Diseño de Sistemas de Control en El Espacio de Estados.

 

: Misb˘k Misb k` ˘ ` mb sistblgs mb n`ktr`h bk bh bspgni` Ngp½ Ng p½ķķtuh t uh` `

mb bstgm`s _grg bh misb˘ misbk` k˘ ` mb sistblgs mb n`ktr`h bk bh bspgni` mb bstgm`s bs kbnbsgri` mbtbrlikgr si bh sistblg g n`ktr`hgr n`ktr`hgr bs n`ktr`hgjhb. n`ktr`hgjhb. Gmini`kghlbktb Gmini`kghlbktb pgrg hg bstilg bstilgni ni½` `k ½ k mbh vbnt`r mb bstgm` bs kbnbsgri` n`ksimbrgr si bh sistblg bs `jsbrvgjhb. Rk sistblg mb n`ktr`h sb mb﬇kb n`l`   sbrv`   sbrv`   si hg sghimg bs misb˘kgmg pgrg sbeuir h` l½ggss nbrngkglbktb p`sijhb ukg sb˘kgh kgh mb rbcbrbknig mgmg. Nugkm` hg sb˘k kgh gh mb rbcbrbknig rbcbr bknig bs n`kstgktb bh sistblg sistblg mb n`ktr`h sb n`k`nb n`l` rbeuhgm`r  n`l`  rbeuhgm`r . N`ksimbrb N`ksim brb uk l`mbh` mb bspgni` bspgni` mb bstgm`s bstgm`s hikbgh mgm` p`r

x˙  )  bs ieugh g k. Hg lgtriz lgtriz  G  n`ktibkb h`s vgh`rbs pr`pi`s mbh sistblg bk hgz` gjibrt`. Hg bnugni `k `½ k ngrgntbr½ ngrgntbr½ķsting ķsting mbh sistblg bk hgz` gjibrt` sb pubmb mbtbrlikgr n`l`

|sI √ G| .. :.>

@jsbrv @jsbrvgm` gm`rb rbss mb bst bstgm` gm`

_grg hg gphingni`k `½ k mb n`ktr`hgm`rbs qub rbquibrgk hg rbghilbktgni `k `½ k mb vgrigjhbs mb bstgm` bs kbnbsgri` n`k`nbr bh vbnt`r mb bstgm`   x   n`lphbtglbktb. ^ik bljgre`, usughlbktb bst` k` bs p`sijhb bk h`s sistblgs rbghbs mbjim` g qub sb kbnbsitg uk sbks`r pgrg ngmg ukg mb hgs vgrigjhbs mb bstgm`. Bs p`r bst` qub sureb hg kbnbsimgm mb bstilgr hgs vgrigjhbs mb bstgm`. Rk `jsbrvgm`r `jsbrvgm`r mb bstgm` bs uk sistblg mik½glin` glin` qub pbrlitb bstilgr hgs vgrigjhbs mb bstgm`. ^i bh `jsbrvgm`r bstilg hgs  k  vgrigjhbs mb bstgm` sb n`k`nb n`l` `jsbrvgm`r mb bstgm` mb `rmbk n`lphbt`, `   `jsbrvgm`r n`lphbt` . ^i bh `jsbrvgm`r bstilg gheukgs mb hgs   k  vgrigjhbs mb bstgm` sb hhglg  `jsbrvgm`r mb bstgm` mb `rmbk rbmunim` . ^i bh `jsbrvgm`r mb `rmbk rbmunim` bs bh l½ l½ķkil` ķkil` p`sijhb sb n`k`nb n`l` `jsbrvgm` n`l`  `jsbrvgm`rr mb `rmbk l½ķķkil`  kil` . Rk `jsbrvgm`r mb bstgm` bstilg  bstgm`  bstilg   hgs vgrigjhbs vgrigjhbs mb bstgm` n`k jgsb bk hgs lbmini`kb lbmini`kbss mbh vbnt` vbnt`rr mb sghimg y y bh vbnt`r vbnt`r mb n`ktr`h n`ktr`h  u . H`s `jsbrvgm`rbs mb bstgm` pubmbk sbr misb kgm`s k˘ gm`s si y s`h` si bh sistb sistblg lg bs bs `jsbrvgjhb   `jsbrvgjhb , bs mbnir si bh rgke` mb hg lgtriz mb `jsbrvgjihimgm @j  bs ieugh g   k. ^b mbk`tg p`r   x˘  bh vbnt`r mb bstgm` bstilgm` p`r bh `jsbrvgm`r `jsbrvgm`r,, y sb utihizg bstb vbnt`r pgrg hg rbghilbktgni½ rbghilbktgni½` `k k hikbgh mb hgs bnugni`kbs mb bstgm`, hg bnugni½`k bnugni `½ k (:.2 ( :.2)) sb n`kvibrtb bk

u < √F ˘  x

 

(:.9)

Rk bsqublg sbknihh` pgrg hg bstilgni½`k `k mbh vbnt`r mb bstgm`   x˘  bs utihizgkm` bh sistblg mik½gglin` lin` mbsnrit` p`r hg bnugni½ bnugni`k `½ k mb bstgm`

 ˘  x˙  ) rbprbsbktg bh `jsbrvgm`r mb bstgm`, y n`k   H   hg lgtriz mb egkgknig mbh `jsbrvgm`r mb milbksi`kbs   k Õ  p. K`tb qub bh `jsbrvgm`r tibkb n`l` bktrgmgs h`s vbnt`rbs  u  y  y , y n`l` sghimg  x˘ . Bh t½brlik` brlik`  y √ Nx˘  bs uk cgnt`r mb n`rrbnni½`k `k bktrb hg sghimg mbh bstilgm`r y hg rbgh. Bk hg ﬇eurg  :.> sb `jsbrvg bh `jsbrvgm`r n`lphbt` n`k bh cgnt`r mb n`rrbni½`k. `k. N`k bh ﬇k mb mbtbrlikgr   H   sb mbjb n`ksimbrgr hg mik½ mikgling g½ ling mbh brr`r bk hg bstilgni bstilgni½`k. `½ k. ^i sb mb﬇kb bh brr`r bk hg bstilgni`k `½ k  b  n`l`

b )) sb tibkb y sb rbstgk hgs bnugni`kbs bnugni`kbs  (:.7  ( :.7)) y (:.7>

x˙   @jsbr @jsbrvgm` vgm`rbs rbs mb bst bstgm`  gm` 

u ( t )

J

+

x ( t )

 

x ( t )

∦

+

y ( t )

 

N

G _HGK]G

H

J

++ +

x ( t )

x ( t )

∦

 

 

G

y ( t )

√ +

N

@J^BZSGM@Z MB @ZMBK @ZM BK N@L_HB N@L_HB]@ ]@

x ( t )

`k k bk hg bstil bstilgni½ gni½`k `k Cieurg :.>1  @jsbrvgm`r n`k n`rrbnni½`

 ˘  x (?), bs mbnir,  x˘  ↚ x. ^b mbjb tbkbr bk nubktg pgrg hg sbhbnni`k `½ k mb h`s vgh`rbs pr`pi`s mb hg lgtriz   (G √ HN)   qub bst`s mbjbk bstgr l½ lggs ½ s g hg izquibrmg bk bh phgk` n`lphba` qub h`s vgh`rbs pr`pi`s mb hg lgtriz  ( G √ JF). Mb bstg c`rlg, bh brr`r bk hg bstilgni½`k `k mbsgpgrbnb l½gs g s r½gpim` g pim` n`lpgrgm` n`k hg mik½ggling ling mbh sistblg. Bh n`lp`rtglibkt` mik½glin` glin` mbh brr`r pubmb sbr mbtbrlikgm` n`k hg sbhbnni½` `k k gmbnugmg mb h`s vgh`rbs mb hg egkgknig mbh `jsbrvgm`r  H .

:. :.>. >.7. 7.

Zbeu Zbeuhg hgni ni½`k `½ k n`k uk `jsbrvgm`r `jsbrvgm`r n`lphbt`

^i sb n`ljikg hg bnugni½`k `k mb bstgm` (:.7 (:.7)) n`k hg bnugni½`k `k (:.9 ( :.9))  sb `jtibkb

 ˘  x˙  ?)

x˙ : <  G:7x7 + G::x: + J: u

 

(:.>7)

½ k  (:.>? :.>?))  sb pubmb `jtbkbr hg bnugni`k `½ k mb sghimg mgmg p`r y mb hg bnugni` `k (

x˙ 7 √ G77x7 √ J7 u :)

m`kmb h`s t½brlik`s brlik`s   x˙ 7 √ G77 x7 √ J7 u  n`rrbsp`kmbk g ngktimgmbs n`k`nimgs. Hg bnugni½` `k k (:.>7) :.>7) sb pubmb rbbsnrijir n`l`

x˙ : <  G::x: + G:7x7 + J: u

 

(:.>>)

m`kmb h`s t½brlik`s brlik`s  G :7 x7 + J: u  s`k ngktimgmbs n`k`nimgs. Hg bnugni½`k `k (:.>7) :.>7) mbsnrijb hg mik½ggling ling mb nglji` mb hgs vgrigjhbs qub k` pubmbk sbr lbmimgs mirbntglbktb. _grg bstgs vgrigjhbs sb pubmb

75

 

˘˘   mb sistblgs : Misb  Misb k` sistblgs mb n`ktr`h bk bh bspgn bspgni` i` mb bstgm`s bstgm`s c`rluhgr uk `jsbrvgm`r n`l`

 ˘  x˙ : <  G:: ˘  x: + G:7x7 + J: u + H y √ N ˘  x:





 

(:.>=)

m`kmb  y  s`k hgs vgrigjhbs vgrigjhbs n`k`nimgs n`k`nimgs mb hg bnugni bnugni`k ( `½ k  (:.>: :.>:))  mgmgs p`r

y <  ˙x7 √ G77x7 √ J7 u 2)

bkt`knbss hg bnugni bnugni`k `½ k pgrg bh `jsbrvgm`r mb `rmbk rbmunim` n`l` y n`k  N 3)

Zbbsnrijibkm` hg bnugni` `k ( ½ k  (:.>3 :.>3)) sb tibkb

 ˘  x˙ : < ( G:: √ HG7: ) ˘  x: + G:7x7 + J: u + H (x˙ 7 √ G77 x7 √ J7 u)

 

(:.>5)

m`km m`kmbb h`s h`s tbrlik`s b½ rlik`s   G:7 x7  +  J : u  +  H (x˙ 7 √ G77 x7 √ J7 u)   s`k s`k n` n`k` k`ni nim`s m`s.. ^ik ^ik bl bljg jgre re`, `, bk hg ½ k (:.>5 :.>5)) sb tibkb bh t½ t brlik` b½ rlik`  ˙x7 . _grg bhilikgrh`, sb rbbsnrijb hg bnugni `k ( `½ k  (:.>5 :.>5)) n`l` bnugni½` bnugni `k

 ˘  x˙ : √ Hx˙ 7 < ( G:: √ HG7:) ˘  x: + (G:7 √ HG77) x7 + (J: √ HJ7 ) u

 

(:.>4)

bnugni` `k ( ½ k  (:.>4 sb hb sulg bh t½ tbrlik` b½ rlik` (G:: √ HG7: ) Hx7  g hg bnugni½ :.>4)) y sb rbbsnrijb n`l`

 ˘  x˙ : √ Hx˙ 7 < ( G:: √ HG7:) ( ˘  x: √ Hx7 )+V(G:7 √ HG77) + (G:: √ HG7: ) H\ x7 + (J: √ HJ7 ) u (:.>9) ½ k (:.>9 y si sb n`ksimbrg  ε˘  < ( ˘  x: √ Hx7 )  sb pubmb phgktbgr phgktbgr hg bnugni bnugni` `k :.>9)) n`l`

˙  < (G:: √ HG7: )  ˘   ˘  ε ε + V(G:7 √ HG77 ) + (G:: √ HG7: ) H\ x7 + (J: √ HJ7 ) u

 

(:.=?)

½ k (:.=? :.=?)) mbsnrijb hg mik½ mikgling g½ ling mbh `jsbrvgm`r mb `rmbk rbmunim`. Bk hg ﬇eurg :.2 ﬇eurg  :.2 sb  sb lubstrg Hg bnugni½ bnugni` `k ( bh bsqublg mbh `jsbrvgm`r mb `rmbk rbmunim` pgrg bh ngs` mb rbeuhgni½ rbeuhgni`k `½ k p`r rbghil rbghilbktgni bktgni½`k `½ k mb vgrigjhbs mb bstgm` mb gnubrm` gnubrm` g hg bnugni½ bnugni` `k ½ k (:.=? :.=?), ), y m`kmb bh vbnt`r mb bstgm`s bstilgm` x˘  sb n`kstruyb n`l`

 ˘  x <

 I ?

 ? I

   +   x7

 ˘  x:

 

(:.=7)

^i sb mb﬇kb bh brr`r n`l`

b )

]  ]  ]  y m`kmb  H  sb nghnuhg bk bh sistblg mugh  G:: H ieugh qub pgrg bh `jsbrvgm`r mb `rmbk n`lphbt`. √ G7:

74

 

½  ½  :.= L`mbh` mb _brturjgni  _brturjgni `k

u ( t )

 

J

++

x ( t )

 

∦

  x ( t )

N

 

y ( t )

G _HGK]G

√F 

@J^BZSGM@Z MB

J: √ HJ7

@ZMBK ZBMRNIM@

 ε x :

⎧? ⎭ ⎪I ⎢ ⎮ ⎣ x

+ +

 ε

∦

++ +

G:7 √ HΗ77

N7√

7

x7

G:: √ HG7:

+ +

H

⎪⎧ ?I ⎢⎭ ⎮ ⎣

x7

Cieurg :.21  @jsbrvgm`r mb `rmbk rbmunim`

:.=.. :.=

L`mbh` L`mbh` mb _b _brtu rturjg rjgni ni½`k `½ k

N`ksimbrb uk sistblg n`k pbrturjgni½ pbrturjgni` `k ½ k  ω  mgm` p`r

x˙   :.7>). ). Bk hg ﬇eurg :.5 ﬇eurg  :.5  sb lubstrg bh bsqublg n`lphbt` pgrg bh `jsbrvgm`r n`k pbrturjgni`kbs y rbeuhgni½`k `k p`r rbghilbktgn rbghilbktgnii`k `½ k hikbgh mb hgs vgrigj vgrigjhbs hbs mb bstgm` bstgm`..

:?

 

½  ½  :.= L`mbh` mb _brturjgni  _brturjgni `k



ω 

∦

Gω 

ω 

++

u ( t )

+

J

+

x ( t )

x ( t )

∦

 

y ( t )

N

G _HGK]G



ω 

+ √



 

ω 

 

∦



ω 

+

+

Hω 

 

Gω 

H u ( t )

+

+

++ +

J

x ( t )

x ( t )

∦

y ( t )

 

N

√ +

√F G @J^BZSGM@Z 

x ( t )

Cieurg :.51  ^istblg mb rbeuhgni `k `½ k n`k `jsbrvgm`r mb pbrturjgni`kbs

:.=.7. :.= .7.

_brtur _brturjgn jgnii`k `½ k n`kstgktb n`kstgktb

˙  < ½ k n` _grg _grg uk sis sistbl tblgg n`k pbrturj pbrturjgni gni½` `k n`ks kstg tgktb ktb sb ti tibk bkbb qu qubb Gω  < ? y  ω  tibkb bh sis sistb tblg lg gu gulb lbkt ktgm gm` `  < ?, sb tibkb mgm` p`r

  

  x˙  G J  < ˙ ? ? ω 

y <   N ?



   

 x

 x

ω 

+

 J ?

u

   

(:.2=) (:.22)

ω 

  :7

 

˘˘   mb sistblgs : Misb  Misb k` sistblgs mb n`ktr`h bk bh bspgn bspgni` i` mb bstgm`s bstgm`s ½ k (:.=4) y n`k hby mb n`ktr`h mb﬇kimg p`r hg bnugni ` `k :.=4) n`k   Fω   <   7, m`kmb bh `jsbrvgm`r n`lphbt` bst bst½g½ mb﬇kim` p`r

   ˘  x˙ ˙˘ ω 

  

 G J  < ? ?

        x˘  ˘ ω 

+

 J ?

u+

 H Hω 

(y √ Nx˘ )

 

(:.23)

y m`kmb bh `jsbrvgm`r pgrg hg pbrturjgni `k `½ k bst½ bstg½ mgm` p`r ˙˘   < x) ω  <  Hω  (y √ N ˘ 

::

 

(:.25)