BSTJ 50: 3. March 1971: On the Control of Linear Multiple Input-Output Systems. (Gopinath, B.)

7.1 MB·English

On the Control of Linear Multiple
Input-Ontput Systems"

by B, GOPINATH
apt movie? ey, 1

The control of Unenr time-invariant sytem is one of the was! bas
rable of wodien avtonsatic control theory, AUhough “eptinal con
troller” which minites area soateeawrinted with contrat can be
Aeterived, sitet applvations “eile controle” snfice, ond are
aon mare deeb The erterie by which those vimple controiers
fre devine ave elsely veined ta the qwoblom of signing he
pvnvalier of the fundamental watris (Ce the poles of the aystem)
ta arbitrary but specihed locations. This goper prezente an epproan®
tthe devin of each eowtrol aateine, Owe approaeh doce not mvotee
uanpotiny eeanpicatad rasonieal forms, as da some previous methods,
Gnd nt the same tine grates sty fo multiapae output eytore
“Lime station of the probiew of esting fuck romtrol ayes
{nate miimune number of 2ynomie vlewente i len preeented.

In zosent yrs fase has been ¢ eoniderable aes a itarest in
the rabies of designing contolleze for near systns, Alihough
rnost of the Uheretval inereet hee eontscesasound optimal «once
fpproachoy itis sonorally know thet in mest srcard eonlrol se
teams, single sank sain aeropimgl eontotereeufce. One of the
folder orcikzas of roronal say i Ui etabiliaing @ nets eonarel
Erstem by sing feedbacks foes Fig. 11. Althewph se prob ae
Ten ave i the single input-oueput exae by mane people, one oF Ue
Art elearecatemnencs was thet oy D. G. Locaberger Ta the ease of
inultiple spat asta, elegans solutions now of reve uri taee Re
22. Alita” of the pablichedsolvices zeeor to ewmeaieal Somos
fund in Cee ule l-nstont ase exe aot convenient 12 worl
wt evens sveveacwnonteat sounsat, MARCHE ent

Tie 1M contrat

“eth, Also, in lio all aes, ace the eps ie often dvoeibed in
fume af varicbler tat ate 9° ver” isces, @ ceanaformetion
esnanisal fom: ennveaient

Tz this iene we present a colsion of the problem inehading the
problem of designing contzoles of minimal dynamic ander tie, a
ontrollo sing the ast mumaber 9 Fynzinie clemen:s), The pees
Ealtion doy vol sare taf woe of eaboniel forms for the detga
‘The aporone? leo belpa 4 eystoncceally exploit thy ations)
feedou (it ebtaingble due tn Fe mu cipeity of che nyt az
butpets. Tr fil there is no previous solution known sm he author
Shieh solves the problem of lesion “minimal lee” ohservers
Frthowt reusing to eomplinne eanoatea! fons

‘We besin ly !tsedueig certs. puliminesis and establishing the
otation wv! chen socing tie poodle of desing waters nd
thservers in Sestions LIL wal TV. Th Seecion V, the probe of design
jag comnales af tw demanste one wala

‘ier tis paper as wun, the author Leearoe awave uf he paper
ay WM Wonka Wonliars cesves Lowen 1 in the dlls. se
wn "The poet given ir Wanhe hawterer roe le ceory of mil
polynoniits, as eempazed tn the yreot gxven in the following setien
Thick uses only the coneept of Hneur spares. Wonhom bizsel hus
Trwmmncnted in his paper thet an abstract ot of hie reeults would
ie very wortnsbsle, The authne sere thus (ie yronf given it Te
fellowing section is an abstract vero, ‘Tie sulle of Seetions TTT
Ugh V ste lc be fond Ex Weshiow's per

"The folloniny defiitcns ooneain evrtcin mdesnet bul awerally
understood covers sich a dynamiesl sydien, ete. Pere m0ce
Mecrfediscueion of these Mas, the nea i eter Uo Hot 4
aveen ronmunvants's sysmeane ss

2 Fanaa Pine Taariant Sytem

A lincur tite-iaveriant eystam ¥ in w dymamieal system governed
by the follorng eatin,

a1) = Fat) + Gat, o
= Het e

whore x() « J is the sate of & and wid) « Z" end vl@) « B° ar the
inputs and oatgute of Erespoetively. Hy G wud Hare m Xm, Xm
fd Xm matsoeszespectivaly, asd ats independ of tne €

‘A "system hateinatar shall denote © Fneae timitarient apse
fo booty.

ana Oya

a oyelie it thorw exits x » Hsieh thes the matrix (@ Fe
fn nome

232 Compas Contraltabiiy

is completely controllable if the rank of (GG +++ FQ) ian
Sea Keto detest,

24 Compile Obenobity

POH ian.

(8 Bis eve; ard
(G8) Bs completely controllable observable,

Most systems ordinary doalt with are eyeti, beesuse, as will be
shown in this section, te condition of not being evaia ie esused BY
having tro identical rubeysteme eribodsed in one patent and. yet
complete desounled ‘rom each other. Lenco itis singular situation
fn tne dears ted winner water fe wmpletaly Rechable ard come
pletely obievab, a alighsarnoant of fellate oun re the ayeten
yale (a Ref. 5.

erating to note thet eaoettheoreis to be gen in thin
oper are dapencent an a simple and hase property of Knearepanes
hie propery ie stated ae the folowing lemma

Lemma 1: Let é,, 6 — Loos my be m dain! linear subepaces of
Uinear space. Let & be a linear space contained im tha ax anon of
1s cs execu ren w

Fate Phen
BSA Jorame je th 2 mh ®
ene & denote Mamtaad

Proof: Tho pit wil be Inve on the principle of nite inuetion
‘Tha loss ebvioualy tess for n~ 1. Now suppose the Tem i
true forall e <a jie, pie that, ven fo, = 12, 5 ey
then & © Oye 4, smplies that & © ¢, for some j =n. It will then be:
fpnwel that tho lentea ie true fur x» which will complete the
Prof of the leeway fee dela,

eeuree ie not coined in tho fet union of ey we tm < wy) of
‘the 9 for if itso coteined, on the lemma tly ho rom
previous pavagrai,Thesefone them esta n: voetors sue. thal

meek sede and gay if ih “
Comider nov say bo of tase ny ventana ya

neon, t4j ek o

(Pon eet of rea umber)

Sines 2 in inaar, + 97) 2.8 9 aja since &

zach ered, for smne ae} © Le 2, 0-7 mL ©

rower, singe there any ony tite number of 43 nile « een sazume
fey relia ftom the wseouulably indie eet Ry chore existe n°”
fuch that for st last twa dieing valet ofa, nezaly ml

ba, mdr bawyes,
‘But this implies that
fay — ade, €4, see 4 now; ®

sree dyin 7g wl at nce, &

Therefore # = y by (ote 2. zy es,, whee a £9, ee Ze,
tind 9, linear” Once sain using (1, «3, = j which concenete
ction (5. QUED,

Wolo: As eon easly be seen, Lemma ¥ dacs not hold in gence for
fn uncountable non o° Tinea spaces,

Definition: A aqsare matte F hes simple stractare if aud only if for
evan miesvanane exstanes loa

ach egenvae bof F there eats ore wed onl one eigenveatoe
hs other words simple if ro two uuzoupied Jerden bocks i the
tearonioal form have the ste cig)

oles AI the eigenveotons ae assanzed to fw ormelined seh that
the fit nonzero cornea FL
Lemme 2: Te siennat that te ater 2 epic sping hat Be
‘spare matric Fn equation (0) has imple acre
‘Proof: Suppaee there exists tro aigenvectors ¢, snd eof P eure
sponding tothe sgenvalue 2. Then 2 to eigenvectors, d, and af
the mutt F eorzesponding tof, where Fis the conjugete teanapose
oF Let 2 be any veetor is A. "Thon suppec 8 the projection of +
fn Bld, dy, the eubspace spread by a ty Lat
PERG, d) such thes Gy O, 20 ®

‘Then since ((e — 1), 2) = 0 because 2 © ACH eh, it fllows that

(2,2) = by equation () end since ex Ries di), 2 = aids = aude

Teter
FPS = lind, + aul, Pa

= Flot nahh

Henao

‘Thussfar the poco i omeplte,

oD.
Desiskon: A subwonce of AY is on variant mbepues af Pit
pains Free. a,(F) devoles an talimensiona invaeinntabapace of

Lema 8: The statment that 3 és oninary implies tha! J an ae A
Cand Be BF, auch tat

a: Gay = oR" = Ta) =

Proofs Notice that the number of invariant eubepces of Fare Bit,
Finoe the number of one-dimensional invariant subspace are fit, (Thie
fellows fom the femiigr structure of invariant subspaces, ose e7a-
‘lon (8) Suppase s« B(O), the apcee spanned by the entumns of @ and

AU sap = 96) <n

Tg ene rt of ny
APL NEL AR Bap rte tw

fox A BA BU
1088 uae wo sree mavanseay, Joma, SMC nL

‘Thersfore # « M66} elon to wows 8(Py 8< 1. Therefor from
Hemme f (6) © 8.18) foranre <n, bic comtrndinte nF=G) = n.
‘Thorens J axa # %, wigh lua gl? : Gal) = 1. Biilaly tie other
wwe. QED.

‘The shure lemma show that 7 a single inout op ayetem eome-
sponding to evary ordinary systom, cueh tant the contenllabiity and
hvervablity of the new stston is ita By that of the old eystem
‘vith mulkpiefapote and outputs. Lesama 4 shows that the weighting
Yeotom a thd g could alvin be ung vecbor in" ae respectively.

emma 4; The statement Bat ® és onbinary, Ant a e BY and 6 « B*,
Simplios hat oF Ga) = fle”: I0'e1) = ako uray *

‘Pew Note the delecainant of [f+ Gut « palyaomial ay and
iby Lemma 8 we have shnom ie nonzero rn as one If the distebu~
tion of doesnot allow nonaeze probability to nny surface of dimension
‘<mten

Probabiiy dat (W:Gel) = nisl. QED.

‘The stability uf %, ad the transient response of % are generally
‘urnoteriagd yt eigenvalues uf P, which in tar are given by the
CCoaraclerise polynomial 41). Tenor loosely by dyaamiee of 3 we
tocar tie ebwsavlerae polyol er tie elgnvalue of F-

‘Giver U-usw apetere 3 mail" to be conarelled the problem
thar pe il conser isthe: of desiening wotheeayscem 3, etch hat
the rosctant “elosed oye" spate hae arbiteney dpnance.

In onder to uuivate the nature of the problem in Septon TV, we
all fret solve the sosalle contrel probiem Which easextially is
Simpliged version of che problem pestulnbel in Seetion IL. The 2
Imstrix in equation (2) i now assumed 29 b0 the "'n” dimensional
Ientity denoted hy [ey iw oluar words, the complete ctate of Y is
tvailable for meusirenvnt, Th thi exe, we show that we need only
Teal back « eettain Jinone foretion 0 the state" Wo aehieve any
ven dynanies forthe elise Toup eystam

‘The problem (orally nslues to the following
Given a lant 3 daserbed by equations (1) ane (2) with fin (2)

sed poe cnn fry ny pk
ity to mane of deal eat

an poegeaes
1069

replaced by Je, Ht 3e requitel tw fad an om 90 moet K refer
tin the following he ivalack gein cue that tbe resultant spetens
das the present charccteistc polynomial (10

er dew um
Lat tke chezacte-ate polynomial x1 be
Sar ay

“thn Fors Hig, 2 the problem rsuees io Faia, K euch thet

we

XP GK 1 Se! a2)

since tho new difrential oqnation is 2 = (F — GKir + Gu, The"
oli enatained in. the felony Una.

Iheorem fs TL mat 3 te 0

red comely nse ten

schy ys ae ml constant,
a wea e+ Dae as,
for amg ofr one stiiving
ay =r GR 0
eS aE KY | | te ORD

(2) Moone then exist a lout one K stiaiying equation (1)
Proofs Tus new tharacteriae polynomial with Seedbvck is

AUP — 0K) = det (of — B+ OR)
1070 tu aka exssise HOAIWTOML IOLENAL, MANOS Je

sale MCL IO, 08
= xl det FGF — BGR
Sinto Kis of sone one, i fellows she: (a7 — PF) °GK tus ratk ones
therefore
deb C4 fl UK) 1 eer FOR
hove br (4) dees the wu He diagonal laments oF A. Therefore
from eatin (15

XP = GK) = AiR OF BY'GR, an

xl = GK)
whieh eine

XA ter AK! OR

aeons Emewe
tide on wii rin he couse pene (66 Re. 8)
—

mur

e000 ve[un Sireen} = an

Now wing tho Coply-Maitan zhsenem (49 Ref), iy wing the
tac that

2)

re forsoo

sand equating euficints of equation (19), we have that the efficient

of eon Re HB. of equation (1) i (sing my 2. #1)
6, fom en
Oban OK to PUK, a BSED

peo

ne EUG,
= tat OK + uw PO,

en

eto tte POOR,
aster MAMAS SRST um

“This proves that if there existe of gunk Lach that eguntiva (14 ie
sisted, then K sai ixes equation (2), al tna iC satis equa
tion (24), then equation (12) ic eats.

Let
teen fa &
ley sad tal

‘where denoies the trarspast, We bse, ceriting equation (2

2 07 GK T
Bo FOR on
ook tea a PMG |

Lat

roo
eo

WNotioe here chat 4 alwey® enets end

Now we wyuuse A — ai! (o an ® ate my % 1 amd 31 vans
reopwoively)ysuen thts AC et rank 1 shen equation (28) brome

[ar Gok
| cee |

WP Ga

em
1072 rpactusvetune snnnsien. sormsan, anc Jer

Since te Mak’ = w'G'F%, vation (28) boeomes

per]

Pi att =a eo)

wot

ut ror Leraea i ie oles tha! gi LP Cal} n for almost al
“Therefore follows tase aqastion (301 use unique gluten: for umost
ny ad i cern the pace.

‘incefre teom sa proof of te above theorem, IL Je ansy 0 ser
now we can fin ener yin wntx. Bates (28) de wer
the elemerte of KTH Tred: inthe mule input-output ease is
tesontia sy one of pickin» Alkwel any solution of equation (38)
‘hich hee rank 1 ll do the job, Nolo shat esting X ts be of
fuk T ek delge tn recuoe the amber of arpliiem implement
fhe eye, for K then ean be zealisad by: iw = 0 — 1) amplitiers
Enewad of (30

In Seti TIL we tan ow # gh mat'x X ool be comted for
thespainm Bovish = 7, Mawersr whe Hx Ze, the sate of 2
Jol devouls observable andl nn nbvonvee to emia tk tats has Yo be
see, Teil beeome elar bi Theorem fives the safation to this
Drublew alt. ‘Uke saluion oornie of vlevgning « Tear eters 2,
Wiel is ousteockad in soe’ way tha ite sate 4 ean ally be ob-
veh aa ena thst thes of 2y lent 19 tse ave of 238 sop
is dies.” (Ube mating il ovr ea in te allowing)

‘Tin syntem 3, wl corset nf a melo: 2 driven by sn input which
ia equa tot sn of Ui ite toa weighed errr tren which i He
siSenenec betes the stale of Band al of 2s

‘Lat B be din by

2a 2 | bitte — 2) + 60 6p
Let tie evor# £2 Thon equationa (1) and (31) imal
b-ve ius ea)

[Now we wall ike # > dosreate to xem ceooding to some dynamic
i the geass that x — 142) shun bw some pected polynomial
1 is obvious that once again the problem 5 lo fil am etek that