Computer aided design of control systems : proceedings of the IFAC Symposium, Zürich, Switzerland, 29-31 August 1979

656 Pages·1980·24.61 MB·English

by  Michel A Cuénod

IFACCo nferPreoncceee dings
ATHERTON: MultivariTaebclhen ologSiycsatle ms
BANKS PRITCHARD: ControoflD istribPuatreadm etSeyrs tems
&
CICHOCKI STRASZAK:S ysteAmnsa lysAipsp licattiooC nosm plePxr ograms
&
CRONHJORT: RealT imeP rogramming
1978
DE GIORGIO ROVEDA: Criteforri aS electAipnpgr opriTaetceh nologuinedseD ri fferent
&
CulturTaelc,h nicaanldS ociCaoln ditions
DUBUISSON:I nformatiaonnd S ystems
GHONAIMY: SysteAmpsp roacfohr Development
HASEGAWA INOUE: UrbanR,e gionaanld N ationPalla nni-nEgn vironmenAtsaple cts
&
LEONHARD: ContrionlP oweErl ectroannidcE sl ectrDirciavle s
MUNDAY: AutomatCiocn trionlS pace
NIEMI:A LinkB etweeSnc ienacned A pplicatoifoA nust omatCiocn trol
NOVAK: SoftwafroerC omputeCro ntrol
OSHIMA: InformatiCoonn trPorlo bleimnsM anufacturTiencgh nology
RIJNSDORP:C aseS tudiiensA utomatiroenl atteodH umanizatoifoW no rk
SAW ARAGI AKASHI:E nvironmenStysatle Pmlsa nninDge,s igann dC ontrol
&
SINGH TITLI:C ontraonld M anagemenotfI ntegraItneddu strCioamlp lexes
&
SMEDEMA: RealT imeP rogramming
1977
NOTICE TO READERS
DeaRre ader
youlri briasnr oyat l reaas dtya ndoirndgec ru stoomres ru bscrtiotb heisrse rimeasyw, e r ecommetnhda t
If
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canb ec ancealtla endyt imwei thonuott ice.
ROBERTM AXWELL
PublisahtPe err gamPorne ss
COMPUTER AIDEDD ESIGNO F
CONTROL SYSTEMS
Proceedoiftn hg/esF C A S ymposz"um,
Ziirz'Scwhz',t ze2r9l-3aA1nu dg,u1 s97t9
Editebdy
M.A.C UENOD
Geneva, Switzerland
Publishfoerd t he
INTERTNIAONAFLE DEARTIONO FA UTOMATICCO NTROL
by
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ofC ontrSoyls teZmsu,ri ch,
1979
Computaeird edde siogfcn o ntsryoslt ems.
-(InternatFieodnearlao tfi oAnu tomatic
ContrCooln.f erepnrcoec eedings).
l. Automactoinct -rDoalt par oces-sing
Congresses
I.T itles II.C uenoMdi,c hel Swiss
Ill.
FederaotfiA ount omaCtoinct rol
629.8'312 1)213 79-42655
ISBN0- 08-024488-2
Thesper oceediwnegrser eproducbeydm eanso ft hep hoto­
offsperto cesussi,n gt hem anuscrisputpsp libeydt hea uthors
oft hed ifferepnatp erTsh.e m anuscriphtasv eb eent ypewdi th
differetnytp ewritaenrdts y pefacTehse.l ayoutth,efi gures
andt abloefss omep aperdsi dn otc onforemx actwliyt ht he
standarredq uiremecnotnss;e quentthley reproduction does
notd isplacyo mpleutneif ormitTyo.e nsurrea pipdu blication
thesien consistecnocuilendso tb e changedn,o rc oultdh e
Englisbhe checkecdo mpleteTlhye.r eaderasr et herefore
asketdo e xcusea nyd eficienicnti heivoss l umew hicmha yh ave
arisferno mt hea bovcea uses.
Editor
The
PrintienGd r eaBtr itabiyAn . W heato&n CoL.t dE,x eter
IFACS YMPOSIOUNM C OMPUTERA IDED
DESIGONF C ONTROSLY STEMS
Orgazneibdy
TheS wisFse deratoifoA nu tomatCiocn tr(oSlG AA/S SPA)
Sponsobrye d
TheI nternatiFoendaelr atoifoA nu tomatCiocn trol
Systems Engineering Committee
Applications Committee
ComputeCro mmittee
EducatiCoonm mittee
InternoantaiPlr ogmr Caommittee
M. Mansou(rS witzerl(aCnhda)i rman)
K.J .As tro(mS weden)
W. Findei(sPeonl and)
R.I serma(nFn. R.G.)
G.A .K orn( U.S.A.)
P.M . Larse(nD enmark)
M. G.S ing(hF rance)
Stre(jCcz echoslovakia)
V.
Unbehaue(nF .R.G.)
H.
A.v anC auwenberg(hBee lgium)
R.v anN autaL emke( Netherlands)
H.
J.H. Westco(tUt. K.)
andm emberosf t heN ationOarlg anizing Committee
NatioOnargla zniinCgo mmittee
W. Schaufelbe(rCgheari rman)
F.C ellier
A. Glattfelder
H.
P.G repper
E.R uosch
M. Steiner
J.We iler
FOREWORD
Theu seo fc opmutertso an alysean d designt hec ontroolf d ifferenptr ocsesesi s
nowa commonp ratcice. Programp ackgaes whichi ncluddeif ferenatl groithms
developiendm oret han thredee cadehasv eb eeni mplemteendo nd igitaclop muters
tog ives tduentansd engineeras p owerftuol olw ithw hictho d esignt he- control
fors ysmtsei nd ifefrenatr eaosf a ppilcatino.
Thea imo ft hisS ypmosiumi st od emnostratteh es ttae-of-thei-nat rhte a reao f
copmuteari dedde signo fc ontrosly tsem.s TheI nternatniaolP rogramC ommittee
hasm adea carefusle lecnt ioofp aperfso r tShyemp osiuman d I hopet hatt he
resulwti llb e benfeicila toc ontroeln gnieerisn teresitne tdh isy oung filed.
Mansour
M.
Chairman
InternatniaolP rogramC ommittee
xiii
SessioDne siagnndA nalysMise thofdso Prl anCtso,n trollers
1.
andC ontrSoyls tems
CADO F MINIAML ORDER
CONTROLLERS
A.B aelstrainndGo . C elentano
IsuttiEotl etttercconoUi,n ivedrisN iatpoalN ia,po lIiat�yl
AbstractI.n thisp apert hep olea ssignmenptr obleims reformula­
teda s a standarpdr obleomf nonlineaprr ogramminPgr.o cedures
forp olea ssignmebnty usingd ynamicco ntrollewrist hm inimaolr ­
dera rer eportebdo thf ort hei nteracticnags ea ndt hen oninter­
actingo ne.N umericaelx ampleasr ed evelopeodn a digitaclo mpu­
ter.
INTRODUCITON
Leta lineart imei nvariandty namic micc ontrolleasr (2.)
systemb e describebdy : If r+m n+1 theser esultasr en ot
<
practicable causneo thinigs saida b­
Ax + Bu ( a1)
x = outt her emaininng+ 1-m-eri genvalues
y Cx ( b1)
= whichm ayr esultw ithr ealp ositive
n r m
whereX ER, UER, yER, andA , B, C part. overcomseu ch
To difficultyi t
arer ealm atricewsi tha ppropriadtie js possible taas si�n1 41a,lw aysb y
mensions�.o reovelre tt hes ystem (1)m eanso f a nondynamciocn troller,
m+
be completerleya chablaen do bservabler -1, 1, eigenvaluaersb itrarily
1 �
withr ankB r, rankC m. closet o m+r-1s pecifiesdy mmetrivca ­
= =
It is wellk nown1 1 that,i f max(r,m)l ues,s o that1 -1d egreeosf freedom
I
= n, by usinga linearn ondynamciocn area vailablteo confinteh er emaining
troller: eigenvaluteos a specifierde giono f
thec omplepxl ane.
u Ku + v , (2)
=
If thisp robleims unsolvabloer if
r.m r
withK ER , VER, it is possiblteo
alle igenvalumeuss tb e exactlays si­
assigna n arbitrarsyy mmetriscp ectrum
gnedt hena dynamic controldleesrc,� i
to thed ynamimca trixA +BKCo f the
bed by:
closedl oops ystemS.i ncei n practice
Ww
+ Dy + Ev (3a)
n independeinntp utso r outputasr e w =
nota lwaysa vailablet,h enc onsidera­ u Hw + Ky + v (3b)
=
blei nteresatr isesi n designincgo n­ v r W,
wherew £R, VER and D, K are
H,
trollerbsy usingo nlyt hea vailable
realc onstanmta tricewsi tha ppropria­
inputsa ndo utputs.
te dimensionmsa,y b e used; s,6J.
In l2,3,J4 it hasb eens hownt hat,f or
By usingt hep rocedureisn J,12,3,J4
almosta llp airs( B,C),m in(n,r+m-1)
thed egreeosf freedoimn thec hoice
eigenvaluceasn b e assigneadr bitrari
of them atrixK in (2)a ren otc omple­
ly closet o min(n,r+m-s1p)e cified
telyu tilizeds,o thats ometimenso n-
symmetrivca luesb y a linearn ondyna-
A.B alesot arnidGn .C eelnatno
2
dynacmoinct roclablnee rsu sed in­y Cx (Sb)
=
steoafdd y naomnsie;cm oreover also
is
witthhp er oceidnu renso atl l A.
IS,16 Asi ti sw elkln oiwfmn a x(=mn , r)
thdee groeffe resed oamv ailwaibtlhe
the pmraobybe al lewmsa oysl v1e d
thdey nacmoinct rolalreuert ili­ 1I
(3) morgeee nralilfmy +,rn +1f,o rm ost
zesdot haetf ficdiyennactmo inct rol­ > cil
alpla i(rBs, iCti) s p osstiobf lien d
lercsab ne dewsiitgrhne edduo credd-
am atrKs iuxct hh atths ep ecotfr um
er.
A+BiKsaC r bitrcalrotisole y
Int hipsa ptehrpe r oceadrugeri evse n A 12,3,
Itm usbten ottehdab tyt hper o­
fodre sigdnyinnagom rii cns, o mpea r­41.
ceduurseeisdn onlmya trKi ces
ticuclaasrne osn dyncaomnitcr,o llers 13,41
witrha nkKa re conIsfsi udcehr ed.
witmhi niomradlse orl,v tihnpego le � 2
constornat ihrneat no kfK i sr emoved
assnimgepnrto balthe amn Tdw.to y pes
itm awye lhla ptpheant t haeeli lg en­
ofd ynacmoinct roalrcleoe nrssi dered,
alumeasby ea ssigenveeidnfm, + r<n+1.
thien teraocnteaissdn, yg n acmoimc­
Moreionvp erra ctthirece eq eumiernt
pnesataonrtdsh ,ne o inn teracting
thate iagleln vcaablneua ersb itrarily
onse,a so bseerrsv.
assicganbnee rd e laixfte hdde e sired
Firstthley apsoslieg pnrmoebniltse m
eirenvaarrleeu aelsi zable.
reformausal n aotneldi near programm­
Geenrailtli ysv erdyi ffitcoeu sl­t
inpgr obtlheemadn, y nacmoinct roller
tabliifas d he sisreeotdfe igenval�
isd eetrmibnyme eda onfas c onvergent
esm abye a ssniegdm;o reiofvt ehri s
algorTihtaehl mg.o rmiatbyhev m i ewed
isp ossainbmdli en (m,irtm) u st
asa ne xtenosfti hoceny clcoiocr dina � 2
be soanl ovneldi snyesawtrie tmnh
tea scmeentth od
equatainomd.n ursn knotwhnisssy; ­
171
The proceb·d enuei rmepsl ehmaevnet ed
steimsd iffitcowu rl�tdt oewe nx pli­
ona d igictoamlp auntsdeo rmn eu meri­
citebleys,it dose osl Tvhe.e rifto re
caelx amaprlreepe so rted.
isc onvetnori eefnotr mtuhpleoa ltee
assigpnrmoebnbltyme ema onfas n on­
THBEA SAILCGI OTRHM
dynacmoinct raosfl ollelro ws.
Solvtihneg apsoslieg pnrmoebnbltye mL eittb e:
ad ynacmoinct roofml ilenrio mradle r
leatdofs i rsetxlaym iinfti hnpegr o­ (6 )
bleimss olvbaybm aleneos fa n ondytnh�ce h aractpeorliysntooifmAc i+ aBlK C
micco ntraosl leTrha el goritahnmd
(2).
develfoosproe ldv tihnipgsr obilse m
d0)
(7 )
alsfou ndamientn htdeae ls oifgd ny n�
micco ntroolfml ienriosmr adle r. thmeo npiocl yntohmreio aooltf ws h i
Let: chg ivtehse smeotr eloevSte( r. ,
A;n
i=,1. .)b ea m teriicnR .
A ={.l\ , . . ,n} (4) Byf indinmgi nitwmhiuetrm sh,ep ect
bea s pecisfeoitfe s dy mmectormipcl ex
tot hmea trKi,ox ft hfeu nction:
numbie.ress.u, c thh aitfA also
i
EA
wherdee notthees ccoomnp-lex
\�EA, * (8 )
j1 ugate. A nonddyensacmri�c controller,
. where
bebdy musbtef insdot hatth e
(2), T
a =
sepctorfut mhd ey nammaitcrA i+xB KC K
()9
oft hcel olsoeodsp y stem: T
d = (0 1)
(+ABKC+B )vx (Sa)
x =
CAD ofM imnailO rderC onotlrrlse 3
ift himsi niimszu remo a, solutTihomena trigxi vitnhmgei niomfu m
K,
oft hpeo ela ssiegnpntrm obilsde em­ e(Kc)a,bn e obbtyau isnitenhdgfe o l
termiinfte hdme;i niimsgu rme atelro wing awlhgioccrahin t vhibmee,w ed
thazne rtohr ese ultnionngd ynamaisac g eenralciyzcelcdio co rdaisn­ate
ntroilslt ehorep tiomnaweli trhCe£ s Ec enmtteh oJd7 J.
ecttot hgei vmetenr ic. ALGORAI.T HM
Tom iniem(iKlz)eei t tb e: Ste1p.L eittb eK =,Kw heKr ies a
s s
startinogf K ,cS h oO=i,a c ned
B= ( bb b) ' (1 )1 s
1 2 r
Ci,i,. .,.i) dai sposoiftt hieo ni n
T 1 2 r
K= ( kk k ') (2 )
1 2 r tege,r2.s,., .r .
1 1
T T Ste2p.M iniem(iKz)e rweistptheo c t
A+B=K( CA +b .k.kC.)=C+ b.
i#L j thvece tokr'si i=,i,. .,.i 'b y
J J l l (1 i 1 2 r
T 3) usi(n2g0a )ns du bstitiunk.t .e
= A.+ b .k.Ci =,.1 .,.r ; k.
StepC ompeu(tKie) fe; l( K)wl <h£e,r e
l l l 3.
thedne,n oting the ch£ai rsaa p cotseirntiuismvtcbeihe cor as pceocnlo ­rdin
ynomoifAa .al s f ollows: gltyo t hreqe uiprreedc itshiaeol n­,
l goristthompo st,h ersweiet(s Kei) n
( 1
S anedx ecSutte2eap g ain.
4) s
ancdh ooasm ientgrS ai sc: Thael gorAii tsch omn verignednete;d
itm abye conassia dnee xrteedn sion
T
S(z,=wI l )z -Iw=l( z-Qw()z-w),
Q oft he ccyocolridcai sncametenett h od
(
5) J7lI.tg ivaes se quoefnm caet riceB
1
wheQri esa n .sny mmeptorsiict iKv,es ucthh acto rresptohnsede i­ngly
h
defimnaittreti hxfe,u nctiiosn queonfce e() Ki sm onodteocnree scent:
()8 h
rewriats(t seePDep n�ei)nx thael gorsittoihpnmcs o rrespondence
:
ofa K 00g ivaiv nagel e u(Kn00e)aa r
e(K=I) J Fk..- Ca.J- d,) C16)
10 miniomrau ms apdodilonefet ( K).
l l l
Ift he abmsionliiumstnu eoma t t tain­
where
edi,ti sa dvistaorb elaep tphlaeyl ­
T
a.= aC. .,. ,.a. ,.a) (7 ) goriAtw himta hn emwa trKi oxrw ith
2 1 s
1
l ni i i a ndeiws posCii,ti,i. o.,n.i) .
1 2 r
F.= (CbC.A b.
Duteot hper opeorftt higeeese n rali­
rl.
l l l l
being zepds eudomiantvreircs,ete sh e
81
a vcetokr.(s 2 0i)=,,12 . ,. I. h ,arvm,ei -
n-i1 ,
nimnaoll ra mm oanlgvl e ctomrisn ik-.
a 0
n-2,i
zin(g1 6t)h;e retfhAoelr gel o rAi thm
= .( 1 leatdoas m atrKi00wx h iicssh u bopti­
Q. 9)
l a 0 0 mawli trhe spteotc hte gniovremn.
1i,
0 0 0
DESIOGFIN N TERADCYTNIANMGI C
Ofc oursmei ntiohmfeeu (mKw )i trhe CONTROOLFML IENRIOSMR ADLE R
spetcott h veet cokr isa ttaiinn eFdo arg ivseent thAel gorAi thm
i
corrnedsepwnoicteah kgivbeyn: magyi ve a(4l )o coslpyo ssteedtm h e
i (5)
poloefws h iacrhne o t iantl hlse e t
k� . Ft= . (-ad.), (20) This mfabayecd tut eot hset ruc
l l l t(4u.) rparlo peorftt hiseey ss (t1e)m;
Wherdee nFotegsee nrtahlepi szee�d
I indneeeecds scaornyd iftoiaror nbsi ­
doinvoeftr hsmeea trFi.x
Jl8. traproyla es sigbnyam n eonntmd iycn a
l
4 A.B aletsroi naGn.dC elteanno
v 1
controlleirs thec ompletree achabili rank(BA B ...A r-· Bl n (28a)
=
ty ando bservabiliotfy t hes ystem( 1)
and
and,m oreovert,h atm .r n
� 191. T T T Tv0-1
rank(CA C ...A n, (28b)
If m.r n certainlsyo mep olec onfigu­ )=
<
rationcsa nb e realizeodn lyb y using respectivelayn, a rbitrarsyy mmetric
a dynamicco ntroller. spectrucma nb e alwaysb e assignetdo
(3)
By usinga dynamicco ntroller the thed ynamimca trix( 22)b y a suitable
6
closedl oops ystemb ecomes: choiceo f K�,w ith\J =min(,vv 0),
v r I I.
Fromt heser emarkist is straightfoward
A+BKC BH x B
to outlinae procedurfeo rs olvintgh e
l= ' + v
[w DC j w polea ssignmenptro blebmy meanso f a
w E (2 1a )
dynamicco ntrolleorf minimaolr der.
y (C 0) (2 1b )
= Leti t be:
v 1:1
wherew e::.R
By rewritintgh ed ynamimca trixa s: a furthesre to f realp oles;t hef ol­
\)
'\, lowingn otationwsi llb e usefuli n the
A A + B K C (22)
v = v v v v
developmetnot f ollow:
where
3)
A = ( 2
v
,n+v ,n+v1-
[:)� = f\ +d1 , vf\ +• . •+ d n +v,n
(3 0)
0
Bv = I (24) 'p\, ( A) AI I= (3 1 )
rv : v = I -A
v
n+v n+v-1
= A +-va , A + +-va
( 2 5) 1 v . . • n+v,v
cv =
[��J T
= ( d ,. .., d ,d ) (32)
n+v, 2, v 1,v
withI thei dentitmya trico f order
v
v, and "' T (33)
av =( an +v,,v. ..., a2, v, a1, v) ,
(2
6)
'\,' \, ( 3
e (K ) =S (a, d ) 4)
v v v v
thep robleims reformulataesd f ollows:
Thep roblemma yb e solvedb y meanso f
finda matrixK withm inimavl
v
thef ollowinagl gorithm.
sucht hatt hed ynamimca trix( 22)s hows
a preassignsepde ctrum. ALGORITHBM.
By a suitablceh oiceo f theo rder v Step�1L.e tt hes tartinvga lueo f v be
it is alwaysp ossiblteo satisftyh e equalt o zero;t hea lgorithAm i s appli
inequality: ed to minimizee (K e(K).
) =
0 0
(m+v')()r +v n+v ( 72) Step_l.I f mine (K ) wheree: :i s a
2: v v e::,
moreoverd,e noteads v andv them i- positivneu mberc hosena ccordingtloy
r o
nimali ntegerssu cht hat ther equirepdr ecisiont,h ea lgorithm
CADo fM iinmaOlr deCro ntlrelors 5
stopso therwisset ep3 is executed. leri s describebdy :
Step·3I.f v=vt hea lgorithsmt ops,
= (F+HCV)+w H y + Ev, (39)
w
otherwissee tv =v+1 andg o to step4 .
u =( G+KCV)+w K y + v, (40)
Step4 . Thea lgorithAm i s applied,
withA =A, B=B, C=C, andt hes tarting where
v v v
value:
AV - VF = BG . (4 1)
K [v-1 : (35)
v, �
: In J 5i tJ i s shownt hatt he po-
n•vl
leso f thec losedl oops ystema reg i-
whereK minimizees <K ), to
v-1 v-1 v-1 venb y thee igenvaluoefs F together
minimizee ( K ); theng o to step2 .
v v witht hee igenvaluoefs t hem atrix:
RemarkI.f thea lgorithBm s topsa t A = A +B (KH ) (42)
v �
[]
step3 thea bsolutmei nimuims nota t­ 0
tainedi;n thisc asei t is advisablteo
reappltyh ea lgorithcmh angintgh ei ni­
tialc onditions. A A +( B (43)
v =
r
DESIGONF NONINTERACTING
so thatt herei s no interactiboent we-
DYNAIMC CONTROLLEORFS
en thep oleso f thec ontrollearnsd t he
MINIMALO RDER
remaininpgo les.
By meanso f thea lgorithBm t hep ole Ifv ;:v: thep rocedurcea nb e conside­
r o
assignmepnrto bleims solvedd esigning reda s leadintgo a reduceodr dero bs-
aninteracticnogn trolleir.,e .t hec lo- erver;i f v v thep roceduries the
r < o
sedl oopp olesa rea ssigneadl lt oget- dualo ne.
her;m oreovegrr eateirs moreo nero­ If v = = min(v, v ) thep olea ssign-
v v r o
us is thea lgorithm. mentp roblecma nb e solveda lsob y
To overcomteh esed ifficultiuesse c an meanso f noninteractcionngt rollers.
be madeo f noninteractcionngt rollers.I ndeedc,h osena set( 29)o f reala nd
If v v then oninteractcionngt rol­ distincpto les,f ors akeo f simplicity,
r� o
leri s describe,d s ee( 3),b y: leti t be:
V V
F =d i'a g( A A ) FER· , (44)
Fw + VBu+ Gy = n+1•···n•+ v'
w =
T n.v
(F+VBH)+w ( G+VBK)+y VBv = V = (VG )= (v, ,v ) , V£R , (45)
= v 1 • • • v
T r.v
= Ww + Ky + Ev, (36) G = v G g1 =' • ( 'gv) ' G£ R ' (46)
· · ·
u Hw + Ky + v, (37) thenf rom( 38)i t results:
=
T T -1 T. .
wherew is a v-vectoarn d = g.( A-A .I ) C =g . (47)
v. n+i c.,
1 1 l l
VA - FV = GC ; (38) i =1 ,2,.. ., v.
if v v then oninteractcionngt rol­Moreoverl eti t be:
r< o