-coupled networks

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