Abstract
In this paper we focus on consistently identifying a transfer function (module) embedded in a dynamic network. When identifying a module embedded in a dynamic network, a critical choice is which variables to include as predictor inputs. In the system identification literature sufficient conditions have been derived. One condition is that there should be no confounding variables. We show that this condition can be relaxed.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 3983-3988 |
| Number of pages | 6 |
| Journal | IFAC-PapersOnLine |
| Volume | 50 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jul 2017 |
| Externally published | Yes |
Bibliographical note
Funding Information:variables in∗ dynamic networks∗∗ ∗ArneDankersPa∗∗∗∗∗ulM.J.VandenHof∗∗ Donatello Materassi ∗∗∗ Harm H. M. Weerts ∗∗ Arne Dankers ∗ Pa∗∗∗ul M. J. Van den Hof ∗∗ ∗∗ DonArneatelloDaMnkaterersasPsai∗∗∗ul MHar. J.mVH.anMden. WHofeerts∗∗ ∗ Donatello Materassi ∗∗∗ Harm H. M. Weerts ∗∗ ∗ Dept. ofDEolnecattreiclalol aMndatCeormaspsuiter EHnagirnmeerHin.g,MU.nWiveeresrittys of Calgary, Dept. of Electrical and Computer Engineering, University of Calgary, ∗Dept.ofElectrCicanadaal and(emComaipul: adankerter Engisn@ehierfieng.ing, Unicomver).sity of Calgary, ∗∗Dept.ofElectrCicanadaal and(emComaipul: adankerter Engisn@ehierfieng.ing, Unicomver).sity of Calgary, ∗∗DDeeppt.t.ooffEElelecctrtricicaallaEnndgCinoemerpiuntge,rEEinndghinoeveerninUg,niUvenrisvietrysiotfyToefcChnaolglaogryy, ∗∗ Canada (email: adankers@hifieng.com). ∗∗ Dept. of Electrical Engineering, Eindhoven University of Technology, ∗∗ Dept. of Electrical Engineering, Eindhoven University of Technology, The Netherlands (email: h.h.m.weerts@tue.nl, ∗∗∗Dept. of Electrical and Computer Engineering, University of Tennessee, Knopx.vmil.lje.,vaUnSdAen(heomf@atiul:ed.nml)a.teras@utk.edu). Tennessee, Knoxville, USA (email: dmateras@utk.edu). Dept. of Electrical and Computer Engineering, University of Tennessee, Knoxville, USA (email: dmateras@utk.edu). Tennessee, Knoxville, USA (email: dmateras@utk.edu). Abstract: In this paper we focus on consistently identifying a transfer function (module) Abstract: In this paper we focus on consistently identifying a transfer function (module) a critical choice is which variables to include as predictor inputs. In the system identification a critical choice is which variables to include as predictor inputs. In the system identification acocnrfiotuicnadlinchgovicaeriaisblwesh.icWhevsahrioawbltehsattothinisclcuodnedaitsiopnrecdainctbore irneplauxtesd. .In the system identification liteconrafoutundirengsuffivarcieiablntesco.nditioWe shnsowhthaavetbthiseencdeonditiorived.nOnecan bceonditiorelaxend.is that there should be no literature sufficient conditions have been derived. One condition is that there should be no confounding variables. We show that this condition can be relaxed. c©on2f0ou171,ndi .IFIAngNCT var(RInOtieDablrnUaesCti.oTnWIaOle FNsehdoerwatithaon oft thisAutocmoanditiotic Ceonmntbcroaelnd) dHbeoedstreinlag axebydd. Eylnseavmieirc Lntde.t wAollr rkig.hStsim repselyrvecdh.oosing wi as 1. INTRODUCTION embedded in a dynamic network. Simply choosing wi as 1. INTRODUCTION theemboenlyddedinputin aadynand wmj aicsnethetwvoark.riableSimtoplybechpreoodicsingtedw(i.eas. 1. INTRODUCTION theemboenlyddedinputin aadynand wmj aicsnethetwvoark.riableSimtoplybechpreoodicsingtedw(iii.ase. Dynamic networks1.aIreNTpRerOvaDsUCTIOive in eNngineering domains tehmeboeudtdpeudt)inwailldgyennaemrailclynneotwt olerakd. Stoimcponlysisctheonotsiensgtimwaitaess Dynamic networks are pervasive in engineering domains the output) will generjjally not lead to consistent estimates DynaDynauchammsicicponnweeettwwroosrksrksysteaamreres,pppipeerrvveaalinessiivvsee,ininandeengngdisineinetributeeeringringddodocommnaatininrossl oofthf eGGo0n..lyCCioonnnspsuiiddteearrntthedheweexxaaasmmtpplehlee vnneaerttiwwaboorrklek tssohhoobwwe nnpriinnedFFigicitge..d11aa(i.ttoeo. such as power systems, pipelines, and distributed control illusof Gtjria. teCothisnsidesrtathetemeexantm. Fpleor nethistwonerktwshork,owntheinreFiga.re1atwtoo systems. These systems cannot be designed, operated or ofG0.ConsidertheexamplenetworkshowninFig.1ato such as power systems, pipelines, and distributed control illusofarGatjlrliea.lteCpoathisntshisdesfrtarottemhemweexnat→m. Fpwoler.nthiseStuwponeproktsweshotrk,ohwanttheoinnrelFyigaw.re1aatwntdoo systems. These systems cannot be designed, operated or paillusratralleltepathisthssfrotatemmwe1nt→. Fwor2.thisSuppneotsweothrk,attheonlyre awr1eantwdo siinnyggsteeeasimassi.eerrTtthooesccoleoslllyeectscttemffrrsoommcantthhneeseostebssyeysstdteeems,smigsn,eaanddn,dopsoseorasystyestsdtemeomr iwllusatrreatkenotwhins.sIttacteamn ebnet.prFoovredthtihsantewtwhoernku, stihnegrewaraesttwhoe maintained without models of the system. Data is becom-opua2trpalulet,l apnadthws fraosmthwe1in→puwt,2a.nSeusptpimosaetethoaftGo0nl+y 2Gw01 Gan0d midaeninttifiaicnaetdionwiitshowuetllmpodiseelsdotfothpelasysatecmri.tiDcaaltaroislebiencotmhe-w are known.1It can b1e prov2ed that when usin2g1 w a113s th31e identification is well poised to play a critical role in the iwsoabrteaiknneodw.nIn.1Iotthcaenrwbeorpdrso,viendsttehaadtowfhidenenutsifiyn0ig1ngwGa003s,tah0n1e iandgvaenacseiemretnot cooflltehcetsefrotemchtnhoelosegiessy.stDemevse,loapnidngsotosoylsstefomr ou2tput, and w as the input, an estimate of G0 +2G201G0 advancement of these technologies. Developing tools for oesuttipmuatt,eaonfdawsumasotfhtehienppuarta,lalenlepsattimhsaitseoobftGai0ne+d.GO00n1Gth31e adadidevvnnaatttnniiififfccyyeeciimmanntggeeionnttranttraiofofnnssswffettehherreseesellffunupnottccieesttiocceihhodnnsnnsololtoogiogieemmplbbes.es.aeeydddedaDDeeecddvvreeiiintllnopopicaddynaiilnnynggroalttmmeooiicololicnssnnetffehorortt-e-isestimobtaateineod.f aIns1umotheofrthewordspara, inslletel paadthsof isideonbtatifyine2i1nd.g GO21n13,thea3n1 awdovraknsciesmaenntacotifvethreesseeatrecchnfioellodgi[eGso.nD¸caevlveelospainndg Wtoaorlsnicfokr, estimateofasumoftheparallelpathsisobtained1.O2n1th3eiosthoebrtahiannedd.,Iintcoatnhearlswoorbdes,shinoswtenatdhoaftiidfebnotitfhyiwngGan21d,wan identifying transfer functions embedded in dynamic net-aesrteimusaetdeoafsaprseudmicotofrthinepuatrsa,lltehlepnatchosnsisistoebnttaiensetdim.Oatnesthoef works is an active research field [Gon¸calves and Warnick, arothee ursedhand,as pitredcainctoralsoinpbuetss,hotwhenn thacontsiifstbenotht estwi1manatdeswo3f 2008, Materassi and Innocenti, 2010, Haber and Verhae-botohtehrGha0nda,nidt cGa0n alrseoobbetashinoewdn. Tthhaetmifabinotphowintainsdthwat wgeonr,k2s0i1s3a,nVanctdiveenrHesoefaertchalfi.,e2ld01[3G].onC¸coanlvdeitsioannsdhWavaernbiecekn, are used21as predi2c3tor inputs, then consistent esti1mates o3f gen, 2013, Van den Hof et al., 2013]. Conditions have been are used21as predi2c3tor inputs, then consistent estimates of 2008, Materassi and Innocenti, 2010, Haber and Verhae-ianrceluudseind0gaws paresdai0cptroerdiinctpourtsin,ptuhtenencaobnlseissttehnetpeosstsimibailtietys of 2p0ro0p8,osMedatteoreansssiuraentdhaIntncooncesnisttie,n2t0e1s0t,imHaatbeserofatnhde tVrearnhsafeer-both G0 a3nd G0 are obtained. The main point is that gen, 2013, Van den Hof et al., 2013]. Conditions have been includin2g1 w as a2p3 redictor input enables the possibility of gen, 2013, Van den Hof et al., 2013]. Conditions have been bcoonthsisGte021ntanaenstddimGGa023tesaaoreref Goobb00tata.ineined.d. TThehe mmaainin ppooiinntt isis thathatt includin2g1 w as a2p3 redicto2r1 input enables the possibility of proposed to ensure that consistent estimates of the transfer incconludingsistentwe3staimats a prees ofdicGto0r1.input enables the possibility of fSSuraalnolaapctppoiaaonskkeaad(,,s)t22015]o0ofe1n5isn]u..trereestthat[Dconanksiestrsenettestal.,imat2016,esMofatthereassitranansferdAcosnsililsutestnrtaetestdi,mantesimopfoGrt210an.tquestionis:whatvariables Sfuualnnapctctiiaononka((,s)s)2015]ofofiinn.ttererestest[[DDanankkeerrssetetalal..,,2016,2016,MMatatererassiassianandd 2121 Sva1lapwa1ka, 2G03115]. v22 w22 GG023 v22 w22 GG023 As illustrated, an important question is: what variables v1 w1 G31 v2 w2 G230 v2 w2 G230 Thuisst qbueeisnticolundheadsabsepernedaidcdtorersisnepduitns tbootehnsDuraenkceornssiestteanlt. v1 w01 G31 v2 0w2 0 G2323 v2 w02 G2323 Thestimisaqteuests oionf ahmasodulebeenemadbderddeessedd ininabotdynah Dmanickneerstwetork?al. G21 3 21 02 21 e2s0ti1m6]ataensdofMaamteordausslie eamndbeSdadleadpainkaa[d2y01n5a]m. iTchneetwpaoprke?r 21 w31 G21 G12 v3 G21 v3 [2016] and Materassi and Salapaka [2015]. The paper w G G v3 G v3 TfhMis aqtueersatsisoinahnads Sbaeleanpaakdadr[2es0s1e5d] ins baostehdDoannkeexrtsenedt ianlg. G210 w3 G210 G120 v3 G210 v3 of[2016]Materanassid ManatderSassialapanakda [S2015]alapakisab[ased2015]on. Thexetenpapdinerg Gw212 Gw03 G21w1G12Gv03 Gw211 Gv03 of2o0tM1io6atn]serandassideveMloanaptededrSaisalnsiaptahanekdafie[S2015]ladlaopfakipsarobb[ased2a0b1i5li]son.ticTexhinetfenepradenpinceger w2 G23 w1 G13 w1 G13 nooftioMatnserdeassiveloanpeddSinalaptheakafie[2015]ld ofipros bbaasedbilisonticexinfetenredncinge oPf eMaral,te2r0a0s0s]i, awnhderSeatlayppaickally[20th1e5]niestwboarskesd tohnateaxrteenddeianlgt w2 G0323 w1 G0313 w1 G0313 [noPetioarl,ns2de000v],elowpheedreintythepicallyfieldtheofneprotwbaorksbilisthatict ainfereredencalte 23 wv 13 wv 13 noitthionarseddeivreclotepdedaciynclitchegrafipehlds (onfeptwroobrkasbiwlisitthicnionfleoroepnsc)e. (a) 23 (b) 13 (c) 13 w[Pithearl,are20di00re],ctwedheacyre tclyipcicgarllyaphthes (nenetwtwoorksrkswthaithtnoarelodeopsa).lt v2(a) (vb1 ) v1(c) TPheearpl,a2p0e0r0]o,fwDhaerneketryspiectallayl.th[2e0n16et]woonrktshtehaothaerre hdaeanldt Fig. 1v.2(a(a)) Example of a(vb1n)etwork, circles devn1(oct)e internal with are directed acyclic graphs (networks with no loops). Fig. 1.(a(a)) Example of a(bn)etwork, circles den(oct)e internal This baespeapd oernofexteDndingankerscloetsed-al.lo[o2016]p idenontifictahtioe otnhmerethohandsd. Fig. 1. (a) Example of a network, circles denote internal The cpoanpdeirtioonfsDfoarnkperresdiecttoarl.in[p2u0t16s]eloenctitohneportehseenrtehdanidn Fig.v1a.eriat(wable)orEksxs,aaamndnapllbyezooexdfesaindenEenotxwateomrtrapk,lenscsirfe1crlaenfuncsdd3etio.notens.in(b),terna(c)l The conditions for predictor input selection presented in variables,andboxesdenotetransferfunctions.(b),(c) Ms atbtaersreassidssoi nannexdteSnaldlapinpgakcalos[2015]e0d1-l5o]oparreidelessnstsifircestasttiroincttimveethwhohdensn. Networks analyzed in Examples 1 and 3. Tohnesicdoenridnigtionnestwfoorkpsrtehdaitctaoreidnipruecttesdeleactyioclnicpgrreaspenhtse,danind WheNnedetwcoidingrksanhalowyzedto iidenEntxiampfy alespa1rticanulad3.r module em- cMoatnserideassiringannedtwSoalrksapathakat[2015]are direarcetedlessacycrestlicrigctraivpehswh, anden When deciding how to identify a particular module em-thaetceoransdsiitiaonnds oSfaDlaapnakkears[2e0t1a5l]. a[2r0e1l6e]ssarreesletrsisctrievsetriwcthievne bedded in a dynamic network, a critical choice is which theconscideonditioringnenstwoforksDanthakerstaeret adirel. [2c0te1d6]aacreycliclessgrarepshstric,taindve bedhdeneddeincidaindgynhaomwictoneidtwenotrikfy, a cpraitritciaclulcahroimceodisulwe heimch-wohnesnidceorinnsgidneertinwgorckasutshaal tnaertwe odrikresctweidthacloyoclpics.gOranpehso,f atnhde Warhieanbledsectiodiinngcluhdoewatsopirdeednicttifoyrainppuarttsi.cSuulaprpmosoedtuhlaetetmhe- wthehencocnditioonsidensringof Dacausnkaelrsneettwaol.rks[20w1ith6] aloreopsles.sOnerestricofttheive vaerdidaebdlesintoaindcylnuadmeiacsnpertewdoicrtko,raincpruittisc.aSlucphpoiocseitshawthtihche kheyedcoiffnedrietniocnessbofetDwaenenketrhseetwaol.p[a2p0e1r6s]iasrheowlescsornefsoturnicdtiinvge obedjedcetdiveinisatodyconnasmisitcennteltywiodrekn,tiafycarittircaanlsfcehrofiucencitsiownhGic00h kweheyndifferconsenidecesringbetcwaeusenalthneettwwoorkspapweriths islohoopsw.coOnenfouondif theng vabrjieacbtilvese tisotioncloundseisatesnptrlyediidcetnotrifiynpauttrsa.nSsufeprpfousnectthioant Gthjei wahreianbcleosnasirdeehrianngdlceadu.sAalcnoentfwouonrkdsinwgivtahrilaobolpesi.sOanneunomf tehae-variablestoincludeaspredictorinputs.Supposethatthji0e varkeyiabldifferesenareceshabndleetwd.eenAtchoenfotwundingo papervsairias hbleowiscoannfouunmndienga-objective is to consistently identify a transfer function G0 kueyrediffvaerrieanbcleestbheatwdeiernectthlyeatffweocptsabpoertshitshheoowutcpountfoaunnddtinhge ⋆obThejectwivoerkisoftoAc.o˘nsaniskteersntislysuppideortedntifyfaiytraMitacsnsferoffuncCanada.tionGThe0ji svarureiabld vesariaareblehathandlet dired. Actlyconfoaffundingects bothvariathebleoutputis anaunmnd theae- The work of A. ˘ankers is supported fiy Mitacs of Canada. Thjei parerdiaibclteosr airnephuatns.dlTedh.eyA acorenfwouenlldsintugdviaerdiaibnletihseanstuantimsteicas-work of P. Van den Hof and H. Weerts is supported fiy the European predictor inputs. They are well studied in the statistics liutereradtvuarreia(bselee tPheaatrdl i[r2e0c0t9ly] afoffrecintsstbaontche)t.hTe houetqpuuetsatinodntwhe RoeTsrekhaeorfcwhPo.rCVkoauonfndcAiel.n(˘HEaRonfCkae)nr,sdAiHsd.vsWaunpecpeeordtrstReidsessfeuiayprpcMhoirttGaecdrsafnioytf tSChYaenSEa˘udYraoN.pTEeahTne, predictor inputs. They are well studied in the statistics under the European Union’s Horizon 2020 research and innovation literature (see Pearl [2009] for instance). The question we unedseearrtcheCEouurnocpiela(nERUCni)o,nA’sdHvaonrcizeodnR2e0s2e0arrcehseGarrcahntanSdYiSn˘nYovNatEioTn, literature (see Pearl [2009] for instance). The question we Research Council (ERC), Advanced Research Grant SYS˘YNET, lit0e1r6a]tubre r(esleaexePdeawrlhe[2n00h9a]nfdolrinignsctoannfcoeu)n. dTinhge qvaureisatbiolens?we unedseearrtcheCEouurnocpiela(nERUCni)o,nA’sdHvaonrcizeodnR2e0s2e0arrcehseGarrcahntanSdYiSn˘nYovNatEioTn, [2016] be relaxed when handling confounding variables? undertheEuropeanUnion’sHorizon2020researchandinnovation a[2ddre016]ssbine rethislaxepadpweher isn: chaanndlingthe cocnditioonfoundingns of Davnariakersbleest?al. programme(grantagreementNo694504).undertheEuropeanUnion’sHorizon2020researchandinnovation [2016] be relaxed when handling confounding variables? Copyright © 2017 IFAC 4056[2016] be relaxed when handling confounding variables?