MATLAB Software for Recursive Identification and Scaling Using a Structured Nonlinear Black-box Model Revision 1

Transcription

1 MATLAB Sofware for Recurive Ideificaio ad Scalig Uig a Srucured Noliear Blac-box Model Reviio Torbjör Wigre Syem ad Corol, Deparme of Iformaio Techology, Uppala Uiveriy, SE-755 Uppala, SWEDEN. orbjor.wigre@i.uu.e. Jauary 25 Abrac Thi repor i ieded a a uer maual for a pacage of MATLAB crip ad fucio, developed for recurive predicio error ideificaio of oliear ae pace yem. The core of he pacage i a implemeaio of a oupu error ideificaio ad calig algorihm. The algorihm i baed o a coiuou ime, rucured blac box ae pace model of a oliear yem. The ofware ca oly be ru off-lie, i.e. o rue real ime operaio i poible. The algorihm i however implemeed o ha rue o-lie operaio ca be obaied by exracio of he mai algorihmic loop. The uer mu he provide he real ime evirome. The ofware pacage coai crip ad fucio ha allow he uer o eiher ipu live meaureme or o geerae e daa by imulaio. The crip ad fucio for he eup ad execuio of he ideificaio algorihm are omewha more geeral ha wha i decribed i he referece. There i e.g. uppor for auomaic re-iiiaio of he algorihm uig he parameer obaied a he ed of a previou ideificaio ru. Thi allow for muliple ru hrough a e of daa, omehig ha i ueful for daa e ha are oo hor o allow covergece of he algorihm.the re-iiiaio ep alo allow he uer o modify he degree of he polyomial model rucure ad o pecify erm ha are o be excluded from he model. Thi mae i poible o ieraively re-fie he eimaed model uig muliple ru. The fucioaliy for diplay of reul iclude crip for ploig of daa, parameer, predicio error, eigevalue ad he codiio umber of he Heia. The eimaed model obaied a he ed of a ru ca be imulaed ad he model oupu ploed, aloe or ogeher wih he daa ued for ideificaio. Model validaio i uppored by wo mehod apar from he diplay fucioaliy. Fir, calculaio of he RPEM lo fucio ca be performed, uig parameer obaied a he ed of a ideificaio ru. Secodly, he accuracy a a fucio of he oupu igal ampliude ca be aeed. Keyword: Ideificaio, Recurive algorihm, Noliear yem, Sae pace model, Sofware, Predicio error mehod, Scalig, MATLAB.

2 Prerequiie Thi repor oly decribe he par of [ ], [ 2 ], [ 3 ] ad [ 4 ] ha are required for he decripio of he ofware. Hece he uer i aumed o have a worig owledge of he algorihm of hee publicaio ad of MATLAB, ee e.g. [ 5 ]. Thi, i ur, require ha he uer ha a worig owledge of yem ideificaio ad i paricular of recurive ideificaio mehod a decribed i e.g. [ 6 ]. Reviio Thi repor decribe reviio. of he accompayig SW. The ofware ha bee eed wih MATLAB 5.3, MATLAB 6.5, ad MATLAB 7. ruig o PC ad UNIX woraio. Iallaio The file SW.zip i copied o he eleced direcory ad uzipped. MATLAB i opeed ad a pah i e up o he eleced direcory uig he pah brower. The ofware i he ready for ue. Noe: Thi repor i wrie wih repec o he ofware, a icluded i he SW.zip file. I may herefore be advaageou o ore he origially upplied ofware for referece purpoe. Error repor Whe error are foud, hee may be repored i a o eiher of he followig addree: orbjor.wigre@i.uu.e. lida.bru@i.uu.e. 2

3 . Iroducio Ideificaio of oliear yem i a acive field of reearch oday. There are everal reao for hi. Fir, may pracical yem how rog oliear effec. Thi i e.g. rue for high agle of aac fligh dyamic, may chemical reacio ad elecromechaical machiery of may id, ee e.g. [ ], [ 2 ], [ 3 ], [ 4 ] ad he referece herei for furher example. Aoher impora reao i perhap ha liear mehod for yem ideificaio are quie well uderood oday, hece i i aural o move he focu o more challegig problem. There are already a umber of ideificaio mehod available for ideificaio of oliear yem. Thee iclude grey-box differeial equaio mehod, where umerical iegraio i combied wih opimizaio i order o opimize he uow phyical parameer ha appear i he differeial equaio. A aleraive approach i o ar wih a dicree ime blac box model, ad o apply exiig mehodology from he liear field o he oluio of he oliear ideificaio problem. Thi i he approach ae i he NARMAX mehod ad i relaed algorihm. There a lea quare formulaio ca ofe be foud, a fac ha faciliae he oluio. Oher mehod apply eural ewor for modelig of oliear dyamic yem. See [ ], [ 2 ] ad he referece herei for a more deailed urvey. Thi repor focue o ofware ha impleme a ew oliear recurive yem ideificaio mehod. Corary o he above mehod, hi blac box mehod eimae coiuou ime parameer i a geeral ae pace model, wih a ow ad poibly oliear meaureme equaio. The ideificaio mehod belog o he cla of recurive predicio error mehod (RPEM) ad he mehod i of oupu error ype. Advaage iclude he fac ha abiliy of he eimaed model i checed by a projecio algorihm i each ieraio ep. The lea quare approache above cao guaraee a reulig able model - hi eed o be checed afer he ideificaio ha bee compleed. A furher advaage i ha he coecio o he phyical parameer ca be reaied o a greaer exe ha if a dicree ime oliear model i ued a he arig poi. There are alo diadvaage. A major diadvaage wih oupu error mehod i ha hey omeime coverge o a local ub-opimal miimum poi of he crierio fucio, meaig ha careful iiializaio i eeded. The effec of local miimum poi i reduced for he mehod decribed i [ ] [ 4 ], by a mehod ha cale he ae, he eimaed parameer of he model ad, mo imporaly, he Heia of he crierio fucio. The calig i implemeed by a calig of he amplig period ued whe ruig he ideificaio algorihm, ee [ ] [ 3 ] for deail. Oe impora apec of hi calig mehod i ha correpodig u-caled parameer value ca be calculaed i a poideificaio ep. 3

4 The oliear ideificaio algorihm i baed o a coiuou ime blac box ae pace model. Thi model i rucured i ha oly oe righ had ide compoe of he ordiary differeial equaio (ODE) model i parameerized a a uow fucio. A how i [ ] ad [ 2 ] hi avoid overparameerizaio. The rericio impoed o he model rucure may eem rericive. However, i i moivaed i [ ] ad [ 2 ] ha he eleced rucure ca alway (locally i he ae) model yem wih more geeral righ had ide, a fac ha exed he applicabiliy of he mehod igificaly. The eleced parameerizaio of he righ had ide fucio of he ODE i a liear-ihe-parameer muli-variae polyomial i he ae ad ipu igal. The approach ae allow for MIMO oliear yem ideificaio. The covariace marix of he meaureme diurbace i eimaed o-lie. Recurive yem ideificaio i a ofware depede echology. Hece, whe publihig ew mehodology i hi field, i i releva o alo provide ueful ofware for applicaio of he preeed algorihm. Thi faciliae a quic pracical exploiaio of ew idea. The developme of he pree MATLAB ofware pacage i moivaed by hi fac. The pree ofware pacage i developed ad eed uig MATLAB 5.3, MATLAB 6.5 ad MATLAB 7.. The ofware pacage doe o rely o ay MATLAB oolboxe. I coi of a umber of crip ad fucio. Briefly, he ofware pacage coi of crip for eup, crip for geeraio or meaureme of daa, crip for execuio of he RPEM ad crip for geeraio ad ploig of reul. There i preely o GUI, he crip mu be ru from he commad widow. Furhermore, ipu parameer eed o be cofigured i oe or everal of he eup crip, a well a whe ruig he crip. I cae of daa geeraio by imulaio, he ODE ha defie he daa geeraig yem mu be pecified i adard MATLAB yle. The ofware ca oly be ru offlie, i.e. here i o uppor for execuio i a real ime evirome. The major par of he algorihmic loop ca however eaily be exraced for uch purpoe. The repor i orgaized accordig o he flow of a a uer ecouer whe applyig he crip of he pacage. Before he ofware i decribed ome baic fac abou he model ad he calig mehod are reviewed. 2. Model The oliear MIMO model o be defied here i ued for eimaio of a uow parameer vecor y m from meaured ipu u( ) ad oupu, give by ( ) ( () () ) ( () () ) = () u u u u u ( ) T y m () = y m, ()... y m, p () T ( ) 4

5 The upercrip ( ) deoe differeiaio ime. The arig poi for he derivaio of he model i he followig :h order ae pace ODE () x x2 x () = x () ( ) ( ) x f x x u u u u,,,,,,,,,, where x = ( x x x ) y c... c x = yp cp... cp x, T... i he ae vecor. The followig polyomial parameerizaio of he righ had ide fucio of (2 ) i ued ( ) ( ) f ( x x u u u u,...,,,...,,...,,...,,) ( 2 ) Here = I x ix = I I x u i = iu = x I u i = u u I i = u I u i u i ( ) i i... u...( u )...( u ) u u u i i i x u x u x x u ix... i i i i i x u... u u = ϕ T ( xu, ). = I I... u u I... I I I... I X u u u u T ( 3 ) ϕ = I I... u u u... ( ) u u u I ( )... u u... u I u I u u I ( ) u... ( x ) I x I I ( x ) ( u ) x u u. T ( 4 ) Pleae ee example 5 below for a low order example of he above parameerizaio. I order o obai a dicree ime model ha i uiable for calig, he Euler iegraio mehod i applied o (2 ). The mai reao for uig he Euler mehod i ha he amplig appear explicily ad liearly i he righ had ide of he reulig differece equaio model ( 5 ). Thi i coveie whe he calig algorihm i iroduced. The reul of he dicreizaio i 5

6 ( + T, ) x S x T x ( + S, ) ( + T, ) S x = x x (, ) (, ) (, ) (, ) x2 + TS x, ( () ) T ϕ,,,,,,, () ( x x u u ) ( 5 ) (, ) y c... c x = yp(, ) cp... cp x (, ) (, ) (, ) = Cx. ( 6 ) I ca be remared ha he Euler mehod may require fa amplig i order o o iroduce igifica dicreizaio error. Thi i foruaely a le impora effec i yem ideificaio applicaio. The reao i ha he miimizaio algorihm ue he parameer a irume o fi he model oupu o he meaured daa, a expreed by he crierio fucio. Eve if a addiioal bia would be iroduced i he eimaed parameer, he ipu-oupu properie of he ideified model ca be expeced o decribe he daa well. 3. Scalig Durig developme of he RPEM decribed i [ ] - [ 4 ], i wa oiced ha problem wih covergece o fale local miimum poi of he crierio were ofe highly relaed o he elecio of he amplig period. The amplig period of coure eed o be hor eough durig meaureme, i order o capure he eeial dyamic of he ideified yem. Hece he meaureme amplig period cao be arbirarily eleced. However, ice he amplig period appear explicily i he model ( 5 ) ad i he correpodig gradie differece equaio, i i raighforward o apply ideificaio algorihm baed o ( 5 ) wih aoher, caled value of he amplig period. Thi idea affec he updaig of he ae, he gradie ad ay projecio algorihm ha i ued o corol he abiliy of he model. A cale facorα appear before he muliplicaio wih he amplig period T S i hoe hree quaiie. To explai he deail, he cale facor α ad he caled amplig period T S Scaled are fir defied a T Scaled S = α TS. ( 7 ) The model ( 5 ), ( 6 ), a applied i he ideificaio algorihm i he raformed io x ( + TS, ) x (, ) x T = x ( ), x T x (, ) ( + S, ) ( +, ) S ( 8 ) 6

7 7 () () + T x x f x x u u S Scaled 2,,,,.,,,,, y y c c c c x x p p p,,......,,, = = Cx ( 9 ) where he upercrip deoe caled quaiie. Noe ha he origial amplig period mu be reaied i all ime argume, o a o refer o he correc meaureme ime. The gradie follow by differeiaio of ( 8 ) ad ( 9 ) d T d d d S x x + =,, + T dx d dx d S Scaled T 2,,,, ϕ x u + T d d d d S Scaled T ϕ x u x x,,, ( ) ψ T d d d d,,, = = y C x. ( ) Noe ha he above chage from o i o o be reaed a a chage of variable i he differeiaio leadig o ( ) ad ( ). The origially derived gradie i applied, bu wih a caled amplig period. The la affeced quaiy of he algorihm i he projecio algorihm ha become (cf. [ ] ) S x x S Scaled T I T d d ϕ = +, ( 2 )

8 D M { ( ( )) = eig S < δ }, δ > ( 3 ) I ( 2 ), S ( ) [ () ] D M () () D = TS () D M M ( 4 ). deoe he liearized yem marix of he model, D M deoe he model e, here defied a he aympoically able model wih a margi δ o he abiliy limi. The la equaio op he updaig of he parameer vecor i cae he updae would reul i value ouide he model e. Oher deail of a RPEM where he calig algorihm i ued ca be foud i [ ] [ 4 ]. Whe a caled value of he amplig period i applied, he algorihm ill aemp o miimize he crierio, hereby obaiig oher miimizig parameer value ha whe he rue amplig period i ued. Whe eig he calig algorihm experimeally, dramaic improveme wa omeime oberved i he algorihmic behavior. Covergece peed could be improved ad iiial value ha lead o divergece ad iabiliy could be made o wor well. The applicaio of he calig algorihm reul i oher eimaed parameer value ha wha would be obaied wihou calig. Foruaely, a how i [ 2 ], he origial parameer ca be calculaed from he eimaed oe. The raformaio i give by a diagoal raformaio marix ha i a fucio of he applied cale facor. The aalyi of he effec of he calig i coiued i [ 3 ], where he effec of he codiioig of he Heia of he crierio fucio i aalyed i deail. Thi how ha he effec of he calig i quie dramaic. Chage of he codiio umber by everal order of magiude wa obaied here, for a imple imulaed ecod order example. 4. Sofware pacage overview The ofware pacage i commad drive, i.e. o GUI i available. I coi of a umber of MATLAB crip ad fucio. Thee are decribed i he ex ubecio. 4. Scrip, fucio ad commad flow Roughly, he crip ad fucio ca be divided io five group: Live daa meaureme. The wo crip of hi group e up ad perform cloced live daa meaureme. The crip are SeupLive.m ad MeauremeMai.m. Simulaed daa geeraio. The four crip of hi group defie a dyamic yem, ha i he ued for geeraio of imulaed daa. The crip ad fucio are SeupDaa.m, f.m, h.m ad GeeraeDaa.m. 8

9 Recurive ideificaio. The five crip of hi group perform he acual ideificaio a, upporig uer ieracio. The crip ad fucio are SeupRPEM.m, RPEM.m, h_m.m, dhdx_m.m ad ReIiiae.m. Supporig fucio - o called by he uer. The four fucio of hi group are called by crip of he previou group. They are all relaed o he implemeaio of he RHS model of he ideified ODE. The crip are GeeraeIdice.m, f_m.m, dfdx_m.m ad dfdhea_m.m. Preparaio ad diplay of reul. There are eleve crip i hi group. They all prepare, compue ad diplay reul of he ideificaio proce. The crip are PloDaa.m, SimulaeModelOupu.m, PloParameer.m, PloPredicioError.m, PloEigevalue.m, PloCodiio.m, CompueRPEMLoFucio.m, PloModelOupu.m, PloSyemAdModelOupu.m, PloReidualError.m ad MeaReidualAalyi.m. Thee group of crip ad fucio eed o be operaed i a paricular order o mae ee. Thi order of execuio bewee crip ad fucio i diplayed wih arrow i Figure. A igle direcioal arrow idicae ha he crip/fucio poied a may be execued oly afer he execuio of he poiig crip/fucio. See Figure for deail. There are hree major way o exploi he five group of crip ad fucio.. I cae he uer ha ipu ad oupu igal available, he fir ep i o defie ad ru he crip SeupLive.m. Thi e baic parameer lie he amplig period. The uer ca he proceed direcly o ue he group Recurive Ideificaio ad Preparaio ad diplay of reul. The daa, which ca be imulaed or live, hould be ored i he (row) marice u ad y. 2. I cae he uer i o perform live meaureme, all he ep of he Live daa meaureme group hould be execued fir. The uer ca he proceed direcly o ue he group Recurive Ideificaio ad Preparaio ad diplay of reul. 3. I cae he uer ied o ue imulaed daa, hi daa ca be geeraed by execuio of he crip ad fucio of he group Simulaed daa geeraio. The uer ca he proceed direcly o ue he group Recurive Ideificaio ad Preparaio ad diplay of reul. 9

10 Live daa meaureme Simulaed daa geeraio Ipu igal u Live meaureme eup SeupLive.m Simulaed meaureme eup SeupDaa.m Syem defiiio - ODE righ had ide f.m Syem defiiio - meaureme equaio h.m Live meaureme MeauremeMai.m Simulaed meaureme geeraio GeeraeDaa.m Recurive ideificaio Recurive ideificaio eup SeupRPEM.m Recurive ideificaio reiiiae.m Recurive ideificaio RPEM.m Model defiio - meaureme equaio h_m.m Model derivaive defiiio - meaureme equaio dhdx_m.m Supporig fucio - o called by he uer Model defiio - ae model f_m.m Model defiio - ae model GeeraeIdice.m Reul evaluaio PloDaa.m Model defiio -ae model dfdhea_m.m Model defiio - ae model dfdx_m.m Preparaio ad diplay of reul Reul evaluaio PloCodiio.m Reul preparaio SimulaeModelOupu.m Reul evaluaio PloParameer.m Reul evaluaio PloPredicioError.m Reul evaluaio PloEigevalue.m Reul evaluaio Compue RPEMLoFucio.m Reul evaluaio PloModelOupu.m Reul evaluaio PloSyemAd ModelOupu.m Reul evaluaio PloReidualError.m Reul evaluaio MeaReidual Aalyi.m Figure : MATLAB crip, fucio ad heir relaio i erm of execuio order.

11 4.2 Rericio The mai rericio of he ofware are The ofware i commad lie drive - o GUI uppor i implemeed. The ofware doe o uppor rue real ime operaio - here i o real ime OS uppor implemeed. The ofware ha bee eed ad ru uig MATLAB 5.3, MATLAB 6.5 ad MATLAB Daa ipu The geeraio of daa begi he ecio where he acual ofware i decribed. Sice he uer ha acce o all ource file, he decripio below do o decribe code relaed iue ad ieral variable. Oly he par ha are required for he ue of he ofware pacage are covered. Whe m- file are reproduced, oly he releva par are icluded, he reader hould be aware ha more iformaio ca be foud i he ource code. Noe ha he eup file are o be reaed a emplae, he uer i hece required o modify righ had ide oly - o addiio or deleio of code hould be ued i he ormal ue of he pacage. 5. Simulaed daa The geeraio of imulaed daa require ha he uer. Modifie he uderlyig ODE model, a give by f.m ad h.m. The fucio f.m impleme he RHS of he ODE, uig a coveioal MATLAB fucio call. Noe ha he buil i ODE olver of MATLAB are o ued. Iead a Euler algorihm i implemeed. The reao i ha hi allow he geeraio of imulaed daa ha ca be exacly decribed by he applied model, hould hi be deired. The fucio h.m impleme he (poibly oliear) meaureme equaio. The fucio allow for addiio of yem oie ad meaureme oie. 2. Provide furher ipu daa i he crip SeupDaa.m. The parameer ha defie he daa geeraio are direcly wrie io hi crip. Thee parameer defie he amplig period, he daa legh, he dimeio of he yem, he ype ad parameer of he ipu igal, he ype ad parameer of he diurbace, a well a he iiial value of he ODE. 3. Execue SeupDaa.m. Thi load he eceary parameer io he MATLAB worpace. 4. Geerae daa by execuio of GeeraeDaa.m. Afer he execuio of hi crip, variable wih amplig iace, ipu igal ad oupu igal are available i he MATLAB worpace. Example : Thi ad he followig example illurae he ue of he ofware pacage for ideificaio of he yem

12 ... () () () 2 () () = () + (). x + x + + u x= u y x e ( 5 ) Thi yem i alo ued i [ 3 ], o ae effec of calig. Thi yem ca be wrie i ae pace form a. x. x 2 () () x2() ( 2 u() ) u() = + x() x2() () = () + () y x e 2. ( 6 ) Noe ha he orderig of ae i o exacly a defied i he model ( 2 ). Thi i ieioal ice uch iuaio are commo i pracical iuaio. I ca be ee ha he yem i ocillaory wih a ipu ampliude depedig reoace frequecy ad dampig. The releva par of he file f.m ad h.m become f.m fucio [f]=f(,x,u,w) f(,)=x(2,)*(2+u(,))-u(,); f(2,)=-x(,)-x(2,); ed h.m fucio [h]=h(,x,u,e); h=x(2,)+e(,); ed Daa i o be geeraed by imulaio uig a amplig period of T =. S. ipu-oupu ample are o be geeraed. The ipu igal i o be eleced wih a uiform diribuio i ampliude, wih a mea of, a rage [, ] ad a cloc period 3.. The meaureme diurbace i o be whie, zero mea wih a adard deviaio of.. The eup crip SeupDaa.m ha perform hi a i SeupDaa.m % dimeio... u=; % Ipu igal dimeio x_=2; % Sae dimeio y=; % Oupu dimeio, ormally 2

13 % Ipu igal relaed...type may be eleced amog: % % IpuType=[ % 'PRBS '; % 'Gauia '; % 'UiformPRBS'; % 'SieWave '; % 'Cuom ']; IpuType=[ 'UiformPRBS']; uampliude=[.]; umea=[ ]; ufrequecy=[.]; ClocPeriod=[ 3];% Cloc period vecor i erm of amplig ime % Syem diurbace relaed...type may be eleced amog: % % DiurbaceTypeSyem=[ % 'WGN '; % 'SieWave'; % 'Cuom ']; DiurbaceTypeSyem=[ 'WGN '; 'WGN ']; wsigma=[.;.]; % Gauia yem oie adard deviaio (dicree ime) wmea=[ ; ]; wsieampliude=[ ; ]; 3

14 wsiefrequecy=[ ; ]; % Meaureme diurbace relaed... Type may be eleced amog: % % DiurbaceTypeMeaureme=[ % 'WGN '; % 'SieWave'; % 'Cuom ']; DiurbaceTypeMeaureme=[ 'WGN ']; esigma=.; % Gauia meaureme oie adard deviaio emea=[ ; ]; esieampliude=[ ]; esiefrequecy=[ ]; % amplig ime ad daa legh T=.; % Samplig ime i ecod N=; % Number of daa poi SampligIace=(T:T:N*T); % ODE relaed... x=[.5 -.]'; % Iiial value The fial ep of he daa geeraio i o execue he file SeupDaa.m ad GeeraeDaa.m. Thi i doe i he MATLAB commad widow a follow» SeupDaa» GeeraeDaa 4

15 perceready =» 5.2 Live daa The geeraio of imulaed daa require ha he uer. I coeced o he yem via MATLAB. The coecio mu be uch ha commad o corol DA-coverer ha geerae ipu igal ca be iued from wihi MATLAB. Similarly, commad ha read AD-coverer ha ample oupu igal mu be available from wihi MATLAB. The crip MeauremeMai.m probably eed modificaio i a few par i order o ierface correcly o he AD- ad DA-coverer of he yem of he uer. 2. Geerae a ipu igal, ha i ored i he marix (row vecor i he oe-dimeioal ipu igal cae) u. 3. Provide furher daa i he crip SeupLive.m. The parameer ha defie he daa geeraio are direcly wrie io hi crip. Thee parameer defie he amplig period, he daa legh ad he dimeio of he yem. 4. Execue SeupLive.m. Thi load he eceary parameer io he MATLAB worpace. 5. Geerae daa by execuio of MeauremeMai.m. Afer he execuio of hi file, variable wih ipu igal ad oupu igal are available i he MATLAB worpace. Thi crip operae a a loop ha coiuouly poll he MATLAB real ime cloc, waiig for he ex amplig iace. Thi mea ha i may o be poible o ue he compuer for oher a durig he daa collecio eio. The reao for hi oluio i ha i avoid he eed for a real ime OS coecio. Noe alo ha he call o AD- ad DA-coverer may be differe o oher yem. Thi crip i hece liely o require ome modificaio. Example 2: The eup crip file SeupLive.m become (empy ice imulaed daa i ued here) % dimeio...noe ha x i o really releva, % i i however required i he RPEM eup o i i e i hi file u=[]; % Ipu igal dimeio x=[]; % Sae dimeio y=[]; % Oupu dimeio, ormally % amplig ime ad daa legh T=[]; % Samplig ime i ecod N=[]; % Number of daa poi 5

16 SampligIace=(T:T:N*T); The meaureme proce i ared by ypig» SeupDaa» GeeraeDaa i he MATLAB commad widow. Durig he meaureme eio, he crip coiuouly diplay he ime, he ipu a well a he meaured oupu, a commaded o DA-coverer ad read by AD-coverer. Afer ermiaio all daa ha i eeded for ideificaio i available i he MATLAB worpace. 5.3 Diplay of daa Afer execuio of eiher oe of he chai of acio of ecio 5. or ecio 5.2, daa ca be ploed. Thi i doe wih. The PloDaa.m crip ha i execued i he MATLAB commad widow. Thi crip mae ue of he dimeio of he yem i order o divide he plo io everal ub-widow, ad i order o provide he axi ex. Example 3: The MATLAB commad widow commad i» PloDaa» The followig plo i geeraed 6

17 2 y() ime [].5 u() ime [] Figure 2: The reul of a PloDaa commad. 6. Recurive Ideificaio A hi poi everyhig i i place for a fir ideificaio ru. 6. RPEM eup The preparaio for he ideificaio ru require ha he uer. Modifie he oupu equaio ad he correpodig derivaive of uderlyig ODE model, a give by h_m.m ad dhdx_m.m. The fucio h_m.m impleme he oupu equaio of he model. Noe ha hi fucio i allowed o be a oliear fucio of he ae ad ipu. The fucio i o allowed o be depede o he eimaed parameer, i mu be ow a priori. Noe alo ha he derivaive of he fucio, wih repec o he eimaed ae, eed o be upplied i he fucio dhdx_m.m. 2. Provide furher ipu daa i he crip SeupRPEM.m. The parameer ha defie he daa geeraio are direcly wrie io hi crip. Thee parameer defie he dimeio of he yem, he iiial value ued i he ODE model, he gai equece µ (), he ize of he iiial value of he R -recurio, he iiial value of he meaureme covariace marix Λ(), he abiliy limi applied by he projecio algorihm, he cale facor, a well a he dow-amplig period ued o avoid oo large log durig log ru wih high degree model. The reader i referred o [ ] - [ 4 ] for deail o hee parameer, a well a o heir ue. 7

18 3. Execue SeupRPEM.m. Thi load he eceary parameer io he MATLAB worpace. Example 4: The yem i o be ideified wih a ecod order model. The projecio algorihm i o ue a abiliy radiu of.975 ad he cale facor i eleced equal o 2. The iiial value of he meaureme covariace marix i eleced equal o.. The iiial value of he R - recurio ( i ivere affecig he iiial algorihmic gai) i eleced equal o. The elecio of he gai equece µ () i a lile more complicaed, ee [ ] for deail. The fucio h_m.m ad dhdx_m.m become h_m.m fucio [h_m]=h_m(x_m,u); h_m=x_m(,); ed dhdx_m.m fucio [dhdx_m]=dhdx_m(x,u); dhdx_m=[ ]; ed The eup crip SeupRPEM.m become x=2; x_m_=[.5 -]'; % % Remaiig iiial value % mufacor=3; % To abilize Gamma ad o reduce he gai if exi('hea ew') mufacor=; ed mu_=5; mu=.9995; y_m_=; iiialnoievariace=.; % Iiial value for he predicio error variace calefacorr=; % he ize of he iiial diagoal approximaio of he Heia % 8

19 % Parameer % abiliylimi=.975; % The liearized pole radiu ued for abiliy checig ad projecio dowsamplig=; % The dowamplig facor ued whe daa from he ru i aved caligt=2; % The calig facor wih which he amplig period i muliplied durig ideificaio Fially he uer execue SeupRPEM.m i he MATLAB commad widow» euprpem» 6.2 RPEM commad widow corol ad eimaed parameer I order o perform a ideificaio ru he uer i required o execue ad provide ipu o he crip RPEM.m. The execuio of hi crip mae ue of four addiioal fucio, implemeig he polyomial model applied for modelig of he RHS of he ODE. Thee fucio are f_m.m, dfdx_m.m, dfdhea_m.m ad GeeraeIdice.m. The laer fucio geerae he expoe of all facor of all erm of he polyomial expaio. The geeraio of hee idice ivolve eed loop. They are herefore calculaed i advace ad ued i repeiive call i he form of a able. To ideify he yem, he uer i required o. Execue he crip RPEM.m 2. Provide he degree of he polyomial model (polyomialorder) whe promped. The polyomialorder variable i a colum vecor wih he fir eleme correpodig o he maximal degree of x, he ecod eleme correpodig o he maximal degree of x 2 ad o o. The la eleme correpod o he maximum degree of he derivaive of highe degree of he la ipu igal compoe. I he pree example, polyomialorder = [ 2 3]' would mea ha he highe degree erm of he polyomial expaio i xxu. 3. Provide a li of idice ha are o o be ued (ouedidice) by he algorihm. The idice exclude erm i he polyomial expaio. Providig a empy marix ([]) idicae ha o erm hall be excluded. The li of o ued idice are o be provided a row i a marix, where he umber of row equal he umber of erm ha are o be excluded from he model. I he pree example ouedidice = [ ; ] would mea ha he erm ad xxu 2 are o be excluded from he model. 9

20 4. Provide he iiial parameer vecor (hea_). Noe ha hi parameer vecor eed o correpod o a liearized yem wih all pole wihi he abiliy radiu idicaed by he crip SeupRPEM.m. If he iiial parameer vecor doe o mee hi crierio he uer i promped for hea_ agai. Oberve ha he cale facor of he amplig period eed o be accoued for - i i a par of he liearized model, cf. [ ]. Noe: A good raegy i o iiialize he algorihm wih a model ha ha ime coa ad a aic gai ha are imilar o hoe of he yem. Example 5: The algorihm i i hi example iiialized wih = ( ) T ( 7 ) Thi correpod o he model ϕ( x,u) = ( u x xu x xu xx xxu) T ( 8 ) I hi example comme ad explaaio have bee added. To diiguih hee from he acual commad he comme The commad equece applied i he MATLAB commad widow i» RPEM a = Ipu polyomialorder ad ouedidice - The crip a for he max degree of ae ad ipu K» polyomialorder=[ ]' polyomialorder = K» ouedidice=[] - The crip a for erm of he polyomial ha are o be excluded ouedidice = [] K» reur allidice = - The crip reur he degree of all icluded erm, ipu degree o he righ 2

21 a = Ipu hea_ - The crip a for a iiial parameer vecor K» hea_=[ ]' hea_ = K» reur LiearizedPoleRadii = - The crip reur he polr-radii of he liearized, iiial model.9.9 perceready = - The crip diplay he fracio of he proceig ha i compleed. a = - The crip diplay he ideified parameer caled parameer o he lef » The eimaed parameer of he lef colum correpod o he oe obaied direcly from he RPEM, i.e. hee are caled parameer. The righ colum coai parameer ha are recompued o correpod o he origial amplig period. Noe ha he exac reul deped o he geeraed ipu igal. Thi may differ bewee yem ad execuio occaio ice he eed for he radom umber geeraor may differ. Hece, ligh variaio of he eimaed parameer are ormal. 2

22 6.3 Re-iiiaio, muliple ru ad ieraive refieme The crip RPEM.m produce a reul for a cerai choice of degree of he righ had polyomial of he ODE (if he abiliy chec i o riggered o ha he algorihm ge uc cloe o he abiliy limi). I cae he reul i o deemed ufficie, he a higher degree righ had ide may be eeded. The oppoie may alo be rue, i.e. he reul i ufficie bu he umber of parameer ued may be uecearily high. So here i a eed o Modify he degree of he polyomial model of he ODE. Remove pecific erm of he polyomial model of he ODE. Reru he RPEM from he previou ed reul, wih a redefied righ had ide polyomial. Exacly hi i uppored by he crip reiiiae.m. Noe: Suppor for eppig alo of he model order would be preferred. Such eppig doe however have complicaed (oliear) abiliy impac. For hi reao he developme of uch fucioaliy i popoed o laer verio of he ofware pacage. I order o perform a ew RPEM ru wih modified degree, he uer i required o. Ru he crip reiiiae.m. Tha crip promp he uer for polyomialorder ad ouedidice. The parameer vecor a he ed of he ru, ogeher wih he previou ad ew degree, are he ued o re-iiiae all releva quaiie of he RPEM. 2. Reru RPEM.m. Noe ha he RPEM doe o eed o promp he uer for ay furher iformaio hi ime. Example 6: I hi example he degree of he ipu igal i icreaed o 2. The MATLAB commad widow commad become» reiiiae a = Ipu polyomialorder ad ouedidice K» polyomialorder=[ 2]' polyomialorder = 2 K» reur» RPEM The ideificaio i ow fialized ad everyhig i e for diplay of he reul. 22

23 7. Diplay of reul The diplay of reul i raighforward. By a udy of he ource code, uer hould be able o ailor available crip ad alo wrie ow oe whe eeded. 7. Parameer I order o plo he parameer he uer i required o. Execue he crip PloParameer.m. The compoe of he parameer vecor are he ploed a a fucio of ime. Noe ha he ime cale i aumed o be ecod. I cae aoher ime cale i required, he figure eed o be edied afer ploig, or he crip eed modificaio. Example 7: The commad i he MATLAB commad widow i» PloParameer» The followig plo i he geeraed.5 parameer ime [] Figure 3: The reul of a PloParameer commad. 7.2 Predicio error I order o plo he parameer he uer i required o 23

24 . Execue he crip PloPredicioError.m. The predicio error are he ploed a a fucio of ime. Noe ha he ime cale i aumed o be ecod. I cae aoher ime cale i required, he figure eed o be edied afer ploig, or he crip eed modificaio. Example 8: The MATLAB commad widow commad i» PloPredicioError» The followig plo i geeraed.5 predicio error () ime [] Figure 4: The reul of a PloPredicioError commad. 7.3 Eigevalue I order o plo he covergece of he eigevalue over ime, he uer i required o. Execue he crip PloEigevalue.m. The eigevalue of he Heia are he ploed a a fucio of ime. Noe ha he ime cale i aumed o be ecod. I cae aoher ime cale i required, he figure eed o be edied afer ploig, or he crip eed modificaio. Example 9: The MATLAB commad widow commad i» PloEigevalue» The followig plo i geeraed 24

25 3 eigevalue of he Heia ime [] Figure 5: The reul of a PloEigevalue commad. 7.4 Codiio umber I order o plo he covergece of he codiio umber over ime, he uer i required o. Execue he crip PloCodiio.m. The codiio umber i he ploed a a fucio of ime. Noe ha he ime cale i aumed o be ecod. I cae aoher ime cale i required, he figure eed o be edied afer ploig, or he crip eed modificaio. Example : The MATLAB commad widow commad i» PloCodiio» The followig plo i geeraed 25

26 4 codiio umber of he Heia ime [] Figure 6: The reul of a PloCodiio commad. 7.5 Ed of ru model imulaio The above plo commad mae ue of logged igal, coverig he raie par of he ideificaio proce. Thi i o alway deired. To compare he ideified mode o he meaured daa i i e.g. more appropriae o imulae he model, uig he parameer obaied a he ed of he ideificaio ru. Hece, o prepare for he remaiig plo commad he uer i required o. Execue he crip SimulaeModelOupu.m. Example: The MATLAB commad widow commad i» SimulaeModelOupu perceready =» The remaiig plo commad ad model validaio commad ca ow be execued. 7.6 Simulaed model oupu I order o plo he imulaed model oupu igal over ime, he uer i required o 26

27 . Execue he crip PloModelOupu.m. The oupu igal of he model are he ploed a a fucio of ime. Noe ha he ime cale i aumed o be ecod. I cae aoher ime cale i required, he figure eed o be edied afer ploig, or he crip eed modificaio. Example 2: The MATLAB commad widow commad i» PloModelOupu» The followig plo i geeraed 2 y m () ime [].5 u() ime [] Figure 7: The reul of a PloModelOupu commad. 7.7 Simulaed model oupu ogeher wih daa I order o plo he imulaed model oupu igal over ime, ogeher wih he correpodig meaured daa, he uer i required o. Execue he crip PloSyemAdModelOupu.m. The oupu igal of he model ad he yem are he ploed a a fucio of ime, i he ame plo. Noe ha he ime cale i aumed o be ecod. I cae aoher ime cale i required, he figure eed o be edied afer ploig, or he crip eed modificaio. Example 3:The MATLAB commad widow commad i» PloSyemAdModelOupu» The followig plo i geeraed 27

28 y() ad y m () ime [].5 u() ime [] Figure 8: The reul of a PloSyemAdModelOupu commad. 7.8 Reidual error I order o plo he predicio error, obaied from he imulaed model oupu igal uig parameer a he ed of he ideificaio ru, he uer i required o. Execue he crip PloReidualError.m. The error are he ploed a a fucio of ime. Noe ha he ime cale i aumed o be ecod. I cae aoher ime cale i required, he figure eed o be edied afer ploig, or he crip eed modificaio. Example 4:The MATLAB commad widow commad i» PloReidualError» The followig plo i geeraed 28

29 .5 reidual error () ime [] Figure 9: The reul of a PloReidualError commad. 8. Model validaio Two crip are provided for model validaio purpoe (i addiio o he commad of ecio 7). The fir mehod udie he value of he lo fucio ha i miimized by he RPEM-algorihm. Sice he meaureme covariace marix i eimaed, he lo fucio coai a addiioal erm o op of he ample average of he quared predicio error. 8. RPEM lo fucio The RPEM lo fucio ha i compued i give by N ( ( S) ) S ( S) T ( it NT ) ( its ( NTS) ) ( ( NTS) ) V NT N,, + = ε ε logde Λ + ( 9 ) 2 i= The uer i referred o [ ], [ 6 ] ad he referece herei for furher deail. I order o compue he lo fucio, uig parameer a he ed of he ideificaio ru, he uer i required o. Execue he crip CompuRPEMLoFucio.m. The lo fucio i he compued. Example 5: The MATLAB commad widow commad i» CompueRPEMLoFucio perceready = 29

30 V = -.794» 8.2 Mea reidual aalyi Mea reidual aalyi i a mehod ha evaluae he obaied aic characeriic of a ideified model of ay id. I operae by orig reidual error io bi, he bi beig decided by he value of he meaured oupu igal wih he ame ime idex a he reidual error. The mea of he reidual are he compued, i each bi, ad ploed agai he rage of he oupu igal. The umber of ample of each bi i alo ploed. The uer i referred o [ 7 ] for furher deail. I order o perform mea reidual aalyi, he uer i required o. Execue he crip MeaReidualAalyi.m. 2. Provide he ierval ued by he mehod whe promped for ierval. Example 6: Thi example perform mea reidual aalyi uig abou 4 ierval, each wih a oupu ampliude widh of.. The MATLAB commad widow commad i» meareidualaalyi a = Ipu ierval for diviio io bi K» ierval=(-2:.:2); K» reur» The followig plo reul 3

31 mea reidual umber of ample y() y() Figure : The reul of a MeaReidualAalyi commad. 9. Summary Thi repor decribe a ofware pacage for ideificaio of oliear yem. Fuure wor i hi field, ha reul i ueful MATLAB rouie, will be iegraed wih he preely available fucioaliy. Updaed verio of hi repor will he be made available.. Referece [ ] T. Wigre, "Recurive Predicio Error Ideificaio of Noliear Sae Space Model", Techical Repor from he deparme of Iformaio Techology 4-24, Uppala Uiveriy, Uppala, Swede, Jauary, 24. [ 2 ] T. Wigre, "Recurive predicio error ideificaio ad calig of oliear ae pace model uig a rericed blac box parameerizaio", ubmied o Auomaica, Jauary, 24. [ 3 ] T. Wigre "Scalig of he amplig period i oliear yem ideificaio", o appear i Proceedig of IEEE ACC 25, Porlad, Orego, U.S.A., Jue 8-, 25. 3

32 [ 4 ] T. Wigre "Recurive ideificaio baed o oliear ae pace model applied o drumboiler dyamic wih oliear oupu equaio", o appear i Proceedig of IEEE ACC 25, Porlad, Orego, U.S.A., Jue 8-, 25. [ 5 ] D. Haelma ad B. Lilefield. Maerig Malab 5 - A Compreheive Tuorial ad Referece. Upper Saddle River, NJ: Preice Hall, 998. [ 6 ] L. Ljug ad T. Söderröm. Theory ad Pracice of Recurive Ideificaio. Cambridge, Ma: MIT Pre, 983. [ 7 ] T. Wigre, "Uer choice ad model validaio i yem ideificaio uig oliear Wieer model", Prep. 3:h IFAC Sympoium o Syem Ideificaio, Roerdam, The Neherlad, pp , Augu 27-29, 23. Ivied eio paper. 32