This is a repository copy of Estimation of generalised frequency response functions.

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1 hs s a rpostory copy of Estmato of gralsd frqucy rspos fuctos. Wht Ros Rsarch Ol URL for ths papr: Moograph: L, L.M. ad Bllgs, S.A. 9 Estmato of gralsd frqucy rspos fuctos. Rsarch Rport. ACSE Rsarch Rport o. 6. Automatc Cotrol ad Systms Egrg, Uvrsty of Shffld Rus Ulss dcatd othrws, fulltt tms ar protctd by copyrght wth all rghts rsrvd. h copyrght cpto scto 9 of th Copyrght, Dsgs ad Patts Act 988 allows th makg of a sgl copy solly for th purpos of o-commrcal rsarch or prvat study wth th lmts of far dalg. h publshr or othr rghts-holdr may allow furthr rproducto ad r-us of ths vrso - rfr to th Wht Ros Rsarch Ol rcord for ths tm. Whr rcords dtfy th publshr as th copyrght holdr, usrs ca vrfy ay spcfc trms of us o th publshr s wbst. akdow If you cosdr cott Wht Ros Rsarch Ol to b brach of UK law, plas otfy us by malg prts@whtros.ac.uk cludg th URL of th rcord ad th raso for th wthdrawal rqust. prts@whtros.ac.uk

2 Estmato of Gralsd Frqucy Rspos Fuctos L.M.L ad S.A.Bllgs Dpartmt of Automatc Cotrol ad Systms Egrg, Uvrsty of Shffld, Shffld Post Bo 6 S JD UK Rsarch Rport No. 6 Dc 9

3 Estmato of Gralzd Frqucy Rspos Fuctos L.M.L ad S.A.Bllgs Abstract Voltrra srs thory has a wd applcato th rprstato, aalyss, dsg ad cotrol of olar systms. A w mthod of stmatg th Voltrra krls th frqucy doma s troducd basd o a o-paramtrc algorthm. Ulk th tradtoal o-paramtrc mthods usg th DF trasformd put-output data, ths w approach uss th tm doma masurmts drctly to stmat th frqucy doma rspos fuctos. Id rms GFRF s, No-paramtrc, OLS, Voltrra srs. I. INRODUCION E Voltrra srs modl, frst proposd by Voltrra [], s a drct gralzato of th lar covoluto tgral ad provds a tutv rprstato a smpl ad asy to apply way. Voltrra thory quckly rcvd a grat dal of attto th fld of lctrcal grg, mchacal grg, ad latr th bologcal fld, as a powrful approach for modlg olar systm bhavors. From th lat 95s, thr has b a cotuous ffort th applcato of Voltrra srs to olar systms thory. Summars of major cotrbutos th applcato of Voltrra srs modlg for th rprstato, aalyss ad dsg of olar systms ca b foud []-[5]. h Voltrra srs s assocatd wth so-calld wakly olar systms, whch ca b wll dscrbd by th frst fw krls wth th hghr ordr krls fallg off rapdly. h frqucy doma vrso of th Voltrra krls, calld gralzd frqucy rspos fuctosgfrf s, whch ca b obtad by takg th multpl Fourr trasform of th Voltrra krls, has also b tsvly studd ad provd to b vry powrful charactrzg olar phoma [6]. Du to th usfulss of th gralzd frqucy rspos fuctos, a umbr of stmato mthods hav b proposd. hr ar grally two classs of mthods for th stmato of GFRF -- paramtrc ad o-paramtrc mthods. For th Mauscrpt rcvd Dc, 9. hs work was supportd by th Egrg ad Physcal Sccs Rsarch CouclEPSRC UK, ad a Europa Rsarch Coucl Advacd Ivstgator Award. L. M. L s wth th Dpartmt of Automatc Cotrol ad Systms Egrg, Uvrsty of Shffld, Shffld S JD, UK -mal: l.l@shffld.ac.uk. S. A. Bllgs s wth th Dpartmt of Automatc Cotrol ad Systms Egrg, Uvrsty of Shffld, Shffld S JD, UK Pho ; Fa: ; mal : s.bllgs@shffld.ac.uk. paramtrc mthod, th put-output data masurmts ar usd to dtfy a NARMAX modl, from whch a NARX modl, cludg purly put-output trms aftr dscardg th trms assocatd wth th os modl, ca b drvd. h th probg mthod [7] ca b appld o ths NARX modl to obta th GFRF s. h o-paramtrc mthod usually maks us of hghr ordr spctral aalyss basd o th frqucy doma Voltrra modl [8]-[]. Boyd t al. [] proposd a o-paramtrc mthod of stmatg th GFRF s basd o th sparato of th cotrbuto of ach Voltrra krl, usg harmoc puts. Du to th computatoal complty assocatd wth o-paramtrc mthods, th Voltrra krls had to b rstrctd to low ordrs, for ampl, up to cubc olarts. hs papr s prmarly cocrd wth th problm of stmatg th GFRF s from harmoc put-output data for cubcally olar systms. By padg th algbrac prsso of th rspos aalyss th frqucy doma, t s show that th GFRF s ca b stmatd drctly from put-output masurmts. h papr s orgazd as follows. Scto stats th prlmars. Scto studs th stmato of th GFRF s usg th smplst Voltrra modl form, that s, quadratc olar systms. Scto 4 dscusss th mor compl cubcally olar systm cas. I scto V th problm of dtrmato of th ctato lvl s addrssd. Fally scto 6, coclusos ar gv. II. PRELIMINARIES Voltrra srs modlg has b wdly studd for th rprstato, aalyss ad dsg of olar systms. For a SISO olar systm, whr ut ad t ar th put ad output rspctvly, th Voltrra srs ca b prssd as y t y t.a ad y t s th -th ordr output of th systm y t h,, u t d.b whr h,, s calld th th-ordr krl or th-ordr mpuls rspos fucto. If =, ths rducs to th famlar lar covoluto tgral. h dscrt tm doma coutrpart of th cotuous tm doma SISO Voltrra prsso s

4 whr y k y k.a y k h,, u k, k.b I practc oly th frst fw krls ar usd o th assumpto that th cotrbuto of th hghr ordr krls falls off rapdly. Systms that ca b adquatly rprstd by a Voltrra srs wth just a fw trms ar calld wakly olar systms. For a wakly olar systm up to thrd ordr Voltrra srs rprstato, th frqucy doma prsso of th dscrt tm Voltrra modl s gv as Y p, q p q l, m, l m l, m, l m p p q, q whr Y ad ar th Fourr rasforms of th output rspos ad put rspctvly, ad,, s calld th th ordr Gralzd Frqucy Rspos Fucto GFRF whch s obtad by takg th mult-dmsoal Fourr trasform of h :,, 4 h j d d,, p h gralzd frqucy rspos fuctos rprst a hrt ad varat proprty of th udrlyg systm, ad hav provd to b a mportat aalyss ad dsg tool for charactrzg olar phoma. I practc, by takg to accout th output masurmt os grally zro-ma wht os, s rplacd by Y Yˆ U p, q U p U q l, m, U l U m U l, m, lm p, q pq h ctatos usd th applcato of o-paramtrc mthods ca b thr Gaussa wht os or o-gaussa harmoc puts. For stac, rfrc [] vstgatd th problm of stmato of GFRF s ad systm dtfcato of a cubcally olar systm basd o 5, subjct to a o-gaussa put. Although 5 s olar btw th spctrum of th masurd put/output uk ad y k,.., U ady, t s lar btw Y ad th ukow GFRF s, ad, ad th stadard Last Squars typ algorthms ca b radly appld to obta dffrt ordrs of trasfr fuctos. h possbl dsadvatags of th abov purly frqucy doma basd approachs ar that larg data sts ar oft dd ad also th frqucy doma osy trm, obtad from tm doma wht Gaussa os k, may o logr b wht, pottally rsultg bas th stmats obtad from a Last Squar procdur. 5 Altratvly, th stady-stat rspos of th olar systm that ca b adquatly rprstd by up to thrd ordr Voltrra krls, ctd by a harmoc sgal at frqucy, s gv [] by k yˆ k k jk A j k AR{ } R{, } A A 6 jk R{, } R{,, } A jk 6 R{,, } k whr A s th ampltud of th put sgal ad k s a zro-ma Gaussa wht os. R rprsts th ral part of a compl umbr. Equato 6 forms th bass of th currt study whch suggsts a altratv o-paramtrc approach of stmatg gralzd frqucy rspos fuctos drctly from osy tm doma data. h study bgs wth quadratc olar systms Scto, followd by cubc olar systms Scto 4. III. ESIMAION OF FREQUENCY RESPONSE FUNCIONS OF QUADRAICALLY NONLINEAR SYSEMS A. Formato of th Estmator Bcaus t s th smplst spcal cas of th ft Voltrra srs modl, th quadratc Voltrra modl class has b studd farly tsvly. By cosdrg th frst two Voltrra krls 6, th output rspos k ca b prssd as k yˆ k k AR{ A Dfg R{ jk A }, j k R{ } k, j k } 7 R ji 8 I whr R ad ar th ral ad magary parts of rspctvly. For smplcty, R ad I wll b wrtt abbrvatd form as R ad I. h th frst trm th rght-had sd of 7 ca b padd as j k AR{ } AR{[ R ji ][cos k j s k]} 9 AR cos k AI s k Smlarly by dfg, R, ji,, R ad otg that s a costat, th scod ad th thrd trms o th rght-had sd of 7 ca b padd as

5 4 A j k A R{, } R{, A A A R cosk I sk R Combg 9 ad ylds k R[ Acos k] I [ As k] j k } A A R [ cosk ] I[ sk ] A R k For k= to N, ca b arragd th matr form as Y X whr Y [ y N ], R I R I R ], [ X [ 4 5 ], wth [ N ], ad [ Acos N Acos ] [ As N As ] A A [ cosn cos ] 4 A A [ sn s ] 4 A A [ ] 5 h stmato of th ukow frqucy rspos fuctos ˆ [R I R I R ] ca ow b drvd from usg a stadard last squar procdur. If th trucato rror s suffctly small, o th assumpto that thrd ad hghr ordrs of Voltrra krls mak a glgbl cotrbuto to th output, th th stmato ˆ wll b ubasd. Ulk th prvous compl stmator basd o 5, th w stmator s th ral doma basd o tm doma masurmts. B. Smulato Eampl ad Dscusso Cosdr a systm dscrbd as y ay by cy u t 5 whr u t Acos t. I th cotuous tm doma, th GFRF s ca b drvd [4] as s s j s as b 6 s, s s j c s s s s s j hs systm 5, whch has a quadratc olar trm y, s ot a quadratcally but ftly olar systm trms of th Voltrra srs rprstato bcaus th olarty s o th output. owvr for a cosdrabl rag of put lvl th systm ca b adquatly rprstd by up to scod ordr Voltrra krls, ad mportatly all th coffcts of th udrlyg systm 5 ca b fully charactrzd by th frst two ordrs of Voltrra krls or th assocatd frqucy rspos fuctos. Oc th stmats of Ĥ ad Ĥ ar avalabl, by usg 6, th orgal cotuous tm systm paramtrs ca b tractd from Ĥ ad Ĥ as aˆ Im[ ] ˆ ˆ b R[ ] ˆ ˆ, b ˆ ˆ, cˆ ˆ ˆ ˆ ˆ ˆ 7 o llustrat th w GFRF stmato algorthm, systm 5 was ctd by a sgl susodal put at A= ad.6 rad / sc for a., b ad c.. A total amout of put ad output data wr collctd at a samplg tm s. sc ad a zro-ma wht os was addd to th output wth SNR =4 db. Bfor procdg usg th w algorthm, th tradtoal paramtrc NARMAX procdur was appld. Frst th dal stuato, os-fr put /output data [5], wr cosdrd th NARX modlg. h rsultg modl was obtad as k k k u k y k- It ca b asly vrfd that th GFRF s from th dscrt tm modl 8 ar actly th sam as th GFRF s from th orgal cotuous tm modl 6, alog th tr frqucy as. hs suggsts that th NARX modlg procdur s th smplst ad most ffct choc for th GFRF s stmato th os-fr stuato. Wh os s prst, howvr, th dtfcato of a NARMAX modl that ca provd satsfactory frqucy doma stmato s ot a trval task. For ampl, a NARMAX modl whos trms ar slctd from a pool of caddat trms, usg th osy masurmts, s gv as k.4 k-.774 k u k- 4.76u k u k y.89 k- u k- k k- 9 whr rprsts th os modl trms. Equato 9 has a vry good modl prdctd output MPO ad modl valdty tst [6] [9] show Fg.. Fg.. Corrlato tst of th NARMAX modllg of 9

6 5 h stmato of ad from 9 at. 6 s show abl I, compard wth th tru valus from th orgal systm 5, ad th rcostructd paramtrs of cotuous tm modl 5 from abl usg 7 ar show abl II. ABLE I ESIMAION OF AND BY E NARMAX MODEL 9 From Eq From Eq 9 5 ru --Estmats, R, j j j j ABLE II ESIMAION OF E PARAMEERS OF E ORIGINAL CONINUOUS IME SYSEM 5 USING 7 a b c ru valu... Estmats from GFRF s by It s clar that thr s bas o th rcostructd stmats of th systms paramtrs abl II. hs suggsts that, though th NARMAX modl 9 s a vry good ft th tm doma as th MPO ad corrlato basd modl valdty tsts dcat, ts ablty of capturg th frqucy doma faturs of th udrlyg systm s ot always rlabl, as dcatd ths smpl ampl. Improvd frqucy rspos fucto stmato s possbl f a largr pool of caddat modl trms s fd to th NARMAX modl stmator. h dffculty s that wthout a frqucy doma valdty gudl, t s somtms ot asy to kow wh to stop th sarch for a modl that s ot oly vald th tm doma but also th frqucy doma. Now, th proposd w procdur 4 s appld to obta th Ĥ ad Ĥ usg a Last Squars stmator. h rsults ar show abl III whch ar vry satsfactory. ABLE III Estmato of ad of systm 5 usg th w approach From Eq From 5 ru w approach R ji R ji, j j j.4 +.j R , a sgfcat mprovmt accuracy compard wth th rsults by th paramtrc mthod abl II. ABLE IV ESIMAION OF E PARAMEERS OF E ORIGINAL CONINUOUS IME SYSEM 5 USING 7 a b c ru valu... Estmats from w approach IV. ESIMAION OF FREQUENCY RESPONSE FUNCIONS OF CUBICALLY NONLINEAR SYSEMS h ltratur assocatd wth th cubc Voltrra srs modl s substatally smallr compard wth that assocatd wth th quadratc Voltrra modl du to th sgfcatly gratr complty ducd. I trms of frqucy rspos fucto stmato for up to thrd ordr Voltrra rprstato, ths procdur s ot as straghtforward as th quadratc cas Scto. I fact, th ma complcato s that, ulk th quadratc Voltrra modl cas whr th frst harmocs th output ar all du to th lar rspos fucto, for a cubc Voltrra modl both frst ad thrd ordr frqucy rspos fuctos mak cotrbutos to th frst harmocs. hrfor addtoal fforts ar dd to solv th sparato of cotrbutos btw ad. Frst, by dfg,, R,, ji,, th fourth trm 6 ca b padd as A jk R{,, } A A R cos k I s k whch maks a cotrbuto purly to th thrd harmocs th rspos. h problm arss wth th ffth trm 6. By dfg,, R,, ji,, th ffth trm 6 ca b padd as A jk 6 R{,, } A A 6 R cos k 6 I s k whch maks a cotrbuto to th frst harmocs, md up wth th cotrbuto from th lar krl, show 9. h ovrall paso of 6 for th frst thr Voltrra krl trms, by combg, ad, s gv as h rcostructd paramtrs of th cotuous tm modl 5 from abl III usg 7 ar show abl IV, whch s

7 6 A k [ AR 6 R]cos k A A [ AI 6 I ][ s k] R [ cosk] 4 A A A I [ sk ] R R cos k A I s k k It s possbl to df A AR 6 R R 5 A AI 6 I I h for k ton, 4 ca b arragd th matr form as Y X 6 whr Y [ y N ], R I R I R R I ], [ X [ ] [ N ], ad [cos N cos ] [ s N s ], wth A A [ cosn cos ] A A [ sn s ] 4 7 A A [ ] 5 A A [ cosn cos ] 6 A A 7 [ sn s ] From whch th ukow frqucy rspos fucto data ˆ [R I R I R R I ] ca b drvd usg a stadard Last Squar procdur. I ordr to sparat 5 th R ad R from R, ad th I ad I from I, two tsts at dffrt put lvls, dotd by A ad A, at th sam frqucy, ar rqurd to obta two sts of stmats, R, R, ad I ad I rspctvly. h th fal stmats of ˆ Rˆ jiˆ ad ˆ,, Rˆ jiˆ ca b calculatd from 5 as A A 6 R 6 R Rˆ A A A 6 A 6 ˆ A R A R R A A A 6 A 6 8a A A 6 I 6 I Iˆ A A A 6 A 6 8b ˆ A I A I I A A A 6 A 6 h abov procdur wll b llustratd usg th wll-kow Duffg oscllator. Cosdr a Duffg oscllator, wth cubc olarty, subjct to a susodal ctato as m y cy k y k y Acos 9 t whr m, c, k ad k ar th mass, th dampg, th lar stffss ad olar stffss rspctvly. h olar stffss paramtr k 9 ds to stay small ordr to b wakly olar for th stc of th Voltrra srs rprstato. h corrspodg GFRF s from 9 ar s s j s cs k s, s s, s, s s j s j s j s j s j k s s s s h systm 9 was ctd at.sc wth th paramtrs s s s.5 rad/sc ad m, c.5, k.5, k. h ctato s chos as A cos t,, whr A ad A. A zro-ma wht os was addd to ach of th outputs wth a SNR =4 db. h lgth of th data was. h stmato rsults for Ĥ ad Ĥ usg th w algorthm ar gv abl V. ABLE V ESIMAION OF AND FOR E DUFFING EQUAION 9 USING E NEW APPROAC Eq 9 ru By usg, th orgal cotuous tm systm paramtrs ca b tractd from Ĥ ad Ĥ as From w approach R ji j j,, R ji,, R ji j j j j

8 7 cˆ Im[ ] ˆ ˆ k R[ ] ˆ kˆ ˆ,, ˆ ˆ ˆ ˆ It s clar from abl VI that th rcostructd cotuous tm systm paramtrs, comparg wth th tru systm paramtrs, ar vry satsfactory. V. E SELECION OF LEVEL OF EXCIAION USING OLS h w algorthm troducd th prvous sctos s basd o th assumpto that th udrlyg systm s wakly olar th ss that t ca b wll dscrbd by th frst fw Voltrra krls, wth th hghr ordr krls fadg off rapdly. h accuracy of th rsults of th w procdur s largly dpdat o ths assumpto as ar all othr o-paramtrc mthods, ad th sgfcac of th olarty of th systm, rflctd by th Voltrra krl ordr, s assocatd wth th lvl of th ctato. Ethr udr-ctato or ovr-ctato may rsult accuracy th stmats. It s thrfor sstal to hav a masur that ca provd a dcato of th ordr of th Voltrra krls udr crta lvls of ctato bfor th fal applcato of th w algorthm. O possbl masur s basd o usg th orthogoal last squars mthod OLS [] [], whch s brfly rvwd blow. Cosdr a systm prssd by th lar--th-paramtrs modl z M p whr,, M ar ukow paramtrs., Rformulatg quato th form of a aulary modl ylds whr M z g w g,,, M ar th aulary paramtrs ad 4 w,,, M ar costructd to b orthogoal ovr th data rcord such that ABLE VI ESIMAION OF PARAMEERS OF ORIGINAL CONINUOUS IME SYSEM 9 USING c k k ru valu from Estmats from N wj t w t, j,,, k t k whr N s th lgth of th data rcord. Multplyg th aulary modl 4 by tslf, usg th orthogoal proprty 5 ad takg th tm avrag gvs Fally df N N z t t N t M N g w t N N t N t 6 g w t t ERR 7 N N z t z t t N t for,,, M. h quatty ERR s calld th Error Rducto Rato ad provds a dcato of whch trms should b cludd th modl accordac wth th cotrbuto ach trm maks to th rgy of th dpdt varabl. rms wth assocatd ERR valus whch ar lss tha a pr-dfd thrshold valu ca b cosdrd to b sgfcat ad glgbl. For smplcty, th quadratc systm 5, wth a., b ad c., was usd as a ampl to llustrat th us of OLS th slcto of th ampltud of ctato A. wo tsts wr coductd at dffrt ctato ampltuds at frqucy rad / sc. Frst, th ampltud of th put was chos at A=. ad th rspos was corruptd by a zro-ma wht os wth SNR =4 db. h OLS was appld to obta th stmats of lar ad quadratc frqucy rspos fuctos, togthr wth th valus ERR, show abl VII. ABLE VII ESIMAION OF AND OF SYSEM 5 A Rsult by Eq 5 ru A. USING OLS Rsult by OLS ERR % R I R I R It ca b s from abl VII that at ths lvl of ctato, th cotrbutos by th quadratc trm,.., R, I ad R, ar trmly small compard wth th cotrbutos from th lar trm. hs mas that th quadratc krl cotrbuto has a vry small SNR, cosqutly th LS stmato rsults for th quadratc trms ar mor affctd by th prsc of os, ladg to urlablty th stmats. h almost glgbl sum of ERR of th quadratc trms,.., from trms R, I ad R, suggsts that th systm ca b rgardd as lar at ths lvl of ctato, thrfor ths ca b cosdrd as udr-ctd. Now th ampltud of put was chos at A5 ad aga a zro-ma wht os was addd to th rspos wth SNR =4 db. h OLS rsults ar gv abl VIII.

9 8 It s clar by lookg at th ERR valus from abl VIII that th ovrall cotrbutos by th quadratc trms, spcally th R trm ar o logr glgbl. As a rsult, th accuracy of th stmats of th quadratc rspos fuctos bcoms vry satsfactory. hrfor trms of GFRF s stmato for systm 5 usg th w approach, th ampltud usd th scod tst, A 5, s a much bttr choc, followg th suggstos of ERR. I addto, th sum of th valus ERR of all th lar ad quadratc trms s 99.4%, dcatg that ths systm at ths ampltud lvl ca b suffctly dscrbd by up to scod ordr Voltrra krls. It ds to b potd out that th approach prstd ths papr ca b tdd to admt -to or mult-to susodal puts. For ampl, f th put s a -to sgal th form u k A cos k 8 ad th systm ca b suffctly dscrbd by frst ad quadratc Voltrra krls, th th sgl-to rspos prsso 7 ca b tdd to j A k j k k A R{ } R{, } A A A j k R{, } 4 R{, } A A j k 4 R{, } k 9 Equato 9 ca b furthr padd to a form smlar to volvg ral ad magary parts of th dvdual GFRF s, from whch,,,,, ad, tc ca b stmatd usg th LS abl VIII Estmato of ad of systm 5 at A5 usg OLS Rsult by Rsult by Eq 5 ERR % OLS ru R I R I procdur llustratd Scto. h oly cocr for th quadratc systm udr a -to susodal put s th sparato of th cotrbutos by, ad,, whch ar two d.c. compots. hs ca b dalt wth by applyg th -to sgal twc at lvls for ach frqucy. VI. CONCLUSIONS A w o-paramtrc algorthm, whch drctly uss th tm doma put-output masurmts but avods th drct dffrtato of ths data, has b drvd to stmat up to th thrd ordr gralsd frqucy rspos fuctos ad subsqutly dtfy th assocatd cotuous tm modl. hs s achvd by padg th algbrac prsso th aalyss of th output rspos usg th ral ad magary parts of ach ordr of Voltrra krls. h w algorthm has th advatag of admttg smallr data sts tha th tradtoal spctral aalyss basd frqucy doma o-paramtrc mthods. Lk may othr gral paramtr stmato problms, th accuracy of th stmato of th GFRF s ths w approach dpds o th fact that th lvl of ctato s approprat. hat s, th rlvat ordr of olarty s adquatly ctd. h OLS mthod has supror umrcal proprts compard wth th ordary LS mthod, th ss that t ca provd vtal formato o th sutablty of th ctato lvl, drctly from th cotrbuto dcators ERR for ach Voltrra ordr. Although th w approach was llustratd usg sgl-to susodal puts, th basc da ca b radly tdd to accommodat -to or v mult-to puts, whch cas th complty of th procdur wll vtably grow dramatcally as th umbr of tos th put ad th ordr of Voltrra krls ar crasd. h cotuous tm modl ca b rcostructd from th stmatd GFRF s. A drct tracto of th cotuous tm modl for th smpl low dyamc ordr systm s possbl from th GFRF s at o sgl frqucy pot, as llustratd ths papr. Wh th cotuous tm modl structur s ot kow a pror, or s a mor complcatd form, ths w procdur ca b rpatd at dffrt ctato frqucs utl a suffct umbr of pots hav b collctd, from whch th dtfcato of th gral form of cotuous tm modl ca b drvd usg th approach []. REFERENCES [] V. Voltrra, hory of Fuctoals, Black ad Sos, 9. [] M. Schtz, h Voltrra ad Wr hors of No-lar Systm, Nw York: Wly, 98. [] W. J. Rugh, Nolar Systm hory h Voltrra/Wr approach, h Joh opks Uvrsty Prss, 98. [4] W. Sadbrg, A Prspctv o Systm hory, IEEE rasactos o Crcuts ad Systms, vol. CAS-, o., pp.88-, 984. [5] F. J. Doyl III, R. K. Parso ad B. A. Oguak, Idtfcato ad Cotrol Usg Voltrra Modls, Sprgr-Vrlag, Lodo,. [6] S. A. Bllgs ad K. M. sag, Spctral aalyss for olar systms, Part II trprtato of olar frqucy rspos fuctos, J. Mch Systms ad Sgal Procssg, vol., o. 4, pp.4-59, 989. [7] J. C. Pyto Jos ad S. A. Bllgs, A Rcursv algorthm for computg th frqucy rspos of a class of o-lar dffrc quato modls, It.J.Cotrol, vol. 5, o. 5, pp.95-94, 989. [8] L. J. ck, h stmato of trasfr fuctos of quadratc systms, chomtrcs, vol., pp , 96. [9] K. I. Km ad E. J. Powrs, A dgtal mthod of modllg quadratcally o-lar systms wth a gral radom put, IEEE rasactos o Acoustcs, Spch, ad Sgal Procssg, vol.6, o., pp , 988.

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