Maximum Likelihood Identification of a Continuous-Time Oscillator Utilizing Sampled Data

Karen González; María Coronel; Rodrigo Carvajal; Pedro Escárate; Juan C. Agüero

doi:10.1016/j.ifacol.2018.09.199

Maximum Likelihood Identification of a Continuous-Time Oscillator Utilizing Sampled Data

Karen González, María Coronel, Rodrigo Carvajal, Pedro Escárate, Juan C. Agüero

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

In this paper we analyze the likelihood function corresponding to a continuous-time oscillator utilizing regular sampling. We analyze the equivalent sampled-data model for two cases i) instantaneous sampling and ii) integrated sampling. We illustrate the behavior of the log-likelihood function via numerical examples showing that it presents several local maxima.

Original language	English
Pages (from-to)	712-717
Number of pages	6
Journal	18th IFAC Symposium on System Identification SYSID 2018: Stockholm, Sweden, 9-11 July 2018
Volume	51
Issue number	15
DOIs	https://doi.org/10.1016/j.ifacol.2018.09.199
State	Published - 1 Jan 2018
Externally published	Yes

Keywords

Continuous Time System Estimation
Maximum Likelihood Methods
Time Series

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10.1016/j.ifacol.2018.09.199

Cite this

@article{f7cd52ff0ffe4164a95ff6a1092e0905,

title = "Maximum Likelihood Identification of a Continuous-Time Oscillator Utilizing Sampled Data",

abstract = "In this paper we analyze the likelihood function corresponding to a continuous-time oscillator utilizing regular sampling. We analyze the equivalent sampled-data model for two cases i) instantaneous sampling and ii) integrated sampling. We illustrate the behavior of the log-likelihood function via numerical examples showing that it presents several local maxima.",

keywords = "Continuous Time System Estimation, Maximum Likelihood Methods, Time Series",

author = "Karen Gonz{\'a}lez and Mar{\'i}a Coronel and Rodrigo Carvajal and Pedro Esc{\'a}rate and Ag{\"u}ero, {Juan C.}",

note = "Funding Information: Maximum Likelihood Identification of a Maximum Likelihood Identification of a Maximum Likelihood Identification of a Continuous-Time Oscillator Utilizing ContinuouSs-amTimpleedOscDatillaato★★r Utilizing Sampled Data ★ Sampled Data Karen Gonz{\'a}lez ∗, Mar{\'i}a Coronel ∗,∗∗, Rodrigo Carvajal ∗, Karen Gonz{\'a}lez∗∗, Mar∗∗∗{\'i}a Coronel ∗,∗∗∗,∗∗, Rodrigo∗,∗∗∗∗Carvajal∗∗, Karen PGeodnrz{\'o}alEeszc{\'a},rMataer{\'i}a Coronel , Rodrigo Carvajal , KarenPGonedroz´alEeszc∗{\'a},raMtaer∗´ı∗a∗CanordonJuelan∗C,∗∗.,AgRo{\"u}derriogo∗,∗C∗∗a∗r.vajal∗, Pedro Esc{\'a}rate∗∗∗ and Juan C. Ag{\"u}ero∗,∗∗∗∗. ∗ Electronics Engineering Department, ∗Electronics Engineering Department, UniversidadETle{\'c}etcrnoincaicsFeEdnegriicnoeeSrainngtaDMepaarr{\'i}tam(eUnSt,M), Chile Universida∗d T{\'e}cnica Federico Santa Mar{\'i}a (USM), Chile Universidad T{\'e}cnica Federico Santa Mar{\'i}a (USM), Chile Universida∗d∗ElT´eeccntriiccaalFEednginericoeeSantaring DMepaarr´ıtma (ent,USM), Chile Electrical Engineering Department, U∗∗niversidad de Los Andes, Venezuela ∗∗∗ Electrical Engineering Department, ∗∗∗ Large BiUnnocivuelarsridTaedledsceoLpeosOAbsnedrevsa,toVreyn,eTz ueclsaon, Arizona ∗∗∗Large Binocular Telescope Observatory, Tucson, Arizona ∗∗∗∗∗∗∗∗∗∗∗TLhaergSechBoiUnionloocverfulEasrleidadcTterliecdesacloELpenosgOiAnbesnedreivnsa,gtoVarenynd,eTzCuuoelcmsaopnu,teArrSizcoiennace, ∗∗∗∗∗∗∗The School of Electrical Engineering and Computer Science, The SchoTohl eofUEnlievcetrrsiictayl oEfnNgienwecearsintlge,aAndusCtroamliaputer Science, ∗∗∗∗The SchoTohel ofUniEleverctrsiictyalofEngiNenwecearsingtle,andAusCtromaliaputer Science, The University of Newcastle, Australia The University of Newcastle, Australia Abstract: In this paper we analyze the likelihood function corresponding to a continuousΦtime Abstract: In this paper we analyze the likelihood function corresponding to a continuuoousΦtime oAsbcislltartaocrt:utIinliztihnigs praegpuerlawr esamnapllyinzeg.thWeelikaenliahlyozoed tfuhencetqiounivcaolerrnetspsaomndpilnegdΦtdoaatacomntoidneuloufosrΦtitmwoe Abstract: In this paper we analyze the likelihood function corresponding to a continuousΦtime cases i) instantaneous sampling and ii) integrated sampling. We illustrate the behavviioor of the oscillator utilizing regular sampling. We analyze the equivalent sampledΦdata model for two logΦlikelihood function via numerical examples showing that it presents seveerral local maxima. logΦlikelihood function via numerical examples showing that it presents several local maxima. l{\textcopyright}og2Φl0i1k8e,l iIhFoAoCd (fIuntnecrtniaotnionvaial FneudmeraetriiocnalofexAaumtopmlaetsics hCoowntirnogl) tHhoasttinitg pbrye Eselsnetvsiesre Lvtedr.a Al lllo rciaghltms raexsiemrvaed.. Keywords: Maximum Likelihood Methods, Continuous Time System Estimation, Time Series. Keywords: Maximum Likelihood Methods, Continuuoous Time System Estimation, Time Series. Keywords: Maximum Likelihood Methods, Continuous Time System Estimation, Time Series. Keywords: Maximum Likelihood Methods, Continuous Time System Estimation, Time Series. 1. INTRODUCTION A typical approach to identify a continuosΦtime system 1. INTRODUCTION A typical approach to identify a continuosΦtime system 1. INTRODUCTION A typical approach to identify a continuosΦtime system is to use an approximation such as the Euler method Identification of osci1. IlNlatTorROsyDstUCTIOems hNas been a topic of Ais totypusicaelaanpproapapcrhoxtoimaidetionntifsyucahcaosnttheinuoEulesΦtimer msysethotemd Identification of oscillator systems has been a topic of (Wahlberg, 1988; Goodwin et al., 2013; Yuz et al., 2011) in Identification of oscillator systems has been a topic of oWrdaehrltboerogb,t1a9in88a;nGaopopdrwoxinimetataeld., 2d0is1c3r;etYeuΦtzimeteaml.,o2d0e1l.1)Foinr rIdecuenrtrifiencatt iinonterestof osciin sellvateerroral rsyesearstemsch arheasas bsueench asa it)opStiatc iofsΦ oWxrdaaemhrpltbloeer,og,ibnt1988;aLinarasnsGoanopoapdnrwiodxniSm{\"o}etdatealerds.,tr2013;d{\"o}ismcre(2tYe0uΦ0tz2imet) etalhme.,op2011)dreolb. lFeominr(Wahlberg,1988;Goodwinetal.,2013;Yuzetal.,2011)in rttiiecsccsur((rHanHenatninnanatner,,es1973;1t9i7n3;sseeRobRvveeorrbaaerellrtteesssseeanaaanrrcddh aSSrppeeanaaanssos,ossu,ch1986;19aa8ss6i;) SKKtuuatbbiiissnnΦΦ ordertoobtainanapproximateddiscreteΦtimemodel.ForordertoobtainanapproximateddiscreteΦtimemodel.For tetiiecutcscaslr.((,rHanHe2na0tn0i5nnn)anatn,er,,iiest)1973;1M9i7na3t;sheeRobRvmeorbaaltereirrcttesearassl panahncyddhsiarSScsppeasana(Bnos,oosusu,jcoh1986;1a9asn86di;)NSKKotiuuatrbbaiiiysΦnn, oxfaemstpimlea,tiinngLtahresspoanraamndetSer{\"o}sdienrsatrc{\"o}omnt(in2u0o0u2)sΦttihmeeparoubtolermeΦ etal.,2005),ii)Mathematicalphysics(BoujoandNoiray, of estimating the parameters in a continuousΦtime autoreΦ tics (Hannan, 1973; Roberts and Spanos, 1986; Kubin of estimating the parameters in a continuousΦtime autoreΦ 2017)etal.,,2005)andii,i)ii)MMobatilheematroboticalic(pTsethysisercsu(Boukoouujetoanald.,Noi2008)ray)., ggreressssiivvee mmoodedell frofromm ddisisccretreteΦeΦtimtimee ssaammpleples,s, rereplaplaccinging thethe 2017), and iii) Mobile robotic (Tsetserukou et al., 2008)). dreersivsiavteivmesobdyeldferlotma adpispcrroextiemΦtaimtioensa,mfoprlmesi,ngrepalalicnienagrtrheeΦ derivatives by delta approximations, forming a linear reΦ ContinuousΦtime oscillators are highly relevant to a broad gressive model from discreteΦtime samples, replacing the ContinuuoousΦtime oscillators are highly relevant to a broad greersivsiaotniv,easnbdyusdienlgtathaeplperaosxtismqautairoenss,mfeotrhmoidngis acolninsiedaerrerdeΦ. CrCaoonnngtteiinnouufooausurseΦΦatimtismoeefoorsseccsillaielalartotcohrsrssiaanrerceehighthigehhylylyarrerelelepvvraaesnnettntottonaaotbrobroonaalddy gderersivsioatn,iveasndbyusdeingltatheappleraosxtimasquatiorenss,mfoethormingd isacolinensidearrered.Φ range of areas of research since they are present not only Inreassdidoint,ioann,ditusiisnwgetlhleklneoawstnsqtuhaartetshmeeptehrofodrimsaconncesidoefrtehde. ContinuousΦtime oscillators are highly relevant to a broad In addition, it is well known that the performance of the irnanngaetuorfeabreuatsaolsforeisneasorcmhestinecchentohleoygiacarel apprepsleicnattinoontso(nselye gression, and using the least squares method is considered. range of areas of research since they are present not only identification methods obtained by this approach provide e.g. Boujo and Noiray (2017); Esscc{\'a}{\'a}rrate et al. (2017) and bdiaensetdificeasttiimonatmeset(hLoadrsssobnteatinaeld., b2y00t7h)i.s approach provide tineh.gena.rBeturefoeurjeonbutcaensdathlsNeoorieinriany)s.o(m2017)e tec;hEnsoclo{\'a}graticaeletapplical. (a2017)tions (sanede bideiasedntificestatioimatn mesetho(Lardsssonobtaetineald.,b2007)y this. approach provide thee.g.reBouferejoncanesdtheNoirrein).ay(2017);Esc{\'a}rateetal.(2017)and bbiiasedased estestiimatmateses ((LarLarssonsson etet alal..,, 2007)2007).. the references therein). On the other hand, in (Kirshner et al., 2011) an exact Osthecillarefetorerncsysestethemsreain).re typically represented using differΦ OnOn ththee oothetherr hahand,nd, inin (Kir(Kirshshnnerer eett aal.,l., 22001111)) aann eexaxacctt Oscillator systems are tyyppiiccally representteed using differΦ evnaluthateioonthoefr thhaendd,iscinret(eKΦdirosmhnaeinr eptowale.r,Φs2p0e1c1t)ruamn euxsiancgt eOnstciiallaettqoourrastyyiosstnesmisnacroentiyynppuiioccuaalslΦytimrreeepp.rreeTshenuttsee,dalugsoirnigthddmiiffsffetroΦ eOnvaluthateionotheofrthahend,discrinet(KireΦdomaishnerneptowal.,erΦsp20ect11)ruamn euxasincgt ential equations in continuuoousΦtime. Thuuss, algorithms to eevvaaplulounataetniionotnialofofBttΦshhpeeliddniiescrsscrietesteΦdedΦderomaioimveadinnanppdoowwaeenrrΦspΦsepsectetcimtrruuatmmionuusisainnpggΦ Oscillator systems are typically represented using differΦ exponential BΦsplines is derived and an estimation apΦ eiednettniiaatlifyeeqcuoantttiiioonnnussouinsΦtciomnetinsyusotuesmΦttsiimmaree. hTihguhsly, arellggleoovrraiittnhhtmm. s to evaluation of the discreteΦdomain powerΦspectrum using ential equations in continuousΦtime. Thus, algorithms to proach that is based on the Whittle{\textquoteright}s likelihood function Onidenthetifyocthonetrinhuaond,usΦtimideentsiysficteatiomsnaorefchigonhlytinureousleΦvtimante. sysΦ iissropprearcehsseennthtteeadd.t.is based on the Whittle{\textquoteright}s likelihood function On the other hand, identification of continuuoousΦtime sysΦ proach that is based on the Whittle{\textquoteright}s likelihood function OtOennmstthhiseetoyttphhiecralhlyanpde,rfiioddreemnettiidffiicbayttiiuoosninogfacpopnrttoiinnxuuimoouastΦettdiimmmeodsyeslsΦ is presented. tems is typically performed by using approxxiimmated models isnptrheesernetceedn.t paper (Chen et al., 2017) an algorithm to On the other hand, identification of continuousΦtime sysΦ In the recent paper (Chen et al., 2017) an algorithm to t((teWWemaaashhhillsbbeerttyyrgg,pp,iicc11988;aa9llll8yy8;peLLjrrjffuunoonrrmmgg eaandnddbbyWWuuiisllillnss,,g22010;aa0pp1pp0r;oGGxxiioommooaddttwwieedinnmeetotddaaleell..s,, idnenthtiefyraecceonnttipnaupoeursΦ(tCimheenARetMaAl.,sy20st1e7m) uansinaglgiorrreitghumlartlyo tems is typically performed by using approximated models identify a continuousΦtime ARMA system using irregularly 2013).Infact,fornonlinearsystemstheredoesnotexist idesdaemnnpttiilffeyydaadccaootnnattiiinnnuutoohusuesΦΦmtimtimaxeeimAAuRRmMAMAliksseysylishtetoemmodusufsringianmgeirreirwroeggrkulaula(vrlyrliya (Wahlberg, 1988; Ljung and Wills, 2010; Goodwin et al., sampled data in the maximum likelihood framework (via an exact representation of the equivaallent sampledΦdata sdaairmmecpleptlemddadadxaimtataiziniantithetohneommftaahxxeiimmliuukmmelihlliiokkoeedlihlihfuoonooddctifraforanmm,aeenwwdoorkrvkia(via(tvhiae 2013). In fact, for nonlinear systems there does not exist direct maximization of the likelihood function, and via the model,andforlinearsystems,themappingfromtheconΦ direEixrpeccettctmmataaiximxoinmΦMizizaaatiotxiiomnnizooafftthetihoenlliiakkleegliholoihriotoohddmfuncf)unisctiotpiorn,nes,eaanndntdedviav.iaTthethhies an exact representation of the equivalent sampledΦdata ExpectationΦMaximization algorithm) is presented. This tinuuoous time parameters to the sampling zeros is highly apxppreocatcahtiocnaΦnMbaexidmirizeacttiloynapaplgloierdithtmo )idiesntpirfyesecnotnetdin.uTouhsisΦ ctionmuopulesx.time parameters to the sampling zeros is highly ExpectationΦMaximization algorithm) is presented. This comptinuouslex.time parameters to the sampling zeros is highly time oscillators. However, the success of the algorithms in The problem of obtaining approximate sampledΦdata timireshosnceirllators. However, the success of the algorithms in Tcomphe lproex.blem of obtaining approxxiimmate sampledΦdata (tKimeirshoscinerllatetorals..,H2011;oweveChr,tenheetsuccessal., 2017)ofthedepalgorendithonmsthine TmThehoedelproprhoablebslemmattrooaffctooedbtabtatiningihneingattaaepppnptirrooonxxiimmofaatetseevsseaarammlpleplreedΦdseΦdadaarctatha (Kirshner et al., 2011; Chen et al., 2017) depend on the model has attracted the attention of seveerral research initia(Kirshliznaertioetn oalf.,the2011;estimChateenoetf theal.,sys2017)tem{\textquoteright}sdeppaenradmonetersth.e moThedelprohasblemattroafc˚˚toedbtatininghe attaepnptironoximofatesevsearalmplereseardΦdactah initialization of the estimate of the system{\textquoteright}s parameters. groups (see e.g. Astr{\"o}{\"o}mm et al. (1984); Blachuta (1999); Innittiahliiszaptaiopnero,fwtheeoebsttaiminatteheofetxhaectsydstisecmre{\textquoteright}stepΦtairmame emteords.el model has attracted the attention of several research In this paper, we obtain the exact discreteΦtime model WWgroeeelllullereprs ete(tsalaeel.. ((e2001)2.g0.01A˚);;stNesiNr{\"o}esmicc anaetnddalLaiL. a(ill1aa98((2002)240);02B);;laCarCcahrurrtascoasc(o19etet99ala)l..; Inforthseiscopnadpoerrd, ewr estoabtitsatiincs)thcoenesxidaecrtindgisfcinreitteeΦstaimmeplemtoidmeel groups (see e.g. Astr{\"o}m et al. (1984); Blachuta (1999); (for second order statistics) considering finite sample time W(W20el1l6er))e.t al. (2001); Nesic and Laila (2002); Carrasco et al. In this paper, we obtain the exact discreteΦtime model (2016)). (foarnsdectohnednocradlecrulsattaitnigstitchse) lcoognΦsliikdeelriihnogofdinfiutencstaimonplfeortitmhee ★((2016)2016))).. (for second order statistics) considering finite sample time ★(2T01h6e)w)o.rk of Juan C. Ag{\"u}ero was partially supported by FONDE∆ system of interest. Our main goal is to gain insight into the ★ The work of Juan C. Ag{\"u}ero was partially supported by FONDE∆ ∆ and then calculating the logΦlikelihood function for the CYTTheThetrwwooourrgkkhoofgfraJJuannutanNCC.o .1AA18gg1uu¨¨1ee5rr8oo. wwTaahsseppartiallywaorrtkialolfyMssuppuapr´pıaoortedrCtoedrobbnyyelFFwOOaNNs pDDEaEr∆∆ difficulties of identifying the parameters of a continuousΦ CYT trough grant No 1181158. The work of Mar{\'i}a Coronel was par∆ system of interest. Our main goal is to gain insight into the tiaYTllThyetsrwuopouprgkohrotgferdaJnubtaynNCCoO.1A1N8gI1C{\"u}1eY5r8To.∆wTPahFseCpwHaorArtki/aDlolofycMstuoaprr{\'a}pıadoorCtNoedraocbniyeolnFwaOla/Ns20pD1aE7r∆∆ tially supported by CONICYT∆PFCHA/Doctorado Nacional/2017∆ difficulties of identifying the parameters of a continuousΦ CYTtroughgrantNo1181158.TheworkofMar{\'i}aCoronelwaspar∆ tihmatetohsecliilklaetloihro.oIdnfuthnectinounmeexrhicibailtsesxeavmerpalleslo,cwalemoabxsiemrvae, 21170804 and the DGIIP at USM. The work was supported by the time oscillator. In the numerical examples, we observe tially supported by CONICYT∆PFCHA/Doctorado Nacional/2017∆ that the likelihood function exhibits several local maxima, Advanced Center for Electrical and Electronic Engineering (AC3E, which helps to understand the difficulties of applying the 21170804ArdovyaenccteodBaCandseanltheteFrBfDGI0or0r0E8Ilectrical)Pe,cCtatrhiciUlaelSMand. TheElectronicwork wEasngineeringsupported(AbyC3theE, which helps to understand the difficulties of applying the Proyecto Basal FB0008), Chile Advanced Center for Electrical and Electronic Engineering (AC3E, which helps to understand the difficulties of applying the 2405-8963 {\textcopyright} 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright {\textcopyright} 2018 IFAC 712 CPoepeyr rriegvhiet w{\textcopyright} u2n0d1e8r IrFeAspConsibility of International Federation of Automa7t1ic2 Control. Copyright {\textcopyright} 2018 IFAC 712 10.1016/j.ifacol.2018.09.199 Copyright {\textcopyright} 2018 IFAC 712 algorith˚s in (Kirshner et al., 2ffi11; ffhen et al., 2ffi17) to this Φroble˚. We analyze the likelihood function for two cases i) instantaneous sa˚Φling and ii) integrating sa˚Φling. Publisher Copyright: {\textcopyright} 2018",

year = "2018",

month = jan,

day = "1",

doi = "10.1016/j.ifacol.2018.09.199",

language = "English",

volume = "51",

pages = "712--717",

journal = "18th IFAC Symposium on System Identification SYSID 2018: Stockholm, Sweden, 9-11 July 2018",

issn = "2405-8963",

number = "15",

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TY - JOUR

T1 - Maximum Likelihood Identification of a Continuous-Time Oscillator Utilizing Sampled Data

AU - González, Karen

AU - Coronel, María

AU - Carvajal, Rodrigo

AU - Escárate, Pedro

AU - Agüero, Juan C.

N1 - Funding Information: Maximum Likelihood Identification of a Maximum Likelihood Identification of a Maximum Likelihood Identification of a Continuous-Time Oscillator Utilizing ContinuouSs-amTimpleedOscDatillaato★★r Utilizing Sampled Data ★ Sampled Data Karen González ∗, María Coronel ∗,∗∗, Rodrigo Carvajal ∗, Karen González∗∗, Mar∗∗∗ía Coronel ∗,∗∗∗,∗∗, Rodrigo∗,∗∗∗∗Carvajal∗∗, Karen PGeodnrzóalEeszcá,rMataería Coronel , Rodrigo Carvajal , KarenPGonedroz´alEeszc∗á,raMtaer∗´ı∗a∗CanordonJuelan∗C,∗∗.,AgRoüderriogo∗,∗C∗∗a∗r.vajal∗, Pedro Escárate∗∗∗ and Juan C. Agüero∗,∗∗∗∗. ∗ Electronics Engineering Department, ∗Electronics Engineering Department, UniversidadETlećetcrnoincaicsFeEdnegriicnoeeSrainngtaDMepaarrítam(eUnSt,M), Chile Universida∗d Técnica Federico Santa María (USM), Chile Universidad Técnica Federico Santa María (USM), Chile Universida∗d∗ElT´eeccntriiccaalFEednginericoeeSantaring DMepaarr´ıtma (ent,USM), Chile Electrical Engineering Department, U∗∗niversidad de Los Andes, Venezuela ∗∗∗ Electrical Engineering Department, ∗∗∗ Large BiUnnocivuelarsridTaedledsceoLpeosOAbsnedrevsa,toVreyn,eTz ueclsaon, Arizona ∗∗∗Large Binocular Telescope Observatory, Tucson, Arizona ∗∗∗∗∗∗∗∗∗∗∗TLhaergSechBoiUnionloocverfulEasrleidadcTterliecdesacloELpenosgOiAnbesnedreivnsa,gtoVarenynd,eTzCuuoelcmsaopnu,teArrSizcoiennace, ∗∗∗∗∗∗∗The School of Electrical Engineering and Computer Science, The SchoTohl eofUEnlievcetrrsiictayl oEfnNgienwecearsintlge,aAndusCtroamliaputer Science, ∗∗∗∗The SchoTohel ofUniEleverctrsiictyalofEngiNenwecearsingtle,andAusCtromaliaputer Science, The University of Newcastle, Australia The University of Newcastle, Australia Abstract: In this paper we analyze the likelihood function corresponding to a continuousΦtime Abstract: In this paper we analyze the likelihood function corresponding to a continuuoousΦtime oAsbcislltartaocrt:utIinliztihnigs praegpuerlawr esamnapllyinzeg.thWeelikaenliahlyozoed tfuhencetqiounivcaolerrnetspsaomndpilnegdΦtdoaatacomntoidneuloufosrΦtitmwoe Abstract: In this paper we analyze the likelihood function corresponding to a continuousΦtime cases i) instantaneous sampling and ii) integrated sampling. We illustrate the behavviioor of the oscillator utilizing regular sampling. We analyze the equivalent sampledΦdata model for two logΦlikelihood function via numerical examples showing that it presents seveerral local maxima. logΦlikelihood function via numerical examples showing that it presents several local maxima. l©og2Φl0i1k8e,l iIhFoAoCd (fIuntnecrtniaotnionvaial FneudmeraetriiocnalofexAaumtopmlaetsics hCoowntirnogl) tHhoasttinitg pbrye Eselsnetvsiesre Lvtedr.a Al lllo rciaghltms raexsiemrvaed.. Keywords: Maximum Likelihood Methods, Continuous Time System Estimation, Time Series. Keywords: Maximum Likelihood Methods, Continuuoous Time System Estimation, Time Series. Keywords: Maximum Likelihood Methods, Continuous Time System Estimation, Time Series. Keywords: Maximum Likelihood Methods, Continuous Time System Estimation, Time Series. 1. INTRODUCTION A typical approach to identify a continuosΦtime system 1. INTRODUCTION A typical approach to identify a continuosΦtime system 1. INTRODUCTION A typical approach to identify a continuosΦtime system is to use an approximation such as the Euler method Identification of osci1. IlNlatTorROsyDstUCTIOems hNas been a topic of Ais totypusicaelaanpproapapcrhoxtoimaidetionntifsyucahcaosnttheinuoEulesΦtimer msysethotemd Identification of oscillator systems has been a topic of (Wahlberg, 1988; Goodwin et al., 2013; Yuz et al., 2011) in Identification of oscillator systems has been a topic of oWrdaehrltboerogb,t1a9in88a;nGaopopdrwoxinimetataeld., 2d0is1c3r;etYeuΦtzimeteaml.,o2d0e1l.1)Foinr rIdecuenrtrifiencatt iinonterestof osciin sellvateerroral rsyesearstemsch arheasas bsueench asa it)opStiatc iofsΦ oWxrdaaemhrpltbloeer,og,ibnt1988;aLinarasnsGoanopoapdnrwiodxniSmöetdatealerds.,tr2013;döismcre(2tYe0uΦ0tz2imet) etalhme.,op2011)dreolb. lFeominr(Wahlberg,1988;Goodwinetal.,2013;Yuzetal.,2011)in rttiiecsccsur((rHanHenatninnanatner,,es1973;1t9i7n3;sseeRobRvveeorrbaaerellrtteesssseeanaaanrrcddh aSSrppeeanaaanssos,ossu,ch1986;19aa8ss6i;) SKKtuuatbbiiissnnΦΦ ordertoobtainanapproximateddiscreteΦtimemodel.ForordertoobtainanapproximateddiscreteΦtimemodel.For tetiiecutcscaslr.((,rHanHe2na0tn0i5nnn)anatn,er,,iiest)1973;1M9i7na3t;sheeRobRvmeorbaaltereirrcttesearassl panahncyddhsiarSScsppeasana(Bnos,oosusu,jcoh1986;1a9asn86di;)NSKKotiuuatrbbaiiiysΦnn, oxfaemstpimlea,tiinngLtahresspoanraamndetSerösdienrsatrcöomnt(in2u0o0u2)sΦttihmeeparoubtolermeΦ etal.,2005),ii)Mathematicalphysics(BoujoandNoiray, of estimating the parameters in a continuousΦtime autoreΦ tics (Hannan, 1973; Roberts and Spanos, 1986; Kubin of estimating the parameters in a continuousΦtime autoreΦ 2017)etal.,,2005)andii,i)ii)MMobatilheematroboticalic(pTsethysisercsu(Boukoouujetoanald.,Noi2008)ray)., ggreressssiivvee mmoodedell frofromm ddisisccretreteΦeΦtimtimee ssaammpleples,s, rereplaplaccinging thethe 2017), and iii) Mobile robotic (Tsetserukou et al., 2008)). dreersivsiavteivmesobdyeldferlotma adpispcrroextiemΦtaimtioensa,mfoprlmesi,ngrepalalicnienagrtrheeΦ derivatives by delta approximations, forming a linear reΦ ContinuousΦtime oscillators are highly relevant to a broad gressive model from discreteΦtime samples, replacing the ContinuuoousΦtime oscillators are highly relevant to a broad greersivsiaotniv,easnbdyusdienlgtathaeplperaosxtismqautairoenss,mfeotrhmoidngis acolninsiedaerrerdeΦ. CrCaoonnngtteiinnouufooausurseΦΦatimtismoeefoorsseccsillaielalartotcohrsrssiaanrerceehighthigehhylylyarrerelelepvvraaesnnettntottonaaotbrobroonaalddy gderersivsioatn,iveasndbyusdeingltatheappleraosxtimasquatiorenss,mfoethormingd isacolinensidearrered.Φ range of areas of research since they are present not only Inreassdidoint,ioann,ditusiisnwgetlhleklneoawstnsqtuhaartetshmeeptehrofodrimsaconncesidoefrtehde. ContinuousΦtime oscillators are highly relevant to a broad In addition, it is well known that the performance of the irnanngaetuorfeabreuatsaolsforeisneasorcmhestinecchentohleoygiacarel apprepsleicnattinoontso(nselye gression, and using the least squares method is considered. range of areas of research since they are present not only identification methods obtained by this approach provide e.g. Boujo and Noiray (2017); Essccáárrate et al. (2017) and bdiaensetdificeasttiimonatmeset(hLoadrsssobnteatinaeld., b2y00t7h)i.s approach provide tineh.gena.rBeturefoeurjeonbutcaensdathlsNeoorieinriany)s.o(m2017)e tec;hEnsocloágraticaeletapplical. (a2017)tions (sanede bideiasedntificestatioimatn mesetho(Lardsssonobtaetineald.,b2007)y this. approach provide thee.g.reBouferejoncanesdtheNoirrein).ay(2017);Escárateetal.(2017)and bbiiasedased estestiimatmateses ((LarLarssonsson etet alal..,, 2007)2007).. the references therein). On the other hand, in (Kirshner et al., 2011) an exact Osthecillarefetorerncsysestethemsreain).re typically represented using differΦ OnOn ththee oothetherr hahand,nd, inin (Kir(Kirshshnnerer eett aal.,l., 22001111)) aann eexaxacctt Oscillator systems are tyyppiiccally representteed using differΦ evnaluthateioonthoefr thhaendd,iscinret(eKΦdirosmhnaeinr eptowale.r,Φs2p0e1c1t)ruamn euxsiancgt eOnstciiallaettqoourrastyyiosstnesmisnacroentiyynppuiioccuaalslΦytimrreeepp.rreeTshenuttsee,dalugsoirnigthddmiiffsffetroΦ eOnvaluthateionotheofrthahend,discrinet(KireΦdomaishnerneptowal.,erΦsp20ect11)ruamn euxasincgt ential equations in continuuoousΦtime. Thuuss, algorithms to eevvaaplulounataetniionotnialofofBttΦshhpeeliddniiescrsscrietesteΦdedΦderomaioimveadinnanppdoowwaeenrrΦspΦsepsectetcimtrruuatmmionuusisainnpggΦ Oscillator systems are typically represented using differΦ exponential BΦsplines is derived and an estimation apΦ eiednettniiaatlifyeeqcuoantttiiioonnnussouinsΦtciomnetinsyusotuesmΦttsiimmaree. hTihguhsly, arellggleoovrraiittnhhtmm. s to evaluation of the discreteΦdomain powerΦspectrum using ential equations in continuousΦtime. Thus, algorithms to proach that is based on the Whittle’s likelihood function Onidenthetifyocthonetrinhuaond,usΦtimideentsiysficteatiomsnaorefchigonhlytinureousleΦvtimante. sysΦ iissropprearcehsseennthtteeadd.t.is based on the Whittle’s likelihood function On the other hand, identification of continuuoousΦtime sysΦ proach that is based on the Whittle’s likelihood function OtOennmstthhiseetoyttphhiecralhlyanpde,rfiioddreemnettiidffiicbayttiiuoosninogfacpopnrttoiinnxuuimoouastΦettdiimmmeodsyeslsΦ is presented. tems is typically performed by using approxxiimmated models isnptrheesernetceedn.t paper (Chen et al., 2017) an algorithm to On the other hand, identification of continuousΦtime sysΦ In the recent paper (Chen et al., 2017) an algorithm to t((teWWemaaashhhillsbbeerttyyrgg,pp,iicc11988;aa9llll8yy8;peLLjrrjffuunoonrrmmgg eaandnddbbyWWuuiisllillnss,,g22010;aa0pp1pp0r;oGGxxiioommooaddttwwieedinnmeetotddaaleell..s,, idnenthtiefyraecceonnttipnaupoeursΦ(tCimheenARetMaAl.,sy20st1e7m) uansinaglgiorrreitghumlartlyo tems is typically performed by using approximated models identify a continuousΦtime ARMA system using irregularly 2013).Infact,fornonlinearsystemstheredoesnotexist idesdaemnnpttiilffeyydaadccaootnnattiiinnnuutoohusuesΦΦmtimtimaxeeimAAuRRmMAMAliksseysylishtetoemmodusufsringianmgeirreirwroeggrkulaula(vrlyrliya (Wahlberg, 1988; Ljung and Wills, 2010; Goodwin et al., sampled data in the maximum likelihood framework (via an exact representation of the equivaallent sampledΦdata sdaairmmecpleptlemddadadxaimtataiziniantithetohneommftaahxxeiimmliuukmmelihlliiokkoeedlihlihfuoonooddctifraforanmm,aeenwwdoorkrvkia(via(tvhiae 2013). In fact, for nonlinear systems there does not exist direct maximization of the likelihood function, and via the model,andforlinearsystems,themappingfromtheconΦ direEixrpeccettctmmataaiximxoinmΦMizizaaatiotxiiomnnizooafftthetihoenlliiakkleegliholoihriotoohddmfuncf)unisctiotpiorn,nes,eaanndntdedviav.iaTthethhies an exact representation of the equivalent sampledΦdata ExpectationΦMaximization algorithm) is presented. This tinuuoous time parameters to the sampling zeros is highly apxppreocatcahtiocnaΦnMbaexidmirizeacttiloynapaplgloierdithtmo )idiesntpirfyesecnotnetdin.uTouhsisΦ ctionmuopulesx.time parameters to the sampling zeros is highly ExpectationΦMaximization algorithm) is presented. This comptinuouslex.time parameters to the sampling zeros is highly time oscillators. However, the success of the algorithms in The problem of obtaining approximate sampledΦdata timireshosnceirllators. However, the success of the algorithms in Tcomphe lproex.blem of obtaining approxxiimmate sampledΦdata (tKimeirshoscinerllatetorals..,H2011;oweveChr,tenheetsuccessal., 2017)ofthedepalgorendithonmsthine TmThehoedelproprhoablebslemmattrooaffctooedbtabtatiningihneingattaaepppnptirrooonxxiimmofaatetseevsseaarammlpleplreedΦdseΦdadaarctatha (Kirshner et al., 2011; Chen et al., 2017) depend on the model has attracted the attention of seveerral research initia(Kirshliznaertioetn oalf.,the2011;estimChateenoetf theal.,sys2017)tem’sdeppaenradmonetersth.e moThedelprohasblemattroafc˚˚toedbtatininghe attaepnptironoximofatesevsearalmplereseardΦdactah initialization of the estimate of the system’s parameters. groups (see e.g. Aströömm et al. (1984); Blachuta (1999); Innittiahliiszaptaiopnero,fwtheeoebsttaiminatteheofetxhaectsydstisecmre’stepΦtairmame emteords.el model has attracted the attention of several research In this paper, we obtain the exact discreteΦtime model WWgroeeelllullereprs ete(tsalaeel.. ((e2001)2.g0.01A˚);;stNesiNröesmicc anaetnddalLaiL. a(ill1aa98((2002)240);02B);;laCarCcahrurrtascoasc(o19etet99ala)l..; Inforthseiscopnadpoerrd, ewr estoabtitsatiincs)thcoenesxidaecrtindgisfcinreitteeΦstaimmeplemtoidmeel groups (see e.g. Aström et al. (1984); Blachuta (1999); (for second order statistics) considering finite sample time W(W20el1l6er))e.t al. (2001); Nesic and Laila (2002); Carrasco et al. In this paper, we obtain the exact discreteΦtime model (2016)). (foarnsdectohnednocradlecrulsattaitnigstitchse) lcoognΦsliikdeelriihnogofdinfiutencstaimonplfeortitmhee ★((2016)2016))).. (for second order statistics) considering finite sample time ★(2T01h6e)w)o.rk of Juan C. Agüero was partially supported by FONDE∆ system of interest. Our main goal is to gain insight into the ★ The work of Juan C. Agüero was partially supported by FONDE∆ ∆ and then calculating the logΦlikelihood function for the CYTTheThetrwwooourrgkkhoofgfraJJuannutanNCC.o .1AA18gg1uu¨¨1ee5rr8oo. wwTaahsseppartiallywaorrtkialolfyMssuppuapr´pıaoortedrCtoedrobbnyyelFFwOOaNNs pDDEaEr∆∆ difficulties of identifying the parameters of a continuousΦ CYT trough grant No 1181158. The work of María Coronel was par∆ system of interest. Our main goal is to gain insight into the tiaYTllThyetsrwuopouprgkohrotgferdaJnubtaynNCCoO.1A1N8gI1Cü1eY5r8To.∆wTPahFseCpwHaorArtki/aDlolofycMstuoaprrápıadoorCtNoedraocbniyeolnFwaOla/Ns20pD1aE7r∆∆ tially supported by CONICYT∆PFCHA/Doctorado Nacional/2017∆ difficulties of identifying the parameters of a continuousΦ CYTtroughgrantNo1181158.TheworkofMaríaCoronelwaspar∆ tihmatetohsecliilklaetloihro.oIdnfuthnectinounmeexrhicibailtsesxeavmerpalleslo,cwalemoabxsiemrvae, 21170804 and the DGIIP at USM. The work was supported by the time oscillator. In the numerical examples, we observe tially supported by CONICYT∆PFCHA/Doctorado Nacional/2017∆ that the likelihood function exhibits several local maxima, Advanced Center for Electrical and Electronic Engineering (AC3E, which helps to understand the difficulties of applying the 21170804ArdovyaenccteodBaCandseanltheteFrBfDGI0or0r0E8Ilectrical)Pe,cCtatrhiciUlaelSMand. TheElectronicwork wEasngineeringsupported(AbyC3theE, which helps to understand the difficulties of applying the Proyecto Basal FB0008), Chile Advanced Center for Electrical and Electronic Engineering (AC3E, which helps to understand the difficulties of applying the 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2018 IFAC 712 CPoepeyr rriegvhiet w© u2n0d1e8r IrFeAspConsibility of International Federation of Automa7t1ic2 Control. Copyright © 2018 IFAC 712 10.1016/j.ifacol.2018.09.199 Copyright © 2018 IFAC 712 algorith˚s in (Kirshner et al., 2ffi11; ffhen et al., 2ffi17) to this Φroble˚. We analyze the likelihood function for two cases i) instantaneous sa˚Φling and ii) integrating sa˚Φling. Publisher Copyright: © 2018

PY - 2018/1/1

Y1 - 2018/1/1

N2 - In this paper we analyze the likelihood function corresponding to a continuous-time oscillator utilizing regular sampling. We analyze the equivalent sampled-data model for two cases i) instantaneous sampling and ii) integrated sampling. We illustrate the behavior of the log-likelihood function via numerical examples showing that it presents several local maxima.

AB - In this paper we analyze the likelihood function corresponding to a continuous-time oscillator utilizing regular sampling. We analyze the equivalent sampled-data model for two cases i) instantaneous sampling and ii) integrated sampling. We illustrate the behavior of the log-likelihood function via numerical examples showing that it presents several local maxima.

KW - Continuous Time System Estimation

KW - Maximum Likelihood Methods

KW - Time Series

UR - http://www.scopus.com/inward/record.url?scp=85054387600&partnerID=8YFLogxK

U2 - 10.1016/j.ifacol.2018.09.199

DO - 10.1016/j.ifacol.2018.09.199

M3 - Article

AN - SCOPUS:85054387600

VL - 51

SP - 712

EP - 717

JO - 18th IFAC Symposium on System Identification SYSID 2018: Stockholm, Sweden, 9-11 July 2018

JF - 18th IFAC Symposium on System Identification SYSID 2018: Stockholm, Sweden, 9-11 July 2018

SN - 2405-8963

IS - 15

ER -

Maximum Likelihood Identification of a Continuous-Time Oscillator Utilizing Sampled Data

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