Outline. Overlook. Controllability measures. Observability measures. Infinite Gramians. MOR: Balanced truncation based on infinite Gramians

Transcription

1 Ouli Ovrlook Corollabiliy masurs Obsrvabiliy masurs Ifii Gramias MOR: alacd rucaio basd o ifii Gramias

2 Ovrlook alacd rucaio: firs balacig h ruca. Giv a I sysm: / y u d d For covic of discussio w do h sysm as a block form: alacig ruca rducd modl h uimpora par is rucad

3 Ovrlook Wha s h uimpora par? h sas which ar difficul o corol ad difficul o obsrv corrspod h uimpora par. I sysm hory h ukow vcor is calld h sa of h sysm. cually h ris i dpic h sysm variabls such as brach currs od volags i h ircoc modl ad hrfor dscrib h sa of h sysm.

4 alacd rucaio ihog Fg

5 alyical soluio of h I Sysm Wh discuss balacd rucaio mhod w limi h I sysm o h followig form: d / d y u I ordr o aalyz corollabiliy obsrvabiliy w d o us h aalyical soluio of h sysm hough w always solv h sysm umrically i.. by umrical mhods ad usig compurs. h aalyical soluio of h sysm: h aalyical rprsaio of.

6 alyical soluio of h I Sysm Wha is h aalyical soluio cid by h ipu u ad sarig wih h iiial sa? s also Chapr 4 scio 4. i [Chi-sog Ch iar Sysm hory ad Dsig 3rd diio 999] Muliplyig o boh sids of yilds d d d / d u u which implis d d u Is igraio from o yilds u d

7 alyical soluio of h I Sysm hus w hav u d caus h ivrs of is ad implis I u d d / d u his is h aalyical soluio of. I is impossibl o plo h wavform of by had w d compurs o compu umrically ad plo a may sampls of im. I is difficul o compu by followig h aalyical formulaio i if is vry larg. W d o solv h I sysm umrically wih som umrical mhods lik backward Eulr...c.

8 Ouli Ovrlook Corollabiliy masurs Obsrvabiliy masurs Ifii Gramias MOR: alacd rucaio basd o ifii Gramias

9 Corollabiliy masur Rachabiliy Dfiiio: Giv a sysm a sa is rachabl from h zro sa if hr is a ipu fucio u of fii rgy such ha ca b obai from h zro sa ad wihi a fii priod of im. u = C y

10 Corollabiliy masur Do rach X h subspac spad by h rachabl sas h X rach X X is h whol sa spac.g. X { : R C } h sysm is rachabl spac is rachabl. X rach X: vry sa i h sa

11 Corollabiliy masur Eampl icur rfrrd o [Chi-sog Ch iar sysm hory ad Dsig 3rd diio Nw York Oford Oford Uivrsiy rss 999] _ V _ 8 dos h volag drop alog h capacior ad is h sa of h sysm. I his circui = a ay im. Coclusio: I his circui sa is a rachabl sa bu ay ozro sa is a urachabl sa! hrfor h whol sysm is urachabl.

12 Corollabiliy masur Eampl is acually h Whaso bridg. + R R 3 R 3 is adjusabl i is adjusd ill V G bcoms zro. I mas hr is o volag drop hrough. hrfor w hav V G V _ R V G R R R R ca b asily masurd by h abov quaio. R R 3 Whaso bridg Whaso bridg is a masurig isrum ivd by Samul Hur Chrisi i 833 ad improvd ad popularizd by Sir Charls Whaso i 843. hp://.wikipdia.org/wiki/whaso_bridg

13 Corollabiliy masur Eampl d / d u + F + _ F + _ y V _ volag drops hrough h wo capaciors. hos sas wih ar rachabl bu hos sas wih ar o rachabl. caus whavr h ipu is h volag drops hrough h wo capaciors ar always idical. hrfor h whol sysm is urachabl.

14 Corollabiliy masur Rachabiliy mari of h sysm: R [ ] y h Cayly-Halmilo horm h rak of h rachabiliy mari ad h spa of is colums ar drmid a mos by h firs rms o h firs colums i... hus for compuaioal purpos h followig fii rachabiliy mari is of imporac: R [ ] Somims R is dircly dfid as h rachabiliy mari. Why i is calld rachabiliy mari? y cocio bw ad rachabiliy? R

15 Corollabiliy masur Noic h aalyical soluio of sysm sa quaio d / d u is u u d h rachabiliy of a sa of h sysm is sd by h zro iiial sa w look a h abov aalyical soluio wih u u d Noic: I!! k k k!

16 !! k k d u d u d u d u d u u which mas a rachabl sa is h liar combiaio of h rms: k hrfor is dfid as h rachabiliy Mari. R Corollabiliy masur

17 Corollabiliy masur cually hr is a horm horm 4.5 i Chapr 4 i [oulas5]: rach horm If X is h subspac spad by h rachabl sas h X rach im R :spac spad by h colums. h horm lls us h subspac spad by all rachabl sas is acly h subspac spad by h colums of h rachabiliy mari R. h fii rachabiliy gramia a im is dfid as : d for

18 Corollabiliy masur Cocio bw rachabiliy mari ad rachabiliy gramias roposiio h fii rachabiliy gramias hav h followig propris: a ad b hir colums spa h rachabiliy subspac i.. im im R. roposiio 4.8 i [ulous 5] roof asir way is o prov im im R whr im im C ad im R im R C W firs prov im im R im w hav d for all

19 Corollabiliy masur k k k I!!!. for all hrfor i i i im R im R W hav provd: im im R

20 Corollabiliy masur N w prov: im R im i im R im R for all i i for all i. im for all why? im symmric ull d

21 Corollabiliy masur ull p p p ad p p p ull p i ull im symmric } spa{ } spa {colums of im p p im im ull im

22 Corollabiliy masur h rlaio im im R provids a way o driv h miimal rgy which ar dd o rach a sa. h sas usig larg miimal rgy ar difficul o rach ad will b rucad durig MOR basd o balacd rucaio. hrfor h miimal rgy for rachig a rachabl sa is a ky cocp for modl ordr rducio basd o balacd rucaio. N w will driv h miimal rgy for rachig a sa.

23 Corollabiliy masur From h aalyical soluio if a sa is rachd a im h wih fii rgy such ha d u How much mus h ipu u b? W hav provd if is rachabl h i.. im ud d d u ad u his mas ca b rachd a im wih ipu u

24 Corollabiliy masur u = C y h ipu u is h ciaio of h sysm is rgy is h rgy rquird o rach h sa. Ergy of a fucio is dfid as: u u u d

25 Corollabiliy masur W s from abov aalysis if is rachabl a im rprsd as: ud ca b y ohr ipu u u ca also rach. Howvr if u u i cao rach a im may d logr im. u u cually h rgy of is h miimal rgy o rach h sa a h giv im priod. roposiio 4. i [ulous 5] Ergy of u : u u u d d rlaio o?

26 Corollabiliy masur sysm is rachabl mas vry sa i h whol sa spac is rachabl. From horm : X rach im R im R hrfor h sysm is rachabl rak R From roposiio : im im R hrfor h sysm is rachabl rak hrfor is osigular for ay if h sysm is rachabl.

27 Corollabiliy masur Ergy of u oic : u u Corollabiliy masur! Oly for rachabl sysms.

28 Corollabiliy masur Rmark : Rachabiliy is a gric propry for I sysms wih h form: d / d u his mas iuiivly ha almos vry I sysm wih h form abov is rachabl. If hr ar ay urachabl sysms hy ar vry rar. h urachabl I sysms lik ampls ar rar. Rmark : h rachabiliy of h sysm ca b mor asily chckd by h criria: h sysm is rachabl rak R

29 Corollabiliy masur cocp which is closly rlad o rachabiliy is ha of corollabiliy. Hr isad of drivig h zro sa o a dsird sa a giv ozro sa is srd o h zro sa. Mor prcisly w hav: Dfiiio of corollabiliy: Giv a I sysm as abov a o-zro sa is corollabl if hr is a ipu u wih fii rgy such ha h sa of h sysm gos o zro from wihi a fii im:.

30 Corollabiliy masur I has b provd ha for im coiuous I sysms as discussd i his lcur h cocps of rachabiliy ad corollabiliy ar quival. horm For im coiuous sysms ulous 5 X rach X cor. horm 4.6 i Similarly X cor is h subspac spad by h corollabl sas. From h propry of rachabl sysm w hav h sysm is corollabl rak R

31 Corollabiliy masur Eampl: laform sysm u Dampig Coffici: dashpo Sprig Cosa: Dampig Coffici: dashpo Sprig Cosa: h sysm is dscribd by h followig liar im ivaria I sysm: assum mass of h plaform is zro ad from Nwo s law: F v k ma u u.5.5 d / d u

32 Corollabiliy masur Is h plaform sysm corollabl? h sysm is corollabl rak R R [ ] ar liarly idpd! rak R hrfor h plaform sysm is corollabl.

33 Corollabiliy masur ssociad wih corollabiliy hr is h cocp of obsrvabiliy. Corollabiliy: ipu u sa. ossibiliy of srig h sa from h ipu. Obsrvabiliy: oupu y sa. ossibiliy of simaig h sa from h oupu.

34 Ouli Ovrlook Corollabiliy masurs Obsrvabiliy masurs Ifii Gramias MOR: alacd rucaio basd o ifii Gramias

35 Obsrvabiliy masur Obsrvabiliy is a masur for how wll iral sas of a sysm ca b simad by kowldg of is ral oupus. Dfiiio of Obsrvabiliy: Giv ay ipu u a sa of h sysm is obsrvabl if sarig wih h sa = ad afr a fii priod of im ca b uiquly drmid by h oupu y. = u C y

36 Obsrvabiliy masur Obsrvabiliy mari? Obsrvabiliy Gramia? Oupu rgy? O

37 Obsrvabiliy masur Drivaio of Obsrvabiliy mari d u From h aalyical soluio of w s ha afr im : u d d / h sysm sarig wih = hrfor d u d h oupu corrspodig o is: d u ad d u d u y

38 Obsrvabiliy masur If is obsrvabl h for ay u ca b uiquly drmid by h corrspodig y : Drivaio of Obsrvabiliy mari y ad u d Sic ca b uiquly drmid by ca b uiquly drmid by y. i is suffici o prov ha us s udr wha codiio ca b uiquly drmid by? y

39 y Diffria h abov quaio o boh sids ad g h drivaivs a =: y y ' y '' y k k ' k k y y y # has a uiqu soluio if has full row rak. # k Obsrvabiliy masur Drivaio of Obsrvabiliy mari

40 k k Q Do: y Q k ' k y y y y ca b uiquly drmid wih k big a mos. if m> h k< if m= k=. m R Obsrvabiliy masur Drivaio of Obsrvabiliy mari

41 Obsrvabiliy mari: O hrfor: O From abov aalysis acually h fii Obsrvabiliy mari is ough o drmi obsrvabiliy: hrfor w dfi h sysm is obsrvabl O rak Obsrvabiliy masur Drivaio of Obsrvabiliy mari

42 Obsrvabiliy masur h oupu rgy associad wih h iiial sa is: y y Q y d d Oupu rgy d. Ergy of obsrvaio producd by a obsrvabl sa.. Obsrvabiliy masur! Fii Obsrvabiliy Gramia a im is dfid as: Q d

43 Obsrvabiliy masur Obsrvabiliy Gramia Rcall h miimal rgy o rach a sa a im is u Noic boh rgis ar rlad o im. u y Q * d Q d Fii rachabiliy corollabiliy Gramia ad obsrvabiliy Gramia will b usd o driv h ifii Gramias which. Mak h wo masurs compuabl.. will b dircly usd for rucaio i MOR.

44 Ouli Ovrlook Corollabiliy masurs Obsrvabiliy masurs Ifii Gramias MOR: alacd rucaio basd o ifii Gramias

45 Ifii Gramias Q Udr which codiio ad ar boudd wh im gos o ifiiy:? mak h wo masurs compuabl d Q d Roughly spakig Q ad ca b boudd wh if is boudd wh.

46 Ifii Gramias mak h wo masurs compuabl is boudd if h ral pars of all h igvalus of ar gaiv. Why? S S b h ig-dcomposiio of S S S S S r im S S r im S r r r r im j im j im j im r i im i i ar igvalus of. i

47 S S S S im r S S r r r r im im im im j j j r i si cos im i im i j j im i boudd Ifii Gramias mak h wo masurs compuabl

48 Ifii Gramias mak h wo masurs compuabl hrfor S S S S S r im S if h ral pars of all h igvalus of ar gaiv. hrfor h follow limis iss if all h igvalus of ar gaiv i.. if h sysm is sabl: lim lim d d Q lim Q lim d d whr ad Q ar h ifii Gramias oly for sabl sysms.

49 Ifii Gramias mak h wo masurs compuabl h ifii Gramias: lim lim Q lim Q lim d d d d From h propry of igral w hav Q Q I h maig of ir produc:

50 Ifii Gramias mak h wo masurs compuabl h miimal rgy cssary for rachig a rachabl sa a im is: u For sabl sysms lowr boud of h miimal rgy cssary for rachig a rachabl sa is: u * bcaus For sabl sysms h uppr boud of h rgy producd by h obsrvabl sa is: * y Q Q bcaus Q Q Compuabl masurs! Oly suiabl for sabl sysms!

51 Ifii Gramias mak h wo masurs compuabl For sabl sysms h miimal rgy cssary for rachig a sa is: * mi u For sabl sysms h maimum rgy producd by a sa is: ma y * Q

52 Ifii Gramias mak h wo masurs compuabl caus h MOR mhod w will iroduc uss ad Q o driv h rducd-ordr modl ad hrfor is oly suiabl for sabl sysms. mi u ma y Q h igspacs of ad Q mak h wo masurms pracically compuabl!

53 Eigspacs of ad Q mak h wo masurs parcically compuabl h sas which ar difficul o rach ar icludd i h subspac spad by hos igvcors of ha corrspods o small igvalus. h sas which ar difficul o obsrv ar icludd i h subspac spad by hos igvcors of Q ha corrspods o small igvalus. why ad how?

54 Eigspacs of ad Q mak h wo masurs pracically compuabl Do as h igvcors of h corrspodig igvalus ar. is symmric i has ral igvalus. C h sa ca hrfor b rprsd by : ar liarly idpd hrfor hy cosiu a basis of h whol spac. mi u If a mari is osigular h is ivrs has h sam igvcors bu h igvalus ar h rciprocals: /

55 u mi * * * u mi idicas h miimal rgy dd o rach h sa hrfor h largr is h mor difficul h sa o rach. mi u mi u hrfor is orhogoal. is symmric ] [ Q Eigspacs of ad Q mak h wo masurs pracically compuabl

56 Eigspacs of ad Q mak h wo masurs pracically compuabl mi u mi u is largr if ad k k k k k k ha if k k ad his mas if is difficul o rach u is larg should hav larg compos i h subspac spad by h igvcors corrspodig o h small igvalus of. Or should almos locas i h subspac spad by h igvcors corrspodig o h small igvalus.

57 Eigspacs of ad Q mak h wo masurs pracically compuabl Similarly if is difficul o obsrv y Q is small should hav larg compos i h subspac spad by h igvcors corrspodig o h small igvalus of Q. Or should almos locas i h subspac spad by h igvcors corrspodig o h small igvalus. k k k i i i i k k Q i i i i k k

58 Eigspacs of ad Q mak h wo masurs pracically compuabl ill ow i sms w could do h rucaio by fidig subspac spad by h igvcors corrspodig o h small igvalus of or Q. Howvr i could happ ha sas which ar difficul o rach produc h maimal rgy of obsrvaio; sas which produc h smalls rgy of obsrvaio ar vrhlss h asis o rach! For such sysm w do o kow which sas o ruca!

59 Eampl: Cosidr h followig I sysm / y u d d Q h wo Gramias ar: hir igvalus ad igvcors ar: Q Q Q Q Eigspacs of ad Q mak h wo masurs pracically compuabl

60 Eigspacs of ad Q mak h wo masurs pracically compuabl.3868 Q.5573 Q Q h agl bw ad is vry small. his mas if S is h subspac spad by h h asily obsrvabl sas may also i S. Q Q - Q I lls us if w ruca h sas which ar difficul o rach h sas loca i S w risk rucaig h sas which ar asy o obsrv produc h maimal rgy of obsrvaio bcaus hy ar also i S.

61 Eigspacs of ad Q mak h wo masurs pracically compuabl Howvr if ad Q hav h sam igvalus ad igvcors h h problms is solvd. h sas i h subspac spad by h igvcors of corrspodig o h small igvalus always i h subspac spad by h igvcors of Q corrspodig o h small igvalus bcaus h igvalus ar h sam ad igvcors ar h sam hrfor h subspacs ar h sam. Ca w achiv his? Ys. W ca achiv i by balacig.

62 Ouli Ovrlook Corollabiliy masurs Obsrvabiliy masurs Ifii Gramias MOR: alacd rucaio basd o ifii Gramias

63 Rcall h alacd rucaio mhod: Giv a I sysm: / y u d d b b balacig ruca rducd modl h uimpora par is rucad MOR: alacd rucaio alacig

64 MOR: alacd rucaio alacig asic ida of balacig rasformaio: Us sa spac rasformaio o g aohr ralizaio of h sam sysm so ha h rasformd Gramias ar diagoal marics. Dfiiio of alacig rasformaio: Fidig a osigular mari such ha Q Q ad Q. Dfiiio of alacd sysm: h rachabl obsrvabl ad sabl I sysm is balacd if is wo Gramias ar qual Q i is pricipal-ais balacd if Q diag.

65 MOR: alacd rucaio alacig asic ida of balacig rasformaio: Us sa spac rasformaio o g aohr ralizaio of h sam sysm so ha h rasformd Gramias ar qual ad ar diagoal marics. I.. Q Q How o cosruc? Rcall ha Q Q. Sic Q w hav Q which mas Q. should b h ivrs of h mari of igvcors of Q.

66 MOR: alacd rucaio alacig Chck :? How o mak? If UU h UU U UU U Hr w mus hav h rlaio U. / / If furhr U h U UU I looks ha w ca compu as Howvr w kow ha w hav o compu h ivrs of / U h mari of U I is h ivrs of h igvcors igvcors if /. I. of Q.Sic Q is o a p.s.d. mari o g. o avoid compuig h ivrs of h mari of igvcors w compu i a diffr way.

67 MOR: alacd rucaio alacig Subsiu / U h lf had sid / / io Q w g / U QU / / U UU QU U QU /. ook a h righ had sid w g / / U QU i.. U QU.hrfor is h ivrs of h mari of igvcors of U QU. Furualy h raspos of U QU is a p.s.d. mari. hrfor h ivrs of h mari of h mari islf.so ha w do o hav o compu h ivrs. igvcors is acly

68 MOR: alacd rucaio alacig h abov aalysis clarly shows ha: Eisc of balacig rasformaio: d / d y Giv a rachabl obsrvabl ad sabl I sysm ad h corrspodig Gramias ad Q a pricipal ais balacig rasformaio is giv as follows: u K U ad UK / / Hr UU is h Cholsky facorizaio of. U QU K is h ig-dcomposiio of U QU. Symmric posiiv smidfii mari has ral o-giv igvalus ad orhogoal igvcors. Hr h Eigvcors i K ar ak as orhoormal K

69 MOR: alacd rucaio alacig Wha is h corrspodig balacd sysm? pply h sa spac asformaio: o h origial ralizaio: d / d y u d / d y u

70 MOR: alacd rucaio alacig : alacig Giv d / d y u Compu Q. Compu UU U QU K K h igvalus ar ordrd from h largs o h smalls d / d u / K U d / d u y / UK y

71 MOR: alacd rucaio ruca balacd sysm: / y u d d. : ar h igvcors of h ui vcors i Q i i i i spa h subspac coaiig asily corollabl ad asily obsrvabl sas. hrfor. is alrady ordrd as : of ssum ha h lms o h diagoal r. mas : sas o - corol - o - obsrv ad difficul - ruca h difficul r r r i h balacd sysm : Rplac wih. : I.. r / y u d d : r z z y u u z z d

72 MOR: alacd rucaio ruca z y u u z z d miimal ralizaio of h sam sysm is: is a o-miimal ralizaio of a sysm. ˆ z y u z dz h rducd-ordr modl ROM hrfor w hav h followig simpl sps for rucaio:

73 MOR: alacd rucaio alacig: ruca d / d y u / K U / UK d / d y u ruca: Small par ad Q Sparad accordig o h sparaio of. dz / d yˆ z z u Rducd modl!

74 MOR: alacd rucaio alacig: d / d y u / K U / UK d / d y u Dos i mak ss if w do modl rducio o h balacd sysm rahr ha h origial sysm? Ys. s a sa rasformaio balacig dos o chag h rasfr Fucio ad h HSVs h balacd sysm is oly a diffr ralizaio of h sysm.

75 Obsrv: MOR: alacd rucaio / y u d d / y u d d ˆ / z y u z d dz W W Y Q Y Y Y :. produc mari h of igvcors h ar : i Hr h colums.. h ad as as spara If Y Y Y Y Y Y Y Y W Y W Y W Y W Y W Y Y W W W W W W W Y Y Y Y Y

76 MOR: alacd rucaio hrfor h wo ROMs ar h sam: dz / d yˆ z z u d / d W Y W y Y u Coclusio: balacd rucaio is rov-galrki projcio as blow: Y d / d y u rov - Galrki Y d / d W y Y usig W Y W u hrfor balacd rucaio is quival o: fidig h ivaria subspac of Q ad rmaiig oly h par Y which corrspods o h largs HSVs squar roo of h igvalus of Q.

77 MOR: alacd rucaio lgorihm alacig:. Compu Q. Giv d / d u y. Compu UU 3. / K U U QU K / UK K 4. alacig ad sparaig accordig o h sparaio of : ruca: 5. Form h rducd modl: dˆ / d yˆ ˆ ˆ u

78 MOR: alacd rucaio compuaioal dails r w rady o g h rducd modl from h abov lgorihm? No y bcaus w do o kow y how o compu ad Q umrically! Rcall: Forualy w hav: d Q roposiio proposiio 4.5 i [oulas 5] ad Q ar h soluio of h followig wo yapuov quaios: Q Q hs wo mari quaios ca b solvd umrically by compur! Of cours by usig som algorihms. d?

79 MOR: alacd rucaio I M us commad: compuaioal dails lyap ' Q lyap '

80 MOR: alacd rucaio Numrical issus h balacig mari is: / K U UU. Compuaio of U may caus umrical isabiliy bcaus U is usually ar sigular. U is usually ar sigular bcaus h mari has umrically low-rak i.. ar sigular. is ar sigular bcaus i may cass is igvalus dcay rapidly o zro som igvalus ar vry clos o zro.g.. Q ad bhavs similarly as. i Howvr i algorihm w d o compu: / / K U UK Ca w avoid compuig U?

81 MOR: alacd rucaio compuaioal dails If usig Cholsky facorizaio of boh Z Z Q Z Obsrv Z p Z QZ Z p Q Z p Z Q Z U V Q Q Z Z p Q Us SVD isad of ig- dcomposiio Comparig wih dfid i lgorihm w immdialy g Q U Z / p o avoid compuig h ivrs of Z Z U V Z V Z p U Q Z p w hav: / V Z Q

82 MOR: alacd rucaio Numrical issus lgorihm SR mhod Gig h rducd modl wihou compuig :. Do Cholsky facorizaio of h wo Gramias: Z Z Q ar lowr riagular marics. Z Z Q Z. Do Sigular valu dcomposiio SVD of mari Z Q i.. hr Z ar wo orhoormal marics U V U U I V V I such ha V Z ZQ U V U U. V W / Q Z V / 3. V Z U. ˆ 4. W V W V. ˆ ˆ Q U Z Q 5. h rducd modl is dˆ / d yˆ ˆ ˆ ˆ ˆ u ˆ Do w hav do balacig?

83 MOR: alacd rucaio Numrical issus W will prov ha h rducd modl w go coms from h abov balacd sysm balacd by /! V Z Q h balacd sysm which is balacd by ad Z U / is: / V Z Q

84 MOR: alacd rucaio Numrical issus h rducd modl w obaid jus ow is: V W V W ˆ ˆ ˆ / V Z W Q / U Z V Q V V U U V U Z Z Q Q Q Z V Z V Z V / / / / / / U Z U Z U Z Us h block forms abov o chck if ˆ ˆ ˆ Ys hy ar qual! h rducd modl is rally from a balacd sysm.

85 MOR: alacd rucaio Numrical issus lgorihm somims cao coiu ihr bcaus h Cholsky facorizaio of Q cao b do. his is bcaus ha i som cass ad Q iclud oo small igvalus lik: which is cosidrd by h algorihm as a sigular mari hrfor Cholsky facorizaio cao b coiud. apr [rq 5] provids a algorihm compuig h umrically full rak facors of ad Q which ar i h forms Z ˆ ˆ R Z R h full rak facors umrically saisfy: Z Z Q Z Z. Q Q ˆ Q [rq 5]. r E.S. Quiaa-Ori Modl rducio basd o spcral projcio mhods. I:. r V.. Mhrma D.C. Sors ds. "Dimsio Rduio of arg-scal Sysms" vol. 45 of cur Nos i Compuaioal Scic ad Egirig pp Sprigr-Vrlag rli/hidlbrg 5. lgorihm 4 i h papr

86 MOR: alacd rucaio Numrical issus lgorihm 3 Gig h rducd modl usig full-rak facors [rq 5]:. Compu full-rak facors of h Gramias:. Compu SVD. Q V V U U V U Z Z / V Z W Q 3.. / U Z V 5. h rducd modl is ˆ ˆ ˆ ˆ ˆ ˆ / ˆ y u d d 4.. ˆ ˆ ˆ V W V W Q Q Z Z Q Z Z. ˆ ˆ ˆ R Z R Z Q

87 MOR: alacd rucaio Error boud horm [rq 5] If h origial I sysm is sabl h h rducd modl obaid by lgorihm lgorihm lgorihm 3 saisfis: h rducd modl is balacd miimal ad sabl. I s Gramias ar qual o h sam diagoal mari. h absolu rror boud proof i [oulas 5] Chapr 7 holds. H s Hˆ s H k r k

88 Rfrcs [].C. oulas pproimaio of larg-scal Sysms SIM ook Sris: dvacs i Dsig ad Corol 5. [] Chi-sog Ch iar sysm hory ad Dsig 3rd diio Nw York Oford Oford Uivrsiy rss 999.