Hence, Consider the linear, time-varying system with state model. y( t) u(t) H(t, )

Transcription

1 Df. h mpul rpo mar of a lar, lumpd, m-varyg ym a mar map H, : RR R rm gv by H, = [h,,, h m, ], whr ach colum h, rpr h rpo of h ym o h mpulv pu u = -, R m, = [ ] occur a h h compo, h m of applcao of h pu ad h obrvao m. Rcall ha f H, for ay gv, ad <, h h ym ocaual. O h ohr had, f H, = for <, h h ym caual. I block dagram form, u H, y Hc, y H, u d Codr h lar, m-varyg ym wh a modl B u, y C D u 95

2 96 horm: h mpul rpo mar for h abov ym gv by Proof: h rpo of h abov m-varyg ym o a pu u gv by Bu, - =, mpl ha L u = -, =[ ], whr h appar a h h locao, h for =,,,m,.. w pck up h corbuo du o h h pu. Rcall ha, D B C H,,,, D dq q q B q C C u u y, D dq q q B q C u u y,, D B C D dq q q B q C η η y dq q q H, y u

3 97 If h ym caual, h H, = for >. hu, for ad y =, for <. If u = -, h Hc, H, = C, B + D -, = < If, B, C ad D ar coa marc, h Codr aga h m-vara ym a modl dq q q H, y u, h y D B C H H H,,, u B D C u y

4 I h -doma, Hc, L X I Y C I I If h ym ally a r,.., =, h Df. h rafr fuco mar of h m-vara a modl gv by Df. h Lvrrr lgorhm. L h polyomal BU C I B D U Y U U C I B D H H L = X Bu X BU I X BU C I B D d I a a b h characrc polyomal of h mar R, h h coffc a, =,,, ca b compud a follow: a N = I, a = - r N = N +a I, a = -/ rn, 98

5 99 N 3 = N +a I, a 3 = -/3 rn 3,... N = N - +a - I, a = -/ rn, ad N + = = N + a I, whr rm h rac of h mar M. Eampl: L a dyamc ym hav h fdback mar b gv by h, N = I, a = -r = --5 = 5, N =N +a I= + a I = 4 3 3, r N r a I a N N

6 ad a3 r N3 r 3 3 o mak ur h calculao ar corrc, chck ha N4 N3 a3i Fally, h characrc polyomal of = Faddv-Lvrrr lgorhm for Compug I - - L dj I R N N N N I d I a a a whr h characrc polyomal of, R h adjo mar of I. Propoo: L {} = I -, whr L {} h Laplac raform opraor. Proof: For h m-vara ca ad u =,,,

7 L{} - = L{}. Bu, = I, hu L{} = I -. Clarly, = L - {I - }. Cayly-Hamlo horm: L b h characrc polyomal of. h h mpl ha Lkw, or a a a I d I a a a a a a a a I a a a a a

8 Obrvao : + alo a lar combao of I,,,, -. Obrvao : Evry polyomal of R ca b prd a a lar combao of I,,,, -,.., f = I Furhrmor, f a polyomal f uch ha dgf > dg, h f = q + h, dg h < f = q + h = q. + h = h f dgf > dg, w ca olv for h drcly from,.., L h , h f h gvalu of ar dc, h j ca b compud from h lar quao f q h h,,,, Eampl: Calcula f = + 3 wh

9 h characrc polyomal of = + + = + = = -. L h = +. h h calar polyomal f = + 3, ad f = f- = = - = -. Bu, f = f! For h rpad gvalu ca, h oluo procdur modfd a follow: L m m Sc dgh, h coffc j, j =,,,, ca ow b obad from h followg of quao: f l = h l, l =,,, - ; =,,, m. Gog back o h prvou ampl, = ad 9 9 f h 3

10 4 W mu olv h quao - = - ad = -7. h mpl ha = -9. Morovr, f = +3 = h = I + = -9I 7 = Suppo aga h gvalu of ar dc. h h paral fraco pao of gv by whr ad h h rdu of mar of Coquly, for R R R R I a a a lm R R R I R L L R I I

11 5 Eampl: Codr h ym. Oba ug h paral fraco pao mhod wh h ym mar gv by Now, ad h characrc polyomal of h rdu marc ar foud a follow: 3 R I d I 3 lm lm R R 3 lm lm R R

12 6 h vr of I- qual o ad h a rao mar of h ym or Suppo ow ha h mar coa rpad gvalu, h h adjo mar R ad may hav commo facor. Codr, for ampl, h mar I L L

13 7 h, W kow from h Cayly-Hamlo horm ha f h characrc polyomal of h mar, h =. Df. polyomal p uch ha p = calld a ahlag polyomal of h mar. Df. h moc polyomal of la dgr whch ahla h mar calld h mmal polyomal of ad dod by. Suppo g a polyomal of arbrary dgr, h p = g alo a ahlag polyomal of. horm: For vry mar, h mmal polyomal dvd h characrc polyomal. Morovr, = f ad oly f a gvalu of, o ha vry roo of = a roo of =. 3 R I

14 Proof: If ahla ad f a moc polyomal of mmum dgr ha ahla, h dg dg. By h Euclda algorhm hr polyomal q ad r uch ha = q+ r ad dgr < dg Bu, = = q + r = q + r r =. Howvr, dgr < dg, ad by dfo h polyomal of mmum dgr uch ha = r dvd. h rul mpl ha vry roo of = a roo of = ad hc vry roo of = a gvalu of. If ad f corrpodg gvcor, h = ad = = =. If ha rpad gvalu,,,, j, j,, j =,,,, h for m +m + +m h mmal polyomal ha h rucur m m m 8

15 9 horm: L R, h whr I h parcular ca, for h a rao mar gv by h ru bcau ˆ m j j j R R R I! lm m j m j m j d R I m j d! m j j j j R { }! m m j j j j j j I R R j L L

16 Eampl: Codr a dyamc ym wh h mar h Ug h Faddv-Lvrrr algorhm, w g N = I N = N + a I = N 3 = N + a I = 4 5 5, 4, 5 4 d 3 3 a a a I

17 So, ad R N N N3 4 5 R N N N3 I I h ca, h mmal polyomal h am a h characrc polyomal,.., = = h bcau hr ar o cacllao. h vr of I R R I whr h rdu marc ar R d d I N N R I I N N 3 3 3! lm lm d d

18 3 I N N 3 R lm I lm I N N I N N3 R lm 4I N N hrfor, for, h a rao mar gv by 3 R Codr ow h followg block dagoal R R Â mar ˆ, whr a ogular mlary raformao. h, characrc polyomal h am a ha of h la ampl,.., ˆ

19 3 I h ca, Bu, provdd ha a ogular ca b corucd. Suppo ha whr ˆ ˆ ˆ J P J J J J J,

20 4 h whr I h ca, mar ad o b h Jorda caocal form. L hav gvalu,,, p, ach wh mulplcy, =,, p, p =. Suppo ha p dpd gvcor aocad wh h gvalu,,, p. If q = - rak I- =, h J J J J P J!...,,, p

21 5 h h of gvcor grad by =,,, p form a ba. horm: h gralzd gvcor of aocad wh h gvalu {,, p } ach wh mulplcy, =,,, p, ar larly dpd. Eampl: L. W alrady kow ha = {-, -, -}.,...,,...,,..., p p p 3 I I I I 4 5,...,,...,,..., p p p

22 6 Clarly, = ad =. Morovr, Clarly, ar wo larly dpd gvcor. Furhrmor, q = - rak I- = 3 - =, ad ad mpl ha h hr gvcor ar larly dpd L u coruc h mlary raformao a follow: I I ad I d

23 7 I ohr word, h w ym mar, Jorda caocal form gv by h mar poal of h quval ym 4 J ˆ J ˆ

24 h mar poal of h orgal ym hrfor gv by J Dcr-m Sym: Codr h lar m-vara dcr-m ym k k Bu k, y k C k Du k akg h o-dd Z raform, yld X z z zi zi BU z y z zc zi C zi B D U z h a rao mar gv by k = k = Z - {zzi- - }. h rafr fuco mar of a dcr-m lar dyamc ym dfd by H z C zi B D 8

25 O h ohr had, h u ampl rpo mar h gv by hk = Z - {Hz}. Df. M R ymmrc f M = M. horm: h gvalu of M = M R ar ral. Proof: L b a gvalu of M ad v rgh gvcor. h Mv = v. Now, f v* h compl cojuga rapo of v, h v*mv = v*v = v*v Bu, v*mv ad v*v ar ral mu b ral, all gvalu of M mu b ral. h ca b vrfd from h fac ha v*mv* = Mv*v = v*m*v = v*m v = v*mv. horm: L M = M R. h hr a orhogoal mar Q = [q q...q ] uch ha M = QDQ or D = Q MQ, whr D a dagoal mar corucd from h gvalu of M, q, =,, h ormalzd vro of gvcor v aocad wh h gvalu of M, =,,,.. q = v / v 9

26 horm: mar M = M R pov df pov m-df f ad oly f a Evry gvalu of M pov zro or pov. b ll ladg prcpal mor of M ar pov all prcpal mor of M ar zro or pov. c hr a ogular mar N a gular mar N or a m, m < mar N uch ha M = N N. W do pov df m-df by M > f M > M f M. Eampl: Codr h followg mar M: 3 M 3 h M = M, {, } = {4, }. h corrpodg gvcor ar: { v, v },

27 h ormalzd gvcor ar gv by hrfor, h Q ad D marc ar gv by Sably of Dyamc Sym Dyamc ym ably a vry mpora propry. I abl h rackg of drd gal or h uppro of udrd gal. Sym ably dcrbd hr rm of pu-oupu ably or rm of ral ably., }, { q q Q 4 D

28 Ipu-oupu Sably: gl pu, gl oupu SISO lar, m-vara LI, couou m dyamc ym boudd-pu, boudd-oupu BIBO abl f ad oly f mpul rpo h aboluly grabl,.., whr M a ral coa. Proof: L h pu u b boudd,.., u k <,. h y boudd. Suppo h o aboluly grabl. h for a caual, lar m-vara ym, wh u = k > ad h >,, wh odcrag vlop. M d h M k d h k d k h d u h d u h y

29 h oupu gv by For, h mpl ha a, y h u d y k h d h d y o boudd v wh u boudd. hu, h mu b aboluly grabl. horm: SISO LI, couou-m dyamc ym BIBO abl f ad oly f vry pol of rafr fuco H l o h lf-half of h -pla. Proof: L H b a propr raoal fuco of, h f vry pol locad a = - p, p >, ha mulplcy, uch ha m 3

30 w g H m j j j p ad h mpul rpo gv by, k m m k j j h L H k jl j j p j j!, whch aboluly grabl. Codr a mulpl-pu, mulpl-oupu MIMO, LI, couou-m dyamc ym dcrbd by h mpul rpo mar H = [h j ]. Such a ym BIBO abl f ad oly f, j, h d j K j lravly, a MIMO, LI, couou-m dyamc ym dcrbd by h propr raoal rafr fuco mar H = L{H} = [H j ] BIBO abl f ad oly f vry pol of H j locad o h lf half of h -pla. p 4

31 w alrady kow, h oluo of h a quao gv by Morovr, h -doma w g Suppo ha h pu dcally zro ad h al a ozro,.., ad h ad ~ ~ Bu~ d X I L H I - - b h ral rafr fuco mar of om couou m LI dyamc ym. h, wh u = ad for om ~ ~ I ~ u ~ or ~ U ~ ~ ~ X I ~ ~ ~ ~ ~ BU ~ k 5

32 w g ~ ~ ad. h uforcd ym margally abl f ad oly f h pol of H or h gvalu of hav hr zro or gav ral par, ad ho wh zro ral par ar mpl roo of h mmal polyomal of. h uforcd ym aympocally abl f ad oly f all h pol of H all gvalu of l rcly o h lf half of h -pla. ympoc ably alo mpl ha ~ lm ~ Obrvao: h wo cocp dal wh ral ably oly. Eampl: L a uforcd dyamc ym b dcrbd by ~ ~ 6

33 7 h h pol of H ar locad a = ad = -5/4 ym margally abl. Suppo ow ha h pu mulu ozro ad ha h al codo zro,.., ad h ad Furhrmor, H ~ ~. ~ ~ u ~ ~ BU I X d Bu ~ ~ Y C I B D U H U

34 Clarly, vry pol of H a gvalu of f vry gvalu of ha a gav ral par, h all pol of H l o h lf-half of h -pla h ym dcrbd by ~ ~ ~ Bu~, ~ ~ y C ~ Du~ BIBO abl. Howvr, bcau of pobl pol-zro cacllao, o vry gvalu of a pol of H. Hc, whl h uforcd ym ~ ~ may b uabl, may o! Eampl: Codr a LI dyamc ym dcrbd by h quao u u y 8

35 I block dagram form, u _ + Σ + + Σ y ~ L ~ ~ u ~ y 9

36 hu, d I d ym rally uabl. ma h h rafr fuco of h ym gv by H C I B Hc, h mpul rpo whr h u u hrfor, h ym BIBO abl. Howvr, a alrady pod ou, h ym rally uabl bcau of h gvalu a =! 3

37 3 I fac, Lyapuov ably Codr h followg LI, couou m, uforcd dyamc ym, R. h h ym aympocally abl f vry gvalu of ha a gav ral par. L I L L

38 3 horm: L = {,, } b h pcrum of. h R{ } <, =,,, f ad oly f for ay gv N = N >, h Lyapuov quao M + M = - N ha a uqu oluo M = M >. Eampl: L h ym mar b dcrbd by h = {-, -, -3}, whch ma ha h ym aympocally abl. L h N = N >, c N = {,, 3}. Solvg h Lyapuov quao, yld N M M

39 h abov mar M pov df c M = {.8,.5, 3.97}. Dcr-m Sym Sably Codr a SISO dcr-m, caual, LI ym, h f h h u ampl rpo ad u h pu appld o, y k h k u k k h k u k horm: Suppo, u K <, =,,. h a SISO dcr m, caual, LI ym BIBO abl f ad oly f h aboluly ummabl,.., h M horm: L a SISO, dcr-m, caual LI ym b dcrbd by a propr raoal rafr fuco Hz = Z{h}, whr Z h z-raform opraor. h h ym BIBO abl f ad oly f vry pol of Hz ha magud rcly l ha l d h u crcl o h z-pla. 33

40 horm Iral Sably: uforcd dcr m, LI dyamc ym dcrbd by k k,. I margally abl f ad oly f h gvalu of hav magud l ha or qual o, ad ho qual o ar mpl roo of h mmal polyomal of. I aympocally abl f ad oly f all gvalu of hav magud l ha. Lyapuov Sably horm: ll gvalu of R hav magud l ha f ad oly f for ay gv N = N > h dcr Lyapuov quao M M = N ha a uqu oluo M = M >. Eampl: Codr a dcr-m, LI dyamc ym dcrbd by k k k

41 h = {-.,.,.5}. Furhrmor, f Q = dag,, 3, h h dcr Lyapuov quao wh N = Q ha h oluo M M Bu, M = {.753, 4.87, 9.437} M > all gvalu of mu l d h u crcl whch w alrady kw. Df. a = R corollabl ovr [, ] f hr a pu u dfd ovr [, ] uch ha Bu d Df. h a modl d/d = + Bu ad o b corollabl complly corollabl f ad oly f vry a R corollabl. 35

42 Df. h of all corollabl a h corollabl ubpac. Propoo: h corollabl ubpac a lar ubpac,.., l V R b h corollabl ubpac, h f ad V, h vcor = + V for ay, R. Lmma: For ay gr p >, rak [B B p- B] = rak [B B - B]. Corollary: L Q = [B B - B]. h for ay Col-p [Q], Col-p [Q],.., h colum pac of Q -vara. L rak[q] = p <, U b a p mar who colum form a ba for h colum pac of Q ad l U b a -p mar who colum oghr wh ho of U form a ba for R,.., Col-p [U U ] = R. Propoo: Gv d/d = + Bu, h a raformao [U U ]z =, yld z z B u z z 36

43 ~ ~ p whr rak [ B B B ] corollabl form. p, ad q. kow a h Kalma Proof: Now, U Col-p [Q] ad U ~ ~ ~ [ u u u p] mpl ha ach ~ colum of U a lar combao of h colum of U or U U for a ~ appropra R pp. Each colum U R c h colum of [U U ] ar a ba R ~ ~, hrfor, hr mu marc ad uch ha ~ ~ ~ U U U ~ U U hu, or [ U U ] z Bu [ U U ] z Bu [ U U U U z U U z Bu z z U U Bu ] z Bu 37

44 By coruco, Col-p [Q] = Col-p [B B - B] Col-p [B], hu, hr a pm mar B uch ha B = U B or B B B U U U U B or B z z u Lmma: h corollably Gramma mar ogular whvr rak [ Q ~ ] = p, whr K ~ Q [ B ~ ~ ~ BB ~ d p B B ] 38

45 Proof: Suppo ha rak [ Q ~ ] = p ad ha hr v, v R p uch ha v K =, h L c v Kv v B B v d ~ B v c c p, h p v Kv c c d c c d f ad oly f c for [, ]. j Now, f c = for [, ] h d c, j j d Clarly, for j = ad =, c = v B =. For j = ad =, dc d ~ ~ v B v ~ B 39

46 For j j d c j ~ j v j B d hu, p p v Q v [ B B B ] B B B v v v h mpl ha rak [ Q ~ ] p, whch coradc h hypoh ha rak [ Q ~ ] = p, hc, K ogular. Eampl: Codr h dyamc ym dcrbd by u Clarly, h ym o corollabl, a h pu u ad do o affc. Now, Q B B rak [Q] = ym o corollabl. L U = [ ]. Sc [ ] pa Col-p [Q], h, f U = [ ], h colum of [U U ] pa R. 4

47 4 Now, ad [U U ] - B = hrfor, h quval ym h Kalma corollabl caocal form dcrbd by horm: h followg am abou corollably ar quval:. h par, B corollabl. Rak [ I- B] = for ach gvalu of 3. Rak [Q] = 4. Rak [ - B] =,.., hr ar larly dpd row fuco of - B for [, ~ ~ ~ ] [ ] [ U U U U u z z

48 4 5. h mar pov df. Furhrmor, h pu rafr o. Proof: 3, c, B corollabl rak [Q] = rak [B B - B] =. Suppo hr uch ha rak [ I B] <, h hr a v R, v, uch ha, v [ I B] = [v I v B] = v I = ad v B =. Bu, v I = v = w *, a lf gvcor of aocad wh. Now, v Q = [v B v B v - B] = [v B v B - v B]= [ ] = Hc, rak [Q] <, whch coradc h hypoh ha rak [Q] =, hc, 3. h proof ha 3 bacally u h am yp of argum. ˆ d BB K ˆ K B u * * * * * * k k k k k k k w w w w w w v

49 um ha rak [ - B],.., hr v uch ha v - B =. k d k k Bu, v B v B, k v Q =, whch coradc k d h aumpo ha rak [Q] =. hu, rak [Q] = rak [ - B] =,.., - B ha larly dpd row. Suppo Kˆ o pov df. From h prvou lmma, Kˆ alo o gav df, hu, Kˆ ca oly b gular pov m-df. hrfor, hr v uch ha v Kˆ = v Kˆ v = or v Kˆ v l = -, h v BB vd ˆ v Kv v BB vd h mpl ha h grad. v B = row of B ar larly dpd whch coradc ha rak[ B] =. hrfor, rak [ B] = pov df. ˆK 43

50 44 Fally, Eampl: Codr h followg lar m-vara dyamcal ym h, d Bu ˆ ˆ ˆˆ BB K d BB d K KK u Q

51 Hc, rak [Q] = 3 =. Morovr, h pcrum of = {-.43, j.37} = {,, 3 }, whch ma ha.5698 rak [ I B] = rak c h hrd colum a lar combao of h fr ad cod colum. Lkw, rak [ I B] = 3, =, 3. 3 Dcr-m Sym L k+ = k + Buk, R ad B R m. Df. a c R corollabl rachabl f ad oly f hr a f N ad a pu quc {u, u,, un-} uch ha f =, h N = c. 45

52 Eampl: Codr h followg lar hf-vara dcr-m dyamc ym h f = [ ] ad u =, = [ ], a corollabl a bcau ca b rachd o p wh N =. Now, Q B B rak [Q] =!.., ym o complly corollabl. Howvr, h ubpac dcrbd by h ba vcor [ ] corollabl. horm: h followg am abou dcr-m corollably ar quval: k k u k. hr a f d N uch ha k = ca b drv o c = k+n by om pu quc {uk, uk+,, uk+n-}.. c Col-p [Q], Q = [B B - B] 46

53 47 3. If c ad c Col-p [Q], h hr N uch ha c k ca b drv o c k+n by om pu quc {uk, uk+,, uk+n-}. Pol Placm By Sa Fdback Codr h lar m-vara couou m dyamc ym dcrbd by L u = u c + u r, whr u c h corol gal ad u r h ral rfrc gal. L u c = F b h a fdback corol gal, h h w clod-loop ym. Gv a pr-pcfd ymmrc pcrum of compl umbr, coruc h fdback mar F uch ha + BF =. If h ym udr codrao h corollabl caocal form,.., u B u a a u r B BF

54 48 h h characrc polyomal gv by Now, f u c = F = [f f ], h ad h clod-loop characrc polyomal gv by +BF = + a - f a - f - + a f. L h drd pcrum b gv by + BF = = { d, d,, d }, h choo F uch ha a a a r r u f a f a Bu BF +BF d

55 Now, a a a +BF d, d, d, d a f a f a f. h, w qua coffc of qual powr,.. o compu h fdback ga a, d a f,,,,. f a a, d,,,,. W lc h drd clod-loop gvalu d,,,,, accordg o h dg prformac crra, whch ca b pd of rpo of h ym, amou of corol ffor, c. Wh h a modl of h phycal ym o h corollabl caocal form, bu complly corollabl, w ca raform o h corollabl caocal form ug a mlary raformao. 49

56 5 Gral Sgl Ipu Ca, B corollabl rak [Q] =, Q R Q -. L v b h la row of Q - ad z = V, whr h mlary raformao V gv by h, h w ym gv by Clam : VB = B c = [ ]. Proof:. V v v v c c VV VBu B u z z = z + B B B B VB v v v v v v

57 5 Bu, c v h la row of Q -. hrfor, VB = B c = [ ]. Clam : VV - = c = Proof: VV - = c V = c V. Now, ] [ ] [ ] [ B B B B B B Q I Q Q v v v v v a a c V V V a a v v v v v v

58 5 Now, From Cayly-Hamlo, = + a a - + a I =. = -a I a a - v = -a v a - v - - a v -. Hc, h rgh-had d h am a h lf-had d. Clarly, h abov a raformao z = V chag ad B o c ad B c, whr c ad B c ar h corollabl caocal form. c a a a a a a V v v v v v v v v

59 53 Dg Procdur: raform, B o c, B c g d = { d, d,, d } by F c,.., c + B c F c = Fd F o ha +BF =,.., +BF = V - c +B c F c V = +BF c V F = F c V h fdback ga mar h phycal doma. Eampl: Codr h ym dcrbd by L h drd pcrum b = {-5, -, -5}. Now, Q = [B B B] = u 4 8

60 Sc Q of full rak, w ca compu vr, Q Morovr, v V v v Hc, c VV c ad h pcrum of gv by = {-, -4, -}. L F [ f f f ]. c c3 c c 54

61 h h clod-loop ym characrc polyomal gv by B F c c c 3 5 fc 54 fc 4 fc or 5 f c = 5 f c = -35; 54 f c = 75 f c = -67; 4 f c3 = 5 f c3 = -46. hu, F c = [ ] ad F = F c V = [ ]. Fally, h clod-loop ym gv by, BF Bu r u r ad pcrum +BF = {-5, -, -5}. 55

62 Obrvably Rvd: Codr h dyamc ym dcrbd by y Bu C Du Df. R obrvabl f ad oly f hr > uch ha h kowldg of y ad u ovr [, ] ad of h ym marc, B, C ad D uffc o drm. Df. h par C, q. complly obrvabl f ad oly f vry R obrvabl. horm: For h ym dcrbd by, h followg ar quval:. h par C, obrvabl C. Rak, for ach gvalu of I 56

63 57 3. Rak [] =, whr h obrvably mar = 4. Rak [C ] =,.., C ha larly dpd colum ach of whch a vcor-valud fuco of m dfd ovr [, 5. h obrvably Gramma mar ogular >. Eampl: Codr h larzd modl of a orbal all dcrbd by C C C, W C C d u.75

64 58 h obrvably mar for h ym gv by = Clarly, rak [] = 4. Morovr, = {,, j.5, -j.5}. For = =, rak, y I C

65 59 Sc Df. Suppo rak [] <, h N[] h uobrvabl ubpac of h a pac. hrfor, f N[] h w cao rcoruc from pu-oupu maurm. Propoo: L N[], h N[]. h uobrvabl ubpac vara..75 C I C

66 C Proof: If N[], h = C = C = C = = C - = C Now, C C = C = C C C C Bu, = -a a I C = -a C a C = N[]. Propoo: L Im[ ], h ca b rcorucd wh cray from pu-oupu maurm ad Im[ ] calld h obrvabl ubpac of h a pac. Propoo: Each R ha h dcompoo = ob + uob, whr ob Im[ ] ad uob N[]. 6

67 6 Corollary: R obrvabl f ad oly f uob =. L h mar form a ba for h obrvabl ubpac ad b a mar who colum oghr wh ho of form a ba for R. h z =, rul h Kalma obrvabl form whr z o obrvabl ad z uo uobrvabl,.., h par, obrvabl. u z z z z uo o uo o uo o uo o B B uo o C o z z y o C o

68 6 Eampl: Codr h dyamc ym dcrbd by h obrvably mar gv by Clarly, h rak of h obrvably mar ym o obrvabl. y u

69 L h obrvabl ubpac b dcrbd by = C ad = [ ]. h, h mlary raformao gv by:. pplcao of h mlary raformao rul h quval ym 3.5 zo zo u z uo z uo 5.5 zo y I, z uo whr I a by dy mar ad z. 63

70 horm: h modl, B, C complly corollabl obrvabl f ad oly f, C, B complly obrvabl corollabl. Proof: If, B, C complly corollabl, h rak [Q] = rak [B B - B] =. Bu, rak B B B = rak [Q ] =, C, B complly obrvabl. 64

71 C If, B, C complly obrvabl, h rak [] = rak C =. C Now, rak [C C - C ] = rak [ ] =, C, B complly corollabl. Suppo ow ha, C, B complly obrvabl, h rak [ ] qual o B rak B = rak [ ] =. Now, rak [B B - B] = rak [ ] = B mpl ha, B, C complly corollabl. Suppo ow ha, C, B complly corollabl, h rak [Q ] = rak [C C - C ] =. 65

72 Bu, rak[q ] = rak C C C =, B, C complly obrvabl. Dyamc Obrvr Dg Drmc Ca Codr h calar a modl u y Suppo alo ha h calar,,, u ad y ar kow ad ha =. W would lk o coruc a ma uch ha a cra. L u ad -, h ε ε ε ε If =,.., =, h = rgardl of. If, o h ohr had,, h dpdg o, may or may o go o zro a cra,.., may or may o covrg aympocally o. 66

73 Mor ralcally, howvr, or y h maurm proporoal o h a. L k y u h, f = - w g ε ε k y ε k k ε ε k ε a f k cho proprly. Suppo ha R, u R m, y R p, ad ha h ym dyamc ar dcrbd by Bu y C h f h par C, obrvabl, w ca aympocally rcoruc wh h obrvr dyamc a maor = K y y Bu K y C Bu KC Ky Bu, whr K R p. 67

74 Schmacally, h dpcd a follow: wh h choc of obrvr rucur, [ - ] = KC [ - ] ad ordr for a cra, h fdback ga mar K mu b cho uch a way ha h gvalu of KC l rcly o h lf half of h -pla. 68

75 horm: h par C, complly obrvabl KC ca b arbrarly agd by h propr choc of K. Proof: C, obrvabl, C corollabl by dualy + C K ca b arbrarly agd by proprly choog K. L K = -K, h KC ca b arbrarly agd bcau + C K = KC. Codr h ym ˆ ˆ C u ˆ. From dualy, f h par C, of h orgal ym obrvabl, h h par, C corollabl ad w ca fd a fdback ga mar K ad uˆ K ˆ uch ha h pcrum of KC ca b arbrarly agd,.. C K KC,,. d d d L ˆ ad ˆ B C, h ˆ ˆ ˆ Bˆ u ˆ ad for dg purpo, raform  o ˆc ad ˆB o B ug h raformao ˆc ˆ Vˆ z, 69

76 whr V v ˆ v v ˆ ad v h la row of h vr of ˆ ˆ ˆ ˆ ˆ Q B B Bˆ. Spcfcally, z ˆ = V VV ˆ zˆ VBˆu = ˆ zˆ Bˆ u or ˆ ˆ ˆ c c z ˆ = ˆ ˆ ˆ ˆ ˆ ˆ cz Bcu = z u. a a a a u ˆ z ˆ z ˆ, h L = -K = - kc, kc, kc, 7

77 z ˆ= z ˆ. a kc, a kc, a kc, a kc, Now, hu, ˆ ˆ B K d d d c c c a a a a, d, d, d, d a k a k a k a k c, c, c, c, a k a k a a c, d c, d a k a k a a c, d c, d a k a k a a c, d c, d 7

78 h, K K V ad w ca mplm h maor c ˆ KC Ky Bu Eampl: Codr h couou-m lar m-vara ym dcrbd by u y. h h a maor dcrbd by K y y u. y 7

79 Now, I ad d I d d Furhrmor, 3 Q 4 B B B 6 ad d Q rak Q 3, C C C ad d 4 rak 3 73

80 Whch ma ha h ym complly corollabl ad obrvabl. Codr h ym ˆ ˆ ˆ ˆ Bu ˆ = ˆ C uˆ ˆ uˆ. h ˆ ˆ ˆ ˆ ˆ ˆ Q B B B C C C, whch mpl ha h ym corollabl bcau h par C, obrvabl. L u coruc h raformao V v ˆ v ˆ v, whr v h la row of ˆQ,.. 74

81 vqˆ v v v 3 v v3 v v v3 v v v3 Solvg h 3 mulaou quao vv3, v v v3 ad v v v3, yld Hc, v V ad V W ca aly how ha ˆ zˆ VV z ˆ + VBu ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ cz Bcu z u. 3 75

82 ˆ ˆ z, ˆ h L u Kc z kc,3 kc, kc, ˆ ˆ c Bc Kc. kc,3 kc, kc, 3 Suppo,, 3 d C K KC. h ˆ 3 ˆ B K c c c mpl ha 3 k c, 6, kc,, ad k c,3 6 3 k k k 3 c, c, c,3 or ha 4 9 K. Fally, c K Kc V ad K Oc aga, o ca aly how ha KC, 3. 76

83 Prformac Evaluao: h followg fgur how how wll h a maor rack h orgal ym a wh.5. : u,.. ad 77

84 78

85 79

86 8

87 8 Eampl: Oba a full-ordr a maor for h d ordr ym y = [ ] uch ha h drd maor pcrum KC = {-5, -5}. =, rak [] = ym obrvabl. Now, h phycal ym pcrum = {-, } ym uabl. u C C

88 8 L h, ad KC = + k + k = +5 = k = 5, or k = 6 ad k =. Hc, L h al codo of h a maor b ad l h rfrc pu b a p pu,... h, h prformac of h a maor how h followg fgur: u y k k ε ε ε ε k k k k k k. 6 K., r u u

89 .7 Full Sa Emaor.6.5 a m c 83

90 .4 Full Sa Emaor. a m c 84

91 85 Fdback From Sa Ema: L a lar m-vara dyamc ym b dcrbd by L h ym dcrbd by b corollabl ad obrvabl, h, f h a varabl ar o avalabl for fdback, w ca dg a a maor ad fdback h ma lu of h acual a varabl. L h pu gal b gv by h h clod-loop ym gv by ad h a maor ow dcrbd by. F r u u B BF r u F B C K r u y u y u D C B

92 I block dagram form, h compl ym how blow u r + S u Dyamc Sym y - F Emaor h calld h corollr-maor cofgurao. 86

93 87 I ca b how ha h abov ym ha h am gvalu ad h am rafr fuco a h ym wh pu gal corol + rfrc gal I augmd form, Codr h followg ogular mlary raformao F r u u r C B B BF KC KC BF y u ε I I I

94 h, h w quval ym gv by BF ε y ε C BF KCε B u r o how ha h maor do o affc h locao of h gvalu of h orgal a fdback ym, w calcula h augmd ym rafr fuco. L ovrall BF BF, KC B ovrall B ad C ovrall C. h, Y C ovrall I B U, ovrall ovrall 88

95 89 whr So, Whch ablh ha h rafr fuco of h orgal clod-loop ym do o chag wh h roduco of h a maor h loop. Eampl: Codr h am ym of h la ampl. h ym uabl wh gvalu a - ad. L h drd clod-loop ym hav gvalu a - j,.., h drd characrc polyomal gv by KC I KC I BF BF I BF I I ovrall. U H BU BF C I Y d d 8 4 f f f f BF I j j BF

96 h mpl ha f = 9 ad f = 4. L u ow apply h fdback corol bad o h ma rahr ha o h acual valu of h a,.., u h, h prformac of h maor-bad clod-loop ym how h followg fgur wh h rfrc pu a u p ad h al valu of h ym a ad ha of h maor ar rpcvly. 9 4 u..5, ad. r 9

97 Sym wh mad a fdback corol.6 Emaor a fdback corolld ym wh u p rfrc pu.4. a ad m c 9

98 Sym wh acual a fdback corol.6 Sa fdback corolld ym wh u p rfrc pu.4. a ad m c 9

99 Fdback Corol Dg for Mulpl-Ipu Lar m Ivara Sym L a op-loop lar m-vara dyamc ym b dcrbd by d d m Bu,, u. Suppo h ym complly corollabl. h a fdback ga mar F ca b foud, uch ha h applcao of h pu u F u rul a clod-loop ym r d d BF Bu,, u r m ha aympocally abl. Now, h characrc polyomal of h clod-loop ym gv by BF I BF. d 93

100 Problm Sam: Fd h fdback ga mar uch ha h quao afd by d d BF d I BF *,,,,. d d d d If quao * ru, h hr a ozro vcor v uch ha Equvally, I BF v # d BF v d v ay ha v a gvcor of h clod-loop ym mar h drd clod-loop gvalu d. BF aocad wh 94

101 Equao # ca b rwr a v di B Fv m. + h ag, boh v ad F ar ukow. Df h ukow vcor m w a w v Fv. h h problm ca b olvd by fdg h of vcor ha afy quao +, amly, I B d m w. w pobly mor ha o h quval o compug h ull pac of I d B. If a uffc umbr of larly dpd vcor foud, h a fdback ga mar F ca b obad. 95

102 No ha h fr lm of w corrpod o h lm of h vcor v ad h rmag m lm of w corrpod o h lm of h vcor Fv. Now, h ull pac of I d B ha mamal dmo m, amly, dcrbd by a of m larly dpd colum of a mar U d wh dmo m m, ohr word, U w w w. d m L w j v j zj, h v v vm V d U d,,,, z z z Z m d, whr zj Fv j. 96

103 Hc, F ca b foud by olvg h mar quao F V V V Z Z Z d d d d d Sc m, h ym of ovrdrmd ad cao b olvd drcly. Howvr, f h ym, B complly corollabl, a of larly dpd colum, o for ach d, ca b foud ach of h marc V d V d V d ad Z Z Z d d d h lcd colum from h lf-had d V V V mar G ad h lcd colum from h rgh-had d Z Z Z d d d d d d, rpcvly. L b h b h mar H, h FG H, or F HG a fdback ga mar ha rul h drd of clod-loop gvalu,,,. d d d d h F mar ju compud o uqu, c h colum of boh marc G ad H ar lcd arbrarly, wh h oly rqurm ha h of colum b larly dpd. hu, may oluo. 97

104 Eampl: L h dyamc of a couou-m LI ym b dcrbd by d d 3 u. h op-loop ym ha gvalu ad 3 ym op-loop uabl. L h of drd clod-loop gvalu b, 3, 5. Now, d d d d di B d 3. If d 3, d 3 d I B dcrbd by h mar ad h ull pac of U

105 If d 5, d 5 d I B dcrbd by ad h ull pac of U L u coruc G ad H by lcg h fr colum of U 3 ad U 5,.. G ad H Boh G ad H hav rak, whch ma ha h vr of G. hu, F HG ad BF 3, 5. 99

106 L u coruc G ad H by lcg h cod colum of U 3 ad U 5,.. G ad H Boh G ad H hav rak, whch ma ha h vr of G. hu, F HG ad BF 3, 5. hr ar may mor fdback ga marc F ha could b compud from lcg dffr colum of U 3 ad U 5 or lar combao of hm o coruc h marc G ad H. a coquc, hr ar vral mhodolog o lc marc G ad H o afy dffr dg crra. For ampl, Malab u a mhod whch lc a fdback ga mar F for a gv drd of gvalu dcrbd by,,, uch ha h clod-loop ym robu rm of rducg h d d d d vy problm aocad wh larg ga.

107 Robu Pol Placm for Mulpl-Ipu Sym m Gv ral marc, B,, B ad a ymmrc of drd clod-loop gvalu,,,, d d d d d, fd a ral mar gvalu of BF ar a v o prurbao a pobl. F m uch ha h L ad y,,,,, b h rgh ad lf gvcor of h clod-loop ym mar M BF aocad wh h gvalu d, amly, M, y M y. d d If M o-dfcv ha a full of larly dpd gvcor, h M dagoalzabl. Morovr, h vy of h gvalu d o prurbao h compo of, B, ad F dpd o h magud of h codo umbr c, whch dfd a c y, y

108 whr h vy of gvalu d, whch dfd a h co of h agl bw h rgh ad lf gvcor corrpodg o d. boud o h v of h gvalu gv by whr c k X X X ma, k X h codo umbr of h mar X of gvcor. Furhrmor, h codo umbr achv h mmum valu of for all,,,,, f ad oly f M a ormal mar,.. M M MM. I h ca h gvcor of M may b cald o gv a orhoormal ba for mar X prfcly codod wh k X., ad h h L u ow r-a h robu pol placm problm a follow: Gv ral marc m, B,, B ad a ymmrc of drd clod-loop gvalu d,,,, d d d d afy h quao, fd a ral mar F m ad a o-gular mar X ha BF X XD,

109 whr D dag,,, d d d robu of h gproblm opmzd. horm: Gv D dag d, d,, d whch a oluo o BF X XD f ad oly f whr B U U U m ad gv by U, uch ha om maur of h codog or ad X o-gular, h hr F Z m U X XD, wh U U U, ad Z a orhogoal mar wh mm o-gular mar. h F plcly F Z U XDX. 3

110 Proof: If B of full rak, h ach colum of B a lar combao of h colum of U ad Z o-gular. Now, BF X XD ca b rwr a BFX XD X. Po mulplyg boh d of h la quao by L u ow pr-mulply h la quao by h lf had d qual o BFX XD X X BF XDX. U,.. U BF U XDX. Z Z ZF U BF U U F F. X, yld h rgh had d qual o U XDX XDX U U XDX. U U XDX 4

111 Hc, ad or F Z U XDX ZF U XDX U XDX. Now, BF XDX mpl ha F f ad oly f XDX B U Im Im Im. Hc, ImXDX NullB ImU. Corollary: h gvcor of h clod-loop ym mar BF aocad wh drd gvalu d dmo m mu blog o h pac S Null U I k, whr k dm Null d B d di. of d 5

112 h robu pol placm problm rduc o h problm of lcg larly dpd vcor S,,,, BF X XD a wll-codod a pobl. uch ha h gproblm Wh h par B, complly corollabl,, ad h mulplcy of h drd gvalu d cao cd m, c h mamum umbr of dpd gvcor whch ca b cho o corrpod o d k d d qual o dm S m. horm: h fdback ga mar F ad h zro-pu u r or u r k a rpo or k of h clod-loop couou- or dcr-m ym d BF or k BF k wh al valu, afy h d qual F ma d k X, whr m B B B m m ad k X ma d k or k k X d ma. 6

113 Eampl: Codr aga h cod-ordr ym of h la ampl,.. d d 3 u, wh drd gvalu, 3, 5. d d d Malab mplm h mhodology o compu h fdback ga mar F ju dcrbd ad u h commad K=plac,B,p, whr p a vcor ha coa h drd gvalu,,, ad K F. d d d Ug h abov Malab commad wh p = [-3, -5], w g h fdback ga mar K 3 8 or F 3 8 ad BF 3, 5. 7

114 PPENDIX Sgular valu dcompoo SVD L H R m ad df M H H R. h M = M. L r b h oal umbr of pov gvalu of M, h w may arrag hm uch ha r r L p = m{m, }, h h { r > = r+ = = p } calld h gular valu of H, whr ad r = rakh. Eampl: L a rcagular mar H b gv by H 4 8

115 9 h Now, di M = gvalu M = {5, 5} ad h gular valu of H ar h quar roo of h gvalu of M,.., {5, 5}. Eampl: L H ow b dcrbd by h ad H H M 4 H 4 6 H H M d d M I

116 whch mpl ha h of gvalu of M {6, 5, } gular valu of H ar {4,5}, c m{m, } = m{, 3} =. lo, rakh =. horm: L H R m, h H = RSQ wh R R = RR = I m, Q Q = QQ = I, ad S R m wh h gular valu of H o ma dagoal ad uch ha Q H HQ = D = S S wh D a dagoal mar wh h quard gular valu of H o ma dagoal. Eampl: L H b gv by 4 H h h gvalu of M = H H {6, 5, } gv r o h ormalzd gvcor q~ ~ q 5 5 ~ q 3 5 5

117 hu, Q q ~ q~ q ~ S 4 4 HQ S S