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