System structure and control : preprints of the IFAC workshop, Prague, Czechoslovakia, 3-5 September 1992

479 Pages·1992·52.853 MB·English

by  Vladimír Strejc, International Federation of Automatic Control

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
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Copyright©IF AC
1992
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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 _