Control Systems (Lecture note #6)

Transcription

1 6.5 Corol Sysms (Lcur o #6 Las Tm: Lar algbra rw Lar algbrac quaos soluos Paramrzao of all soluos Smlary rasformao: compao form Egalus ad gcors dagoal form bg pcur: o brach of h cours Vcor spacs marcs lgbrac quaos Egalus Egcors Dagoal form Caocal form Modlg (BCD Soluos o : x x Bu; y Cx Du Marx fucos such as Thr ar mor brachs maly drd from lar algbra. Rw: sysm of quaos: x = y a a a m a a a m m a a a m x x x x y y y y m m L h h colum of b a =[a a a ] h x x x x a a a a x a x. a x Th xsc of soluo dpds o h rlaoshp bw ( ad ([ y]

2 Summary: f ( ([ : y] ( y R( h h quaos ar coss ad hr s o soluo f ( = ([ : y] h a las o soluo f ( = ([ : y] < ( ( > h hr ar f umbr of soluos f ( = ([ : y] = ( ( = h hr s a uqu soluo For a marx x = y has a uqu soluo y R m ff - xss or Th ras of marcs play a mpora rol. Egalus gcors ad dagoal form scalar s calld a galu of C f a ozro x C such ha x = x ad x s h gcor assocad wh. Cas : ll galus ar dsc Thorm: h gcors {. } ar L. L Q=[ ] h - Q Q : : : 4

3 O h ohr had f hr xs a osgular Q ad a dagoal marx : : : such ha Q - Q = h s ar h galus of ad h colums of Q ar h gcors: Q - Q = Q=Q = Howr hr ar suaos whr hr xss o such Q o ma Q - Q a dagoal marx. Ths cas wll b cord oday. 5 Today: W ar gog o sudy Gralzd gcors Jorda form Polyomal fucos of a squar marx Mor gral fucos such as (s- - Tools for solg a sa-spac quao x x Bu; y Cx Du G x( ad u( for wha s x( ad y(? Nx m w wll b abl o do hs. 6

4 4 7 Cas : Egalus wh Mulplcy > Wha may happ wh h mulplcy of a galu s grar ha? Th marx may o b dagoalzabl Exampl. ( 8 Wha s? Rcall = W xpc o ha { } L { } L Howr from ( -= ( -=-=. Wha dos hs ma? Th ull spac of - has dmso ; Thr dos xs L { } s ( - = ( - = f w a = { } ar o L ad cao b usd as a bass

5 5 9 Ha o h somhg dffr for ad W sll choos as h soluo o (- = For suppos ha sasfs ~ Dffr from h prous Th (- (- = (- = for som d w ca jus choos = (- Th { } ar L (w jus accp hs. f w a Q=[ ] h Ā=Q - Q ca b dagoal. Bu wha dos loo l? W d o fd Ā such ha Q=QĀ. Obsr ha From ha W ] [ ] [ Q No dagoal bu clos

6 6 For hs parcular xampl how o g such ha ( Q Q Q Q ~ as xpcd Ths xampl jus show h complxy ha may ars wh w ha rpad galus. To hadl such a suao sysmacally w d o df h gralzd gcors.

7 Dfo. cor s a gralzd gcor of grad assocad wh f bu Do Wha s h w rprsao w.r {. }? [ ] = [ ]Ā : Jorda bloc : Exampl: Δ( Oly o L - ( - ( s = Ha o us Frs pc such ha (- = bu (- gralzd Nd (. ( 6-4 gcors. 9-6 ca b ayhg bu (-! Pc h ( Q Q 6 4 Q Q

8 alra approach: Δ( ( - ( Nd o fd such ha (- = ad (- = You ca also f frs h sol (- = o g. 6-4 (- = = From ( o uqu 9-6 / or or -/ / Q or or / Q Q for ay of h abo Q 5 Exampl: Fd Jorda form for s asy o s ha h galus of ar Dos h marx ha gralzd gcor for? L us chc h ully of Cao fd wo L gcors for Mus ha gralzd gcors for 6 8

9 Nd o fd such ha pproach : Fd frs h l should sasfy Pc Th For Bass for h ull spac: Bass for h ull spac: = 7 = Wha s? Basd o h propry of gralzd galu mus ha Dd g ryhg rgh? Chc f? 8 9

10 Nd o fd such ha pproach : Fd frs h sol for. should sasfy Bass for h ull spac: L. ds o sasfy or for ay 9 = Thorm. Th gralzd gcors assocad wh a parcular galu ar L Thorm. Th gralzd gcors assocad wh dffr galus ar L Th gcors ad gralzd gcors spa C good bass ~ s h Jorda Caocal Form L L L : : L L L L mp m (

11 marx wh rpad galus could sll b dagoalzabl ohr cas: For h sam galu may ha mor ha o Jorda blocs such as Exampl.. (. : ~ L gcors! s dagoalzabl wh rpad galus. :

12 summary w ha h followg cass: ll galus of ar dsc dagoalzabl Thr ar rpad galus.g wh mulplcy. f (- = - (- = hr xs L soluos o (- = ad hy ar all gcors. f hs s h cas for all rpad galus dagoalzabl f (- = - (- < hr xs gralzd gcors o dagoalzabl hr xs Jorda blocs Today: Lar algbra (coud Gralzd gcors Jorda form Polyomal fucos of a squar marx Expoal fuco of a squar marx Vcor spacs marcs lgbrac quaos Egalus Egcors Dagoal form Caocal form Modlg (BCD Soluos o : x x Bu; y Cx Du Marx fucos such as 4

13 Fucos of a Squar Marx Polyomals of a Squar Marx Exampl. Wha s???? gral suppos : C C = = = = rms = L f( b a polyomal.g. f( = Wha s f(? f( =

14 s hr a asr way o compu f(? Would h procss b asr for a dagoal or bloc dagoal marx? How o procd? QQ QQ QQ f( = Q Q Q Q QQ Q Q QQ Q Q 5Q Q 4Q Q 7QQ Q5 4 7 Q Q f Q Q Q 7 4 Exampl (Coud Q f Q Q5 4 7 Q Q5 4 7 Q ~ as xpcd 8 4

15 gral f ( α QQ Q Q Qf Q daags o us dagoal or Jorda caocal form? Q Q f f ( f f 9 Cayly Hamlo Thorm m m Δ( Δ( Thr s somhg spcal abou (. Frs cosdr a dagoalzabl. Δ( Q(Q Δ( Δ( Q Q Δ( Ths s ru f has Jorda blocs. Cayly-Hamlo Thorm: ( = 5

16 6 W oly d o cosdr a Jorda bloc. For xampl Q Δ( Δ( Δ( Q (Q Q Δ( For a Jorda bloc Ā (Ā = j j j h rm coas Π Δ( No Cocluso: (= Π Δ( Π Δ( L α. α α Δ( α. α α Δ( Ha mplcao: α. α α - - ca b xprssd as lar combao of -. ducly ca b xprssd as lar combao of hs rms for all gr Furhrmor all polyomals of ca b xprssd so. summary:

17 y polyomal of a squar marx ca b xprssd as a polyomal of h sam marx of dgr - f hr s a polyomal ( of dgr m < such ha ( = h ay polyomal ca b xprssd as a polyomal of dgr m- Th mmal polyomal ( of s h moc polyomal (wh hghs powr coffc = of las dgr such ha ( = Exampl: Moao for a gral problm f (. Fd f ( How o sol hs problm? ( W should b abl o rprs f( as 85 = + = g( ~ Much asr o compu Wha s?? How o oba hm? gral problm: Fd g( ha s qual o f( bu smplr o alua 4 7

18 Udr wha codos would f( = g(? Thorm. G C ad a polyomal f L h dsc galus of b =.m ach wh mulplcy ( m =. L g( β Th f(=g( ff f (l ( = g (l ( l = - = m l d f whr f f ( f( l β β ( l d Udr h abo codo h coffcs s ca b drmd 5 Dfo. {f (l ( l = - = m} ar calld h alus of f o h spcrum of y wo polyomals hag h sam alus o h spcrum of df h sam marx fuco 6 8

19 Exampl (coud f (. Fd f ( for ( ( f ( ( f ( Wha s a good g( = +? g ( ( = + = + g ( ( = + = + f ad g hag h sam alus o h spcrum of rqurs g ( ( =f ( ( + = 85 g ( ( =f ( ( + = = 85 - = - 85 g( = ( ( 85-7 g( = ( ( O way o compu f(: Form ( ad fd { } ad {f (l ( } Cosruc a ( - h ordr polyomal g( = s f ad g ha h sam alus o h spcrum of : for all f( = g(

20 Exampl: Compu for ohr words: G f(= fd f(. Frs (=(+ has o dsc galu = - wh mulplcy. L g(= + o h spcrum of ha f( g( ( β β ; 99 f '( g'( ( β β β β 99 g( 99 No: f( g( β β β β Exampl: Compu for f( g( ( β ββ ; f'( g'( ββ ; f "( g"( (- β has galu wh mulplcy. Cosdr f( = g(= + + ( - ( - β β - ; β.5.5 ( - g( (.5.5 ( 4

21 4 ( - (.5 (.5 g( ( - (.5 (.5 g( / ( ( - (.5 (.5 g( / ( g( f( 4 Exampl: Compu for

22 Gral Fucos of a Squar Marx Polyomals of a squar marx ar aurally dfd. How abou o-polyomal fucos? Suppos f( = s or /(s -. Wha s f(? Two dfos By mas of a polyomal g( hag h sam alus o h spcrum of By Taylor xpaso Ths wo ur ou o b qual. W wll ha a lo of dscussos o f(=. Th soluo of a LT sysm rls o hs fuco. 4 Dfo: G C. L h dsc galus of b =.m ach wh mulplcy ( m =. L f( b a gral fuco wh {f (l ( } wll dfd. Suppos ha g( s a polyomal sasfyg f (l ( = g (l ( l = - = m Th f( g(. Grally g s a polyomal of dgr -. Exampl: f(. Fd f( wh ( = = f ( ( = f ( ( = 4 44

23 Now l g( = + g ( ( = + = (=f ( ( g ( ( = + = (=f ( ( = - = - = - g( = ( - + (- + f( = g( = ( - + ( Sps o calcula f( g f( ad : Form ( ad fd { } ad f (l ( Cosruc a ( - h ordr polyomal g( such ha g (l ( = f (l ( for all ad l f( = g( Dfo. L f( = wh h radus of corgc. Th f( = f j < for all j. ca b show ha Dfos ad ar qual 46

24 Exampl. Fd for a dagoal ad for Jorda caocal form f ( : :! : : f (! : : : :! f ( : : :! : : : 47 Now suppos ha s a Jorda bloc. Fd f( ( ( = ( - 4 wh of mulplcy 4 f ( ( = f ( ( = f ( ( = f ( ( = g( = + ( - + ( - + ( - g ( ( = = (= f ( g ( ( = = (=f ( ( g ( ( = = ( =f ( g ( ( = 6 = (=f ( ( f f ( ( ( f ( f ( ( ( Dra wh rspc o o

25 5 49 = = = / = /6 g( = + ( - + ( - / + (- /6 f( = g( = + ( - + ( - / + ( - /6!!! Compos: - 5!!! For lowr ordr submarcs of For hghr ordr marcs you ca xd from h par

26 6 5 For marcs Jorda caocal form For a gral marx : QQ Qf (Q f (QQ f ( Q Q Th smlar rasformao mas hgs asr. 5 Exampl: Compu for pproach : hrough h dagoal form. 6 Q Q ( ( ( 6 Q Q 6 4 6

27 7 5 pproach : hrough h alus of f( = a h spcrum of. ( ( ( 4 6 L g(=a +b+c c b 4a c a( ( g( c c b( ( g( c b a ( a(- g( a a c b c /6 4 ( b /6 ( a 6 4 ( 6 ( 54 Proprs of ; Soluo o a couous-m sysm Du Cx y Bu; x x Nx Tm: Du[] Cx[] y[] Bu[]; [] ] x[ Soluo o h dscr-m sysm Equal sa quaos Dalg wh complx galus

28 Homwor s #6. Fd Jorda-form rprsaos Ā ad rasformao marx Q for h followg marcs: Cosdr h marcs Problm. Fd such ha. L f( b a polyomal. Suppos ha s a gcor of wh corrspodg galu show ha s also a gcor of f( wh corrspodg galu f(. No: Show h dald procdur Compu for h followg marcs: 4 ; ;