New trends in design of control systems : a postprint volume from the IFAC workshop, Smolenice, Slovak Republic, 7-10 September 1994

421 Pages·1995·23.302 MB·English

by  M Huba, J Miklěs, International Federation of Automatic Control

NEWT RENDISN D ESIGN
OFC ONTROSLY STEMS
Postprivnotl umfreo mt he Workshop,
A IFA C
SmoleniScleo,v aRke publi-c1,0 September1994
7
by
Edited
M.HU BA andJ .M IKLES
SlovTaekc hnUinciavle risnBi rtayt i,s lava
SlovRaekp ublic
Publifosrth heed
INTERNATIONFAELD ERATIOOFAN U TOMATCIOCN TROL
by
PERGAMON
AnI mprofiE nlts eSvciieern ce
UK ElsevSiceire nLctde, Th eB oulevarLadn,g forLda neK,i dlingOtxonf,ro dO, X15GB ,U K
USA Elsevier SIncci.e660,n cWhiete PlaiRnoasd ,T anytown,N ewY ork 1059U1S-A 5153,
JAPAN Elsevier SJcaipeanTncs,eu nashBiumial dinAgn nex3, -2Y0us-hi1mBa2u,n kyo-kTuok,y o 11Ja3p,an
Copyrigh1t9©IF9 A5C
AllRi ghRtse servNeodpa .r to ft hipsu blicamatyi obner eproducsetdo,r ienda retr�syvsatle omr
transmititnae ndy f oromr b ya nym eans: electronic, electrotsatpameetc,iha cn,i cmaagln,e tic
photocopryeicnogr,d oirno gt herwiwsiet,h opuJe rmissiionwn r itfrionmg thceo pyrihgohldte rs.
Firesdti ti1o9n 95
LibraorfyC ongreCssa taloginPgu blicaDtiaotan
In
A catalogue forreth ciosbor odk iasv ailafbrolmeth e Libraryo fC ongress
BritiLsihb raCrayt aloguIinPn ugb lication Data
A catalorgecuoer fdo rt hiboso k iasv ailafbrolmet hBeri tiLsihb rary
ISBN0 -08-042316 7
Thisv olume was reprodubyc emde anso ft hpeh oto-opffrsoecte ussisn gt hmaen uscripstupsp l�bdy t he
authorosft hdeif efrepantp ersT.h ema nuscriphtasv bee etny peusdi ngd ifefrentty pewriatnedr s
typefacTehsel. a y-ouJ,fiagnurdte asb loefss o mepa persd idn ota greceo mpletewliytt hh set andard
requirements: constehqreW!e ntplryo ductdiooenns o td isplayc omplete unifoTrome insturye.r apid
publicattihoisdn i screpcaonulcdyn otb ec hangendo:rc oultdh Een glisbhe c heckceodmp letely.
Thereftohrreee, a dearrsea sketdoex cusea nyd eficienocfit ehsip su blicawthiiocnh mayd ill!t ot he
be
aboveme ntionerde asons.
TheEd itors
PrintienGd r eaBtr itain
IFAWCO RKSHOPO NN EWT RENDISN D ESIGN
OFC ONTROSLY STEMS
Sponsobreyd
InternatFieodnearlao tfAi uotno maCtoinct (rIoFlA C)
Organibzye d
SlovSaokc ioeftC yy berneatincIdsn fonnatics
Supportbeyd
FacuolfEt lye ctErnigcianle eSrlionvTgae,kc hnUinciavle rBsriattyi,s lava
GemmaS,l ovakia
CameroSnl,o vakia
HenneSsl,o vakia
HirocReomh oznSflko,v akia
NafGtbae lSyl,o kviaa
TennorSelgo,v akia
InternatPiroongarla Cmommem ittee
J.M ikl(eS�L OVAKI(AC)h ainnan) G.PR.a o( IND)
M.H ub(aS LOVAKIA()C o-Chainnan) F.C apko(vSiL�O VAKIA)
M.A lex(fSkL OVAKIA) S.E nge(lDl)
G.C ont(eI ) B.F ranko(vSLiOtV AKIA)
J.F og(eSlL OVAKIA) G.H ulk(6S LOVAKIA)
K.SH.i n(dUiK ) H.J 6r(gAl)
V.H ru�(aSkL OVAKIA) ¥·K amy( CZECHIA)
L.J uri�(iScLaO VAKIA) S.K oza(kS LOVAKIA)
L.K evic(zHk)y J.M urg(aS�L OVAKIA)
V.K uce(rCaZE CHIA) A.d eP ao(rI RL)
D.P apadoup(oGRu)l os G.S chm(iDd)
M.M 'Saad G.S oti(rBoGv)
(F)
J.S amovs(kSyL OVAKIA) H.U nbeha(uDe)n
J.S uch(yS LOVAKIA) A.W einma(nAn)
A.PV.o schi(nRiUnS SIA) V.V ese(lSyL OVAKIA)
atioOnragla niCzomimnigt tee
JY
S.K oza(kC hainnan)
P.B istak
M.M olnar
B.H n1z
M.P ostulkova
P.K ulha
K.Z akova
M.B udinsky
Copyri©g IFhAtCN ewT renidnDs e siogfCn o ntSryoslt ems.
SmolenSilcoev.Rak ep ubl1i9c94,
TOOLSI NN ONLINEARS YSTEMT HEORY.
OUTPUTI NJECTIONASN D CANONICAL
FORMS
C.H.M oog*, M.P erdon*G*.C ,o nte***
A.
*LaboradAt'uotiorem atdieNq aunetE ecsoC,len et radleNe a nt1eru sed, e l aN oe,
NantFeransc,e
*DiMpa.t em"aV.tV ioclat eUTTan"i,vt eadr'Ais ncionVaiB,are eBciea n6c0h1e3,1
AnocnIaat,l y
**DEilpte.t roneAi uctaa otmiUcnai,sv iedtriAa 'no cnaV,iB are eBciea n6c0h1e9,1
AnocnIaat,l y
AbstrFaucrtt.rh eseurlo tnts h ceo nstroufac c tainoonn foircmfoa rln onlri nea
systseimmisl,ta otr h Meo rCsaen onFiocriamnlt hlei nceasaerar, eo btaiTnheed .
probcloenms icdoesrieisdnpt asr,t iicnsu tluard,ty hieenx gei sntcoefa g eneralized
outpiuntj etcthtiaac ohni eavd eessi rbeldod cekc oupAls iunffigc.ic eonntd iitsi on
givientn e romfass eparacboinldiittyii ontn u,ri ncst o hmaptl cehtaerlayc btye rized
usidnigff erfoernmsta inaltd h epirro pe.r ties
Key wordNso.n lisnyesatrce amnso;n ical forfomrsm;s differential
Introudction serbvloaeb,s eravnardbe laceh a,ub nloebservable.
1
Int hceo nstroufsc utcaihc o ann onfoircmana, l
The stucdayn onofoifrc malws i trhe spteoc t
opepnr obcloenmc tehrpenos ssb iiloifdt eyc ou­
classoesfs uibtltear ansformarteiporneass ents
plibnymg e,a nasn o ouft piuntjo enc,t uitnhoeb ­
vlauatbolioenla nalybsasiipncrg o peorfdt yi­es
servsaubblsey frsotmte hmre e maipnaritnT.og
namicsayls temnso.n lFisonyres atrie nm sp,a r­
thriesg aarp da,r trieaswluas lp tr eseinn[t ,2e ]d
ticulatrr,a onurssmfianttgih oganetsn ertahlei zes
showtihnatgth e decoaucphliileneivg fa btihse
classnioctailo ns ooffc oocrhda,nis gnteaa ttese
dynamoiftc hsue n obsepravrattbh,lai etng en­
feedbaancdka ddiotuitvpeiu ntj ecotniceoa nn ,
eriasla fu ncotnoi fa ltlh set avtraei abclane s,
obtai(2n[) [,21 )a)c anolnf iromcs aimitlota hre
bee xpresass efuadn ctoifto hnoe b sebrlvveaa ri­
Morse Cano(ni[c1fa1roll ) i)nF seoyarsrmt ems.
ableso faafu nndc toifto hnue n obseornveasb.l e
Int hwiasy i ti sp ossitbodl ies ptlhaseyy stem
Howe,v epnrroo cefodrcu hreec tkhipanrtgo perty
structthuairtsie n varwiiatrnhest p etcott h e
wasg iven.
mentitornaeondrs mfatasi doonnsie nt, h lei near
case,b yt hMeo rCsaen onFiocrawmli trhes pectI nt hpiasp weer ,i ingvaemtsoetrd ee epslucyah
tocha ngoesfc oordisntaafetteeedas bc,ka sna dd ­ separacboinldiiattnywid eo ns nteacteesa snadr y
ditoiuvtepi untj ecAt nioonnlsis.ny esatrree mp ­sufficcioenndti tthiaasotsn usrfo rea ,g ivmeenr o­
resenint tehdce an onifocramw lea rdee alwiintgh moprhifucn ction anfodr a g ivienn­
turonustt ob el inearfoirmzt ehdie n put/outtepghuew tri t1::h :; ! (h6.t::,,.he:; ene). x istoeffu nnccet ions
n,
poionfvt i .e wMoree,ori vtisn tevrraniala bles and sucthh at
resutlot pbaer titiinottnoher bedle o cckosr,r eefi­(...e..,i {,h -i.Vi((e.)..h, e n,)
!6(,. ,ne) = efi(.e,.i{.,h -·1.. ,,n ){tJi)(.{ h,
sporrdtiosn ugb systthaeartmser esipveecoltby- Thirse sguilvtteh pseos sbiiloiftt eys tiinpn rga,c -
tictehs,eu fficcioenndtfoi rtd ieocnob luiiptlyta hceas se thseu per(sIwc)irb lieolp m ti t­
= I
staitn[e 2dT) h.ei ntf,h c eo ndiissta itoins fiteeaddn,)dd e fiasn efodl loTwhsve.e cstpcoaer
oncea onb tbayim ne,ao nfass uitoaubtlpeu£ t i tsh e ssppaanconeve edbr yt heel ements
injeacc tanioonnri,ec pearnlet satthicaoltno seolfdy C w,h edrCie st he IC isneofitfsn yimtbeo ls
resetmhbaalcteh si ebvyta hbMelo erp,sr eo ced­C {deE, Ce} .F osr� 2,£( )•i tsh sep ace
=
durietn, hl ei near case. spannebdyt hoeevl eermo eNfn- t1dswC h,e re
A'-1idtsCICh i en fisneiottfs e y mbAo'l-1sd C
=
Thpea pioesrr g aasnfo ilzleoIdSnw e sc.2t t ihoen { /\ ../\. eiE C ;i 1.,,. s} .,m odutlhoe
tonoelesfod rce adr royntin hcgeo nstrauncdt ediq6ou nisvrdf. ale,;e lnasctpe=ia onbnny e edtq huea lities
ford eritvhrieesn ugpl rtess ietnnh tee das reeq uel
deisbcerSde.c3 ti idsoe nv ottsoe udm matrhiez e
resouf[l 1ta2sn)[ d 2a )b otuhcteo nstorafu ction
canofonrifomcr nal o nlsiynsetwaeirrmt esshp ecwth eriest hsei gnattuhpreee r mutation
tgoe neralizedi notvlrvtaihnsnestg foa rt(mea t. <T i o.i, n) .s of
�
andt hien ptuhte, oanudtt hpeduietrr ivati1v .e s.
InS ec4tw ieos nt aast mee,n tiaobnoaevnded , T hJ1ee l. e.m oe£f3in1( a)t rsce a lsl-efodBr ymms e.a ns
prove trheeso uftlm htape ia np eSre.c t5i on
otf hoep erda,dt eofirbn ye d
contsaoimenexs pa lmes.
dF({x,,u>}?) = L7=(1F8/8dxx;i )+
2 DiffeerntiaFlo rms Ljl=,., .mk.?_; (o8/F8u j-,.>k))d u]
Thdey mniacsayls tweecms o nshiedareredre e ­iitps o sstioasb sloec tioaa nteyel emoefn t
scrbiyabs eyds otfifer mos rtdd eierffr e�nqtuiaaa­lo ne-nfoarmmea,l y. voef£c, tt howarit bICl el
tioonfts lifo er m refedrt roase thdei fferoefFn.t A inae ll ement
vE £w iblecl a lalnee xdao cnte -ifvof r dmF
{i: =f( x+)g ( xu) (1)f osro mFEe =
h(x) IC.
y =
Given twvo £Ee( )l3a enmdeE n £t(s• '>,
womouhrptethyprE fhis ue'JRucetPt n an acx Edt t( 'JRiei"etoa, .tnnh reeshteeio. xre nff pi pg,ureh,u asesaE 'JRt rsr m m a,seet toe rhfidoe o­ wwn =aipL ltvro; =hwd ; e Lu;i,ictwvs i t;Eh i e. eIC , Ee VE ;(iIC A,s e 'd+iE1es -nN'wt1- o)bd1ht-yCdveeefoa ,C xi/\d rntr med r ­
analfyutnicictn ai noo npseb na olfl )v raeraila­ nrde preis£ne( +n••tb'ey)Ld iJ V ieW/\ij e ;w.
ables.
Sinncoce o nfusiaornit shmseeay, ym dbw oilll
Assutmhitan ngao nlcionnestayrrso tloe fm
albseuo s foerdd enotthoiepn etgro aratc ibneg­
E
thfeo r(m1i )gs i vaenndd,e nobtyi anngd
respeeltcyht,ei vdiomfie tsnstsn a itoens tpwaec£e(e)n•a n£d( •+d1e>fi nbyed d(i Lvi e=i )
ma,n odfi tisn psupta lceeut,sc onstihidene ­rL id iv e/\i ·A o ne-vfofo rrwm h idcv=h 0 is
saicdl osIeitdws .e ll-tkhneaoxtwafo ncr tm s are
finitoefr esiaenltd eintaetCre =ms { x,ii = clowsheicdll,eo fo sremad ro en ly locally exact.
1.,. .u�.k)j, 1, k �O }T. hsee t
. n,; = ...,m. ,.
ofal lt hmee roormpfhuincc itanifi onninstu em­ Propos1i( tFiroobne nTihueso r[e1m])L, e t
beorfi ndeteorfCmfo irnmaasfi t eesld de,n otVe s {v i.,.,V .r }b eas upbascoef £T.h en
by whose elemoefntdteesnn oswtiiemld­lV =h bapaaesnbA: :asciosn siosfct lionfsgoe rdim afsn d
plICb,yyF ( {x ,iu ;k).)T }h fiee lcda bnee ndoweodnli dyfiv /\ vi./\. /\. V r= 0 foarli l I =., .. r .,
wiat hed rieffsnttriuarcleIC t lutarotte he ed system
Ani mpocrotnasnetoq tfuh eaenb coev e Theorem
byd efinadi enrgi ovpaetriaasvt feoo rl: l ows
6
E itsh aao tnfo er vmfo rw hidc/\v h v 0 ilso cally
=
6ix= / ax()+ g (,x)<u0fo>ra l=lIi.,... . , n otfh foer vm= F dGfo,rF , G E IC.
cu;"/cu)?+ l)fo rk0� an dfo ra ll1j,
= = •••Im 3 CanonicaFlo rm
6F({x,u�,k))} "£':1:,(8F/8x+, )6Ixdn,e awliintnohgn lsiynsetoartef hm foesr (m) 1,
=
E;:...,imk�,;( o 8/F8u�-,.u»-,.;6> itithius ans tve (no4f[olu)o[tvl,n)5 tet l]oc hys oe tn atstthireeda, en irsnt fophruemt a tanido ns
whe8r/e8a xn,8d /u 8;rke) prteshueesn utpa alr ­outbpuuattl ,asfi on intuem boedfer r ivoaft ives
tidaelar tiiovvpee rators. botihn pauntod u ttp.Iu np artiscuuiltara,b le
notoifoge nnse rasltiazteedt- rasnpsafcoerm ation
Iwti luls ebifuetnl h ,foe l lotwcoio nngs,ti hdee arn odfre g ulgaern ersatlaiftzeeee dd cbaabncek
vecstpoaorcv rese denobtye£fod(r •� ) (ing ivasefo nl l[o)2w.]s (
IC s I
2
Definiti1onA generasltiazteedt- rasnpsafcoer S­omree maarbkotsuh tre e epsrentaotbi­on
E'
matioanm eromorphic map taianbeodav rneeo iwno rder[ 2fo)(rm s oeree
is . dtela)si.T hmea trAii,c 2eA asr ienJ orn da
T :(x ,u• .,.u,.,(ul: ,). . .) -+ (,C, u,.u.,,.(u ,.l:. .) fo)r man dt hmea trBi,Cc aersie nb lodcika go­
nafolr mw,i etnht croinessoi nsoltfOyi ' nasgn d
sucthh aftos ro mien tegtehrfe o llocu:oinndgi ­
l's.sTuhbes ydsetsecmbrsyit bheed t fiwros t
tiohnos:l d le
blotchkraset,p rtehsoeeb nste rpvaoarftbt hl ee
i)spa nK:{�} s�p K:a{nxdd,u d,u. ,., d. u <")};
systaerimen, v awriirtaehns ttp oge ecnte ralized
iis}p and�d sxp}a ndd{,du.,.,dd. uu ,( ")}.
state-tsrpaancsefo ramnradet giguoelnnaser r al­
DefinitionA regulgaern eralsitzafetedee d­izsetdaf teeec dkbTsah.me a tAr1id xi sptlhaey s
backa m e2r omorpmhaipc informaabtoituohatnel gesbtrrauiacctit nu­re
is finiotfy t htca orretsotp hoantd sc ontained in
F:(x ,uu.,, .,u.< ,". )..( xv,ti,.,. ,v.< .".))., E,
) thlei sottf h Meo rCsaen oFnojmrc.Ta hlme a ­
-+
sucthh aftos ro mien tegtehfreo l lowicnogn dit­rAbc2Iid isypstl hiaen formraetpiroenis fe nted,
E
tiohnosl: d le ilsi nbeyta hrMe,o rlsi/e3s. It it is m potrot ant
i
i}spa nK:{vd}� s panddx,.d.,u.tPd, )d}u;, notthetta he vbralioacbkl aednsv )c a,n
6,6 �
iis)p and�ds up}a nddx,.d.,v.d, vd(ti",) }b.y c onstrbuec teixoipnntr, ee rsomsfysea dn d
itdse rivatives.
The genternrrsaafolrimzaeitdni tornosd uced
abocvnaeb es ease npa a rtrci ascoeufq lu aasiI­ntrondouawcc lionasfgos u tipjnucett idoen­s,
stafteiecd ibantc hske e nosf[e 3L ]o.c atlhlayts, c resisbeendtas i tarlalnyts ifootnrhmsmao atd ­
definyiiteilodns :
ifyt hdey naomfib cyms e aonasfss ignomfe nts
E
thfoer m 9/((+z )g (xy)y,u., ,. , y. ( ,ri )it)s
ui= tPi(vx.,,. ,v . ( a;i) 1)=., . ,. m
poss:Ci 1-+b le,as suunmdpettrodi eoscnuosiu,tp albel e
Vi 1/J=i (ux,., . .(l1 ;),) iu 1= ,., . m i
. thvera ia6b ilnEe f' r o6,m6 anvd�) a nidt s
Int he orfte hpseat p weerw ,ic loln soindleydr e rtiivvaesl.enTtght ein,
threeg ulsairt uianwt hiiocnh ra=n k(8¢/8v)
{= 9(/(x)+g,(. x.,).yu <,ry),)y
rank(=8dtf;i/mu8 u=d)ivm . E= y:C = h(x).
Conceoruntipiunntjg e catr ieoasn,o nablebgete hsneye srot­betma biyan pepdl yiatn rga ntso­
E
alizaattit ohpnio,si onftt h,ne o tuisoeindn formaotfit ohknei dnde scarbio,bav enedd de­
thlei nferaarm eiwosob rtkab iycn oends idenroitbniygyn ,[gtl: hJd ee rivoafyat liovtnehtgser a­
fora s ystoeftm h foer m( 1t)r,a nsformjaetcitooofnrf:swi,e ce assn t foatrem atlhfolely l owing
E
thtma odify thbeym edaoynfans da dmiitcDisev fien .i tion
assignomfte hnet s for(m/ (+x g)( x+) u Definiiton3 An assiogfnt mhefeno tr m
9(.y.,,.<y r )),i nvonl:Co vni1-+l nfuignn cetaoirfyo ns9 (/+(g xx()) u,,y.y,., .y (irsa) n)o utpi:i:un ectj1-+­
andi tdrsei vaAtcitlvuletaysht,.ern asformations
tiointf h feol lowicnogn dihtoilodn s :
oft hism achyla asuvsne sd ieprreodp esritnicees ,
i)9 (,y y,., . ,y. ( r{)):i:i sg eneriincvaelrltyi ble;
theyc hcaotnuhgloeedbr sveaboitflh sieyt s,yt em
i8i)s:C pandddyyV,[,. 1 ,..d y[d"udl,u,. , .} . �
antdh ien trtoiodofusn co mree strciocndtiispa­vnK:e{d yd,y. ,., d. y (-l:l)+d,ru d,u. ,.} .;
tiointssh erreefrqoeurHdieo. w euvseigrne,gn ­
iiis)p anddyd ,y. ,., d. y (dl:ud),u,. ., }. �
eralsitzasetpdeata crnes formraetgiguoelnnas­rs,pa nK:{d,dy Cy11.,.. d ,y [-l:+11rd,u d,u .,.} ..
eralsitzfaeteedec dkabsna pda rtoiuctuiplnuar­t
Outipnujte octfth aiebo onkvsie cn adbn ep roved
jectoifto hnes d keisncdar biobtvehtedda , on ot
alttheoerb serovtfah bseiy lsiitittepsy mo ,s si(bsl[ee1e t0 p]o)r eserve ottfhh oeeb sdeirmv­ension
tgoe atn onlciannoenarfoi rcasmas lt aittneh de abislpiatiycnt e h see nosf[e 1 4T]h.er ye pernets
theraec folrasoesft rnasformtahtdaioton no t
folloPwrionpgo (ssi[et1e[i2 2o]fo],nr c oan struc­
givet pora itsheo ulnodgeiscpiahlre,an bolmee na.
tipvreoo) f.
PropositioGni vean s ysteomft hfeo rm Goibnagct kot hree prtesieonnts aeeinn
(1s)u ch 2g teehnraitc raalnlgEkyx( )= ma nd Propos2i,tt hpiero onb oldfe emcE o'u{p 3frl oimn g
ran(ko h(x)/po ,xt )h eerex ias gte ner6­,{ 2an dV 1b ym eaonafsn o utput dionejse ction
aliszteadt e-tsrpaancsef ormaa gteinoenra,l inzeosedte mt ob et racitnga ebnleeIri tats lh .e re­
=
stafteee dbaancdak na ddiotuitvpeiu ntj ectifoorcne o nvenliiemtnihattet tteotn ott hiceoas ne
thattra nsfoirmn ttohf eol lowifnogrm iwnh ifch has parptriocpueNlrmaateril ieyts, .
E
wibleasl s umedfo ret ahccaohtm ,p o/ino eff,n t
theerxeif sutn c</Jt(ii onvis., ., . '>v , and
6,6 , � )
tPi6(,v2 )s uch that z
6 =
=
I:' y = /i tP=i 1(, {{v2i.,.. , . v,' ) v2)).
{ �� : l1/Ji({s,
3
ProoAfss.u mteht ad vA v= 0h oldTsh.e n,
Ino thteerr wmesas ,s umteht ta heerxeis setp a•b yt hFreo benTiheuosr, et mheerxeil sotc aal ly
ratfiunocnt iooftn hsoe b servvaarbilaaenb dl efusn cttf;i(oh{.n,. ,e. n a)n d a Ff suunctchhit ao n
thuen obseornveasb.l e v= Fdtf;B. y0 = df( ddA6 A ..A. e -h1) =
d(vA1Ad . f..A.e- hi) d=F /\ dtf; dAf.1 A. .A.{- hl1
Theonn,he as t hfeo llotwwion gc ases: oenihetas ht ehtra dFe peonndlosyne 1.,., .e lha -nd
Iid oesn odte peonned s an do nV '2in othero ntf; . Thebny,d f= Li-5h(-)l(f/8 8{i+) df.;
tertPmis{ v2{)sc ,a bne c hoseeqnut aol0 ,o r F(6,. .,.{- 1h,t/J({··h·,·{n,o) n)hedas tf;t haft
(8<Pi/::/=8 0t/J.iN )o was, alrreeacday6l ,6l ed,d epeonndlosyne ,1. .,.e 1ha -nodn tf; .
i
anvd) c abne e xpreins tseerdom fys a nidt s Conversely,
�
dertiivvHaese.n iciteps ,o sstiodb elfieinn te h,e if/6( ,. .,.{ =n )¢ (1,{. .,.{- 1h,t/J(.e··he·)n·,)
casie nw hific dho esn odte peonn{d 3v 2,, onheas v = (8¢/8tf;a)nddtf; ,h devAn vc= e ,
(8¢/8tf;)d/\(d 8tf;¢A d/ tf;8 tf;0).
BI· I·(I y f, I· y I ••.( •• --•) f,I)I ·- :/ ..i.'l'I..( ..'l i ..2 'I v 11 • •• I <11>v )• Applytihrneeg s oufPl rto =p os3itt oti hoceno Im ­
pone/ino tfts hd ey namfii nc so ncea dne cide
Int hceas ei nw hi8c¢h; /::/=8 tf;;0t ,hI em plicit E',
whetihiteps ro sstiopb rloec ientd hece o nstruc­
FuncTthieoonry eeml ds
tiooftn h cean oniforcmaas l d escraitt bheeend d
tP=i g(i;f, 6, 6,vi ,. .,.v � )' ) ofS ect3i.o n
andi itps os sitbodl eefi ne 5 Examples
O;(;fy,y, ., .. , y •<>) gi=(fi ,,66, v1 ,.. ,v.� ' )>. 5.1 Example
Usinnogt atcioonnssiw sitttehhsn oeot f t hper e­
cedSiencgt lieoutnsc s o,n sider the system
Thmea p0= (18,. .O.i. ,.) .d e finaesno utput
,
injeacctcioorntd oDi enfigin oi3nt,t' h tat rans- V1
form{.s=3 /6(,6,6v,1,. .,.1( v ,•v )2 m.)t o6. = ele1+6v+2e r
g({,sv2),w hegr= e {t/;1,1t/J .i.),,. a ciheving 6
thdees idered ad•n•e•y dci oeulapdm liainxngig m al
loosfas c cessibility.
Thdey namoiftcs hn eo nobsepravriatgsb i lvee n
Thsee paracboinldisiettaeyibn oo niv osen sluyf ­bya f unc/t6(i1o3{n, vfo2r)w ,h iocnhhe as v =
ficnitfoe ras surtihnpego ssibdielciotueyp3 l oi(fn8 /g/86 )df. 3+ (8//8 vd2v=)2 2{i{sd6+ 6dv2.
bym eaonfas n o uptuitn je.cN teivoenrtihte lBesycs o,pm utdivn/\ vg o neeas igley0t h,se ntchee
deseoruviren st eirnve isrtot ftu here e soufl tt hsee paracboinldiiittssyi a otni. s Ifinfe adclte,t ­
nexSte cttihoacnth, a trearctihzfueen ctifoorn st it/Jn(g{ sV,2= ) e �+ V2o,n heas / e(i{s., V 2=)
whitchhse e parabciolnidtihyto ilodns . tf;({v2s),1+{ el.T hebny,a ppltyhioenu gt put
ijnectB(if,yo )=n (f- e?)6/,o noeb tatihnes
system
SeparabiliCtoyn dition
4
Letu sc onsindoewfr,r, o am g eneproailn t
ofv ietwh per obolfe m chefocrak g iinvge,n
meromofrupnhci/tc{i 1{o.,n. , .{ nw)i teh E
5.2 Example
C anda giviennt ehg weirt 1h h
theex tiesnocffue n cti¢o(1;n{. s$.,{ . -h 1($,) Fotrh sey stem
na, ndtf; ({.h.,,.e) n,s ucthh a/t(1 1{ ,n{) =
¢(1{.,. ,.{ h,-t/J1(e··he,·nT·)h)e.• ••Pn reoxpto si­ {1= V1
tion parw oafvyoi srdo elsvt ihpnerg o b,l ebmased 6 = {�{i+ V2
ont hteo oilnst roidnSu eccet2d.i on {y = ei
Prospiot3i Goinv ean m eromoirphcf unction
f(ei..,.e) n,w it{h E C,a ndgi veanni nte­thneo nobspearhrvastaa dl yemn iacs/6 (,6v,2 )
.
gerh wiht1 h theerex ilsotc fuanlcl­y forw hivc h=( o/f86 )d f.+3 ( 8//8 vd2v)=2
tio¢n6(s, . $., .{- $h,1 (n), andt/J (e.h.,,.e ns)u ch 2e1{3df.+3 vd.Ane asyc omputsahtoitwohstna
2
th1a(6t, e)n=<P (ei..., .{ 1htf;,-( {.h.,,{. n ))d vA vi sn oetq utao0l . H encase c,a anl bseo
iafn odn .l.i.•y f,w rigtv i n= i->Lh(8//8 {i)ondef. ip,r ovdeidr e(cst[ele2yt ] h)se,e paracboinldiit­y
hadsvA v = 0. tiionsno sta tiansdfia ecudta lilnty h,ci asse t,h e
4
nonobsebrlvocacabknl nebo edt e coufrpolme d[ 1)Y4 .FZ.h engL .Ca aon,dTr anssfetrr ucture
three .s t ofn onelaisrny st,Te emsc.Rh epE.as t China
NormaUln i,v1 .9 93
Referneces
[1M]. C hoqBureutth-,Ma .D eWitt-Morette,
Anal,y msainsifoalnddp hysics, North­
Holland, 1989
[2G) .C .o ntAe.,MP .e rdCo.nMH,o. o Tgo,w ard
an otoifco ann onfoirmcfo arnl o nlisnyesa­r
tems,P uhlP.o lAicsahSd c.ti o.a ,pp ear
[3·]ED .e laalneda u M. AnFa lligeesibsnr,­a ic
ter}>roaftt hsieto rnu alcgtourriewt ihtamhn
.papilcattioo nf eeddeubcpaolciNkno gn,l in­
eaCro ntSryoslt Deemssi gn IFAC Sympo­
siuBmo,r de1a9u9px2p,4,. 8 9-494
[4M).F liNeosnsl,i cnoenattrrh oelao nrddyi f­
fereanletgbiraPalr, o1 1c...A A..CS o.nference
onM odelalnAiddn agp CtoinrvtoeSl o,p ron,
Hunrgya1,9 86
[5M).F liAeustsaot,qmi ueetc ordpiesffr entiels,
ForuMmat hetmiac1u1,m9 ,8 9
[6M]. F ilesGse,n eirsalatnioonnl idnee ailrae
formcea nondiecq oumeam ndeetl inseaa­ri
tipoanbr o uclCaR.g.eA,c aPda.rS3ic0si8,.,
1899
[)7M .F liesGesn,e raclorinoztlecldae nro nical
forfmso lri naenadnr o nlidnyemnaiarc s,
I.E.TrEa.nEsA.u. .t C otnr3.5,1, 9 90, pp.
99140-01
[8A].I sidNoornil,ic noenatsrry sotl,Se emscond
Edit,Ei doSsnp reirn-glVaeBgre,r l18i99n ,
[9.].R M arinWo.,R esponadnde Ak. .Jv an
deSrh afEtq,u ivaolfne nonlnceieas ry stems
toi pnut-opurtipmuet tfoao prpmeosan,r
SIAJMC. o ntOrpotli mization
[1C0H.). M ooFg.,P lesGt.a nC,ao nnAdt. eM .
Per,dO oncn anonfoircmaoslfn onlsiynse­ar
temPsr,oE cC.C9 3G,ro nin,1g 9e9n3
[11) A.SSt.r uMcoItrnusvrareai,ola f Ln itnse ar
MulatriivaSbyelsmets S,I AJMo. f C ontrol
andO pt.,,1 1917p3p4, 4 6-465
[1A2.)M P.e rdGo.nC ,o natned C .HM.o og,
Somcea nonical npornolpisenyresta­ire s of
temResa,l tiizoaannM do delilnSi ynsgt em
TheorPyr,o McT.NS 8 ,9A mstemr,d1a 989,
pp8.9 -96
[1M3.)Z eCiatnzo,n foircmfoasrl n onlisnyesa­r
tems,
in GeometTrhioecro yf NonCloinnteraorl
SysteBms.J, ka ubcWz.yR kes,p onadnedk
K.T chEodn.s W,ro cwl,Wa idawnPio­ctwo
litechWrnoicwklsiakj i1,e9 85
5
Copyrig©h IFtACN ewT rendsDes igno fC ontSryoslt ems,
in
Smolenice,R epSulbol1vi9akc9 ,4
OUTPUT TRACKING FOR .A. CL.A.SS OF SINGLE-INPUT SINGLE­
OUTPUT NONLINEAR SYSTEMS: C.A.SE OF POLYNOMIAL REF­
ERENCE SIGN.A.LS
•
AlessaPnasdtrooar neAd*n toTnorinoa mbe**
*DipartimenAtuot omatiec Ian formatica,
di
PolitecniTcoor incoo,r sDou cad eglAib ruzz2i4 ,1 012T9o rinIot,a ly
di
**DipartimeMnetcoc aniecA au tomatica,
di
TerzUan iveraitRao ma, C.Se gr2e, 0 014R6o ma,I taly
di via
.AbstractT. hipsa pdeera wlistt hhp er obolfte hmge l oobualt pturta cfokria n g
ofsi ngle-siinnpgulten,-o onultispnyuestat wreh misac rhge l oblalilnye arisable
classs tafteee dabnacdco ko rdtirnasaftoersm Tahrteei foensr.ie gntncoabe le ts r acked
bvia yt hoeu tpaurpter odubcyae ned x tenronnall igneenaerr( attehoxero sysItte m).
shotwhna utn,d ceerr tsauiffinc cioenntd itthipeor nosb,l seoml vaanbdil,ne
sis ucach as eap, r ocefodrtu hrdeee soifagc no mpenspartoiso proN some idn.i mum
phasree quirenmeeendfotera d p pltyhipenr go ceisd ure.
is
Keywords.G lobal convergence, sNyosntlienmNesoa,nr m inimum-phsasyesdt ems
1 INTRODUCTION Undtehrase s umptthitaohtnse i ngles-iinngpleu­t
Thper obolfte hmase y mpttortaigcca k nidn outspyustt gelmo bdalilft'ye omtooar plhiince ar
turbarnecjee cotfai noonn lisnyesatsrdis­eu mb ­systtehasmeyis, m pttortaiccok fair negf etrrean­ce
jecttou nmeasurdaibslteu rwbasa sntcuesd ijeedc tgoernye,rb aytt heeedx osysotbetmai,n ed
byH irsc(h1o9r8Hn1i )rs,c ahnoDdra nv( i1s9 87b)ym, e aonfast wo-sptreopc edisu re:
Isi(d1o9r,8iI 9 s)iadnoBdry ir n(es10 9a)9n Tdor ­(it)h neo nlimnaepea,xr p retshsreie nfge rence
namb(e1 99T1h)fie.r asptp rowaecrhteehi sen ­outpasu ftau ncotfti horene fesrteanfotcrtee h e
sertoifto hinen veorfts hseey stientm hc el oseldi nseyasrt eomn,- liinnveet rhtreodtu hgeh ob­
loop s(yHsitresmc1 h9o8r1an)n,d,t h ien puts­erpvreorpis bo ysN eidc oesatil (a.1 99(11a9)9,l b),
outpluitn ear(iIzsait1di9oo8rn9iT ),h. em ain (129)9t,h ereby asyemspttiomttaihtdceieal n­lgy
probwliettmhh a ec tiumapll emeonftt haetsiseoi nr edt rsatjaectcoetr orreys tpoto hndedien sgi red
algoritthhasems su mpttihoatnths ey stemo utpturta jectory;
minimpisuh mas oeb:v iosuuscaclh oy n,d intisoi to n thdee sisrteatdtr ea jetchtuoosbr tyai ned
(ii)
necessary. is usefodri mplemtehntetr ianccgko inntlgra fowiso l r
A genetrhale owrasyd evelboyIp seidda onrdi t hlei nseyasrt em.
Byrn(e1s09, )9u ndtehrase s umptthitaohtnre e f­ 2 NOATTIOANN DB ASIC
erenscieg ntalobs e as ymptottircaalcalkryee d BACKGROUMNAGT E­
generbayat nee xdo syHsotweemsv.ue crath, h e­ RIAL
Letth foel lowviencgdt ioffre reeqnutaibtaeil o n
oralyl oawc so mpentsoba etd oers igwnheidc,h
ensuarl eosc al conpvreorypg.ee rntce COllliaittd iemtrEe eR d + :
!(th)( l!t,t ), (1)
Int hpiasp ietrc ,o nsitdheterr aecdpk rionbgl wehme !r=( eEt R)l , EN, thset avteeca ttto irm e
fora r estrcisi lcasotsfse idn gles-iinngpluet- output
tE R+h,·(·,)En C 0(Rnis x R+,Rn as uitable
nonlinear( tsshytesa ftteeeem dsab nacdco ko rdi­ )
vecftuonrc tainoRd +n ,t hfoel isl owsiunbgs et
cnlaastotsesfrr asefofremrslaeiitngnicngeoealean snor enirbezaaysaant )bnadee ll dxae tr egoref­R : R+ := {rE R : ris� O}T.h foel lowing
twasos umptaiboonust (s1ya)srs teue pmp toos ed
naln onligneenaerr( attheoexr o sysUtnedme)r.
holtdh routghhsioesuc tt ion, without mentioning
sufficcioenndti gitvieoinnt,sp osstiodb eslieg n
theemx plicitly.
ad ynacmoimcp entshaaeist tno sruu rnesd,ae drd i.A­ssumptio1n: E quat{i1oa}nd mi!t(at )
tiocnoaln diatg ilooncbsoal,n verpgreeonrpctey .V tE R+ aaa ni aaotleadou lti(oin.e e = .=0 ,
Thper ocepdruorpeoh esreecd a rroiuwetid t ho­,a ni aoleaqtiuebiidrlu pmo ionfet q ua(t1i)o)n.
I
oumta kianngmy i nimpuishm as ases umptoino ilnl
thsey staenimdt an extenotfti hopanrt o pos.Asesdu mption Foera cihn ittiimateloE R +
2:
byT ornam(b1ise9 93). anfdro e acihn ictoinadli! t(itooen)oe ,oE a n
equa{t1ih}oa naau niqauouelt i!=o (ntd )e,p enng di
I
on atnoedo ' w hiicadh e finferoad l tEl R +I�t
•This work beens upportebdy ofMini stero
dell'Universieht dasae llaRi cercaS cientfunidsfi ca. t.o
7
Iti su sefutlo r ecalfrlo m Hahn (19,6a 7n)d 3 PRELIMINARY
Vidyasag(ar1 99so3m)ed efinititohnast be RESULTS
useind thes equel. will Considtehrefo llowinnognl inesayrs teamtt ime
Definitio1n:L eICt b ed efinaeafd o llowa:t RE+:
IC := {¢(E·c° )( R+,R:+ (0)¢=) O ; e(t)= ae((t)), (6a)
4>(1)r> ¢ (2)rV,1r ,rE2R +:r i>r 2·} z(t=) c(e(t)), (6b)
A real-vfuanlcute¢id·(o ) ben l ongtioan eICgt i aa wheree (tE) R nn, E N, ist hes tatvee ctor
functioofnc lasICs. att imet ,z (tE) Ri st heo utpuatt t imet ,
DefinitionL etbe definefdo llowa:a (E·C" )+ l(RnR,n)an dc (·E)c n+1(Ranr1eR )
2: l aa suitafbulnec tioTnhsr.o ughotuhtis se ctiAons­,
{er(E·c° )( R+,R:+ )
l := sumptio1na sn d2 a res upposteodh ol(dw ithout
er(a$ier )( )1\>',21 1E1 1+R2: 8 1> a2; mentiontihnegmex plicit�ltyh)e ,q uati(o1nr )e­
er(a)O }. placbeyde quati(o6na )L.e tt hev ectzoer( EtR )n
•lim-+ oo = att imte E bRed e+fi neasd fo llows:
A real-fuvnacltuerie(odb·n e) l ontgoai entig aa
functioofnc lass l et() [z(tz<1)>,( .t.),., n - zl)((tj,)T (7)
Definitio3n:A rela. l-vafulnucetdil o(·n·) E, whZerze:1 =<>( ti)= ,·1 .,. ,. - n1 ,is t hei -tthi me
C°(R+ R+,R )t haata titahfifeeo al lowidnegr ivtawtoifozv ·(e )a t t imet .T heo bservability
x
relat:i ona map ofs yste(m6 ).B,e (E· C°)( R",Rinsd) e,­
l(·E,ICa,)V aER+, (2)fi nedas, in Isido(r1i9 8in9 t)he,fo l lowiwnagy :
l(rE,l·,)V rER+, (3)
.Bee)(:= [ce()L,ac (e).,. ,L. : 1-ce()f, (8)
iaaf unctioofnc lasICs.C . for eE R " wherLe� c(e)i, 0.,.I .n 1,is
DefinitionT heeq uilpoiibnret=i 0uo mfa y at­heali l-t Lhi ed eI rivaattievE e R =" o ffu nct-iocn( ·)
te(m1 )i ua4n :i forml(yw iretahp etcott hi £n itial
alongt hev ector fuan(c·t)iF.oo nre achs olution
timteos )ta blief f,oe ra cpho airetainlvu em bee(rtt ,,ooe)of ( 6aa)t t imet RE+ ,?t tofr,om
E E R,E > 0,t heerez iaap toaa iretainlvu em ­thei nittiialm te oE Ra n+dfr om thei nitsitaalt e
becSr( EeR ), cS (>e O), d epenodnEe, an utc thh aet( t)o e,oeoE nR,it i se asyt oc hectkh at the
thfeo llowinhgo lfpdoroaarpl t elhri etn yi tial t=i mea
followipnrgo perhtoyl ds:
toER +:
eoE R nIl e llol le(l,tto ,eo)ll (4) ze(= t.B)ee ((t)V)t,E +R, tt o?. (9)
<cS (e) <E ,
foarl tli mte RE+ ,?==>t tow,h eere( tto,,eoi)a L etJt? .e)(:= 8t?0je> bet heJ acobimaant rioxft he
thaeo luotfai yoant{ e1am}tt imeR +tt, ? E toob,s ervabmialpi 8tyo fs yste(m6 )a,te E R ".
from thien ittiimtaeola nfrodm thien iattiaatle ()
.Assumptio3n: T hef ollowinhgo lfredolara tion
e(t)o e.o aompoea itreiavnleu mbeErR , > 0:
=
Dteefim(ni 1ti)iag o5l :no Tbhaeleulq nyui fiolrpmil(obywiire tn=i re ht0uao mpefa c yta ­tIod et.[e()JJ?It? fJ, vefJE R n.fJ (10)
thien ittiimtaeola ) tt ractiivffe,o e ra pacihr Roefm ark1 :U ndeArs sumpti3o,tnh ereexi stas
poairetainlvu em bMe,r aE R,M , > 0,t here
E E nx n matBreiex)(s ucthh atth efo llowimnagt rix,
eziaafi tnait tiemT e(M , E +R, d epenodne nt
E) E
anMd, a ucthh tahtfe o llopwroipnehgro tlyfd oar
altlhi en ittiimateloEa R + : Mee():= Jt?.e)(Bee)( (11)
eoER nIl e llolM le(ltt,oe ,)ol l isd efiniptoes itfoirv e eE nR,an dt hefo llowing
< <E ,( 5) all
foarl tli mte RE+ ==>t, ?t o+ T(M, wherep roperty holds:
e),
e(,tt,oeoi)at haeo luotfia oyna t{e1ma} t t ime ? 1, (12)
t RE+ t, ?t ofro,m thien ittiimtaeola nfrdo m >.!n
Dttehefiie{mnni 1 tii}iatago 6tli :noa Ttabthllaeelet =ulq onytoui ).fo i rlmpliy(ob wiiretn=i re ht0uao mpfa et ycoat Fdw­ eoh[ter>.e,r x:Ie-a>. !mM:pne(lae e) pJ,=o sois}}in .cebfhl R..oee{ i m ocIDfeBe t {e1e,.)(.. , ..i }s t {>.h1ei ­n
thien ittiimtaeola )s y mptotically isiftit aab lv ee rsoefm atriJt?x .e)(i,. e.
locualnliyf o(rmwlireyta hp etcott h ien ittiimael Bee)(:= Ji.1e(), (13)
toa)t aabnlgdel obuanlilfyo( rmwiltreyha petcot
whoseexi stenicsee n surbeyd 1 0.
thien ittiimtaeola ) t tractive. ()
Lteem{mm1a i 1}ga: T l hoeebu qanuliilflyoapi rmaoblyireymni= ptu 0t mo oa fatt yiaacbNt­alohleett il rctayehn astapJ ossIiem.) poC lfceeJ ht?ro .ei)(co,mefBa yeey )(i,eslau d cm hasa ­
iafn odn lityfh eerez iaafu tnac tli(o·-n)o, fc lautr iMx ee()t hati sd efiniptoes itfoivre al le E
IC.Ca ucthh atthf eo llopwroipenrhgto yl fdoaar l lR n,bu tt hadto enso ts atisrfeyl at(io1n2 U)n-.
thien ittiimateloEa R + : dert heas sumptiotnh at>.:: ,.,::. "# 0,w ith
le(l,te ,ot ol$)l l (lello,tl - to),
foarlt li mte R E+ ,?t toa,n fdo arle loE R n, >.::,.,::. := inefR,.e{ minte{1,... , n}{>.i: de[t>.;I-
Wheere(t t Ioe o)t h8e0 liuotOnfa yat(1e )am t Jt?.e)(JJ.(e)J= O}}a ,po ssibcihloei cofeB ee)(
timte R IE + t, U?t ofro,m thien ittiimtaeola nds atisfpyrionpge r{1ty2i )sB e(:e= --:;-1:;)r Jl(e.).
from itnhieat tiaea(t lte =o e)o . >... fl..efl .e ..
�
8