STEP-INVARIANT TRANSFORM FROM Z- TO S-DOMAIN A General Framework

Size: px

Start display at page:

Download "STEP-INVARIANT TRANSFORM FROM Z- TO S-DOMAIN A General Framework"

Preston Hardy
5 years ago
Views:

1 IEEE Instrumntaton and Masurmnt Tchnology Confrnc, IMTC/ May -4,, Baltmor, MD, USA STEP-IVARIAT TRASFORM FROM Z- TO S-DOMAI A Gnral Framwork JÓZSEF G. ÉMET and ISTVÁ KOLLÁR DEPARTMET OF MEASUREMET AD IFORMATIO SYSTEMS BUDAPEST UIVERSITY OF TECOLOGY AD ECOOMICS BUDAPEST, UGARY, -5 S: {EMET,KOLLAR}@MIT.BME.U WWW: TTP:// POE: , FAX: Abstract Th convrson from dscrt-tm to contnuous-tm modls s somtms qut troublsom. Ths papr answrs rlatd pndng qustons concrnng z-doman pols at th orgn, and rlatng thm to s-doman modls wth tm dlay. Spcal attnton s dvotd to th tradoff btwn fractonal dlay and drct fd-through. Th papr placs ths qustons nto a gnral framwork of modl mappng btwn z-doman transfr functons and s-doman transfr functons, mayb wth nput dlay. An algorthm to convrt z-doman modls wth a pol at th orgn s xpland. A MATLAB routn has bn mplmntd. Th routn allows to xcut th convrson thr by dfnng th dlay or th allowabl drct fd-through. Ths routn s mad avalabl through th WWW. Kywords MATLAB, ZO transform, dscrt-tocontnuous transform, dc, stp-nvarant transform, dlay systm, fractonal dlay. I. Introducton Whn th transfr functon of a systm was dntfd n th z-doman (somtms th natur of data rqurs ths, s [], t may nd to b transformd to th s-doman,. g. to gv physcal ntrprtaton of th rsults. owvr, whl transform from s-doman to z- doman s unqu, th othr drcton s somtms ambguous, somtms thr s vn no s-doman quvalnt n th dsrd modl st. Th wll-known cas s th prsnc of ngatv ral pols n th z-doman modl. In [] an xtra manpulaton was suggstd to ovrcom th problm. By addng a canclng pol/zro par to th z-doman modl th transformaton bcoms possbl, but thn ambguty appars. Th undtrmnd part n th mpuls rspons of th rsultng s-doman systm, as t was shown, s a sgnal wth zro crossngs xactly at th samplng nstants. Ths hddn oscllatons can b lmnatd f w apply a parsmony prncpl, by whch th soluton s unqu. Basd on ths rsults, n ths papr w sk a gnral framwork for modl convrson, by tratng othr cass whr wavforms ar nvolvd, whch ar hddn from samplng. W pay spcal attnton to systms wth fractonal dlay. Th natural soluton would b to sk transformaton btwn propr transfr functons, that s, btwn ratos of polynomals n z-doman and s-doman. Wth th soluton of [], almost all cass ar tratd, xcpt z-doman systms wth pols at th orgn. In ths cas, a dlay must appar n th s-doman. If w can dtrmn th assocatd modls, w hav s-doman countrparts of all z- doman transfr functon modls. ( z st D( z D( s Th approachs and algorthms prsntd hr also fll th gap among th routns n MATLAB, stablshng th Ths work s supportd by th Blgan atonal Fund for Scntfc Rsarch, th Flmsh govrnmnt (GOA-IMMI, and BIL99/8, and th Blgan govrnmnt as a part of th Blgan programm on Intrunvrsty Pols of attracton (IUAP4/ ntatd by th Blgan Stat, Prm Mnstr s Offc, Scnc Polcy Programmng.

2 symmtry of convrsons (prsntly s-doman modls wth arbtrary tm dlay can b transformd to z-doman, but th rsults cannot b transformd back f th dlay s not an ntgr multpl of th samplng ntrval, s th routns cd and dc. As a scond bnft, thy gv furthr nsght nto a gnral and nhrnt ambvalnc: th bhavor of th rconstructd contnuos-tm modl can b dtrmnd btwn samplng nstants only by makng rstrctons n th modl class and/or gvng xtra paramtrs or ruls for th convrson. Frst, w xpos th thortcal problm of nvrtng back th ZO transform of contnuous-tm systms, wth spcal car to th cas of fractonal dlay. W dscuss a numrcal algorthm prsntd n [4] that addrsss ths topc. Thn, w ar gong to us th dcomposton nto partal fractons and prsnt a nw algorthm for th convrson. Last, w draw th attnton to problms mrgng wth masurd data and nsurng paramtrc modls. II. Prlmnars Idntfcaton of contnuous-tm lnar systms can b don thr n tm doman or n frquncy doman. Tmdoman mthods usually yld a dscrt-tm modl of th systm, whl frquncy doman mthods can dntfy thr a dscrt-tm (z-doman or a contnuous-tm (s-doman modl. In ordr to achv accurat rsults, as n valuatng modls from masurmnts, th xprmnt st-up must mt th rqurmnts prscrbd by th appld stmaton mthod []. Thrfor, t may occur that th consqunc of th masurmnt arrangmnt s that w stmat a z-doman modl whch thn nds to b mappd to th s-doman. Th ZO transform provds such a mappng btwn modls of th two domans, assumng that data sampls of th xctaton ar nput to a zro-ordr-hold trm, and th output sgnal s sampld wthout xtra band-lmtng [,]: ZO { } ( ( s z Z Samplng L s ( Th nvrtd Laplac transform n th scond nnr braclt s th stp rspons of th systm, whch dfns th valus at th samplng nstants for th rsultng z- doman systm (Samplng{.}. nc, Eq.. s also calld stp-nvarant transform and th quaton can b rarrangd to show th nvaranc of th stp-rspons n samplng nstants: Z z z ZO SamplngL s Gvn a samplng rat, all transformatons nvolvd n Eq. -. ar wll-dfnd, thus th convrson xsts and s unqu for all transfr functons n th s-doman. In partcular, pols can b matchd as z, whr z pt and p ar th th pol n th z-doman and s-doman, rspctvly. owvr, convrtng from th z-plan to th s-plan may caus problms, bcaus ntrpolaton whch s th nvrt opraton to samplng, can only b carrd out proprly f som xtra rul s gvn:.g., for band-lmtd sgnals th yqust-crtron s mt (ths s crtanly not th cas snc th stp rspons s nvr band-lmtd. Thr ar at last two cass, whn ths lattr crtron s strongly volatd, causng th followng problms:. Whn th z-doman modl contans a ngatv ral pol, th complx logarthm of ths pol s nthr unqu, nor lads to ral coffcnts n th s-doman transfr functon []: ( p ( ln z + jπm, m ( T. If th z-doman modl (n z has pol(s n th orgn, th modl contans pur tm-dlay. Yt, th amount of dlay cannot b dtrmnd unambguously. (In fact, th xponntal coffcnt n th s-doman transfr functon rprsntng th dlay can b xpandd nto an nfnt srs ladng to an nfnt numbr of pols, wth th poston not rstrctd. Ths ndcats that th yqust crtron s gong to b volatd. As t was shown n [], addng a canclng pol-zro par to th z-doman modl can crcumvnt th frst problm (although lads to ambguty. As far as th scond problm s concrnd, [] xplans a straghtforward way to convrt s-doman transfr functons wth fractonal tm-dlay nto th z-doman, whras [4] gvs a smpl MATLAB cod for convrsons n both drctons. owvr, th convrson from z to s rls on non-lnar, zro-ordr approxmaton, and th artcl dos not trat th possblty of drct fd-through, w wll trat latr. Instad, t xcluds th possblty of drct fd-through from th nput to th output of th systm. In th cas of a ral-world systm, ths possblty should not b a prory xcludd. III. Illustraton of th problm of a dlay To llustrat th problm of systms wth tm dlay, lt us consdr th followng xampl:

3 -.st 4s + 5 s + + (4a s th transfr functon of a contnuous tm systm wth a tm dlay of.t, whr T s gong to b th samplng prod and T s. W can calculat th ZO quvalnt for th sampld systm (ths can b don.g. by th command cd n MATLAB:. 9 z. 9 z 5. (4b z 47. z + 5. z If w factor out z from th dnomnator as a unt dlay, thn th rsultng xprsson wll contan an mpropr fracton (a drct fd-through.. 9 z. 9 z 5. z (4c z 47. z + 5. Ths mpropr fracton n Eq. 4c can b ntrprtd n two radcally dffrnt ways: thr w consdr th ntgr part as a drct fdthrough from th systm nput onto ts output, multpld by z - (ladng to th modl n Eq. 4d or w can account for t as a ngatv fractonal tm dlay n th contnuous-tm modl (supprssng th drct fd-through n that doman. In ths lattr cas th ntgr dlay z and th ngatv fractonal dlay rsults n a postv fractonal dlay (ladng back to th orgnal modl n Eq. 4a. As a thrd possblty, w can combn th two ntrprtatons n a trad-off, thus gttng nfnt numbr of s- doman modls for a sngl transfr functon n z. MATLAB s dc routn cuts th quston short, by smply not lttng to convrt pols nar to th orgn. Thrfor, w may only choos to lmnat th pol n from (z, transform t to th s-doman, thn complmnt t wth th unt dlay that w prvously omttd: -st.9 s s + 5 s (4d s + s + ( Thus w got two systms ( (s and (s whch ar dffrnt, vn though thy produc xactly th sam valus at th samplng nstants. Although vn mor complcatd cass xst, ths llustrats th gnral problm. amly, at samplng nstants th dffrnt transforms produc th sam valus, whl nsd th ntrvals btwn sampls th outputs dffr. Ths ntr-sampl bhavor can b of dffrnt natur. In th cas of pols at th yqust frquncy band-lmt, th dffrnc taks th form of hddn oscllatons, whl by tradng th amount of th fractonal dlay vs. th drct fdthrough trm, th dffrnc can b concntratd n th frst samplng ntrval of th stp rspons. In Fgs. - th stp rsponss, th mpuls rsponss, and th Bod dagrams ar dpctd, rspctvly, for th transfr functons and. Aftr th frst samplng nstant, th output s dntcal for th stp and th mpuls rsponss vn btwn th sampls. Ths s, of cours, not th cas for a gnral pc-ws constant xctaton, whch can b rprsntd as a wghtd sum of shftd stp functons. In th gnral cas th nput changs from sampl to sampl and th rsponss of th two systms ar dffrnt btwn sampls at any nstant. Th Bod dagrams llustrat th consquncs of ths dffrnc. At hghr frquncs th xctaton contans gratr stps, and th dffrnc s mor strssd. Th ampltud charactrstc of th systm ( s convrgs to th amount of th drct-fd-through, whl that of rolls off abov th pol frquncy ω P. (Th samplng radan frquncy s ωs 6.8 rad/s. Th xponntally dcrasng curvs n th log-ln phas plots corrspond to th lnar phas shft ntroducd by th dlay. Clarly, th gratr th dffrnc n th dlay of th two systms and gratr th dampng, th gratr s th dffrnc n th frquncy bhavor. Incrasng th samplng rat can rduc th uncrtanty of th dlay and th dffrnc n th Bod plots. Fg. Stp rspons of (s (contnuous ln and (s (crosss and th Drac dlta Fg. Impuls rspons of (s and (s

4 db dg -4 ω P -6 - Frquncy (rad/sc Fg. Bod dagrams of (s & (s a b c z sτ ( t b sτ sτ ( t T b ( t τ b st b s b s b s + a s b s ( s + a s... + b a... + b a Fg. 4 a Systms consdrd ntally; b Ambvalnc of th dlay for th drct trm (stp rsponss: wthout dlay, wth fractonal dlay, and ntgr dlay c Class of systms that could b modlld xplotng th ambvalnc t t t ω S Consquncs So far w hav consdrd two xtrms: a dynamc systm contanng a fractonal tm-dlay and an ntgr dlay trm followd by a systm wth drct fd-through,.. a dynamc systm n paralll wth a statc gan. W may argu for th frst cas, but may not smply xclud th scond on. Furthrmor, thr s th odd da to combn ths two solutons (Fg. 4a. Thr s, howvr, a furthr cas of ambvalnc: snc th xctaton s pc-ws constant, th dlay of th drct fd-through trm s only dtrmnd up to th samplng prod. In othr words, th fact that th stp functon s constant aftr, causs anothr ambvalnc (Fg 4b. By takng ths nto account, th class of systms ncludd n th modl st can b qut wd (Fg. 4c. A rasonabl ntrprtaton s whn th dlay s th sam for th two branchs (Eq.5b. Ths rstrcton also nsurs unqunss f thr b or τ s gvn as paramtrs to th mappng (s latr. Rvstng th framwork, th class of systm modls s: whr d R, c R, dz dz dz + dz cz z c z cz bs bs b s -s + b s τ as s a s as + d c dz cz b bs a as (5a (5b whr b R, a R, < τ and th pols ar nsd th yqust band. (Ths rsolvs th ambguty comng from hddn oscllatons; th tchnqu dscrbd n [] can b appld f ncssary. W wll assum that th ordr of th dnomnator s fnt, thrfor th pols ar ncssarly gvn as th complx logarthms of th z-doman pols. Ths statmnt s ncssary bcaus th curv ntrpolatng th sampls of th z-doman stp-rspons s only vald aftr th ntrng pont of th sgnal dscrbd by th ratonal fracton part of th s-doman functon. If a fractonal dlay s allowd, thn xtrapolaton can only b mad for th sampl ntrval bfor th frst non-zro sampl n th stp-rspons. Ths xtrapolaton wll b unqu f w consdr an ntrpolaton (rspctng th yqust band wth an addtonal pont nstad of th zro. Ths addtonal nformaton s assumd to b spcfd f w mak th rstrcton for th pols. For th sak of smplcty, n most of th dscusson, w consdr that th ntgr part of fractonal dlays can b tratd sparatly. owvr, n th nxt scton w ar gong to show that ths s a furthr rstrcton, whch s a possbl opton to avod furthr ambguts. IV. Furthr ambguts In fact, by omttng th fractonal dlay n Eq. 4a, th ambvalnc stll xsts, bcaus th transfr functon obtand by mappng can always b xtndd by a canclng pol-zro par n th orgn, and w can factor out from th dnomnator an ntgr dlay. As a rsult w gt a systm contanng an ntgr dlay and a drct fdthrough n th ratonal fracton part of th transfr functon (Fg. 5. 4

5 Fg. 5 Stp rsponss of a systm wth no dlay and ts ZO alas contanng a drct fd-through n th ratonal fracton part of th transfr functon. All th mor, vn f a drct fd-through s not allowd n th ratonal transfr functon by our dfnton of causalty, w can mt ambguty. Smply consdr th cas whn w modfy Eq. 4b so that only th frst-ordr trm of th numrator s dffrnt from zro, and w smplfy by z (Eq. 6. m z z. 9z 47. z + 5. z z + 5. In ths cas, th ntrpolaton curv of th stprspons has a zro-crossng at th samplng nstant. Thus, dpndng on whthr w factor out a dlay (by multplyng by z/z or not, w can gt through mappng to two dffrnt contnuous-tm systms contanng no fractonal dlay and no drct fd-through (Fg. 6. Fg. 6 Stp rsponss of a systm wth no dlay and ts ZO alas contanng no drct fd-through. V. Soluton by partal fractons Consdrng th s-to-z transform of a systm wth fractonal dlay, t s straghtforward that f a z-doman modl was obtand by ZO-transform, ts s-doman countrpart must xst. Th mnmum goal can b to fnd ths path, and add t to MATLAB s dc. owvr w must count (6 wth ambguty, and w should b awar of all th possbl s-doman modls. Furthrmor, w should b abl to dtrmn whthr a gvn z-doman systm could b obtand from an s-doman systm by ZO-transform, or not. Dcomposton of th z-doman modl nto partal fractons, and transform trm by trm s th smplst way to brak down th problm nto th transformaton of small buldng blocks. Snc th stp rspons s zro bfor t, th dlay must b dcomposd nto an ntgr dlay and a ngatv fractonal dlay yldng a noncausal systm ([]: λts lts mts +, whr λ >, l and < m < r w only consdr th cas whn l, hnc < λ <, bcaus n th gnral cas th rmanng ntgr dlay can b tratd sparatly. Ts Aftr sparatng th ntgr dlay th pr-shftd stp rspons must b transformd. (By usng polynomals n z -, th fracton bcoms propr. Sngl pols can b transformd as: mts r s s ( k+ m T r s r mts z z Doubl pols can b transformd wth a bt mor car: mts r + r ( s s ( s s r s (6 (7 ( k + m T s ( ( k+ mt + r k + m T s ( k + m T s ( k + mt ( r + r ( m T + r ( k + T mts mts ( r + r ( m T Tz r + (8 z z ( z z Trms contanng pols of hghr multplcty can b handld n a smlar way, so w do not dscuss t hr. For sngl pols: ( s ( s whr b ( and ( b mts r and R r (9 a s a s ar th numrator and th drvatv of th dnomnator of th stp rspons at s, rspctvly. R dnots th rsdual of th corrspondng trm n th dscrt transfr functon (n z comprsng th dlay. For doubl pols: ( s b( s a ( s ( s / ( s b a r and r ( a a b( s ( s / 5

6 mts ( r + r ( m T mts R and R Tzr ( All th quatons ar lnar n th coffcnts of b, so thy can b solvd by standard mthods. Th only paramtr whch nds to b sought by nonlnar optmzaton s m. VI. umrcal rsults by partal fractons Th algorthm basd on partal fractons s avalabl n [7]. It gvs vry good accuracy, vn wth hghrordr modls and doubl ral pols. Wth doubl complx pols, howvr, th accuracy droppd sgnfcantly. In gnral, whn thr ar many pols, th startng of th stp rspons s vry smooth. Thus dntfyng th dlay s mor dffcult on th on hand, but th rlvanc of th quston dmnshs on th othr hand (xcpt th cas whn th dlay s th paramtr to b rtrvd, but thn spcal tchnqus mght b appld. Th algorthm handls complx pols th sam way as ral pols. umrcal stablty can b sgnfcantly mprovd by unfyng conjugat pols nto ral valud rsonator trms. As anothr way for mprovng numrcal stablty of th algorthm s dvson of th rows of th quatons by th rght-hand sds of th quatons. By ths, w can gt a last squars stmator wth balancd wghts. It s vn possbl that othr wghtngs ar mor advantagous. VII. Applcaton to dntfcaton In practcal problms th modls com from masurd data, and th followng qustons nd to b tackld: Is th xctaton (wth good approxmaton pcws constant? Is th systm lnar wth good approxmaton? What can b consdrd as zro? (On wll nvr gt pols xactly at zro, or concdng roots, or canclng pol-zro pars. A modl of th uncrtanty should b ncludd. o to th frst group of qustons may rsult n bas. Th scond group of qustons should b consdrd by takng nto account confdnc bounds. For nstanc, f a confdnc bound ovrlaps th orgn, thn th pol may b consdrd as qual to zro. owvr, snc a pol at zro lads to a structurally dffrnt modl, a rlatd quston s whch s th propr modl for a gvn systm, and whch choc s justfd by th data. Mor complx qustons ar how confdnc bounds (or othr masurs of uncrtanty can b transformd along wth modl paramtrs. Ths bcoms ncssary onc w want to compar modls of contnuous-tm systms obtand by s-doman and z-doman mthods. VIII. Conclusons Modl mappng from th z-doman to th s-doman has bn studd. Ambguous cass hav bn pontd out. Th common lmnt n ths cass s th prsnc of a dlay n th s-doman stp-rspons (or drct fdthrough wth on-stp dlay. For physcal rasons w cannot xclud th possblty of systms wth drct fd-through from th st of causal systms. As t has bn dmonstratd: drct fd-through combnd wth any knd of contnuous-tm dlay can b mstakn for fractonal dlay. Th dffrncs btwn rsultng s-doman modls can b qut mportant vn n smpl cass. W gav an algorthm to convrt z-doman modls contanng a pol n zro (cas not supportd by dc n MATLAB, whch also allows a drct fd-through n th ratonal part of th s-doman transfr functon. Qustons whch stll hav to b answrd: Dos vry z-doman transfr functon hav a corrspondng s-doman quvalnt or not? Ar thr any mor ambguts than numratd n ths papr? Is th transform unambguous whn drct fd-through s xcludd? What s th consqunc of nosy masurmnts, that s, uncrtan z-doman modls? ow can w dcd f a coffcnt s zro or not? IX. Rfrncs [] Schoukns, J., R. Pntlon and. Van hamm, Idntfcaton of Lnar Dynamc Systms usng Pcws Constant Exctatons: Us, Msus and Altrnatvs, Automatca, Vol., o. 7, 994, pp [] Kollár I., G. Frankln and Rk Pntlon, On th quvalnc of z-doman and s-doman modls n systm dntfcaton, Proc. IEEE Instrumntaton and Masurmnt Tchnology Confrnc, Brussls, Blgum, Jun 4-6, 996. pp [] Frankln, G. F., J. D. Powll and M. L. Workman, Dgtal control of dynamc systms. Radng, MA: Addson- Wsly,. d., 99. [4] Kuzntsov, A. G. and D. W. Clark, Smpl numrcal algorthms for contnuous-to-dscrt and dscrt-tocontnuous convrson of th systms wth tm dlay, SYSID 94, th IFAC Symposum on Systm Idntfcaton, Copnhagn, July 4-6, 994, Vol.., pp [5] Evans, C., D. Rs, L. Jons and D. ll, Tm and frquncy doman dntfcaton of jt ngn dynamcs: Problms and solutons, SYSID 94, th IFAC Symposum on Systm Idntfcaton, Copnhagn, July 4-6, 994, Vol.., pp [6] Golub, G. and C. F. Van Loan, Matrx computatons, Johns opkns Unvrsty Prss, 989. [7] 6

Lecture 3: Phasor notation, Transfer Functions. Context

Lecture 3: Phasor notation, Transfer Functions. Context EECS 5 Fall 4, ctur 3 ctur 3: Phasor notaton, Transfr Functons EECS 5 Fall 3, ctur 3 Contxt In th last lctur, w dscussd: how to convrt a lnar crcut nto a st of dffrntal quatons, How to convrt th st of