Dinámica y control de un sistema masa-resorte-amortiguador

UNIVERSIDAD UNIVERS IDAD AUTÓNOMA DE YUCATÁN YUCATÁN FACULT FACULTAD DE INGENIERÍA INGENIE RÍA QUÍMICA INGENIERÍA EN BIOTECNOLOGÍA

DINÁMICA Y CONTROL DE PROCESOS

DINÁMICA Y CONTROL DE UN SISTEMA MASA-RESORTE-AMORTIGUADOR  TRABAJO QUE PRESENTAN:

LUIS ALBERTO BARRALES ROJAS ORLANDO ENRIQUE CASANOVA CARVAJAL CARLO ALEJANDRO CAUICH CÓRDOVA TANA TANA HERNÁNDEZ BARRUETA MELISSA DIANELA MERCADO RUBIO

CURSO ESCOLAR  ENERO-JUNIO 20!

M"RIDA# YUCATÁN YUCATÁN A VIERNES VIERNES 2 DE JUNIO DE 20!

Contenido $ SISTEMA MASA-RESORTE-AMORTIGUADOR$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$% 2$ FUNCIÓN DE TRANSFERENCIA DEL PROCESO G p(s)$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$& $$$$$$$$$$$$$$$$& '$ SISTEMA MASA-RESORTE-AMORTIGUADOR EN LAZO ABIERTO$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$& '$$ ESTABILIDAD DEL SISTEMA EN LAZO ABIERTO$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$( '$$$ DETERMINACIÓN DE POLOS Y CEROS DEL SISTEMA EN LAZO ABIERTO$$$$ ABIERTO$$$$$$$ $$$(( '$$2$ RESPUESTA TEMPORAL DEL SISTEMA EN LAZO ABIERTO$$$$$$$$$$ ABIERTO$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$ $$$$$$$$$$ $$$$$$$$ $$$(( %$ SISTEMA MASA-RESORTE-AMORTIGUADOR EN LAZO CERRADO$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$) %$$ SENSOR$$$$$$$$$$$$ SENSOR$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ) %$2$ CIRCUITO DE RETROALIMENTACIÓN H*+,$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ %$'$ DIAGRAMA DE POLOS POLOS Y CEROS DEL SISTEMA EN LAZO LAZO CERRADO$$$$$$$$$$$$$$$ CERRADO$$$$$$$$$$ $$$$$$$$$$ $$$$$$ $ %$%$ ESTABILIDAD DEL SISTEMA EN LAZO CERRADO$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ %$'$$ %$'$$ ROUTH-HUR ROUTH-HUR.IT .ITZ$$$$$$$$$$ Z$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$2 $$$$$$$$$$$$$$$$$$$$2 %$'$2$ LUGAR GEOM"TRICO DE LAS RAÍCES$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$2 %$'$'$ RESPUESTA TEMPORAL DEL SISTEMA EN LAZO CERRADO$$$$$$$$$$$$$$$$$$$$$$$$$ CERRADO$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$ $$$$' ' !$ CONTROLADO CONTROLADORES$$$$$ RES$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$ % !$$ SINTONIZAC SINTONIZACIÓN IÓN DE CONTROLAD CONTROLADORES$$$ ORES$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ % !$$$ M"TODO DE ZIEGEL-NICHOLS EN LAZO ABIERTO$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$% !$2$ CONTROL CONTROL P Y PID$$$$$$$$$$$$$$$$$$ PID$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$! ! &$ APLICACIÓN DE UN SISTEMA MASA-RESORTE-AMORTIGUADOR: EL SISMÓMETRO $$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$& & ($ AN ANE/OS$$ E/OS$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$) $$$$$$$$$$$$$$$) ($$ MAT MATLAB$$$$$$$$$$$$$$$$$ LAB$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ) ($$$ DETERMINACIÓN DE POLOS Y CEROS DEL SISTEMA EN LAZO ABIERTO$$$ ABIERTO$$$$$) $$) ($$2$ RESPUESTA TEMPORAL DEL SISTEMA EN LAZO ABIERTO$$$$$$$$$$ ABIERTO$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$ $$$$$$$$$ $$$$$$$) $$$) ($$'$ DETERMINACIÓN DE POLOS Y CEROS DEL SISTEMA EN LAZO CERRADO$$$) CERRADO$$ $) ($$%$ LUGAR GEOM"TRICO DE LAS RAICES DEL SISTEMA EN LAZO CERRADO$ $

($$!$ ($$!$ SINTO SINTONIZ NIZACI ACIÓN ÓN POR EL M"TOD M"TODO O DE ZIEGE ZIEGEL-N L-NICH ICHOL OLS S O CURV CURVA DE RESPUEST RESPUESTA$ A$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$ $$$$$$  ($2$ SIMULIN$$$ SIMULIN$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 20 ($2$$ RESPUESTA TEMPORAL DEL SISTEMA EN LAZO CERRADO$$$$$$$$$$$$$$$$$$$$$$$$$ CERRADO$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$ $$$$20 20 ($2$2$ CURVAS RESPUESTAS$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$20 ($2$2$ ($2$2$ CONTROLA CONTROLADORES DORES P Y PID$$$$$$$$$$$$$$$$$$$ PID$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 20 )$ REFERENCIA REFERENCIAS$$$$$$$$$$ S$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$20 $$$$$$$$$$$$$$$$$$$$$$$$$$20

1. SISTEMA MASA-RESORTE-AMORTIGUADOR  E1 3 45+16 657879 + +6;<=75> 37 <=1>?=@7 < ;1 +=+6?7 ?7+7-5+56-7?56=;7<5 ; + ;1 6=4 < +=+6?7 ?@>1=@ +=?43# 3 @;73 + 5= 45 37 S;1<7 L < N61$ U1 ?6< +=+6?>6=@ 4757 8615 @;7@=1+ <3 +=+6?7 + 3 +=;=16: $ S <=11 4+=@=1+ @1 +16=<+ <=5@@=173+ 4757 @7<7 ?7+7 <3 +=+6?7$ 2$ S <=8;97 ;1 <=757?7 < @;54 3=85 4757 @7<7 ;17 < 37+ ?7+7+# 45+71< 37+ ;57+ ; 7@671 +85 337+ 1 65?=1+ < 4+=@=1+ < ?7+7 U1 +=+6?7 ?7+7-5+56-7?56=;7<5 +6> @?4;+6 45 ;17 ?7+7 37 @;73 +6> ;1=<7 7 ;1 5+56  ;1 7?56=;7<5# +6 36=? 6=1 3 @6 < 5<;@=5 37 3@=<7< < 37 ?7+7 @;71< 3 +=+6?7 + 1@;1657 =8571< *545+167@=K1 < 37+ @757@65+6=@7+ < <=+=47@=K1  45<=<7 < 157 < ;17 +65;@6;57 +65;@6;57 573,$ A<?>+ A<?>+ < 37 ?7+7 + @1+=<57 @1+=<57 37 @1+6716 @1+6716 <3 5+56 5+56 k   B ; + 3 @=@=16 < 7?56=;7?=16$ E3 <=757?7 < @;54 3=85 <3 +=+6?7 < 45+167 1 37 Figura 1.

?7+7-5+56-7?56=;7<5  ;7<5  Figura 1. D=757?7 ?7+7-5+56-7?56=

E3 7?56=;7<5 95@ ;17 ;57 <41<=16 < 37 3@=<7< < 37 ?7+7 165 ?75 +7 37 3@=<7<# ?75 + 37 ;57 ; 95@$ P5 +=?43=@=<7< +;41<5?+ ; 37 ;57 +# 1 ?71=6;<# 4545@=173 7 37 574=<# + <@=5

| F  A|= B|v ( t )|

  <1< B > 0  + 37 @1+6716 <

 4545@=173=<7<$  4545@=173=<7<$ E161@+ 37 ;57 ; 95@ 3 7?56=;7<5 7?56=;7<5 +

 F  A =−Bv ( t ) =−B

 dy dt 

D1< 3 +=1 176= =1<=@7 ; 37 ;57 < 7?56=;7@=K1 7 1 +16=< @16575= 7 37 3@=<7< <3 @;54$ L7 ;57 6673 95@=<7 +85 37 ?7+7 +# 161@+:  F = F  R + F  A =−kx − B

 dy dt 

D1<  F  R  + 37 ;57 <3 5+56$ S= + 5-+@5=8 +6 @? ;17 @;7@=K1 <=51@=73# ;<7: 2

f  (  ( t )= )= M  d  y2 + B dy + k y dt  d t 

*,

L7 @;7@=K1 *, ?<37 3 ?=?=16 7?56=;7< < 37 ?7+7$ E1 +6 @7+# 37 ;57 < 7?56=;7@=K1 45<;@ ;17 45<=<7 < 157 1 3 +=+6?7$ E+ < 1675 ; 6<+ 3+ 475>?65+ <3 ?<3 *M# k  B, +1 @716=<7<+ 4+=6=7+$ L7 @;7@=K1 <=51@=73 861=<7 *, + ;17 E@;7@=K1 D=51@=73 O5<=175=7 *EDO, 4;+ @16=1 1=@7?16 <5=7<7+ <  y 5+4@6 <3 6=?4 7<?>+ + @37+==@7 @? < +;1< 5<1 4;+ 3 5<1 < 37 ?75 <5=7<7 + 2#  < 45=?5 57< <8=< 7 ; 37 461@=7 <3 <=51@=73 < ?75 57< + $ E1 37 Ta!a 1 + ?;+6571 37+ 75=783+# @=@=16+  ;1=<7<+ <3 S=+6?7 I16517@=173 4757 ;1 +=+6?7 ?7+7-7?56=;7<5-5+56# ?7+7-7?56=;7<5-5+56# 67?8=1 @1@=< @? +=+6?7 ?@>1=@ < 657+37@=K1$

?7+7-7?56=;7<5-5+56 56 Ta!a 1. S=+6?7 ?7+7-7?56=;7<5-5+

SÍMBOLOS  y  M  k   B  f  a v

VARIABLES Y COEFICIENTES D=+671@=7 M7+7 C1+6716 <3 5+56 C=@=16 < 7?56=;7?=16 F;57 A@357@=K1 V3@=<7<

SIU ?  + 2 + ?+2 ?+2 ?+

". FUNCI#N DE D E TRANSFERENCIA DEL $ROCESO G  p(s) A @16=1;7@=K1 + 45@< 7 <65?=175 37 6571+5?7<7 < L7 P37@ < 37 @;7@=K1 *, +;41=1< ; 37 @1<=@=K1 =1=@=73 Y  ( ( s )= 0 2

 F ( s )= M s Y ( s )+ BsY ( s )+ kY ( s )  F ( s )=Y  ( ( s ) [ M s + Bs + k ] 2

P5 36=? + 1@;1657 37 ;1@=K1 < 6571+51@=7 G p(s): Y ( s ) 1 G p ( s )= = 2  F ( s )  M s + Bs + k 

P757 +6 657879 + @1+=<575> ;1 +=+6?7 @1 ;17 ?7+7 <    37+ @1+6716+ <3 5+56  <3 7?56=;7<5 < !  ) ;1=<7<+# 5+4@6=7?16 *R7?5# 200,$ E+ <@=5# 37 ;1@=K1 < 6571+51@=7 G p(s) ;<7: G p ( s )=

%.

1 2

s + 8 s + 15

SIST SISTE EMA

MASA MASA-R -RE ESOR SORTE-AM E-AMOR ORT TIGUA IGUADO DOR R

EN

LA&O A&O

A'IERTO U17  ; + @1@ @? + @?4567 3 45@+# + <=8;97 3 <=757?7 <3 45@+ 1 37 78=56  < +6 ?< + <65?=17 @K? + @?4567 3 +=+6?7$ E1 37 Figura " + 45+167 3 <=757?7 <3 +=+6?7 ?7+7-5+56-7?56=;7<5 4757 ;1 +=+6?7 1 37 78=56$ P757 3 +=+6?7  454;+6 1 +6 657879# 37 1657<7 <3 +=+6?7  F(s) + 37 ;57 743=@7<7 7 37 ?7+7  37 +73=<7 Y(s) + 37 <=+671@=7 ; + <+4377 37 ?7+7 45 @6 < 37 ;57 743=@7<7$

G(s) F(s)

  

Y(s)

?7+7-5+56-7?56=;7<5 1 37 78=56 Figura ". D=757?7 < ;1 +=+6?7 ?7+7-5+56-7?56=;7<5

%.1. ESTA'ILIDAD DEL SISTEMA EN LA&O A'IERTO L7 +678=3=<7< <3 +=+6?7 1 37 78=56 + <65?=1K ?<=716 3 <=757?7 < 43+  @5+ < 37 ;1@=K1 G*+,$

%.1.1. DETERMINACI#N DE $OLOS Y CEROS DEL SISTEMA EN LA&O A'IERTO T73 @? + 45+167 1 37 Figura %( 3 +=+6?7 ?7+7-5+56-7?56=;7<5 ?7+7-5+56-7?56=;7<5 1 6=1 @5+$ @5+$ S=1 ?875 ?875 45+1 45+167 67 < <++ 43+ 43+ *-!  -', ;8=@7<+ ;8=@7<+ 7 37 =;=5< =;=5<77 <  jw ?=+?+ ; 3 @1=51 +678=3=<7< 73 +=+6?7$

Figura %$ P3+ <3 +=+6?7 ?7+7-5+56-7?56=;7<5 1 37 78=56

G ( s )=

1

s

2

+ 8 s + 15

%.1.". RES$UESTA TEM$ORAL DEL SISTEMA EN LA&O A'IERTO U17 @757@65+6=@7 < 3+ +=+6?7+ +6783+ + ;# @;71< + 743=@7 ;17 1657<7 @1@=<7# 37 5+4;+67 <8 615 3 ?=+? @?4567?=16$ C;71< 73 +=+6?7 ?7+7-5+56-7?56=;7<5 1 37 78=56 ; + 4541 1 +6 657879 + 3 =165<;@ ;17 1657<7 < 6=4 +@73K1  57?47# 37 5+4;+67 <3 +=+6?7 6=1 3 ?=+? @?4567?=16 4757 @7<7 @7+# @? + 8+57 1 37 Figura )$

Figura 4. Respuesta temporal de un sistema masa-resorte-amortiguador en lazo abierto

).

SIST SISTE EMA

MASA MASA-R -RE ESOR SORTE-AM E-AMOR ORT TIGUA IGUADO DOR R

EN

LA&O A&O

CERRADO A @16=1;7@=K1 + 4541 3 +=+6?7 @1 37 @557<# @1 37 45?=+7 < ; 3 +=+6?7 +;5=5>  456;587@=1+$ P757 4<5 78375 < ;1 +=+6?7 1 37 @557<# + 1@+75= <65?=175 3 6=4 < +1+5 ; + ;6=3=75>  +6 <41< < 37 75=783 5+4;+67 <3 +=+6?7$ E1 +6 @7+ 37 75=783 5+4;+67 Y(s)# ; + 3 <+4377?=16 < 37 ?7+7# 6=1 ;1=<7<+ < <=+671@=7 *?65+, ?=1657+ ; 37 75=78 75=783 3 < 1657< 1657<77  F(s)# ; + 37 ;57 743=@7<7 73 +=+6?7# 6=1 ;1=<7<+ < ;57 *N61+,$

).1. SENSOR  P757 4<5 56573=?1675 3 +=+6?7 @1 37 75=783 ; + 7 7 @165375# Y(s), + ;6=3=75> ;1 +1+5 ; ?=<7 3 <+4377?=16 @1 87+ 1 3 45=1@=4= < F757<7# < ;17 88=17

; + ?; <165 < ;1 @7?4 ?716=@  ; 157 ;1 3679# @? + ?;+657 1 37 Figura *$

Figura *. D=+ @1+65;@6= < ;1 +1+5 4757 ?<=5 <+4377?=16$ N# S: 43+ 156  +;5 < ;1 =?>1$ I: @55=16 3@65=@7$ V ;6: 3679 < +73=<7 < 37 88=17$ Y(t): <+4377?=16 <3 +=+6?7

E3 3679 < +73=<7 *V ;6, <3 +1+5 + 4545@=173 7 3+ +=;=16+ 7@65+: $ 2$ '$ %$

L7 =16 =161+ 1+=<7 =<7<< <3 <3 @7?4 @7?4 ?71 ?716= 6=@ @ N? N ?5 5 < < +4= +4=57 57++ < 37 37 88= 88=17 17 V3@=<7< 3@=<7< 7 37 ; 37 88=1 88=177 @5;7 @5;7 3 @7?4 @7?4 ?716=@ ?716=@ V75=7@=K1 5=7@=K1 < 37 <=+671 <=+671@=7 @=7 1657 1657 37 88=17 88=17  3 @7?4 @7?4 ?716= ?716=@ @

P757 3 +1+5 + ?7161<5>1 =9+ 3+ 45=?5+ 65+ 7@65+# < ?< ; 3 3679 ?<=< <41<7 1=@7?16 < 37 75=7@=K1 < 37 <=+671@=7 165 37 88=17  3 @7?4 ?716=@$

).". CIRCUITO DE RETROALIMENTA RETROALIMENTACI#N CI#N +, S <65?=1K ;# 4757 =1+ 45>@6=@+  H(s) 6=1 ;1 735 < 0 ;1=<7<+ +=?45 ;: $ E3 +1+ +1+5 5 1 15 5 ;1 ;177 +73= +73=<7 <7 <  V @;71 @;71< < 3 +=+6 +=+6? ?77 ?7+7 ?7+7-5 -5+ +5 56 6-7? -7?5 56= 6=; ;7< 7<5  5  73@71@ +; ?>=?7 317@=K1 2$ L7 +7 +733 < < 5 55 51 1@= @=77 F(s) 75 1 ;1 571 < 0 7 0 ;1=<7<+

E3 +=+6?7 ?7+7-5+56-7?56=;7<5 @1 56573=?167@=K1 1 ;1=675=7 + ?;+657 1 37

Figura /$ E1 <1< G(s) 545+167 37 ;1@=K1 < 6571+51@=7 <3 +=+6?7# + <@=5# 37 ;1@=K1 < 6571+51@=7 <3 45@+ G p(s) ?;36=43=@7< 45 37 7171@=7  K  G(s)  

F(s)

K 

Y(s)

+ -

10

?7+7-5+56-7?56=;7<5 @1 56573=?167@=K1 56573=?167@=K1 1 ;1=675=7 Figura /. S=+6?7 ?7+7-5+56-7?56=;7<5

L7 ;1@=K1 < 6571+51@=7 1 37 @557< <3 +=+6?7 7165=5# ;<7 < 37 5?7:

( )=

T  s

K  2

s

+ 8 s + 15 + KH  ( s )

S;+6=6;1<  H(s)  45 3 735 =97< < 0 ;1=<7<+# 37 ;1@=K1 < 6571+51@=7 1 37 @557< ;<7 @? + =1<=@7 7 @16=1;7@=K1: T ( s )=

K  2

s

+ 8 s + 15 + 10 K 

).%. DIAGRAMA DE $OLOS Y CEROS DEL SISTEMA EN LA&O CERRADO

E1 37 37 Figura 0 + 8+57 ; 3 <=757?7 < 43+  @5+ <3 +=+6?7 1 37 @557< ;6=3=71< <+ 735+ < 7171@=7 *  %,$ E1 7?87+ 5>=@7+ + 8+571 43+ @?439+ @19;7<+ 3 ; =1<=@7 ; 3 +=+6?7 + ?75=173?16 +6783$ A3 8+575 8+575 3 475 < 57@+ @?4397 @?4397++ @19;7<7+ @19;7<7+ + 4;< <<;@=5 <<;@=5 ; 3 +=+6?7 + +;87? +;87?56= 56=;7 ;7<# <# + <@=5 <@=5## ; 3 +=+6? +=+6?77 45+1 45+1675 675>> 571< 571<++ +@=37 +@=37@= @=1+ 1+ 1 37 7+ 7+ 6571+=65=7$ A+ ?=+?# @;71< 3 735 <  7;?167 <  7 %# 37 4756 =?7=175=7 < 37 57 67?8=1 7;?167$

Figura 0. D=757?7 < 43+

 @5+ <3 +=+6?7 ?7+7-5+56-7?56=;7<5 ?7+7-5+56-7?56= ;7<5 1 37 @557<$ I;=5<7 $ D5@7 %

).). ESTA'ILIDAD DEL SISTEMA EN LA&O CERRADO A57 ; + @1@ 37 ;1@=K1 < 6571+51@=7 <3 +=+6?7 1 37 @557< T(s) + 45@< 7 <65?=175 37 +678=3=<7< <3 +=+6?7 ;6=3=71< 3 @5=65= < R;6-H;5=6  3 ?6< <3 3;75 ?65=@ < 37+ 57@+$

).%.1. ROUT+-+URIT& E+6 ?6< ;6=3=7 37 @;7@=K1 @757@65+6=@7 <3 +=+6?7

1+G

( s ) H ( s )=0  4757

<65?=175 37 +678=3=<7< <3 ?=+?$ E3 7553 4757 <65?=175 37 +678=3=<7< <3 +=+6?7 + 45+167 7 @16=1;7@=K1: 2

[

s  1 1 s 8 0 s b1

b1=

15

]

+ 10 K  0 0

( 8 ) ( 15 + 10  K )−( 1 ) ( 0 ) 8

$

b1=15 + 10 K 

+

15 10 K ≥ 0

 K ≥−1.5

C1 +6 + 8+57 ; 3 +=+6?7 + +6783 @;71< 37 7171@=7  K  +  + =;73  ?75  7 -$! ;1=<7<+$

).%.". LUGAR GEOM2TRICO DE LAS RA3CES E3 3;75 3;75  ?6 ?65=@ 5=@ < 37+ 57@+ 57@+ *LGR, *LGR, + ;17 557?=1 557?=167 67 ; +=5 4757 <65?=175 6<7+ 37+ 4+=83+ 57@+ < ;17 @;7@=K1 @757@65+6=@7 <   G*+,H*+,  0 @;71< 757 731 475>?65 *1 45=1@=4=# 37 7171@=7  < ;1 +=+6?7,  + ;6=3=7 4757 @1@5 3 @?4567?=16 6673 <3 +=+6?7 < 37 @557< 1 5=?1 6571+=65= *H51>1<# 200,$ T73 @? + 8+57 1 37 Figura 4( 3 LG =1=@=7 1 3+ 43+ *-!  -', ?=+?+ ; + ;8=@71 +85 3 9 573  65?=171 1 3 =1=1=6# 1 +6 @7+# <8=< 7 37 7;+1@=7 < @5+$

?7+7-5+56-7?56=;7<5 1 37 @557< Figura 4. L;75 ?65=@ < 37+ 57@+ <3 +=+6?7 ?7+7-5+56-7?56=;7<5

).%.%. RES$UESTA TEM$ORAL DEL SISTEMA EN LA&O CERRADO E1 37 Figura 5 + 8+57 37 5+4;+67 <3 +=+6?7 7 ;17 1657<7 6=4 +@73K1  57?47 4757 7?8+ @7++# 673 @? + <<;9 <3 @5=65= < +678=3=<7< < R;6H;5=6# 3 +=+6?7 45?71@ +6783 1 7?8+ 735+ < $

%



%



Figura 5. R+4;+67 <3 +=+6?7 ?7+7-5+56-7?56=;7<5# 1 37 @557<# 7 ;17 1657<7 @1@=<7$ A55=87: 1657<7 +@73K1$ A879: 1657<7 57?47$ R+4T?4573%$?<3

K=1

*. CONTROLADORES *.1. SINTONI&ACI#N DE CONTROLADORES C1 37 45?= 45?=+7 +7 < ; ;## @6= @6=7? 7?16 16# # 3 +=+6? +=+6?77 ?7+7-5 ?7+7-5+ +56 56-7? -7?56 56=; =;7< 7<55 +;5=5 +;5=5>>  456;587@=1+ <;5716 3 +67< +67@=175= + =?456716 = ?456716 7575 ;1 @16537<5$ @16537<5$ E1 +6 657879 + @?47575> 3 @16537<5 P545@=173 *P,  3 P545@=173-I16573-D5=76= *PID, +6 + <@=<=K <8=< 7 ; ;17 < 37+ 743=@7@=1+ <3 +=+6?7 ?7+7-5+567?56=;7<5# + 1 3+ +=+?K57+$ N + @1+=<5K ;6=3=75 ;1 @16537<5 PI 7 ; +6 1 6?7 1 @;167 37 3@=<7< 7 37 ; @5@ 3 555 *@;+6=K1 =?456716 @;71< + 65767 < ;1 +=+?,  3 @16537<5 PD 1 + @747 < ?<=5 3 555 *=161+=<7< <3 +=+?,$ A16+ < 4<5 7575 3 @16537<5 73 +=+6?7 7 ; <65?=175 3 735 < 3+  475>?65+ < @7<7 ;1$

*.1.1. M2TODO DE &IEGEL-NIC+OLS EN LA&O A'IERTO E1 +6 +6 657879 657879 + ;6=3= ;6=3=KK 3 ?6< ?6< < Z=3 Z=3-N= -N=@ @3+ 3+ 1 37 37 78=56 78=56 4757 4757  4+65=5?16 @73@;375 3+ 475>?65+ < 3+ @16537<5+ 4757 <+ 7171@=7+ <=+6=1 <=+6=167+ 67+ * *  %,$ %,$ E1 37 Figura Figura 16 + ?;+6571 37+ @;57+ < 5+4;+67 73 +=+6 +=+6? ?77 1 37 37 78= 78=56 56# # 7 4756 4756=5 =5 < 37 @;73 @;73 + 86; 86;= =5 51 1 3+ 3+ 73 735 5++ < T a=0.0625

  ?=;6 ?=;6+ +##

τ =0.5187   ?=1; ?=1;6 6++ 

 K = 0.04125

4757 4757 ;17 ;17

71 7171@= 71@=77 < $ Y 4757 4757 ;1 ;177 717 7171@ 1@=7 =7 < %# + 86; 86;= =5 51 1 3+ 3+ 73 735 5++ < T a=0.025

  ?=; ?=;6+#

τ =0.3325   ?=1 ?=1;6 ;6+ 

 K = 0.082

# 3+ @;73+ @;73+

+=5=51 4757 @73@;375 3+ 475>?65+ < 3+ @16537<5+ P  PID ; + ?;+6571 1 37 Ta!a "$

5+56-?7+7-7?56=;7<5 4757  *=;=5<7,  % *<5@7,$ Figura 16. C;57+ < 5+4;+67 <3 +=+6?7 5+56-?7+7-7?56=;7<5 C;57+R+4%$?<3

Ta!a ". P75>?65+ 4757 37 +=161=7@=K1 < @16537<5+ P  PID Cont r ol ador       1    =       K

P PI D

      4    =       K

Kp

Ti

Td

0. 1252

0. 0313

0. 0500

0. 0125

2 0 0. 8 90 7 241. 0688

P

1 6 2. 1 95 1

PI D

194. 6341

*.". CONTROL $ Y $ID A @16=1; @16=1;7@= 7@=K1# K1# 1 37 Figura 11  + ?;+657 3 <=757?7 < 83;+ <3 +=+6?7 ?7+75+56-7?56=;7<5 5+56-7?56=;7<5 @;71< + 3 757 ;1 @16537<5 < 6=4 P  PID:

Gc(s)

Gs(s)  

F(s)

Y(s)

K 

+ -

10

Gc(s)

Gs(s)  

F(s)

K 

Y(s)

+ -

10

Figura 11. D=757?7 < 83;+ < ;1 +=+6?7 ?7+7-5+56-7?56=;7<5 @1 ;1 @16537<5 P  PID @1 7171@=7 

A 4756=5 < +6+ 475>?65+ @73@;37<+ 1 M76378  =165<;@=<+ 1 S=?;3=1 + 15K 37 5+4;+67 <3 +=+6?7 @1 ;1 @16537<5 P  PID @1 7171@=7 <   %# 673 @? + 8+57 1 37 Figura 1"$

Figura 1". R+4;+67 <3

+=+6?7 7 ;17

1657<7

+@73K1

;1=675=   ̶

 P   ̶   ̶ 

PID$ A55=87:

$ A879: %$

Contro!adore781.9d!: Contro!adore7 81.9d!: Contro!adore78).9 Contro!adore78).9d! d!

E1 +6 @7+ + 8+57 ;# @;71< 37 7171@=7 7;?167# 3 6=?4 ; 3 +=+6?7 675<7 1 3375 7 37 7+ +6783 + 3=57?16 ?75$ A<?>+# 37 5@;1@=7 7+67 37 ; 337 3 +=+6?7 @;71< + ;6=3=7 ;1 @16537<5 P + ?15 ; 3 PID 45 +; 5+4;+67 + ?>+ 3167$ L7 <@=+=K1 < +@5 ;1 ; 65 @16537<5 <41<5> < 37 743=@7@=K1 4757 37 ; + 1@+=6$ E1 +6 @7+ ; + 4541 ;6=3=75+ 1 ;1 +=+?K?65# 3 @16537<5 PID + 3 =<73 7 ; 6=1 ;17 5+4;+67 ?>+ 5>4=<7  1 + 1@+75= ; 37 5@;1@=7 +8547+ @=56 735 3?=6$

/.

A$LICACI#N

DE

UN

SISTEMA

MASA-RESORTE-

AMORTIGUADOR; EL SISM#METRO E3 +=+?K57  +=+?K?65 + ;1 =1+6 =1+65; 5;? ?16 16 4757 4757 ?<= ?<=5  5 655?6+ 655?6+  4 4; ;+ + 6?83 6?835+ 5+  45@7<+ 45 3+ ?=?=16+ ?=?=16+ < 37+ 437@7+ 437@7+ 6@6K1=@7+$ E+6 747576# 1 +;+ =1=@=+# @1+=+67 1 ;1 41<;3 ; 45 +; ?7+7 45?71@7 =1?K=3 <8=< 7 37 =15@=7# ?=1657+ 6< 7 +; 735<<5 + ?7 <=@ 41<;3 33787 ;1 4;1K1 ; =87

+@5=8=1< +85 ;1 5<=33 < 4743 47;67< 1 6=?4# < ?< ; 73 ?475 37 =857@=K1 + 5=+65787 3 ?=?=16 1 3 4743# @1+6=6;1< +67 545+167@=K1 5>=@7 3 <1?=17< +=+?57?7$ L+ ?<51+ +=+?K?65+ < 871<7 71@7 @1+=+61 < ;17 4;7 ?7+7 < 45;87 @1=17<7  45 ;57+ 3@65=@7+$ C;71< 37 T=557 + ?;# 3@65K1=@7?16 + 65767 < ?71615 37 ?7+7 =97 7 657+ < 37 56573=?167@=K1 <3 @=5@;=6$ L7 @716=<7< < ;57 1@+75=7 4757 @1+;=5 +6 + 161@+ 5=+657<7$

Figura 1%. M@71=+? < 7@@=K1 < ;1 +=+?K?65 @1 87+ 1 ;1 +=+6?7 ?7+7-7?56=;7<5-5+56

0. ANE
0.1.". RES$UESTA TEM$ORAL DEL SISTEMA EN LA&O A'IERTO

0.1. 0. 1.%. %. DE DETER TERMI MINA NACI CI#N #N DE $O $OLOS LOS Y CE CERO ROS S DE DEL L SI SIST STEM EMA A EN LA LA&O &O CERRADO

0.1. 0. 1.). ). LU LUGA GAR R GE GEOM OM2T 2TRI RICO CO DE LA LAS S RA RAIC ICES ES DE DEL L SI SISTE STEMA MA EN LA LA&O &O CERRADO

0.1.*. SINTONI&ACI#N $OR EL M2TODO DE &IEGEL-NIC+OLS O CUR=A DE RES$UESTA

0.". SIMULIN8  0.".1. RES$UESTA TEM$ORAL DEL SISTEMA EN LA&O CERRADO R+4T?4573%$?<3

0.".". CUR= CUR=AS AS RES$UESTAS C;57+R+4%$?<3

0.".". CONTROLADORES $ Y $ID C16537<5+$?<3 C16537<5+%$?<3

4. REFERENCIAS H51>1<# R$ *200,$  Introducción  Introducción a os sist!"as sist!"as d! contro# contro#  M=@: P75+1$ R7?5#

A$

*200,$

$ist!" $ist!"a a

"asa%r "asa%r!s !sort ort!%a !%a"or "orti& ti&uad uador or##

O861=<

664:6541$83$@?;437<+&%&%(2%<@1653-45@6=@7-'2$4< 

<