SYSTESMT RUCTUARNED CONTROL Preprionftt hse WorkshPorpa,g uCez,e choslovakia, IFAC -5September1992 3 by Edited V. STREJC InstoiIftn uftoerT mahteiaoonrynAd u tomation, PragCuzee,c hoslovakia Published forth e INTERNATIONAFLE DERATIOON F AUTOMATICC ONTROL by PERGAMONP RESS NEW OXFORD • YORK• SEOUL • TOKYO UK PergaPmrone psiseH ,e adinHgitllHon a lOlx,f oOrXd3O BWE,n gland USA PergaPmreons sI,nc 6«J. ,W hitPel aRioma d, York1 059U1S,A Tarrytown, New KOREA PergamPorne Ksosrea ,K PO Box3 15S,e oul1 10-603K,or ea JAPAN PergaPmreons Jsa pan, Anne3lt-,2 0Y-u1s2h iBmunak,y o·Tkouk,y1 1o3 J,a pan Trunashima Building Copyright©IF AC 1992 AlRli gRh!.st servNeopad r.ot ft hpisub licamatyib oern e produced, asr teotrrseiydes votarel m trallSmiliJlatn eyd ofrbo yar nm y electdreocnticr,oma sgtnaetticime,cc hatnaicpael,, in photocorpeycinogor,rdo tinhgmuJnse : rwwiistepeh,ro muJis siworniJ frilolmgt he c opyrhcilgtkhrts . in Firesdti tion 1992 LibraorryC ongreCssa talogIinPn ugb licaDtaitoan A catalogue thriubosoo rkdi sa vfaoirl farbo!lmhe Lie b raoryfC ongress British Library iCnPa utballiocgauDtiaintogan A catalrougourfedo th ri boso k isa vailfarbo!hlmee B ritiLsihb rary ISBN0 -08-0-12057-5 Thesper epriwnetrser eproducbeyd oft hpeho to·opf!rsocee/s us sitnehma gn uscriptssupp lideb y thea uJhorosf tdheif! ereparJ perTm£asnsh ma.en uschraivbepeet t sy puseidnd gife fretynptew riaunr.tis typecfseaThef ay·ouJ,fiagn.tiw etsa bloefss omepap as didrtr t0ta grac ompfetdwyi tthh es tandard requir.e rr'.ceonrl.rs:e qutehr.Jerl eyp roductdiooenns o td isplcaoymp letuer .:furTmoi ry. rapid p•<hlictahtiidsoi r1s crepcanocuyln do tb ec ng norc oultdh Een glibsehc heckeonismpdurl euc iy Therefotrheer, e aders asketdoe xchaausneyded e fi:c eincieosft hipsu blicawrhiloc11h be deuet. ot he al:>ovmee r.Jionreeda asroe!IS . milY TheF ,,ditor Printeidn Initituot(eI nformatiTohnr ory Automa.tion Czechoilo\'Aacka demyo (S cienacnud Czecho:Slovakia. Pra.gue 18208 8, IFAWCO RKSHOPO NS YSTESMT RUCTUARNED CONTROL Sponsobrye d International AutomatCiocn tr(oIlF AC) Federatoifo n -TechniCcoamlm ittoene Theory Organized by InstiotfuI tnef ormatTihoeno ry AutomatioCnz,e chosloAvcaakd emoyf S ciences and onbe haloff C zechosloNvaatki onCaolm mittoefeA utomatCiocn trol InternatPiroongarla mCmoem mittee Kucer(aC S) (Chairman) Karcani(aUsK ) V. (PL) N. A.B anaszuk H.K wakernaa(kN L) L. B.R .B armish (USA) Lewi(sU SA) F. M.A .C hristodo(uGlRo)u M.M ansou(rC H) G.C onte K.O z<;aldi(rTaRn) Dav(isIo()nC D N) B.P ears(oUnS A) ED.Jei: .c:u sse JM..S t ern(aSd) (F) J. P.A . Fuhrmann '1s-,k1(\_, '·'.)-). y. (lL) M. Grimbl(eU K) ,\S.te,ceth(H:'aC: S ) HJip.p (eD ) JC..W ille(mNsL ) P. Hrusa(kC S) J. J. NationOarlg anizCionmgm ittee S. (Chairman) Celikovsky u D. KDusckoervaa V. M.S uc a o kov Strejc V. M. Sebek A. Vanecek P. Zagalak SOMER EMARKSO N A NE\A/C HARACTERIZATOIFO N LINEACRO NTROLLABILITY ?.I.ichFeLlI ESS LaboratodiersSe i gnauextS ystemeCs.,N .R.SE..-S.E. PlatedaeuM oulon, Gif-sur-YvFertatnec,e 91192 e-mail:fliess@bfirtenseet5 l. module-otrheetaipcp roacthol inecaorn trollliatbiyis s hownt ob ee quivalteonW ti llems' tArbasjtercatccotAhr:a yr acterizWaet ailosnso.k etcah c omputationadle-rfirveaet oifoB nr unokvys'csa nonicfoarlm ands howh owt or elatthee n otioonfn umericsatla te-sptaooc uer a lgebrsatiacn dpoint. Lineasry stemcso,n trollability, trreaajcehcatboiBrlriiuetnsyo,,v skcya'nso nicfoarlm ,m odule tKheeyowrofiyrl,dt sr:a tions. Introduction -Ina lolu rp asptu baltiicodnesa lwiintgsh t ate­ 1 variawbeld einsdo, it n trotdhusect ea te-Wsep ace. A module-tfrhaemoerwehotarirskce cebnetelinyn ­ clotsheig sa pa nds hotwh eeq uivableetnwceee n trodu[c6foe]rcd o nsatantndit m ey-ivnlagir near rseyasc­haabnicdlo inttyri oll.il tayb temIsit.ms o rgee netrhaatlnh o en weh ihcahbs e en knowsnin Kcael man nootn lcoym pristehse (14] stavtaer iabbulatel ssta,soh i w eth oolfte h sey stIetm . short review of the 2 A halse atdot hfeo lloawdivnagn ces: n.1.oudl e-theoreatpipcr oach -Anyl indeyanra mciacansl wabyegs i vaec nl assic Kalmsatna vtaer iraebplree se[n6t]a.t ion Lekt b ea g ivdeinf fergernotufiineadlDl de .n ote -Controlalnaodbb isleirtvya[ b6ti]ul roinut ttyo b e byk [fit]h rei nogfl indeiaffre roepnetriaaotlfto hres expreassns aetdu mroadlu plreo perCtoinetsr.o l2l.a1 ­ form bil,mi otryeoavpepre,ta or bse indepoefan ndye nt denominoaftt hiseoy ns tveamr iables. da �a - -Polzeesr,ao nshd i ddmeond easrm eo ssti mpdley­ 0 acfta fin[e7d] . finei t k[fii]is n g enenroanlc oumtmatiivcseo m(miutt ative -Thei ndoefxi mpltiicmiet- vlairnyseiyansrgt ems getacs l eara-lcguotr idtehfmiin[ci1 t1i]o.n onlikyfi asfi elodfc onstaNnetvse)r.t kh[efili]es s s, -Input-ionuvtepruttii csbha irlaicttyeb ryai r zaendk ap rinciidperaaillna gn tdh e mparionp eorflt eifets condi[t4i[]o5,n] S .i lversmtarnu'castl ugroer itkh[mfi ]-moadruaeln easl otgoto huosos fme o duolveesr correstpoao snudist maobdluefil let ra[t2i[]o3,n) .c ommutativei dpedraoilmn aci(inspseea e.l,g[ .1,5 ]). Thilsa tatpeprr oach yieflodrsm u[il3ana]nvs de rRseicoatnlh laa tfi nitgeelnye rlaetkfe[tdfi ]-module thpeo ssiobfdl eictoyu pplrionpgre irg ht-inviestr htdeii brlseeuc mot fi ttso rsmioodnuT la en odfa f !IrIe e systbeyma quasi-sstatta aet-ifce e(d2bw]ah,ci kc hm oduliee ..M, T = doenso nte cessaintyia nttee groafdt iiffoenr ential<I> , ©<I>. equation. 2.N2o tatiTohnels e ft k[fis]p-arnnobndyeuad l e Thep urpoofts heci osm muniicsat thiroene folds:e t { EI }i sd enobtye[ d Ther anokfa w w;ji w]. -Wefi rswta nttoc onvitnhrceee a dtehrao tu arp ­ mod=u Mlies w ritrtkMe .n proaicsihn timateltyoW_. i,l.rleeslmtasat'ne ddp oint, ' whichb giartvtoseh"e vefroauln daptaipoen(ra1sl7 )­2.A3 l inesayrs tL:e mi safi nitgeelnye rated left (21H]i.gs e ometroirciaelanlntyae ldiy sbs aissoe nd k[fi]-moAd,wu hlieic shc altlhesedy stmeomd ule, thneo toitfo rna jectormioesntrsat.[ 8th]a ht is ort hmeo duolfLe · We d chaacrteriozfca otnitornoe lmlaaybb eeri e lgairtdayed s thgee ometcoruinctearlpoa frt th ef reeneosfs h e withW ill(e1m7s) -(l2i1n)e,a r t temmo ndl e. sys­ s2y.s4Et xecamom rprlceTs apkoen,dt iou gfi nite { .a .,. Wq} -Amontgh eva rioucsa nonicfoarlm s,t heo ne of whiacrher elatbeyd fia nisteseoet tfwl =i nwde1ia,ffre rential Brunov(s1ik]syp erhathpesm o stp rofound see equatoiveorkn :s a ( alsKoa ilathS on(t1(a31g]6 ,)\ )V.he e rsek etch a computaalt-idfoernreiev vaitfaii olnt r(actoimopnasr e 0 Ek) witFhu hrmaanndWni ll(e1m2s] ). Ea(= wL) a 0µv= w}:(la aµv finei t s SYSTEM '2SnTdl lUICFT/\UCH\E " .'lANHD- Y.oCnsO hNuTpHO L -Septe1nber 1002, PRAGUE, CZECHOSLOVAKIA 3 5 Let bet hfer elee fkt[ )fi- modsuplaen nbeywd t2� +co. {wF1. ,., w. 9 }. D enobtye£ thseu bmodsuplaen= nedA tratjoerocy! 1=( tt12,i) s a m appiTn(gt) i: , t, byt he 'ss ucthh at A -+ C00(tt1c, )o n categoofrIR y[ fi]-modules. ea ThusT,( t21), tin the elemeonfAt af unction inC 00(t1,aTtsh2sie)sg .dno sf t troa jaenyc otfo2:ro ine( st 1,t'.!) isi nt hicso ntetxhtes eHto m(A,C00(ti1n, t2)) tlK Theq uotimeondtu Al=e F/E.i st he module.s amcea tegao.sar byo ve. Ther esiwd ue{w 1.,.. },i .ets.hyc,esa tnemo niimc­al ageof w sat=i stfiheeesq , u aWq tiEoawn()s = 0. T32(.S t3)e,:-At,co -+ t �Ct 001(� t tt22<3,i )t s 3� t.ist a�obi+ edaco r .Ae t tsrriacjtoeicfot no ry 2.T5h es ystLe mi sc onrtollabilfe,[ o6an)nl idyf , at rajeTc(t,tt)o41 ry C00(t1,il,fa.1 )n odn liyf , itmso duAl ei sfr ee. foarn ,\y EA,t hfue:nA -+c tiTo(nt4s1) ,a 1nTd(2 ,t13J{.·\) J(.•\ coincoin(dt e,2t 3). 2.A6 l inedaynra mi(c6sD) is a l inear ssuycsthe m thaittms o duAl ceo ntaas ienuts {u1 ,. ..; m }U 33.S e-t oo � t1I t2I t3 I t4� +I oo.Tr a•J ec• tones ofi npvuatir aebsvl eirfytihnfoegl =l owing thqeu otimeondtu Al/e[ ui]st orsiTohnpei.r n odpeep:re tnyTa­ (r'1'ttse '2 )a : Ati -odb+ eC c 00o(<m�t ,pta<; at)nii< df7'.b a ,l('3nt 'e4. t1do) n:Al -iy+ft C ,h00e (er;t,xe�ti )s ts denocfoef t hien pcuth aninwsee qilulsil v atlote hinfetnr ee­at r• aJerc(;tti:,o:)rA y-+ COO (It 1I ,stu4c)th h ar(t;tt, �) ness[ uT]h.i isn dependebneca es sumetdh e sequel. andr( '3tt' .) a rree striocftT( i'tot' )n.s .. 11 2 2.7F oarn yo fs ucdhy namDi,ct sh eerxei sact lsa ssi3c. r4(' tt'2 ) ( resT(pt3.''t'.. ) )i st hep as(tr esfpu.t ure trajl'e ctory). Kalmasnt avtaer iarbelper esen[t6a]t ion Definition systiessm a itdob ec ontlrloable A linear (1) la\ Villeimfsa,"n do nliyfp ,a satn df ututrraej ec­ (::) (:: ) (, ) �:t" iotr iaer�asel wacyos m patible. A � �; + JJ 3.T5 heoreTmh el inear Lsy issct oenmt arbollel where } laW illeimfsa,"n do nliyfi ,tm so duAl ei sf ree. -thdei mensoifto hnsset ate{ ando f "a thqeu otimeondtu Ale/ [vuix) ee= wdasx ,1 .ak.. -; xv 11ecstpoarc e equal; 3.6F irst atshsaAut mi esf reLee.bt (b,. .,.b m) are = 1 - A a basiosf Anyt rajectory andB armea triocveeksro fa pproprsiiaztees .C be"" (tt12,i) s o,Af. c oursceo,m pelleTdyt(e rrt,)2,e r:Am ib� nye d (1) thken owleodfgTe( ti.{t(rtoi).bt o).b.,.dm }i . Thec ontlraobliolfi tyi se quivalent = = \Veil-kpnroowpne rotfCi "e"sf uncmtailko,ie n tps o s­ t2.ot8 Rh efemr areke noefAs ,is . ,et .ot hceo ntrolloaftb hiel ity i systaesms ocitaoAt .e d sibtloed efinfoer,a ny 1.,. . a function 2.A9n o utpyu t (y1.,.., p )Yi sa fi nisteeot f e le­fi(tE) c =(1t,t1 �s)u cthh a=t , m, = mentisnD .T heo utpmuatp reads f(it){ T( 1I,t t2I) 'J ·(t)t �< t< t ; u1 = 7('tt' ) fi' (Jt ),t � t t'.i dcr + 3'• ' < < L Dadta () S eTt( 'tt' 4) = f; 't<'1t' 1) a n::?dT ( '3'tt' ) a ..r ceo mpatible. finite l' : Um 37. L the k Assumneo wt hat isc ontrollalbal e wherCe a nd Da'asr mea triocveesro fa ppro­ Willembust"t haAt i sn otf ree. rTehse"tnatr hicet pristiez es. trajecttoot rhiee st omrosdiuoTln oe f A .T ake E T. i; AsT isfi nite-doinmaaelsna sk i-vecstpoarc en,e ces­ i; a diffetrieanl sarsialtyi shfioemso genleionuesa r equa­ On Wille11t1rsa'j ectocrhya r­ tion 3 acterizatoifo nc ontrollabil�(­n)+ C1(n�-l+) ..C.n �+0 ( c1..,., eEn IR ) = ity 3.8A nyt rajec't(o.trtiyis )a�ti stfhiaeeb so ve tionA. ssumteh atth ep astt rajectoreyq ua­ (Scieles[ o8 ]) 'ci;,r�)� 3.1T hec hoiocfte h ef ucntioannaall ytiatc e of thef ututrraej ectory )�cl on otl ioen t h and s�1111e thetr ajectocrainebs e m anifoldc.a nso urf integcruarlv Aes.d 'i<3fr'rf ''' t e s roelnutdioon noset The c= functihoonwse,v aeprp,e atrobs e s implaslnedn atu­secTt(',tt'2 ) a ndT( '3tt' '4) a rien compa:tT ihbiesln lyees r-- ralD.e nobtyeC 00(t1,tth2eIR)[ fi-]-modoufCl 00e s ternm o1d1 ulem usbte f ere. A fnuctioonnt sh ien ter(vtat1l2, ) w,h ere t1< � -co RemarIkth asb eedne monstirnta htceeo dn ­where t3.e9x otft hcel assstiavcta er iraebplree sentatioann Gdt ahrameta troivceekors f a ppropsriizaetse, - F Willecmosn't rol[l2a0b]i,al(ni2dti1 yt] ms o dule-­Gi isn vertible. theorceotuinct erpaarrebt o tehq vuailetnot [6] Kalmacno'nsc eHpotw.e Ytehrre,e cotuors sueci hn Theh eigohfta ne lemeEn Ut_ 1i st hlee ast indiprreocwotof u hladv oeb scutrherede lation4is.n4ht iepg� e rs1 u cthl w.tE U z Iti sno ws imple wew antteoud n cover. toc onI\. strBurcutuo zv("s)kb ya_1's.€si = s (1€,.. ,.€ m,) ofA sucthha at: (i€ isb asiosfA , ) 4 On Brunovsksy 'canonical(i )it hcae o mponoef(nb tesl otnoUg _ a n1td h cea non­ iciamla ogfei€ nG (A)i sb asoifts h lea tgtreard ed­ form fremeo dule, a Assoctioa tteh ev asrtiraaetbpelr ee se(1n),t at(iiiotinhh ee igvh1ot f( 1i msa ximal, ) 4ot.fht1s e hus ebt dmauotTle[he xoefo ]fA l lsopwarinebnnsghyeut o dlhl ctedo sm:p oneni(ttihsosve n )ic6 gvei,h.+l . t.i,€ sm ia a rxdeie mtaelr.m( ii+1ni essd u,c thh at x. LemmaT hen extth rceoen diatrieeoq nuasil vent: (i andc oincide, 4.O5n oeb tatihcneos n trollablv1e indices (i)[ir xk] = A antdh.B er unovcsaknyo nfoircmaw,lh e�r e. .. � ()ir ik[Bi =x ) ]mm ., (V;m: x;1, Thet hicrodn diitmipolnasi ewse,lk ln owtnh,a t thneu bmero fi ndepeinndpecunhtta nniesll ess s thaonre qmu taotl h dei mensoifto hnse t aWthee.n onoef t hotsher eeqeu ivcano lnednitit ssi aotniss fied, thweh oilnep iuustt iliztehddey namDiicsssa id and tob ew elflo rmed. State-spaacned r eachabiyl it 5 Takes ucah d ynamaincdas s sume, mor5e.Lo1iv keiernS , e ctIiIwo,enas sumfoer,t hsea koefs im­ 4i.t2cs o ntarboilllAistsyo.c tioa( t1te)h fiel tratiopnl ictihtakyt= , J R. X = {.-1.'Pd}epfi nbeyda descencdhianXigPn = {xCPx)CP+,l1.,.} .o fs ubmoedsou fAl. 52. Thes tate-sopfta hcdeey namDi wcisti hn put ist hdeu aJlR -vescptaoXcr e Hom(U-1,J )Ro f LemmaF ora ny 0,t heq uotimeondtu leUu _1i n( 2)T.h idse finwihtii= cohn ,p teorr emciotvse r � XP/X (p+li)st orps iIotdnsi. m enassia o kn- vecttohre noofnt uimoenr viaclauclea sbn,e s eeasn ar e­ spaicase l ways striocftt ihnoeon t iooftn r ajecitnSo ercit3eis.o n Thec orrme. spgornaddiemndog d uliesG (A)= ffip>xoP/(x (P+l)). 53. Ther eacbhialiptryo pearsut syu,ia sle ,q uiva­ 1 Le�nmaG (A)i sgr aded-free lentt hpteoo s siobfji oliinttiywn pogo iintnX.s \ Ve assumlei,ik neS ect3i,to hna tth ceo ntvrraoila bles Thei npustt atfiel trat[i2Uo] n = { Ur}r e of arCe00 . 4A. a3s soctioa(1 )ti edsde fibnyae d de scenczdh ianign ofk -vescutbosrp aces 5.P4r opotsiioRne achaibsei qluiitvyta olc eonnt­ trollability. {Ur= { O}i,f� -2 i)A ssutmhea Dt i sn octo ntrolTlhaeb le. (2) U_1= spadrnx },i f= -1 KParlo(mouafnn c oonltlrasbulbes pwahciecc,oh r responds Ur= spadnx,u .,.. r (r,)} ,iu f 0 tot hteo rsmioodnuo lfAe ,o bvioduosenlsoy vt e rify � r three achapbriolpietryt y. A statsitca te-fe[e2di]bnt,a h cciko sn tiegsxi tv,e n (i)iA ssunmoew t,h aDt i sc ontrolJloaibnlien.g by neiwn pvu t=v 1.,. . sucthh atthfi el tration twpoo initnXs i seq vuailteonfit n diant gr ajectory asa socasi aabtoetvdoe a ,V nm vdc oinwciitdUhe.O ne x A__. C00[t1,t1:.2 ]1t ,t-2oof:. +oo whetrhee easirleyc otvhecerl sas sfoircm ula ficp:r sv;td erivaotfipi (o)€n{ i(<s = ((11,( m)i st he BrunobvassikosyfS ect4i.)o,5tn a kfo••er• =t t1a nd t t2g ivveanl uTehsic.sa bne a chiienvi endfi nitely m=a nmya nneTrhsec. o mponoefn twsh iacrhe linceoamrb inattihoe'n sas n adf u ,i nniutmeb /e;;r­ ( of thedierr ivaatriievm em;se ,d idaetdeulcye d. of 1I tm eanst hee xisteonfca e b asis , 7Jmo)f thatfo,r thec omponen7Jo t=f s ( 7JI ,b• •e. lontgo X AP such CONCLUSIOTNh icso mmuniciasat fiuornt her thei maganeoy f 7J p(P Oi)>,n X P /P +1X i sa ba7Jsi(ospf t) h ilsa ttveerac ntdo r steipne mbodytihsnetg u odfly i nseyasrt weimtsh in spac(es eea,l so[,1 5)). 10 modutlhee oNroyot.n l�yl egiagsna cien wehdi,c h( 1M0.]F liJe.Lsve si,n P.eM ,a rtainPnd. Ro ucohn, prehaipsasm attoefar se thecthiocibaculaet l, s o Ond ffierteinafllalntyo nlisnyesat,rPe rmsoc. sevneerrwael s aurloetb ste a.diD nisctriemcteaebn e IFASCy m.Np OLCOSB'o9r2d,e 1a9u9x2,. simlinile[as9r F] i.n ,an lolttyihcoaeut r along treated (1M1.]F lsisJe,.L eviannPed. R ouchIonndo,ef x understoafnn odnilnisgny esatmrea myas l bseon efit ani mlpitciimte -vlaeiranydrii nfgf eerqeunat­ial [1f0r]ot mhn iespw e rescvptei. tioAn n:o nmcmoutaltiinvaeela bgrre aaipc­ pora,cL hinAelagAr .p ptloa. p,p ear. References [1P2A.].F uhramnann Jd. WCi.l lFeacmtrszo,ia tion inedsai tci nfinfiortray t imoantafrlnui cxt ions, [1P].B urnovAsc klyas,s tiifioocfnla inceoanrt rol­ ItnegErqaula tOipeornasTt hoero r1y9,7p 9p,. lable systems, pKp1y.7b 3e-r1n8e7t.i ka, Q., 1970, .2_, 287-301. [2E]. DealnaMdl. Fe laiuAe nsa sl,g eibnrrtapeirce ­ (1T3.]K ail,La itnhSeayrs tPermesn,-t HlialEc,ne­ tatoitfoh nse t ruacltguorrei tanha mp pwliitcha ­ glewColoidNff .sJ9,.8 ,01. titoonfe edbdaeccko upliInFgAS,Cy mPpr.o c. NOLCOSB'o9r2d,e 1a9u9x2,. (1R4E.]. K almAadnav,n cTehde oorfLy i nSeyasr ­ (3S]E. l A smain Md. F lsi,sFe ormudl'evsie nrsion,t emisn",T o piicnMs a themSaytsitceaml The­ in"A naloyfCs otinrso Dlylneaidmc Sasylt ems",o ryR"..,EK almPa.nFL,al. b a n:tldL AA.br ib, B.B ondna,Br . BridGeat,uh ria Jen.dIP .. Eds.M,c GrHaiwlN-lew,Y orckh,.a 1 p01,9 69, pp2.3 5-339. KupkEad.,sB, i ruksh,ea r 199p1p2,.0 1- Boston, 210. (1L5.].H R oweRni,nT gho er,vy ola.n d Aca­ I II, demPirce SsasDn,i e1g9o8,8 . (4]F liAeustso,m aettci oqrdupiesff eerl,eFs not­i ruMma t.h,1 98p9p,2. 2 7-238. l\.f. [1E6.) DS.o e l , nt r (5] FlieGseesnr,a lliizneeasdry stweimtsh Sprgeirna-g,V ,e'.\ fralYtaohgrmka,tC iocn2.tlrT ohlo y, Ilvuf.m poerdd i sbturtpeiadr amaentddei rffse re(n1­J7.]WC i.l lFeorimfmsot w,i mseeHJr \iJlUe.is n seyasrt ems to tivaelc stpoarcI enJst.C,. o not,l4r 91,9 8p9p,. -ParIt F: i niitmnees-idolniantlei amiren variant 198199-99. sysmtsAe,u totmiac22a1,,9 8p6p,5. 6 1-580. (6M]. F lsi,se Sombea sic structuprraolp ertie(s 1J8.] CW.il le,mF sormt imsee rtioel si nseyasr­ ofg enerlailnisezyaesrdt Seymsst,e ms Control Exact tems- PartI I modelliAnugt.,o matica, Lett1.9,9 p0p,3. 9 1-396. lQ., 221,9 8p6p,6. :7 5-694. (7M].F lsi,se sAi mdpfelien iotfi on hiddmeond es, [1J9.]WC i.l lFermostm,i msee rtiole isn ear systems posla enzde r,Ko ysberna2t7i1,k9 a9p,1p 1,.8 6- -PartA pprmoaxtie Automatica, III modelling, 189. 231,9 8p: 7p8,. 7 -115. (8M]. F ije,s srAe maornkW illemtsr'a ojreyc t [2J0.]C W.i llems,fo rd MyondacemsDli,ys n amics charaacttioeolfrn eii anzr Systems controllability, Repor1t9e8dp9 p,1. 7 1-269. CotnrLoeltt .1,91 ,9 92. .2_, (2J1.]C W.i ll"ePmasr,ga mdsai ndp uzzilnte hse [9M]. F lieRsesv,ie brlslei neaanrdn onlinear theoofdr yymn aicsayls teImEsETr"Ea, n .sA u­ dicsreted-ytniiamcmIesE ,E TEr ans. Automat. tomat. 361,9 9p1p,2. 5 9-294. Control, Cotnr3.71,,9 92. 11 THE DIRECT (INVERSE) NYQUlST ARRAY AND QUANTITATIVE FEEDBACK THEORY APPROACHES TO MULTIVARIABLE FEEDBACK DESIGN : AS TRUCTURA L ASSESSMENT FOR 2-INPUT2 --0UTPUT SYSTE}..15 J.O 'REILLY W.E.L EITHEAD Dept.o f ElectronicEsl ectricEangl. , Depto.f ElectroniEcl ectricEangl. , Univresitoy&f of& S trathclyde, GLASGOGW1 2 U.K. GLASCO'C.Il' lQE, Glagsow University SQQ, U.K. AbstractA. new appilctaions-oientedap proach-Inidviudal Design- to multivariafbeleed back controwla s presetned inO 'Reilralnyd Leithea(d1 99a1n)d LeiCthheada annnd Oe'Rleil ly (19911a9,9a ,2b). There itw as shown thatI ndividuCahla nnelD esigins a transparefnlte,xi balned supportive desgin methodology whichd irectaliym st o meett heu ser'cso ntrorle quiremeanntsd i sw ell�uitteodt hee ngineercionngt ext. Thisp aperi sa ddressetdo t heq uestiaosn t o what ther elationsihpo f IndividuaClh anneDle sigtno earliefrr equency-dommaeitnh odsf or mu!tivariadbeslisie g ns ucha s the Direc(tI nversNey)q uisAtr ray (Rosenb1r9o6c9k1, 9 74a)n dQ uantitatiFveee dbacTkh eory( OFT) (Horowitz;1 99).1 Moreg enerally, given that IndividCuhaaln neDle signi sa frameworfko rth e evaluatoifo na ny lineatri me-invarimaunltl ivariab!e - controlltehri,sp aperp rviodes an interpretaatnido anss essmento f these two importafnrte uqencydomain methods;i th ighlighttsh eiard vantagesa nd establishefso rw hatc lassoefs m u!tivariparobblleem s theya re , valid.F ore ase of expostiion thei nvestigaftoicouns eosn 2-inpu2t- outpumtu !tivarisaybs!tee m(sO 'eRiyl l and Leithea1d9 9):1 thec onclusioenxst endw,i tht hen ecessary changest,o generaml- inputm -output system(sL citheaandd 'Reilyl 1992c). 0 Keywords. Multivariasbysltee msF;re quency domainD;i rec(ti nversNey)q uisatrr ay;Q uantitatifveee dback theory;I ndividcuhaalnn eld esig.n 1. INTRODUCTION 2. REVIEW OF INDIVIDUAL CHANNEL DESIGN Single-loforepq unecy-response methodsp,r icniaplly of the Ith asb eens hown in 'Reillayn d. L.ci�head t(h1at9t h9e1 ) Nyquist-Botdyep eh aveb eeni mmenselsyu ccessffuolrf eedback multvairiable control d0es ign pirsoi bnhleerme lnyt interactiinv e controsly stemdse signf rom the outsetT.h e extensioann d nautre with an intpelary betweenpl ant attributceuss,t omer similltion intod esiznr r::cto:fc fer eqL·c;::.:y:.·i-:C1. ::a-p,pircacshtpse cific·J! ;otnh ed esigpnr ocesist self. conscq�entchee, taos mullivarisaybsltee ms sy-stemwsi thm ore thano ne inpuatn d m-inptu m-oruntpdu dte sigpnr oblem ism ost In moret hano neo utpu-t hasbe en lesdsr amatic. of channels thati s,e achp lant outpuits cp laieerda srttol aay tr inee ftdee rrenrc.eo s Formal freuqency-domani approachetso multivariablienp ut. ; feedback control tihne cDliurdeecst( InversNey)q uistA rray Each individucahla nneils e nclosewdi thian feedbaclko op methodo f Rosenbroc{k1 96199,7 4)T.h e basiicd eab ehintdh e wiht a compensatwohri chm ust designetdo m eett hec hannel (InverseN)y quist Array method is the use of specificatiLone.t us againfbe o cusa ttentioonn the 2-input Dcpnrooiemnr-itcneroaocpmleslt nylc sseatatr'oinbesn em i( aniptfnpe elrciaeecssdtis itmaooirtn lyao at) nrro e tadhmueocN ubeonu ntt-w n Ihonettree e lrisaimcngitClnioeann-ottgtloehp r,e o l 2ma-anodtu rLtieKpxi(u tsmth)u (eO la(t'dR1i e9vill)ay9,r 1a tin hapdibrL sl2oe e-bii nltpehutmew a12i.-t9doh9 u d1tia)gpAsou.nm ta usl !h tcoiowvnnita nrr Ooic'laoRlbgneelatrieir l nol yl case. problecm anb e decomposedi ntot het wo equivalnt single-input A second frequency-domapipnr oach to multivariablsei ngle-out(pSuItS Oi)n dividucahla nnelnsa,m elyce hannelc 1 and 1Qcuo9an7nt9rt1oi,s9lty 8 ast2t1Fie.ev9md ees9ed 1sbYi,aagc nncki,u v ar rnTedhn eHto(otlrrQhoyyFew a Ts itz)u b((jH1eorc9ootw8f im6 twu)zhc) 1ihc 9ihn6t ta0ek1,re9 esisi6 tsts3 , , Canhadn LCneehlain Ctn2he.le a1d19C 9ha1s)t h e open-loop SlSOt ransmitt(an0'c Ree illy name fromt hes tateadi m of meetinrge sponsspee cifaitcoins in thef aceo f plant parameteurn certainitnis epse cifireadn geas nd disturbasnectes . (I) More recentlay n,e w applicatoins-oierntedf requency-domainw here approach - Individual CDheasninge-nl to multivariacbo!nteor l systems naalyiss andd esigins p resented in 'Reilalnyd Leithcad gJ2321 k2322 (1991a)nd Leitheaadn d O'Reill(y1 90a9 ,1 1 99a,2b,c). By A --- and h2(sA) ---- (2) invokicnugs tomepre rformanscpece icfaition on different channels,r (s)g"'1 g1n highly successfsuiln gle-insipnuglte utput classicaanlay lsis,o f the tydpees cribaebdo ve,i sm ade-o possible. thiswa y, a andi ss ubjectto t hed isturbance transpareflnetx,i blaen d supporitve multivarIinaa bnlael ysiasn d desigmne thodoloigsyd evelopewdh ichd irectaliym st o meett he g12 usersc'o ntrorelq uriements and isw ell�uitedt o thee ngineering dl(s) (3) contextI.t i st o be notedt hatIn diviudal ChannelD esign isn ot g22 a high-gaidne signm ethod FurthermorIen,d ividuCahla nnel Designi s nfootr ia n dveeissgtnim gea.attn hidon dugp n edersr es tbauntda ti hnlegog bp aolts enttriauanlcd t urCLehaialtn hneelC1a2 9d h9 )a1.s t heo pen-looSpJ SOt ransmtiatnce (O'Reilanldy ltfiirmmiaet-moainetsniw vfaororcirkoa mn nutt!r toilvlaeFrroi.ra ifbne!seetd baanccakde n es,oi ngni aogofn aaln cyo nltirnoelalre r (4) designebdy theI nverse Nyquits Array metho-ddf ora n automotive gas turbincea ses tudyi s evaluatiend 'Reilyl and Leitheadw here (919)1. k1 g11 This papiesra ddresstedo theq ues0iton what ist he h1(s) ---- relationsohfi Ipn dividuCahla nnelD esignt o tahse Ditroe c(tI nevres) � (5) NyquistA rray (Rosenbroc1k9 69,1 974)a nd Quantitative FeedbacTkh eory (Horo'>.itz 199)1. !-.r1e ognerally, given and iss ubejctt o thed isturbance thatl ndividCuhJ(a!nO nFTe)lD e sign isa frameworfko ret hee valuation of any time-ivnariant multivariab!e contthrioslpl aperr, pro,icleisni nteearpret atioann d assessmeonft t heset wo importaent frequeanncy -dormmeatihno ds;i t highlights theiard vantagse and (6) estabislhesf orw hatc lasseosf multivariapbrolbel mes theya re valid.F or easeo f expositio,n the nivestigation focuseso n No ssmuptiosn arem ade as regards then aturoef any of the 2-inpu2t- output 'Rielly and Leitehad traansffeurn ctiognis(j s)o r hi()s; thati s,t heyn eed not be 1ge9n9)e1:r at mlh- ei cnopnumctl- muousuitlopetnuxsistty vesnatdersw,mi(iy stLas ehbtti hlte(ehem ne0 saea ndc d e Os'sRaerciyhl alln9y g9 e2st,co ) . stabFlouerr t mhienrit,mh uemp mhua!stevia.ri able natureo f the channelsi s '2nd IFAC W'!rkshop STllUCTUllE on 12 - SYSTEM PRAGUE,A NCDZ E•C.H;OuSNLTOlVlAOKLI A 3 5 September llJ02, characterbiys etdh e complci<-ircqumcunlcuyv ariabsltcr�t uure G(s) is diagonaclo lumn( row)d ominant11 io1110n i:.. functi-yo(ns )(,O 'Reilalnyd L eithea1d9 91)w:h ent hem agnitude relationsbheitpwse etnh e magnitudoefs the indtinvicd turaaln sfer of- y(si)s· muchl estsh ano net hel oops ignailn teractiislo onw , functieolne menotfsG hold: otherwisleo,o ps ignailn teractiisoh ni gh. It isi m�ortantto (7) emphasitshea tI ndividCuhaaln neDle signi s a s:nglc-loop nor design-loionpt eractiaorncs p reserveidn individucahla nnels. 4 Some usefurle sulftosrS ectio3n sa nd r asf ellows. ac (s ) 19 EquaticJ's implyt hatt hem agnitudoef (Resu3l.t2 o f Leithead a). A t:-i.se R2c-:iunl?�t2t - olut pusty stemh as ar.Od' Regiali9ln2ym a rgins multivariasb(tlr7eu) c tufaruecn .ctdi o')n(s{S S) )( g12£21 fixed "'i:ht heo pen-loocph annpdi: !:::.'.�:�: .:i;.t�::j: oc-es stricltelsyts h ano ne;i .e., )!(Jg 1!" '.'2) a;hsjs)o,� izted k;&ii(l 1-y(sI) 1< (9) i,j 1,2( i" j),w hichp erfortmh e = the phasea nd gainm arginfl.:sar: scso:ciicar.t\esldi: t ht he final (7), (8) Alsob,y defini:ion them atrix (i)o pen-loroop ll-oofff e achc hannealr ep rimei nc!icatoofr s thet ransideynnta micbaelh a,iooufrt hec losed-lsoyos;it em; [ (iit)h ep hasea nd gainm arginisn dicawthee thetrh e correct l/k1 + &!! &12 numbero f encirclembeyn ttsh eN yquisptl otosf the( -1,0) K-l+G- (10) pmoaiinntta,as i rnieenqdt u hierp beryde stehneocc felc ohsaendg-elisson tyo hpse tc eosmnt tarboillliaetrrye., &!2 l/k2+g22 A on Resul1t i sR esul2t . isd iagoncaoll umn( rowd)o minanitf caveat (Resul4t. 2o f Leithcaadn d O'Reill1y9 92a). • For a� e2s-iunlp1u2t - o2u.tp usty stemr,o bustneosfs t hec losed-loop systems tabiltiot yc hangeisn the controldloeer s imply and robustnoefs st hec losed-lsooyps tesmt abiltiot uyn cneor1ta inty in thep lant. The necessacroyn ditiofnosr robustneasrse provided by 6.1 4.2 (11) Result of Leitheaadn d 'Reilly (1o9r9 R2eas)u lt of Diagonaclo lumnd ominancien and (12)r espectiivmepllyy (1991) 'Reilalnyd L eithead r0es tatehde rea sR esul3t. that 0 The phasea nd gainm arginass socia"'tietdh t he oacp l2eo-nRslie eondsop-pucul2s3 thol-y .aootsnp ut n teetspmlrtus aatynb ssitmlteiiamottp r tylea kma nenicuatgens sucu- e(rr-elotysjf a)h ri,ponirbt,=o uyjvls i,t d2nte;t(eohdjisfa )s tttf, hho eer 1-&&1211h tI -l-&& 1211(-- l+-ktk&1!&!)1 I 1- < 11 3() tl":� cf m�Jtivuianbn!:eC '.'-:�f t•r.ct•i1jOChnsS)d o S:.;;pc�•C''.cSt exceptc: t $i;;•.iiicanatnldy gc:rh-.ea.an:tnee:tr.rlch .oa :n[·t O h e:( gf1a:ci, n0c r)o sspo�vf'err:e oqtufe :nu:;coi'.fe. s"'c c:i: ocfl eea$ch -&12h 21 - &- 12 (kig22) I - g12 II< 1 1 4() 1&22 1&22 --l+-k2&22 (l/k2g+2)2 3. THE DIRECT NYQUIST ARRAY In othewro rdsd,i agoncaoll umnd ominancoef K-1 + Gi mplies thatt hei nteractions, (INVERSE) Perhaptsh eb estk nowno f allc las.sicailnlsyp irseidn gle-loogp 21 &! 2 methodfso rm ultivaricaobnlter osly stemdse sigins t heI nverse h1r1 h1r2, NyquisAtr r·a y( !NAme)t hodd uet oR osenbro(c1k 96199,7 4)i;t g11 and& 22 respectiavfefleyIc ntg attempttos reducae multivarifaebeldeb acdke sigtno a seto f classiScIaSlO c ontrdoels ignsT.h e INA ist hef irsatp proactho ChanneCl2 andC 1, ares malli nm agnitudet;h icso rroborates (197p4l7, 7 ) multivaridaesbilgen t hate nvisagtehse d esigans an interactivReo senbrocrke'ssu lt thatc hangeisn theo pen-loop process:m atchinpge rformancsep ecificataigoanisn sdte sign transmittCain(csed) u e to changesi n theo therl oopc ontroller constraitnhtrso ugthh em ediumo f compu1t9eg6rr 9a)ph.i cdails ploafy kj(j;tia)r cw ithianp pro1p,r i(aOtset rowskbio)u nds.T hus,i n the an arraoyf N yquisdti agram(sR osenbrock Whatt heI NA spiroiftt he! CD Result iti se stablisthheadtt heD NA (INA) and theD irecNty quisAtr ray (DNsAe)e kt o achievies the methodi s robuswti thr espectto changeisn the controller; reductioofn s ysteimn teracttioo ann amountw hereS ISOd esign namelyt,h ep hasea ndg ainm arginosf t hec hanneClis( s)i,= l, 2, methodcsa n be deployerda,t hetrh ant hec ompleteel iminatioinn dicawthee thetrh ec orrecntu mbero f encirclemebnyt tsh e of interactaiso ni n non-interacctoinntgr ol.T he requisiteN9 y.q1u isptl otosf th(ec1 r9i7t4ipc)oa,il n(t -1.(o()- 1j,/Ok)i n Theorem measuroef r'e ducesdy steimn teractiisdo ina gondaolm inance. of Rosenbrock as requirfeodr c losed-lsotoapb ility, For easeo f expositi.,.o.ne,f ocusa ttentioonn t heD NA arem aintainiendt hep resenocfec hangeisn t hec ontroller. 1974p,l7 4), method( Rosenbrock and thatf or2 -inpu2t- output So far so good. But what of the more important DNA(INA) systemsa;l lc onclusiaopnpsl ye qua(lIltNyoA t)h ec ompanio!nN A robustnoesfs t he 2 methodt op lanutn certaintFyo?r method.T he essencoef theD NA approacihs t hatb y iti sr ecallferdo mR esult thatr obustnoefs st hec losed-loop meanso f a suitabclhyo sepnr e-compensa(tpoor-sc tompensator)s ystems tabiltiot yc hangeisn the controldloeers n ot imply matrixP (s), the resultipnrge -compensatpelda ntQ (s) is robustneosfs t·h ec losed-lsoyospt esmt abiltiot uyn certainty dk2i(asg)co anna ldlotymh ibenena anptsp,ol tihewiadt ths oiuntgp lree-jlucodooitnpcot e tr hoelcl kle1o(r ssse) ad n-dl ooptr heegp ullaantctoa.rsT esh tiusfd aoycf t Lw eaist hhiegaahnd-d lO i'gRhebtyie lt(dlh 1ye 9l i9n2eaaq)ru .ad riant ic stabiloiftt yh em ultivarisaybsltee m. Now, itf olwlso immediateflryo me quatio(n7s) ,( 8)(,9 ), But what of the effectso f changeis n the controller( 10)t hatt hem agnitudoefs t hem ultivarisatbrluec tufruen ctions kj(s),i,,2j(=il;< ojn) thet ransmittabenlc'i'eJ :.�.n_U!l�.r�aI_!n<:.Lli._q -yhj(s)a res tricltelsyts h ano ne;i .e., wouatyp oufYti 7 t heR torsaennsbfrfeou(rcn1 kc9 t7i4po,beln 7 st5 w)ec eonn sidaentrdhs iY isq ueisotnt bhye (15) feedbalcok opk jo pen,a nda l!o therl oopqks j (i;tjc)l owisthed T.h is isp reci1ste,hl2eyI, n(d1i vidCuhaaln netlr ansmittCain(cse)= g(l - wheret hem ultivarisatbrluec tfuurnec ti-oyn( sa)n d (5ths)eu b system ")'ohpj)e :i1,-j= l otorpa n;tsjm)i.tJ tnCao inr(cdsee)d tr uo er teos tcrhtiahcneti;c ehisan n tghieen c t ohnueti r-otlhl etTrr haunss,bf ye (r1-5fau)nnh djcR( tesis)aou rnl3e t,r w ees pheacvteti hdveeef floiyl..n li..eobndyg r esualntd. ( 2). Stkjh)(esS m)tta os•t tr!"'aiimbKxt i- hl1i(Ri ansotp )syp+e,rQn o(bpsrwr)ohi,(cea k1rt 9eeK7 (4Ts,Oh) sie tboscru raonc wdrd9ss.ik .1afi 1go ror)en c qalculoois nrettedrhs-oa lltol oepdrt ihaeg RDoeNnAsa dul4oll.(my tiI F nNoaArnKd )mt-ie 1 at( gsho)on+daG adl(solstsmy)hoi ,ecnp iahanpattlwse iaeadtn n hmtd a gttahriein mco Gaepr(segs ni)a-n nlisdon o p shouladl sboe diagonadlolmyi �ant. transmittances ·,-hjj ),i1,.2(i a 2-i�put - = " m:.tLreitu xs,! i CnDte.r prtehti sd esigpnr cced�re withitnh e 2-outpusty s:kei£amiir e( mle asureosf rob�stneojfs) st he for ccn-exotf Specificaclolnys,i detrh e c!f!.�ognchneadrlo m inance systesml 3tiltiotp yl anutn certainty. closcd-:oop conditioonn sQ and K-1 + Q for2 x2s ystemesr ratherf,o r In 5hortf,o rd iagonadlolmyi nanptl anttsh,� �otationraela sonass,s umeG (s)i s alreaddyi ai;onald!oKym i-1n a+n t; methodi sr o1.Justto plantu ncertainTthyi.s s 1.1:oDef N :Af f(a!ii:"rsAs ) 1.ec.o nsidtehre d iagondaolm inancceo nditioonn sG and illustrated grianp Fhiigcuarlel y Jn particuliatir s,o bserved G. inF igur1e fora typicNaylq uI.is ptl oto f -yhj(s)t hatt heD NA 19691,9 74), By definition (Rosenbrock the2 x2p lanmta trix (!NA)m ethodi s highlcyo nservatiinv teh att he magnitude 11 restnclu')oj('nsh J) < removferso mc onsidermaatniyo nt haotf C hannCe2lw henche2 (s) ovetrh eb andwidth tchoent rsoyflsr taewmmhesiw caoalhrrl tker h oabituss trt oe p qluainufrtnre codem Rr etsauiW3lni tftt oyhr.i nrC ehparnenCsetbel.n yg t I2en2lda ( - dd;-i)It niCsohtnaoe,n=f a n& 1cde22 l2 t (-r!;-a h1n)s mgi2tc2t(a asnbcni elcn ec e :er:>i st hatth eN yquisto fp;- lhj(ost)s, 1j2, , shounlodt thes ubsysttreamn sffuenrc thi1o( ns ) roll= edi-onf tfh is rianodpci1uposcfr1tucaa!::hclrcc(< ,:; e,1. 0s cchp)s.avho er.i c eanlsttf ft rrcaeerbqqoluuwsieehsslnnaohcectveiisit edsesehir =s s an o. nt r ue iqtuoiatbvlovite ohlueys ;fcm rrieodtssq oouh fverirenagercnhq yg-iu effe-i)r n ttiec hqisyfaun uitetrs neit,crfah oiyv seft aisCr lsuh f mauitenrshdnt aC eah1ti'2 lseJ i sr nu.<cm t lehiudIan dt ttt ihhh neeeg unnacmSeeolr tythf,a�o iwttredn h iteay g ond(aoIlmNliAynm) ape nlttah notsidF.s o r rop bl;;ur.stt:tohs pa altra en t iplspa rinestdc iiasgeol�nyi> &lhl-yf Irndct ocqhmouihinesindn giachtnyaeit ssro u-nmf erbdrey aq nu·,,een,. c-y hich notd iagondaolmDliNynA a wnet h,a vea lreaodbys cr thatth e (1979fo)r 2 1· L.:cc ;.2-'ocutt pustys temTs.h ea ns-,.Hutc o rcwitz epsres-ecnoocftm ehp eDe Nn(Asp (aoItsNitAoa-)npc pormopiaesctmn hoasc tahrtoiisouxos..sc n- �:e-i):�e t j fh oarttom hf e cqouensi.�dtn1 (uo�onr: t�oc1, '9r.7:ec9rpoc,e8n n:8s.ta)t s rti aontwi hhneept tr ehtseheriohn sficu genh c-efrrttchac;ei� netnyc,y pre-compepnlsaanittsed di agondaolmliyn anItti. s !: erteh at· nom atitsea nr desitgenc hniiqsuu see do,r i sd uet ot his theI NAm ethoadn,db yi mplictahteDi NoAn m ethohda,sb een speci(QfFwiThc)da ets ign teicsth hnaiittqi uste h el atter. Namely, critifcoirsa e lda cokf r obustbnyeD sosy laen dS tei(n1 981) thed iscregp2a2n(c-ly')' h1)-g22-(;-l) g22;-inC hannce2l throutghheu seo fa simpelxea mpleI.nt hiesx ampla e, int hem id-htiog rhe-qfueinnctye rovfC=a hla nnce2l,a ssociated diagonalailsmionssgti, n guplraer-,a ndp ost-compeinss atwiiottnhh eH orow(i1t9z7 9) methoidsk, e pwitt hibonu ndbsy musueldt wihviacslrhtai rcauksbrc loftebuu unesrscsett s;-niiConsnic)t se ch leNo ystqeotu kipes(l t,1o 0 ot)pf ot ihnet tmhorede id aigaogdnooanmdlaio lnmlaiymQnn aFTcatg enhn tei( 2 t cuxod n2e)d iptllia')'onItn<t h elI e- sttshh ee (Leithead and1 929aO)'.R Meoirleloyv tehri,ds i agonalidsiisncgr eipnat nhcehy i gh-banCdhwaindntceh2l . compensoaftD iooynla en dS tei(1n9 8i1s)i nvaalsia d b asoifs Thusf arw,e havee xamintehdeb asisct rucotuftr hee channterla nsmitotfta hnecH eosr ow(i1t9z7 Q9)F Tm ethofdo r critiocfit shmeI NA( DNA)m ethosdi ncReo scnbr(o1c9k69 ) 2-inp2u-to utpsuyts tewmist htihne f ramewoofrJ kC Da nd explicwiatrlnysbo th againtshte use of diagonalising shownt hatth eayr ef acilibtyaa t beadn dwisdetpha raatnido n compensaatsibe oinn gu ndesir(ap9bl3l 1e)a ndt heu seo f somceo ndistuicoahns H orowi(t1z9'7sh9 i)g h-fredqiuaegnocnya l transforwmhaitciahor ne(s a lmossitn)g uilnat rh es teasdtya te (p9l32 ). dominance co.n dit.i Tohne,f ollo·wtiwnoogb servations InL eithaenadd 'Rei(l1l29yb9 ),i tw as poionuttte hda t onm ultivaQrFiT·a·a b rleae p propriate. Npyreq-uciosmtop fetp nshlshemoao0 uttu l niltooditbn v e aa srptipralubitcleoftped uul nracenft toisrwo hnii scc 1lh(o sst)e(h ie) Fo oftr rh'oem b� lu�ussvst.Ia n nnth. aQe·bFT f l ae(c oIemf) ue ntshtoRrdeu,sc put3llu ratrenu eqtndu c ietrrhetasat i nty to the( 1,0p)o int.B utp re-compenissa; t·iaolnfi odr theN yquipslto tosf t hem ultlvasrtiracbulfteuu nrcet ions non-diagdoonmailnlpayln at(n t clotsoeo nef)o rw hicthh e 1h1( s)a nd"'Y h2(sd)o notg o neart he( 1,0p)o inatt Nyquipsltoo tf t hem ultiv1 aJsr tiraubclteu re 1·f(uisns)cn toito n frequen;ceisset sh ant heg ainc rossofvreerq ueonfc y i' neatrh (e,1 0 )p oint. ChannCe2l - atf·r equeanbocvieea sn di nclucdrionsgs over frequeonfcC yh annCe2l, R esulti ss atisbfyi tehde 3 multivaQrFiTa(bIal)se s umpotfiI o')'1 n <I . QU/-ST:TATIVC f'EEDBACK THEORY Horo-.9dit9z) Y,a niavn d (1987) 4. (ii()1 99p1r)(o l, iidnef ifnrietqeHu oeccnoocn•y:d ,i,tf:io:<otr .nh ;se;e; x-di; sct,e-n-c,e. ,·.-;;z QuantitFaeteidvbeaT chke or(yQ FTi)s a contrdoels ign ofl inetairm e-invcaornitarnottl hlaeatrc sh ieavreb itrarily methcoudr retnhtesl uyb joefca t g redaeta olf i nteresatn. smasleln sitiTvhiert oylo.ef t hecsoen ditiisco lnasr iifni ed engineedreisnitggh ne odreyv otteodt hep ractdiecsailg n of Resu7l2.t o fL eithaenaddO 'Rei(l1l9y9 1wah)e rfeu rther As feedbcaocnkt rsoyls tetmhse,s tated ofaQ iFmT ist om eet ligihstt hrowno n thee xamploefYs a niavn dH orowitz responsspee cificiant tihoenf sa ceo f planpta rameter (1987). uncertaiinns tpieecsi rfainegdae nsd d isturbsaents(c Heo rowitz 199)1. Althoumgohs otf t heQ FT literactounrceev ranrsi ous MultivaQrFTi(a2)b lHe-o rowi(t1z92 )8,Y aoiavn dH oroo.itz 4.l. clasosfeS sI SOs ystesmesv,e ral exttote hnesm iuolntsi -in(p1u9t8 6) multi-opurtopbulte m fdraotmae s earlays 1 960. These multivaerxitaebnlseai roedn ess criibnHe odr owi(1t9z6 0, 1963)C,o nsiadgeari tnh e2- inp2u-to utpmuutl tivacroinatbrlocl Shaked (1976a)n,dm orre ecenitnHl oyr ow(i1t9z7 199,28 ) probleTmh.e raer et wod esisgtne tpost hes ecomnudl tivariable andY aetn iavn dH orowi(t19z8 6)A. revioefwt hiwso rki s QFTm etho(dH orow1i92t8,z Y aniavn dH orowi1t9z8 6)T.h e al presenitnHe odr ow(i1t9z9 1)I.nS ecti4o,n a nds ect4i.o;l.,n firdsets isgtnei pst hes amaes t hef irdsets isgtne opf H orowitz we examiwniet htihne f ramewoofrl kC DJ t.h em ostr ecent( 197d9)i scuisnsSe edc ti4o.n1u sinFgi gu3r;en ameldye,s ign developmteomn utlst ivacroinatbrlroeel p resebnyt Heodr owitzt hec ontrokl1(l se)or n theb asiosf then omincahla nnel (197199,28 ) andY aniavn dH orow(i1t9z8 6). transmiPtU11n cgel (1- ;-). Thes econd desiisg n step differietni tn;v= oltvheeds es ig�nf thec ontrokl2l(eosrn) t he ·4.M1u ltivarQiFaTb(ll-e)H orowi1tz9(79) mbaansiiopsftu hleiatt tt riuaornnonsf1,usfu t ten hrca ttt0hi 2eno2 on·m Aifnttarelaar nl sfifutenatrcll tegi eobnr a'ic Althotuhgehay p petaor similtahre, feeddebsaicogkfn fort hes econdde sigsnt epi si n factth eC hanndeC2hl Horowi(1t9z7 9i)ss uffibecd iiefnftelfryre onmtt h aotf H orowitzt ransmitntaamnecley,: (192)8 and YaniHvo raonwid(t 1z9 86a)st ow arrasnetp arate treatments, Q<:2 2 g12(l - 1h1) (1 6) Consiadgeari tnh e2- inp2u-to utpmuutl tivariable control prob.l eAmftesro mea lgebrmaainci pulaHtoiroonw,i( t1z9 79, Thu-sf,o 2r- inp2u-to utpsuyts tetmhset, wo desisgtne opfs p84a)s setrhtsa fto,r t hes tatpeudr poosfem eetitnogl eranHcoer ow(i1t9z7 a9)n dY aniavn dH orow(i1t9z8 a6)r es imitloa r specifiocnae taicoohnf t hef ouerl emetnutv's o tfh e(2 2) onep articular toad pepsriowgainct hh tihne ! CDf ramework closed-slyosotpet mr ansfer-fmuantcrtitixho,en p robxl oefm namelwyh,e np erformsapneccei fiicmaptoisoaen sb andwidth suitadbeslieog ftn h ceo ntroklt(ls ea)rn sdk 2(csa)n be recaasst separat'iRoeni alnldU:y i the1a9d9) 1. sucht,h ef ollo"'ing foucrl osedS-llSopoOrp o bldeempsi citnFe di gu3r.e U sintgh e thrpeoei (ns0th so uld be notedAs. d�finitfioorgin jsa ndPi ji nF igu3r,ei ·ti sr eadily tshhaot"' n wnht hek tf eedbalcoko opp ent,h ec ontrokl1(l sei)rs d esigne(di Thipsa rticduelsaiprgr no ced(ubruent o ·ta l!lC Dd esign on_! he basiosf then omintarla nsmitP1ta ncgel 1 ) proc��uirse!l sJ)o rpee rtinwehnetnt heb andwidth of kg2l2 f& �e2ge21d obrla oco(kopI p- e')'ni),sld: ie kseiwgtoinhnseect ed oh nebt arsookiIlf2st l ( hesv.enr)i o ,tm thih nea l CChhaannnnCCee2tll issis nictgehn eic foinlctearsnotsgthl alaliynetkn rhi e b( aCnhda\>oninCode2ftl h transmi&t&2t!I2 a (-n1c ).e bandwiidstr he)l athiivgeahln yd t hef irst .nominal channel Thesneo mintarla nsmitt(a1n ce1)s a nd& 22(1- ;-) transmiitstw aenlrclee p resbeyn ted- 1mRhaH2yPbe) Pa 'naqsdnug din dt(Piel-f ' f)1;-e ohf)rf1rt.e oh ntmIetnh n oedcm& ehalinle ntn etdlhar,- telsa r tr.arsnsumcmittiu!Jl:rn(emc:n eaa&sunly Imdc ( beeilor-sf f f e(ri iFto)ohf tre rNh oyembq uuulsipttsilintnovte taohsQsrfesfF it aTah2c()beome l mfu uee lnt tsh&ito!IvRr(d aeuI,srs c ptiu"Yt3lrall·u aurb trnceul ettqfned uuuc rinetercrh ettasait io nntsy frotmh es trucotfut rheec hannterla r.srr.iA1 c1ooin:tcrcosl.l er') '2(hsa)n d1h 1( sd)o notg on eatrh e(1 ,0p)o ine>t: cep2tt desigonnet RHdh ebZ asscifsa wronsgt rucitsuu rnel ikteolm ye et frequensciigenisf icgarnetaltyhea nr the& ain thep lanpta rameter toOlbesrearntvcheea stt.r .Hco rowitz frequeonfcC iheasn nce1l a ndC htannCe2l r especcrctsisovveelr y. 2(-1o9u7t9mp)usu lytts itveiamsrsQb ,iFa Tas bdeloedens ai gfcnoh rae nancehkil , fotrr2 -aintnshpmauittt ( tainiAs-ci fe)ir nce oqmumeceno(ncinityd) o i ftSo iefoHc notsri4 oo.wn1ti( ,h1t e9rz) 8o 2alonefd tY haein niafvni dn ite �ssu,2m e(i.ns1f i�)n ,)ic �itnntt rhoel gloaeltirohnk joe (pro.T hri h =i iss im no( r5ep)a e nrd(2t )i,n ent(H 1o9r9o1wia(t)1z.9 8169,8 7i)s c lariifniL eedi thaenaddO 'Reyi ll I w1 h= ens,a yt.,,,_h eC hannCetl b anwdidtihss ignifilceastnsht alny I J _