Sa pac modl: linar: y or in om : Sa q : f, u Oupu q : y h, u u Du F Gu y H Ju whr: u: inpu y: oupu : a vcor,,, D ar con maric
Eampl " $ & ' " $ & 'u y " & * * * * [ ],, D
H D I " $ " & $ ' " & $ ' " & & $ ' ' " & $ ' " $ " & $ ' " & $ ' " $ " & $ ' " & $ ' " $ " & $ ' " $ " & $ '
Sa raniion, mari ponnial Scalr ca : a bu Homogniou : a oluion : a Mari ca : u Homogniou : oluion :, by linariy icalldh a raniion mari
Sa raniion mari: oluion : I i an nn mari L - I- -, or L I- - d d i invribl: - - I... n n n...
Eampl, u I I
I/O modl o a pac Infini many oluion, all quivaln. onrollr canonical form: d n d n y a n d n d n y a $ y b b b $ n d d y a y b n " " " " " a a a n & ' []u d n d u b n & & u ' $ ' d d u b u
I/O modl o a pac onrollr canonical form i no uniqu Thi i alo conrollr canonical form d n d n y a n d n d n y a d d y a y d b n n d u b d n d u b u a n a n a a " " " " $ y b n b n b b $ & ' []u & ' $ & u '
Eampl d d d y d 4 d y 4 d d y dy 5 y ydτ d d d y d y dy 5 y d d d : dr d r n4 a a a a b b b b " * $ $ * $ * $ * * 5 y " & " ' $ ' $ ' $ ' $ & ' ' u ' ' &
haracriic valu har. q of a ym i di- h polynomial di- i calld char. pol. h roo of char. q. ar char. valu hy ar alo h ign-valu of.g. i h char. pol. i h char. q. -, -, - ar char. valu or ignvalu u d d I
I u u u u L
can S I No can a : y, d d???
Soluion of a pac modl u y Du Rcall: X-XU I-XU XI- - UI- - L - I- - *u L - I- - y -τ uτd τ -τ uτd τ Du
u don u ho for hand calculaion u:xi- - UI- - L - {I- - U}{L - I- - } & YI- - UDUI- - y L - {I- - UDU}{L - I- - }.g. [ ] y u, If u uni p X
u u u [ ] DU X Y u u y No: T.F.D I- -
Eignvalu, ignvcor Givn a nn quar mari, nonzro vcor p i calld an ignvcor of if p p i.... p p i an ignvalu of Eampl:, L, p i an -vcor, & h -valu L, p i alo an -vcor, aoc. wih h - p p p p p
Eignvalu, ignvcor For a givn nn mari, if, p i an ign-pair, hn p p p-p Ip-p I-p p di- i a oluion o h char. q of : di- char. pol. of nn ha dgn ha n ign-valu..g., di--, -
If hn h corrponding p, p, will b linarly indpndn, i.., h mari [p p p n ] will b invribl. Thn: p p p p [p p ][p p ] [ p p ] [p p ] " " " $ & & & & &
Λ - Λdiag,, If ha n linarly indpndn Eignvcor, hn can b diagonalizd. No: No all quar maric can b diagonalizd.
Eampl work or : for, d d I I I
[ ] work : for I
, diag Λ, diag, diag
In Malab >> [ ; ; 4]; >> [,D]ig.68.77.5.68 -.77.5 -.4597 -..888 p p p D.679. 4.7
If do no hav n linarly indpndn ign-vcor om of h ignvalu ar idnical, hn can no b diagonalizd E.g. di- 4 56 5 4768-8 -6-6 4-6 by olving I- 9 9 4 9 6 4 4 4 4 4 4 8, p p Thr ar only wo linarly indpndn ign-vcor
>> [- 8-4 -4; 4-4 4 4; - -6 - ; 9-4 9-9] - 8-4 -4 4-4 4 4 - -6-9 -4 9-9 >> [,D]ig -.77..i. -.i. -..447 -.i.447.i -.447.77 -..i -. -.i -. -..8944.8944 -.8944 D -8. -6..i -6. -.i -6.
Should u: >>[,J]jordan J.75.65 8 4 -.75.75 6 9-8 -6-6 -6 a Jordan block aociad wih -6
Mor Malab Eampl >> ym''; >> [ ;- -]; >> d*y- an ^* I >> facoran an *
>> [,D]ig.77 -.447 -.77.8944 D - -, >> [,D]jordan - -.77.77 cal o.447.8944 cal o D - -
- - >> p an..78.5.498 >> pm an.64.5 -.465 -.97 >> ym'' >> pm* an [ -p-**p-, p--p-*] [ -*p-*p-*, *p-*-p-]
d d 4 4 } { } { 4 4 : : chck
Similariy ranformaion D D Du y Du y u u u Du y u,,,, w l If am ym a
Eampl [ ] [ ], l y u u y u y u diagonalizd dcoupld
Invarianc: d I d I d d d d I I d I d char.poly or char.q. no changd afr ranformaion char. valu & ignvalu no changd u ignvcor changd
] [ Tranfr funcion : H I D I D I D D I D I D H
onrollabiliy: " u y Du i complly conrollabl if any, conrol u which can bring o in fini im. Thm : c.c. iff rank[ n ] n or d[ n ] if i n or rank [ I - ] n
Eampl: or d rank linarly ind. rank ] [, n
In Malab: >> Scrb, >> rranks S [ ].g. rank If S i quar whn i n >> ds
" u y Du i complly obrrabl if Wihou loof Thm : c.o. or iff Obrvabiliy ovr a fini im can nabl u o drmin. rank n I - rank h knowldg of gnraliy, can u n, n, or d n u, y if i n
Eampl: [ ] [ ] c.o. d, n
In Malab: >> Vobv, >> rrankv rank mu n V Lookfor conrollabiliy Lookfor obrvabiliy Or if ingl oupu i V i quar, can u >> dv d mu b nonzro
[ ] ; oupu ; ;,, I c I b D u y y r d d a r y y y r y d d y d d d d y y y r y d dy d y d d y d - - L L φ φ
Rcall linar ranformaion: u u y Du y Du onrollabiliybing abl o u u o driv any a o origin in fini im n S [ ] ha rank n Obrvabiliybing abl o compur any from obrvd y fr ranformaion, ignvalu, char. poly, char. q, char. valu, T.F., pol, zro unchangd, bu ignvcor changd,,, D D V n O ha full rank n
onrollabiliy i invarian undr ranf. { } { } rank,rank min,rank rank min rank ] [ ] [ ] [ ] [ afr ] [ : bfor roof n n n n n
{ } { } onrollabiliy no changd rank rank rank rank,rank min,rank rank min rank rank u rank rank n
Obrvabiliy invarian undr ranf. afr, : bfor roof O O O O O O
{ } { } Obrvabiliy no changd rank rank,rank rank min rank rank rank,rank rank min rank O O O O O O O O
Sa Fdback u Givn y Du h law : u k r i calld a a fdback conrol law D r u - K y fdback from a o conrol u
clod - loop a pacquaion u k r k r k r y Du only h Mari changd o k ignvalu/char. valu changd by a fdback o ho of k
Thm : If h ym i complly conrollabl hn a fdback can chang h char.valu or ignvalu o any arbirary locaion. Th convr i alo ru. i.. rank n ignvalu of k can b changd o any by choic of k
In Malab: Givn,,,D ompu crb, hck rank If i i n, hn Slc any n ignvalumu b in compl conjuga pair v[ ; ; ; ; n ] 4ompu: Kplac,,v k will hav ignvalu a
Thm: onrollabiliy i unchangd afr a fdback. u obrvabiliy may chang