absecver cinenteal? fom roll ction. Deen Re abrerver tr Re mewm state equation => Select Ke elgerraLuee 4 Ace 1 Ca) = Once Aoeio de, => a esttneded. + X =T.& > X yeady fr stecte feedback. 9 oan t Sree? |e, Al-L.G=| * las 2 % G [ooo o1] Oey M 44, 6,ee => -observer desian & achieved. Dem es. Ay ot ake Aystern b doervalle => We can Cocke a ye 4 Be observer amnparkere we want. ze Hew te aelet the desired aboerver poles? i ee cy mp he cflrtier ie expen @ {A-1¢) ave te the Gf 7 the foster EF canvergea te ee camitartly Hrandornckion j i preserve the eigencalten. ae Hhatever ypuriing Re exenrabucs tthe a lA-Le) Ate the Gf amey creche: peoblerve > aS 5 af | Aa] Hb rg => Re minguitide 4b wa be Naegn (Berge deosie aaned « FON) fe the | modal ( hineac model) may becowe racnvelecl * thigh gain amplef, cae We Bowe te Letateon Post Local) Hee 4 -aeabitay « cbincever design must Exccklify @ Congpromise - —. ee puoh * ecgewraluce” cv2 for ak Pp NB tothe ofr / f a they sohadd nat be too parte the Abt +e tnduce Risk “gaia. : Tonptad Remon observer denipn and stake. feted ar drdepedet 4 eack Aker. i 4 —p Corkroller clang 2 Seheit*-eipemelto ~bkk —? observer ee te 7 SEL Ae) The tpt’ 4 server ae eatin BR {Hota ccb - tam weauiren” Compare chier vabthiir, ? ata, pte “cil a Complete hace vabily a A He ucdea ave presedt Bitte octpat (enough! etpct nfs te vecenatrut the Stal —povkeeD chrervabiltt, = Natal the ameden are prs inthe wk thrte dhermine which modes are preset ond which cre at e free we faa ol 2 REAL4BU _ X(5)=(ST-A) Kio HSM Bu) : y= Ox ee : 1 1 4 WS (st, 2 alg Oe eae Ee trast! Cer co s Sete tern Be petvancted | deanene rapidly « A Wy Js) = C(SL-A) Butsy= C [3-35] 845 = \eCuaehy ces =i ats= [SRPaT= eh + cag. vcd A ee fron the excl ¢ Che Bao. Ber Edad chro a Kien, He mode CME hibdpeara PRE b cots fe (MeB=0) o Dy fe nit obserralle (cigs =o) ov beth. . ape 2 Where dod the Mode porom doe . aferranbuedge 2 We wl a pee/ ro cancellldton. Ys) =TUS) US) = C(St-Ay* Buy 2 = -¢ Ag(SE = _£ Remark: BS Godt 3 Alo: He Bigenvalee bast ecoasygi omen ace mecrnarh ‘exzennalves but the Convece & mt tue. 9 Tp the system & LG Corbi bfe and el ke) aang ane oe 2 a stake fldlabh design Fas the -farm U==K Xe cherrnVe Sie have confralence betwen pbaF gers zBye bike jules ) Herve cleat oy ee aoa br preter devin “tbe inty acecnit “Fe yr i elpenvebien ace nd “abt 3 syston Oe) ey table Re Li's patton! ens yt a Atete spree cheery ion Prem _" nee oserigshin ; A cee orl teminivinal niall and off PE on » Mere wreble and castrrbhbe. Gib. {ms eps — cancollto) 2 “ foto ae | athe Met BU. Po ae Yee CX Zizi ' To] ober 2 Go Xe Soe gene al Comments 4 he obser ver discamed Concern dee rnakniatie. = Ax BY SphEona { Nas Cee . LUENBERGER . icbtenrer a lake Attest édturNance «Tee cold: Be stede & Ot ae pec Lae = ANCE) EAH) ey ate 4 YH = CXlf)+el) obpad Hee Eo. 1 gee x wire Xe) ; ka ‘i an oatmihe fe Lees wed ae ia i shan he te ALMAN FIL a =X- $e Xx seleded te miednige 2 hears Seyuare ee E{ée} f EeMere disscuks ba full order observer, ll the Bre eittmcted. Ta ase coves, Some Ailes ave ovnclabte , Sey p Sete, ence We meed citinatte (np) States = Bin veduced srde- suet RAK orkip between cansrteal fermst Me Ax+ey a . Te bee Cx del jeokineroce (pee enn Mees an axpeorectiva f ee ee Joc ee Cr Xn : = vhere thy =A Aa TG! Ay =m naan -4 TAM eV AcY Peev eV => MAM A ‘ VME wi tag's A E a c oS) Xo} Xe = pei apcipulek : . ban expen valve 4A { (er Ag or Art) es “Er => ban eipewrelitar 9 phe. | 4) ar Ag Dn Vander and =| % a + mone a Ae At: wepoted => Vi singer =p oe peneed Bee cade etry: Seppe Agee repeated mp 7 Vanclermade - Rao the Perm. Hines + limit : oe ke ae er Pe - peho ot . fy nbytdy - Aséume thet steteo. are oa Bui. pick, -4, -3, -3t ey ao olterver prtu Y(5) = GU) UG) f > Rove Se fer comdered only Combinaetas times ese Hine optrwn: ———— deat ee doing Con piter Bos pe from Contin stb Dike cattrollt- . dwotn— yt ue Gis) yt) Sampling. proce: Sig Spel ‘ : Sere AXW HUH Euppore the OTS & cesodtbee! GTS) yy cx) CW. Soy STS apg Seampbing RC Pak ef he ST Sah he J ADC pa Ye ah “Be. Penrod f° sampling B T(t) DAC ——> # ib TX tatenpalictin probtow-—b> DAC Se oS => Zot. sanple g ( er order Kod rae ee grocer | plrntd (ak) Beye he vabee - veya pew order “ AO mek sere.a 2ern ‘esas FNC oad Pee -— kt T Ale) be Kis exw (e Butelae (dl dP tee state be Baca Eee any enstact +t (xt) ) 5 ace ety compile KUte) | ty ty oe av ew sonore ee eal: XUn) =p oo % Xtb) = @ x th wef e Ule)de. j Bey HAT te (one si )T : Cleat) XCar) = ar Pee Ee Bulode Xie] = Sauna efter un xfer 6 Xie) + VE ued aa => (xted =}@ x94 F UB as yc = (ore Cc XCAI Pa ree he es = 6 xte)+ Ue) vec (4=C XJ seat AléaT- EGS) rfe=e" (ieee BUT).a Range 4 wartahle T= (k)T-T (gar oa oe a cs 4 Pete RUE) _ Bde et ae er “oa Cas aS ee jaeg | G12 qe (Sas ae TeAT 2 (ATI cmxpilet ‘ os ( a A» (SI-A) > (st-ay* : ‘(61-4 ce f(sta) MSE Bopler mitfsd: (ST-A) ts (s-4)isa)- GA. Ka see MA Kee ig : Roe re Eo se i? Se Act ack St oe eo nie ee a Kao me 3 Kio Spt bet» “(saa)? (SA +2 é ill dedabre 8 Dotty BM deriabooe iA) een ae "GE i ot (81-4) = iH eyEUR Af ov He speom $4, fra ne ® vrepectted mite exit Hao me pe ae ie Hines Brite / PIAL & Any pancried f deguce (n-1) 4 | di Tet ly fly feat, | > aa ee, “ Le | pep! foci eines eo RA) = RA fA) = PIA = oh A" AS | eC ie nek > db od deg n-da4 C~pitte. - Re = FF PINS A+B xs 2A ol Aer frye oe fia) ares 1 a Hayne a as PP oa: Nate be ee fa - ef = efh sod hee ope CoE | xefh ee ee iene abiig fF par nee? Hake spmco. egvetios i Xlem) = 6 x(e) + Putea Jfey= C XH)- Uce) = -KXIED. we Can gee > | ( w& cod? System Pn) (Skanner “hearem Satis sped). b> Sanplnes per riod affect the caitiel design’ nutoes He Lede Tehility ¢ bTs 1CIS% XN Ene Cadrobi Me ay, Ko Ay (MG Fj ar eid & z - sy 2B Convetosya are (2S teen : re ce z Dif tes steutite 2 rg Yis) = G(8) U(S)- 6h) c fats (pact 18s} Ti0) 2G) bee, a ; tt oe esnceoparcing banc canonical ferns— eee oe be . : athe ie G19) Jak fl 21% e Zi Gis) = “a a (1-Z7) 2 (ia) Yee Sai be Std >uUMTI- é “UtaT) = WS aed > Uwe “Utes,Sy Stem Structures tultinmndi Given @ MIMO system ES r ewe ae te devel: TY ewe SISO a at In ie by Appeceubied equacin opeciel ste make the clesign easier, cote’ state chperver Gow keno dtemaystte dynamics ave uh evhtaS For a even pyatenn governed a nf order ) X= Ax + 8u | cx. le cA Vis) GLY Ul s)| tes pxm (stock feedback ye Px. pense equations (n-2} (a) (het) = Jur +O, YH 4% You + ow aa *y eshte Space UN age Lepesertatio: <5 =I, 202 = Me: Bye! Ee ss Gah (net Yn= hc Fon =i: Kn ; : My = 4h = cr) +J Ub MPG Heyer 2. 22 hn =- 4 Byrn - < Na 4 = Ky ae CO - ; yy 6 | Kn} [240-44 4) [Aa} |g | CitWer canted) X= RABY z pre — Je(1o - = wx secuie a Hat a, Yet) =Uof Xa AXE “TE: T(5) = C(SI-A) 8. " { ¥=(0e a a ales) S 4a o +. 44.9 ben ( Metis). as SUVS) b ay) SYS) tote - 4 25 Yshig Ba 169. 4 ae Uts) aaa NR eas + an A Aches J-MIMO Syat emat MiMo ayteme are dynamic syaems desen'bed a Ve ctor apecntin ES e 7 ~ Cavside- the sytem described’ ye 1 re Xai +A, Xie A - taxi = UM Where. AZ ERA) and & eR _§ yState_Space Deociphin, 2Tramfrfochin Deserigtion ? : a r Dobine a stele ved SC REE rade ae western ele xX aye x=Xe wis X= X =X Bx X= Xs 2 X= X= Xe Nos Xr XdX (lm Vex = [In One ee in MIMO Stette feedback < 2- Con Se ieee fron genvrnl form to Blob o\enlcat form. dock Ce tre lle fe mm. Cle. = MX+BuU i ysl X o A Ts Blok controle ee &aAx+8u 7 Tie en cancion Deae tn 62 ap FarGerw : Initial Corditine? S*X(s) + ASX) + (x5 Ap See Xs) = Tm+[ 1,5 MS + - Ae) - -- the X(s) = Uls) +h Sth ]X()= Us) eet ii) (CASS ths as ey =! +h, StF JUG). =p eeRe a “ TAke opel px Cymontel were inthe Cad Te eck % cus MATRIX POL YMOMIAL a : a wkere coefhiueits are matrices. Remark» ) j Ee wate polynamial) TSA, SH - hes Ba properties shel ave diffecest” fron the Poopectice 1 he Seaton pody revel pncdoe “4 prysowl > Tn © Denemenabr * poly ” a Sot AS = +h S+A CasdieNor ite he Nor yh Axe - FARE = Ul) Define “the ak pit any a ——_ at i (AY Grog) oe. Yisi = Cox" 4, Xa, Yenc SSI. Sala Conmpatting TPs The SKF ASE --- tAXo= UC) Visi = CL XOFGS H+ - (f+ Ast +4) X@=HO be 2) Mis) = Fers 44a s+ - - +0r) XIS)ye = am = [ers g 84 = rTP TSH St xn Ave pm eee nen Re marks Hatt. elope vat comme mall ire Sh scr dys) = TETAS Bee 4 Ap Tis ye Nels Dg) =PReeowee™ Vaart Pei Citra aaa bende rin (RMF D) aan -l . Tish 25 ts) N ty LMPD yy + Bes as L L - eae R- TINeN, I =e (5) = Nel5) Deals). RM FD. NCE Nels) Consider Re MIMO syiten deserled. be Ios Ay XO, Ae xt = 8 (48, ue) Px (ia = Lae Ey Kot his) os the XS] = ee Bs SUF 33h espe 2 tBUS e £4ss +A" +. +he)Xt)2[ 8154 gst .- AB)UG, a Nts) = Prstacet the) (BS te 3 =), 6) 41 — LN Fd t appoee: : x =Ax+BuU ein he . Va cx tbh U=-k X 4 X= (A-BKLX Y= exwe wack Ks man unknswne ve Rewe “to form UKE) fe Rane clenival myn -uriesina to déteenine KK. Minto. system: n equecttion Aas aefiesty {sole > chesoe i Arvel® gain NEO) ola ; : Cr [keke K] Ay \ Ke : Sega “Ae -hy Ne = j oa be ; a oe Gun | ete bu a =C XIN U(cr con pen <> Bhook Ce — es X= ‘te X $B U N= Ce Xe an Be Con ltr Ga - - Ge sto Hn back Pat m | te. con vite Com onpesn dine dire dl iby ‘nape clin fe = R MPSoe b ; as; ‘s, (sy ok Cokoller fee: é _ Rip Madne Fraction: Seoeefobin Kemah: s+ akeodl be T boectidl tact Hee behantor J He Sytem depends om Re ckeradténahe plrnontel D,lsy = The mate -plynowS will determine Re be haven $ “The system, Sas Del) HT the Sete eguction > Hock osatroller fon => Alber stoke peclack stk cored lop aysdene 0 wboo in Sole cll .- Sarte be Panter je bared foo system depend Will a 2 depend a the charadenstic poly ndzmcel A te bored ayer > e 7 we Aled 4 derived charattensre amie php we can acheeve the den'red behavior we Wank. a ye Block observer form: : a % =hoX +804 O- i 7 [ Y= GX. Hee nce pe] ONC Soleo.) NON Corespanding tothe block olrener ° a7 am. a uwitte divectby by inspection he matase | “fer Pantin in LM FO form: OEE Bo- Grn) 2, =i (A +8) j ha ae Gly = FO mEs) Dy é THe hecattertstic equction & terete pyrantel Ds) => pry dG! whi deterntie He behuner The syste.a al Rrasec econ SSS a eae eee feed back ae v7 —+5 Block obsecver foe ousefil for Ae i observer destem . Jn SISO. systems: sth Raracte nate equation. polignonel bce aeclar elyromial > Slate space form 5 da (Ajeet (AT-A), go> he dtp Go1= LG) AA) 4 CU “dtsy The rote 4 olt- (AT~ A). whicare epowrlves | A tb dcernin o The bekomo J Stateag = x vost dts)” aelzinne He hekara- j the Sytten eZ) : 2 TP We move te MIMO Syattiwe: “the charcte. poids a Metrix potynsmi a yes re) no 8 gy } pete’ mx Me soe Ast fer Des) & mat vel defied » Li a ret A defied & aa cop unde I sk DAV =On => wrest tnig eT ovale. 1 S+2 Sie : en )s + i) e M0 _ red * a =e SHY A Pe racy wt hee ay of . Sunde Aad Afr bev Paps «sd $/50. i 1 7 —> we May tovider a vost an a modvix Dy => « bhodk vootr Role SobveXs : Gren he matrix pllinemial D(S) ae « © Ds) = 1s4+d, 34 2 hy, See im cee ostvick Rid ier mad Se ne rata 2 + BR+ by =m —phdt Sobek: A tlt sled J Ds) bony mx wie oh sotto fying ( ly Ee Eek oe EL ape, Remarks, 4 A Given metvix polynomial DIA) may or ray nat have »Solvenk (Raoke] Gt). - an RPE soblet [when eacit ) i cobatdlered Ao a ee Bealtvodt 5 At oedin er 7 ae Tighe tyal=o 3 A cocplmnber A saab yng del 6 called a latent wat { N(A)- . Assocteded watRabatet vost Ags ht howe ea Lede veetor Xe Sotinfying D(z) Xe LS mx memo : es Rephit Bevel ~ ‘ Es ie Bi Bn 1 Ss - ne Likert voit ~ c = % As Vector “The Concept” 2 Erpeanlhe /Eponreite- ib only a porticalar care fertic Heoy J Latest vosts/ LeKenk veckoa (Ax) & ACSA 8r + (Qi-*) X =e =) eipeavelste —e Ad (AT-A) =o. eigenvectors —> #kEAT-A)xX=oaf R fo a welt Solent Y tte mectvix pulyrornid DUA)

Dia) = (AT-L) G (A) . ke wack det D(a) = datQ (A) det (AT) =p AL? ae velues 4 the ightroteent” ave batherk vot ¢ DUM 4 Same tig ferkeft acl at. dt Bia) = det (AT-LpG IA), => ACY cigenred 4) RAS Lafrookveat, ave Letect vols 4 DAA) i ied Exe es fao 4f-'° ee Be : vee Jee Comrmtder the matrices 4% On ) oe we = Be ke (2 3 a Bee Mai ty block veate d Dia} 2 serach iE oe 3 te = Cathly for— oe ae ; oe ‘ e (seh aie | : 5 2 aber eg Mis & 3 gle bfok wot 4 DA) § oe : (s+ 6-39) = MGje Est BD se Be. = m4 D, mt O NCD) felt) Ds) = HP + thd, + De =e Ona. SUPe at Yaya ae 2 [°, 2 eli adel #| aire fi, a eight bebock «vest > - 4) 2 Mg 2 6 Ce Serge Ri yetee (iat HOSE Leaked 4 D(A? ‘ 2 Del GUS) a Ng (sheds GI = 409 Af de®! Bt dD Oo j ‘ Dees Cater ro 4 OA). eben He be heavier 4 Later lt > . etponnlenf Blob ® fren baton Z dea detnd Lited~ pot = ested ewdls Loach woits —> -coutruel €hraradlenste auctric pbynon aah > dcernine Vee requcred Keb feeadldere gain ar “Re. required compuceton< : Suge oe fever bloke contro Mar form: <> : ° Ona eae 2 : e.\ “ 2 ‘ Xow On Uu \ cf eee y= [Ce Geese = Be laraTanch, rune phys) (D,01 = 7 SHe eg kasee : er spp Yn (5) Rao - te 120 be Jerk zi > 14 f < e SqS0 . BAN Np cVe + Vp ela a, | ete ae! An ume eT lee se Re & ae ee wf Say oe prP Ra 3 ag 8 eS Decal . a x Ce 1 o oY 0 Se. 0 a aN i R inh Toulon when Pte truce and Ma con clots, btu mo elven = Aone : a shes s det MA) eon relat: “2k if re beste) YA) Last exo, 'Consider the mectrix polynoncall AA =TAHD, A AG - -tRA +), > Rian eng ht pbves 4 DIA) eS ees, at Ag ban eigen rele { P= Aba lace natal nay : SA) ake (Ag T -k) =e cet NA) HIT GA) AMAT A=! eigen value £ right nbedk 4 DA) e4 Ltekviat £ ba) Ts an eigen valiar PR aloo a latent vector ft DA)? xX 1 an eigenvector 4 Ry aasectated uth the eee Ai > (AT RIX EO " Xm valle Fem DIA) = GIA) TALLIS HAO AER] - > DIA) X= QUAD [ acE-8 J X¢ = B . ea dD Ka dan perce { D(A) ff Mes eee raph soloed Ref BB) Bees Ae 6 fatest vod 4 INA) Fp IP we Amy a mate "Ri te a vght ae 4 DA) Sth 4. effin ey sua Lures eee expenvectons Roo dated | coats and Caken vectwo q dia). ae | a Ae poled 4 DA) SINWAE = + D244 + Or. =p[ pea =tat-L)@ (A) 2y Dy fo an expenvabae f L = det (AzI-L) =o: da bd = dt (AT-L) at @ lac =o oy hp 8 a fest vate DIA), wey cgeclue Palft ot A DIA) « Fae iat YF be a & “pool {Ly, "(aeFL)=o > YE uM) = Val a ao Br, Migs a ut a jb ee OF en _veikir 4 bray. DA) . Eo, nig a Gk look 45 DIAL og alt a. wih Lath expenvellts vol Bef epanvedtt Oe Lacs ou cand) Apt Latest vedere FOAL Deiter If bhock Cskwller. reform: On Tm Out OL) es chararttenstte. mat he = On % Im Q, polynomial fo then by Be ‘ - \ : inopection : ie E & af Drapeanrans “ahah phe“ - Suprne at DIA) fas a « comple ot | riphtt solves Shoe eae jes & & Re foie a Conplite ait of right sobre Ire jg U Erte) = © (Ac) @ sR AT (Ay) 6 freA strate dA )—> 2 > a eiperrebie. { R ales an exper valle ¢ Ae: Ne Ae Ve = IN aie = cS ec. de (Al-Ac) = det (AIA) a AE ASS EES 2 4 2 OTA ET tee, (6) | ap dE) = det (ATAe] ee 2) | > Sd (AT ero AER) ALR ) > Ary espenwralue ¢ Ry ean aspen vedue fhe: : = Pas an eipenvedy af Re aloo an eipemetor fe | Re Aimer Yo Ne ( beeaure of ivan) a bs the releckionahy ¢ an Scouve bor a asdootakid wth eipeave le A (She (AT-Ae |X =. eh 7 Ar Aye i> -& a. 2) Poi (a Oa a In iQ *I! Ow “ Tah ae | Bate LeheS Dy eg SO = Mya AX > Ayexgs xb XZ=AML= A, i =? AX, -X- = 8 aa EAN CA Py 4 ArXyt Ber Ke ~~ + hy Xeg + (Athy) Xr =O AcKyt Aes AX 4 9. ot A Naas (nth %=8 Stik i. wth” YM 28 ap Xa bo nple lit dba): Mae Concbuman we can castincl etpenectir f fe Pro a eget atest vedi gf DIA)» ie . Given Xq a fotoat vedtor! DAN, OWA - OY Mehr Dwain ae “Ab apoient f NA). : a hee pn Late te q_ripht Aaoct vector ) 2 a> Aseipning « a. a / Delp Hehe eipenvedt - f the block cortrbler | forin mdknx - a octal rena: Tua A AN, eh ie oe Akar Vp pb mosingular.pea i V, = & @ ee Block Fender Wmodo mri ' oy a ge Ta pele ins - } bi, oR) Oe must make Aue. eck yO & novsingular, BS each: Since “tke atgen atraatire {tie Anluost{ Blo opt) determines Re cipenatructure d D(A) =p Hes” can we > contind «2 maker aby mamta - wilt @ desired cipoustaire, Desired beRawior ay dentved feikeat veéta [Latent-vedtor Jowred eipenatiryctue => desived mekeiy pelyrondl Er ea ege eenalerl lure (det charac matrix 7 ). asionmend 4 ‘red wnobverits 4 «pe lication aS: a A a fe an pigeon DAE TATE AEE. De A+Dr - fim & od deaived ripples {hake a = x Ri ba elven 4 DiAla> - ERE D, Ree eH Dea Ret Dr =e ro Aaah ae Ri, & aE DAF 8 ae RE = +0 gtd O ohn oT Reh, . nn. Peete =F be ese for as = +b, B¢ =-h e i Deed P+ - — 40,8 skyba If ' Rr ; \ Re =-[er ry. ay? eo g 4 {harkee- Rr} ave Selected. 5.1 po renee ==> The rasta untoue palechin 0% (dr d., Bb) / =—[& B- ae) Ver repeecesh odorivel ack felt x uy Om Tn - : O. oer ee Ne tone heh, Ae e U=z-Ke Xe Xax[he~ Beke) Ker! Ke =[ Ker Kept ke}. © Aer Be Ke =| Om dm yan pee ee Twn o Ar Ask, ~ rks “he Dares Loop systems fas asa C.M,P ( chovtd retviy phi Dy) = TAKS ke + Art ed)” fe, — Beh > costncet Dy a) « DraysTATH OAT Aa + Koc = ba dns A tbe Art key abr+ exparvedtor Yo ou Coweaponing by ee A =AY") me ivy yy axp axe ee: G -he] il ie SE Sony ye oa Op Op Thy Ye AY enrtyt 3 Sey AAT HY BY Ama A = z Yr aye = Ss YT Bf Latewves ae BA Et bake wredter 4 UA) 2 He fir pot Cpebnet) lithe epewecto die « ttene 3 Sike bkent sructure { DA) determines pie (hbed voter, teB.cA) Conrpletely the dpe stuctuee 4 fic [A) ‘Ne Hock clever form: , F Nee hoXe + BU Op O Ay { if ax Ae Tp “Ana N =/Co Ko \ coz lO & Digs charectennnie. matrix, ph paniek pia erate Agat*+ B55 And in Bide equal i. Vb et = | AL) | ere (0). Ag F : +A thy es 4 Thike ie Al where Nie i - o4 4 | ga Bn LE ‘i ee Ge F ‘4 Coeds ane b we Cone & comfete ok A tae s ie no poke Alas ber er vis a ee ge ee a avin + ee ae a ay: if Dua) ae eigenrne : A ra (aToa) = 8k (TEAR G) 7 ipenveue a a) d& (Ap Li Hoe KX ip an espe a Ais an capen vale ja » % ape é the pdieit L: a abe Bot we : Y ated - D(A) and a a baker . | . an eigen Astinier dosiga: dura ( Box Co) See Agel a ) Ao-1 15 |fopap 4). Gy he Rar atershic Ming | penal d Re denied Cp Op : Tp Of 4. (Ac-LCa)= Geta : | ' system a Qe ~ Oo T4y pt ee oe (Agel Be | uc eer a Dyla) =tab+ Ay Ae ap fe, pobtest” 4 Dita) => 4 M+ Ai. - aN =. Lt wh + LY hd, + = thy, =o t qa ee + he Aye Ns +Ad, =e ee pa ek age : Be +b Ay, BP" Ady = LE ee a TE a 7 are Le Alga} | be li 1 LLG 15°] | A, iEd Vb eringular~ =the — polation; My LE as). fe A, y B -\ Aa | i ji ni —> 4 BLock Arvenve tera: tegen vale able oe BEERS vals Be cali” from Siso. : : *[ O-tie SaeD 4 : . A, =| ge ae A * CA a | x” A> Ack = AX ‘ \ xd 4 ee Ga Tn — On ©), Ln as ei : trebrn) © : bel Re ees vga Aes ee] Os THAME. +A AtAr.25 ses AOE y = Ack ae oes ~ eon ey; iw | AeXs = eke | E..- a oT idea & ae te ATER =X e > NTE =EXR : on | Si AX =XR. ae : “Can a iat pie block cas socslee imtcblee a fle becin | bheck i BLOCK FPARTTAL FeAcTiony : a “sted Tr Si80 aystewn, (st-Ay" Ady } Ax SEA) ee a er ke oy Gk (5-Ay) (Say - — (S-2n) (SA) Gay ae ea ~ (S-Ad) et. ne EG ole oeFoon we generale fr Book vats. a cle: Nee \ Vp AN =/\ ‘a ig iS q,., =|"*R (0 Ae, oh re ( b) * fh = |e, Br [st eu Bie “[srey ( st-Vy'A y= [Ve (81-Ae) vay ye i (STA) Ve = [Sl-te} = Ve (SF ROS" Nea a 4 : ie = he Me . 2 ft Meio eee 1 (STe hey | Ty RAS soe : “ot Ree Bes ce p= RR: Ree i Ri A RS wv (sty! i ple 85] Ce Ur-eT Jy J:fsx- he) = xX, (Sta-k, y Yt aK, (STRID Ye e PN, US Tek) Ve Set Det tatian \es Bhs xe (stadt Y (teqst-Ae Jax GIR) Ye. ese Aes ae v, es = DK (st AY VT “OG (S5~Raf 7 >- Book PFE - csr- AD 2% 52 B Nz se i . Nog eee {Me «ta Va] Lat >[si-RATS ] = 5 ee “Cre amet" =f x: sx-eeT* Ye.A ee eh Rte Eee ye Ye ME TLC ae Contin +X, oft y. 4 Right sobledt hy. Tf A i tamfsvnndls te bloke -cocbrotler form. and JR Ri ave the vighl Mock wots A the Raradctershe matrix prlynowicl yA . “Ther eh Slee ette deconpercd waky @ Sum i cocbii bit, tener ett gt oy Suqaet ne = AX+6U : x, Pic RU i > A a= Y= Coxe Op Op -A By Ay =| %P OP hl bas ete Re @ =[9 Oj Med ts no. oF %| rs o Ah Ba 1 = yr or AA, RY: th a+ ig aud “Sapiane teate bytes Ag | nace Gf Weck cvite | OF Adied q DIA) : Ee ie euip Mpa] * : 41 5 | ce fay (0) | ' VAY = ( ; y F lg te 4. Hy 0| ond Vp =[%%- - =X) ; Sil ye Ne : ¥ (tpt) ( {st-Aj= (0 i ae bs \ | (st-yavit] =[Yieave ] 2 \ (sEajiy (sr Say ee eae ea : NC ~ i Oye |M. (0) “(Stp-ty) [(st-n4j" - Fi Ye] : (stty | S[XuX%er X Ye Be) ij (sag) | Yo (si- Ay =X (St- Le Yad Xa ST-Lay y+ BENG ST-by NG =) Xi (st-LiJ Ye. é* = ey, +x e y, he = Hide Tie 2 + Yokeratl xe \,Be eS ested > Ae = [re (st-A aye 2 Dx (se ba Y, Al! Finally [5T-AY =Te [ix ($1- bs Ae Te Lt BEE (a a) % (etn Ye At v Me CS Ast gayi ay Lp bm eall } He aude Ag) incbo Vom, , eack cording 1m eipenvaluca.BLOCK SPECTRAL eee =; POT [ytacdy = Ap Meslay a Be See ee te 8 Gee wortee tile A a 4 " A. = Va Ave y, ck ee > 4 y f we write vie Y, Vi alXe% 7X % Ace WANs [x108 -- Xd oe fe) “ \ Rag Beack sat (cottnoan exponent] ( 0) iSr [EXm=nA ek Yr Ag = APM AX he ett Xe BEY) tet Aa ae ih in HE pet a decompottion B he inte ry matix , where eack coadainam eigenvalues. : X man 1 ALS ATS As ee! aie» A= poexe ReeJE= DH RE VeVe'= Tn — => yxyet : of. Age Gos Vii Tet + pf foe 04 Sr wala: YO oe spat” “ fied a Ac HDX RVs ue Some =F, 5 Ye Xe= -\f a eA YF emy = Tvexeeeve = RY; => Yj bec ole fags eacies and Ceewing coy thane B RGAX, = Le Ri Ye Xj = XU A] { sone ao prosioal I-A tF Xefsr TY: & ley = yee = ity, fe : fect f Yi inte eliminecte AM Wbock modes in oft Leaning oly the mode erik . Bina pode of (gj )o eed oral => pik a Kic y Hoe a Kin | S-Dj, SMe SA ot = Kis ie one en - Ei val Ps wef ¥ [erAay'= ibe: de eet welt Yee fa? %: é AY) ‘“ fe =AX+8U XK Acke + BU WISIGKY 2s y Ye Se Ce Xe 69) =C (5-4 B= C8748, red LOT [ix Brey ¥18. Bae =)" CeXi ee [st- ealjeed yc (s1-R38; | at Ys) = GISUIS) ‘once eEas aYntYet - +e Us) wae BE “the system—te be decomposed tobe 4 Rat are parrathl (independerton cack ae elt ol J x; ekty, whee Me of % i fA the a ne i bbock ve \ X= AcXe + Be ¥ i Xe - X.l4 = eM Xi + Je ee gure) dt Xo) = [ix ar Txt ) #f [ix Bett-t) ‘edna =[ 3% & “ee ai fe Et Ph guide Remarks The Beacle mode EM i, affected by the catid signal ol FY; he => fence Yi Car cheek the Blacle cash hee ae roa. ae Yrete Eid PV I X orb, 3 fué Y; B.Ulaide =[Yax% Fy.) xX (o) e xX; el uate x>.¢,Xizo = eft wil oat be prevent at Iie ortpat. =P =o pitt b mat chservable at te witpal sp = We can chede olpervabildy, f cash bbock rat Remarks Im SISO syitena the metry veitluee» Ky seispien the. pecper eo Y Ke = : owtigouly) Kzkj =o <#] keke oy ct matrix op KY are pajectien The mii veridues can be couteucted and vighk and oo ni See eigenvectond) G Ki=XeVo Nz ( taf eigenvectors) re CX= 8 —» nocokre = cae ee catty denablly YY; R =O 5 mockenalle. ,og ) X= A XFBU Vac x paatpral ne pespec Sar ea iosteser =D nerxm (Aor, omxme Deine thet (A, 8) i> block extrbialle dé Crdex ve Wy =[B/ AB) MOL AS} lA mate contin G=b ffwgen. Gy, 202, 48:88, Ane] fo maenaingrslar Bleck eafattal’ radeeve lt AG frat eee condition tac) aktafied we Onn conitrusted a piovika ty tra mation “he covert the ee prem ge gener! fem —to eock cardi bler from vere x =EX = sitem foe TE we wait Uvdenion a atte fecdbecl cvitndler for Pio Bia bene UEHKX (ary cigenvalher) een ale ne ee expanvertratay Repke oS (A Catect nde) —p es vedtira, _inekr 9-5 abe Ag a dy =[8 Pa} -Laad) ‘e cS ee c pee! 2 3 ete i Tes|t4 2b —p pels 204] > Ac 2343 APOE 3-2 4 3 $344 [a4 4-7/4 0 od) T5 2461 froad Ac =|4124//2049}/5 2otlelooo 2343] /4-4 ola -1 281 “las 3 4 547 3}lo4 olasaal [oa Goo Ses C,=8 TE =f Ob 6 a) 25 05 o oo >} " @ 6 " Sead O00,reife et PSB 6s 4 “AL - Dae Mer Aap Ady 4 dosived eg envailon Yuen Age S48 +f (a pete f Asysinant ps) Rav-5 , Aga-6 Rards we cnt Ry 4 Re Re -tt] = Ry = “5-6 - en : male \ peamate = Rel Z ‘| “A 1 fom iinagions < “ a5 Perms pe ce oe Vee eee poavkiletic. ke imporve the npttews nse. wince ane stransfe roncion deze aad cha ny At mitons penal —p Zapfaa] f fod et P __prame prbveis 48 enmre Levecahast we wail. ak oes ee 7 Pane tme + ap fe my perturketin anh ae omying peeturbchemen ih eigewvalvene Swe chasse Pate make perturlutins a2 pth ao pone ( Bp2remnance) Sg rope dieagannll form = psabbihes —e bebe Ry tm | cae fae aboecver fore eras fla gare nee st. finding Ke —p and mex KK { Ag tKes = 4a =Ra A, +key = Ded = ReTRANS FORMATION TO BLOCK OBSERVER ORM: (p Bega ae iteper {Pel toe capiitremar Q@ obser ver 4 index 4 from Gas feo (4,8) sil . 5 “A 5 ty Block chsev ver fer 5 CA WwW Ww, -|eaee ge ie rp eles pleat (= fel laa wt Cel aan napin cA tRe syste G index bitty, SG) Remarlee The system to cbserre We { tadexcvn” (Pal “tke anja fs -chotrra fe 4 fede tg [Becele obserrb&t) X HTX, Oka TeX, > X, aT =e (Ax+Bu) AOE Aas Y=[a+- 9) x. pA We can uvite inspection Hi Tenwafe Punction LH FD Giving _ The macy Taf Uhetee rf a a é ‘ A “yj itp” 7 5 a oe E(9) = Co[ST-Ao] 8 = c[s1-A)8= Doh a Dio=T sta sty - --+A, stg N,(5) = By 54+ BR sre i ne he Sl ereafoe meion | Te, tor cenvert tnty Blake Pom lebaewer form) can be chtntncd by duct: Bp 8) eles | ya Ercel f ei Tr = hom (Bae AB) =r (BAR. = 7-3. a em wedelt need to check —> AG vrestopatl 6. dence: tence — cscbulletelily machi = [8 AB -- AB) Yr (lan —> wytem bn coitrrbalte. % 7 B= fbribgr = ben Jeb alk bs Ah --- Abs DE paute vectors —s pa jn 4 eae re e veckw. D> Cand io let Banged 2 fa [euler ete Bs. bea]Moa Ke ha Ce “acim he eulne Depinttion “The con lai index ky the mumber hinearly Crelopenclecd” cobms & CP cesrecivtec! 4 ae colur g AAS with the Atk column be Since & faa m cbumna —> We have m Cortrellalvliy indices . The sytem > compltely (fully) cortivblee 2 K= n ¥ y Ky& 1 apten Sod ply contnblhlle (tt Can be” portly centioblhe)). ‘whe. fick stop ox cotnk dorgn voto compibe-tte controll, Wht indr'cea Te conpitahin | Ms can be orgenigal mrt tng tela [ Youn. Biagranr ] Beeb 4X de CA ndepedeil 0 t Ldepandet Ke fy £m