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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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UNIVERSIDAD TECNICA DE ORUROFACULTAD NACIONAL DE INGENIERÍA
INGENIERÍA ELÉCTRICA – ELECTRÓNICA
Materia: ELT 2590 SISTEMAS DE CONTROL IA!i"iar: E#i"$ E"e%a Ri&era T'&arD'(e%te: I%)* Ra#ir' Fra%+ A"ie%,re Gar(-a
RESOLUCIÓN PRÁCTICA 2II/2015
. E%(e%tre "a/ %(i'%e/ ,e tra%/ere%(ia1 G ( s )=V o ( s )
V i, ara (a,a re, 3e /e
#e/tra e% "a 4)ra* M6t',' ,e %','/1 #6t',' ,e #a""a/7
SOLUCIÓN:
I%(i/' a7
8'r %','/
N',' a:
i2
i.
i
a
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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i1=i
2+i
3
i L1=i L2+i R
1 L
1∫ ( vi−v R ) dt = 1 L
2∫ ( v R−vC ) dt + 1 R v R m /m(derivando)
1
L1
( vi−v R )= 1
L2
( v R−vC )+ 1
R v́ R
1
L1
vi− 1
L1
v R= 1
L2
v R− 1
L2
vC + 1
R v́ R
1
L1
v i= 1
L2
v R+ 1
L1
v R+ 1
R v́ R−
1
L2
vC m/ m(aplicando T . L)
1
L1
V i =( 1 L2
+ 1
L1
+ 1
R s)V R− 1 L
2
V C
1
2V
i=( 13 + 12+ 11 s )V R−13V C
1
2V
i=( 13 + 12+ 11 s )V R−13V C
1
2V i=( 56 +s)V R−13 V C
V i=(53 +2 s )V R−23 V C (1)
Ta#i6% /ae#'/ 3e "a ('rrie%te e% %a #i/#a ra#a e/ i)a"1 e%t'%(e/:
i L2=iC
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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1
L2
∫ ( v R−vC ) dt =C ( v́C ) m/m(derivando )
1
L2
( v R−vC )=C ´(vC )m/m(aplicando T . L)
1
L2
V R=(C s2+ 1 L2 )V C
1
3V
R=(0.5s2+ 13 )V C
V R
=
(3
2s2+1
)V
C
(2)
A,e#;/ ,e" )r;4('1 te%e#'/ 3e:
vo= 1
L2
∫ ( v R−vC ) dt
v́o= 1
L2
( v R −vC )
s V o= 1
L2
V R− 1
L2
V C
s V o=1
3V R−
1
3V C (3)
Ree#"a+a%,' 27 e% .7
V i=(5
3 +2 s )(3
2 s
2
+1)V C −
2
3 V C
V i=(3 s3+ 52 s2+2 s+1)V C (4)
Ree#"a+a%,' 27 e% 7
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
4/23
s V o=1
3 (32 s2+1)V C −13 V C
s V o=
(
1
2s2
)V C
V o=1
2s V C
2
s V o=V C (5)
Ree#"a+a%,' 57 e%
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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8'r #a""a/
Ma""a .
v1=2 í1+1 (i1−i2 )
V 1=(2 s +1 ) I
1− I
2(1 )
Ma""a 2
0=3 í2+2∫ i2 dt +1(i2−i1 )
0=3 í2+2 i
2+1 (í2−í1 )
0=(3 s2+s+2 ) I 2−s I 1
I 1=(3 s2+s+2 ) I 2
s (2)
De" r'"e#a1 te%e#'/:
vo=3 í2
V o=3 s I 2 (3 )
I 2=V o
3 s
Ree#"a+a%,' 2 7 e% .7:
V 1=(2 s +1 )
(3 s2+s +2 ) I 2s
− I 2
i2i.
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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V 1=( (2 s+1 ) (3s
2+s+2 )s
−1) I 2
V 1=
(6 s
3+5 s2+4 s +2
s
)V o
3 s
G ( s )=V o
V 1= 3
s2
6 s3+5 s2+4 s+2
2 Ote%)a "a %(i=% ,e tra%/ere%(ia Eo(s)/ Ei(s) ,e" (ir(it' e"6(tri(' 3e
/e #e/tra* M6t','/ ,e "'/ %','/ $ #a""a/7
M6t','/ ,e "'/ %','/:
N',' a
i1=i
3+i
2
ei−ea R1
=ea−eo
R2+1
L∫ eadt
R2 L éi− R2 L éa= R1 L éa− R1 L éo+ R1 R2 ea
R2 Ls E1=(( R2+ R1) Ls+ R1 R2 ) Ea− R1 Ls Eo(1)
Le)':
i R2=iC =i3
i
i2
i.
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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ea−eo R2
=C éo
ea−eo ¿ R2C éo
Ea= Eo ( R2Cs+1 )(2)
Ree#"a+a%,' 27 e% .7:
R2 Ls E1=(( R2+ R1) Ls+ R1 R2 ) E o ( R2 Cs+1)− R1 Ls E o
R2 Ls E
1=[( ( R2+ R1 ) Ls + R1 R2) ( R2Cs+1 )− R1 Ls ] Eo
Eo
E1=
R2 Ls
( ( R2+ R1 ) Ls+ R1 R2 ) ( R2Cs+1 )− R1 Ls
Eo
E1=
R2 Ls
R2
CL ( R1+ R2 ) s2+( L R2+ R1 R2
2C )s + R1 R2
R1+ R
LC (¿ ¿2)s2+(C R1 R
2+ L)s + R
1
Eo
E i=
Ls¿
M6t',' ,e #a""a/
Ma""a .
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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ei= R1i1+ L (í1−í2)
Ei=( Ls+ R1 ) I 1− Ls I 2(1)
Ma""a 2
0= L (í2−í1 )+ R2 i2+ 1
C ∫ i2 dt
0= Lí2− L í1+ R2 í2+ 1
C i2
0= I 2( L s2+ R2 s+ 1C )− L s2 I 1
I 1=CLs
2+CR2 s+1
CL s2
I 2(2)
Ree#"a+a%,' 27 e% .7:
Ei=[ ( Ls+ R1 ) CL s2+CR
2s+1
CLs2
− Ls] I 2(3)
D'%,e:
eo= 1
C ∫i2 dt
Cs Eo= I 2
Ree#"a+a%,' e% 7:
Ei=[ ( Ls+ R1 )CL s2+CR2 s+1
CLs2
− Ls]Cs Eo Eo
E i=
Ls
C L2
s3+ LC R
2s2+ Ls+ R
1CL s
2+C R1 R
2s+ R
1−C L2 s3
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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R1+ R
LC (¿ ¿2)s2+(C R1 R
2+ L)s + R
1
Eo
E i=
Ls¿
E% "a 4)ra /e #e/tra % (ir(it' e"6(tri('* Ot6%)a/e % /i/te#a ,ee(a(i'%e/ -%te)r'>,iere%(ia"e/ /i#"t;%ea/ 3e rere/e%te% a" (ir(it'*M6t',' ,e %','/1 #6t',' #a""a/7
M6t',' %','/:
N',' a:
v−va R1
=C 1
d
dt (va−vb )(1)
N',' :
i1=i
2+i
3
C 1
d
dt (va−vb )=
vb−vd R3
+ 1
L1∫ vb−vc dt (2)
N',' (:
1 L1∫ vb−vc dt = v
c
R2(3 )
N',' ,:
vb−vd R3
=C 2
d
dt ( vd )(4)
,
(
a
i2
i
i.
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
10/23
M6t','/ ,e #a""a/:
Ma""a .:
v (t )= R1i1+
1
C ∫ i1 dt + L1
d
dt (i1−i2 )+ R2 (i1−i2) (1
)
Ma""a 2:
0= L1d
dt (i2−i1 )+ R2 (i2−i1 )+ R3 (i2−i1 )+
1
C 2
∫ i2 dt (2)
< %a re, ,e e%te e% T /e ti"i+a re(e%te#e%te e% /i/te#a/ ,e ('%tr'" ,e(a ('#' %a re, ,e 4"tr'* E% "a 4)ra /e #e/tra e" (ir(it' ,e %a re, ,ee%te e% T* De#6/tre/e 3e "a %(i=% ,e tra%/ere%(ia ,e "a re, e/
V o(s)V ¿(s)
= 1+2 R1 Cs+ R1 R2C
2s2
1+(2 R1+ R2 )Cs+ R1 R2 C 2
s2
Di?@e/e e" ,ia)ra#a ,e '"'/>(er'/ (a%,' R1=0.5, R2=1 $ C =0.5 *
M6t',' ,e %','/:
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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N',' a:
C ( v́¿− v́a )= 1
R1
( va )+C ( v́a− v́o )
R1C v́¿= R1C v́a+ R1C v́a+va− R1C v́o
R1
Cs V ¿=(2 R1Cs+1 )V a− R1Cs V o(1)
N',' :
C ( v́a− v́o )= 1
R1
(vo−v¿)
R1Cs V a=(Cs+1 )V o−V ¿
V a=
(Cs+1 )V o−V ¿ R
1Cs
(2)
Ree#"a+a%,' 27 e% .7:
R1
Cs V ¿=(2 R1Cs+1 ) (Cs+1) V o−V ¿
R1Cs − R
1Cs V o
De/e@a%,' $ /i#"i4(a%,' te%e#'/:
V o(s)V ¿(s)
= 1+2 R1 Cs+ R1 R2C
2s2
1+(2 R1+ R2 )Cs+ R1 R2 C 2
s2
Ree#"a+a%,' ,at'/:
V o(s)V
¿
(s)=
1+0.5s+0.125 s2
1+s+0.125 s2 =
( s+2−2 j ) (s+2+2 j )(s+6.8)(s+1.2)
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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Maa ,e '"'/ $ (er'/ ,e" /i/te#a:
5 Ote%)a "a %(i=% ,e tra%/ere%(ia X 1(s)/ U (s) $ X 2(s)/ U (s) ,e"
/i/te#a #e(;%i(' 3e /e #e/tra e% "a 4)ra*
S'"(i=%:
u−fk 1−fb1−fk 3=m1 ´
1
u−k 1 1−b1 ´
1−k 3 ( 1− 2 )=m1 ´ 1
>.*2>*B
>@2
@2
Re
I#
FF.
F.
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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U =(m1 s2+b1 s +k 1+k 3) X 1−k 3 X 2(1)
fk 3−fk 2−fb2=m2 ´
2
k 3 ( 1− 2)−k 2 2−b2 ´ 2=m2 ´ 2
k 3 X 1=(m2 s
2+b2 s +k 2+k 3) X 2(2)
De/e@a%,' 2 ,e 27:
X 2=
k 3
(m2 s2+b2s+k 2+k 3)
X 1
Ree#"a+a%,' e% .7:
U =(m1 s2+b1 s +k 1+k 3) X 1−k 3 k 3(m2 s
2+b2 s +k 2+k 3) X 1
U =[ (m1 s2+b1 s+k 1+k 3)− k 32
(m2 s2+b2 s+k 2+k 3 ) ] X 1
X 1
U =
(m2 s2+b2 s+k 2+k 3)
(m1 s2+b1 s+k 1+k 3) (m2 s
2+b2 s+k 2+k 3)−k 32
Ree#"a+a%,' 27 e% .7:
U =(m1 s2+b1s+k 1+k 3) (m2 s
2+b2 s+k 2+k 3 ) X 2k 3
−k 3 X 2
F2
F2F
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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U =[ (m1 s2+b1 s+k 1+k 3) (m2 s2+ b2 s+ k 2+k 3)
k 3−k 3 ] X 2
X 2
U =
k 3
(m1 s2+b1 s+k 1+k 3) (m2 s
2+b2 s+k 2+k 3 )−k 32
8ara e" /i/te#a ,e "a 4)ra1 e%(e%tre "a %(i=% ,e tra%/ere%(ia
G ( s )= X 1(s)/ ! (s)
S'"(i=%:
f −fv 3−fv2−fk 2=m2 ´
2
f −v 3 ´ 2−v 2 ( ´ 2− ´ 1 )−k 2( 2− 1 )=m2 ´ 2
! =(m2 s2+(v 3+v2)s+k 2) X 2−(v 2 s+k 2) X 1
! =(s2+2 s+1 ) X 2−( s+1 ) X 1(1)
F&
F
F&2
F2
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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fv 2+ fk 2−fk 1−fv1=m1 ´
1
v 2 ( ´ 2− ´ 1 )+k 2 ( 2− 1 )−k 1 1−v1 ´ 1=m1 ´ 1
0=(m1 s2+( v1+v2 ) s +k 2+k 1) X 1−(v2 s +k 2) X 2
0=(s2+3 s+2) X 1−( s+1 ) X 2
X 2=
s2+3 s +2
s+1 X
1(2)
Ree#"a+a%,' 27 e% .7:
! =(s2+2 s+1 ) s2
+3 s +2s+1 X 1−(s +1 ) X 1
X 1
! =
s+1( s2+2 s+1 ) ( s2+3 s+2 )− (s+1 )2
X 1
! =
1
s3+4 s2+4 s+1
8ara e" /i/te#a #e(;%i(' r'ta(i'%a" 3e /e #e/tra e% "a 4)ra e%(e%tre "a
%(i=% ,e tra%/ere%(ia G ( s )="
2( s)
T (s ) .
F&.
F.
F&2
F2
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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S'"(i=%:
# "́1=T −$
1"́1−$
2 ( "́1− "́2 )− % ( "1−"2 )
T =( s2+2 s+1 )"1−(s +1 )"2(1)
0=$2 ( "́1− "́2 )+ % ("1−"2)−$3 "́2
(s +1 )"1=(2 s+1 )"2
"1=
(2 s+1 )"2
s+1(2)
Ree#"a+a%,' 27 e% .7:
T =( s2+2 s+1 ) (2 s +1 )"
2
s+1−( s+1 ) "
2
2
.
T. T2"1
T2 T "2
T
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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"2
T =
s +1( s2+2 s +1) (2 s+1 )−(s+1)2
"2
T
= 1
2 s2
+2 s
B 8ara e" /i/te#a r'ta(i'%a" 3e /e #e/tra e% "a 4)ra1 e%(e%tre "a %(i=%
,e tra%/ere%(ia1 G ( s )=" L(s)/ T (s) *
S'"(i=%:
# "́2=T
2−T $10=T $1−T % 0=T % −T 30=T L−T $2
# "́2=T
2−$
1 ( "́2− "́3 ) $1 ( "́2− "́3 )= % ("3−"4 ) % ("3−"4 )=T 3 T L=$2"́ L
(# s2+$1 s)"2−$1 s "3=T 2 $1 s "2=( $1 s+ % )"3− % "4 T 3= % "3− % "4 T L=$2 s " L
A,e#;/ /ae#'/ 3e:
T
T 2= & 1
& 2= 10
100
T 3
T L=
" L
"4=
& 3
& 4=
50
10
T T2
"4"
3"
2
T. T. T T T T T2
T2
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
18/23
Le)' te%e#'/:
"4=0.2" L T L=0.2T 3T 2=10T
Re/'"&ie%,' e" /i/te#a ,e "a/ e(a(i'%e/ 3e e/t;% /ra$a,a/ e%('%tra#'/
"a %(i=% ,e tra%/ere%(ia*
"3=
5$2
s+0.2 %
% " L
"2=
($1 s+ % )(5 $2 s+0.2 % )−0.2 % 2
$1s%
" L
10T =
{(# s
2
+$1 s) [ ( $1 s+ % )( 5$2 s+0.2 % )−0.2 %
2
$1
s%
]−$1 s [
5$2
s+0.2 %
%
]}" L
Ree#"a+a%,' ,at'/:
s+¿(¿ (0.2s+0.2 )−0.2 ]−s2 (0.2s+0.2)
¿( s2+s )¿10T =¿
" L
T =
10s
0.2s4+0.4s3+0.2 s2
" L
T =
50
s3+2 s2+s
" L
T =
50
s (s +1 )2
9 Da,' e" /i/te#a ('#i%a,' tra/"a(i'%a" $ r'ta(i'%a" 3e /e #e/tra e% "a
4)ra1 e%(e%tre "a %(i=% ,e tra%/ere%(ia1 G ( s )= X (s)/T (s) .
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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S'"(i=%:
T −T % 1=# 1 "́
T =# 1
"́+ % 1("−"
1)
T =(# 1 s2+ %
1)"− % 1"1
T % 1−T ! −T ' 3=# 2 "́1
% 1 ("−"1 )−T ! − '3( "́1− "́2)=# 2 "́1
% 1
"+ '3
s "2=(# 2 s
2+ '3
s + % 1 )"1+ !r
T.
TF TD T.
TD
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
20/23
T '3=# 3 "́2
'3( "́
1− "́
2)=#
3"́2
'3
s "1=(# 3 s
2+ '3 s)"2
! − ! fv− ! k 2− ()= ( ́
! − ()= ( ´ +f v ´ + % 2
! =( ( s2+ f v s + % 2 ) X + ()
s
A,e#;/:
X ="1
r
Re/'"&ie%,' e" /i/te#a ,e e(a(i'%e/1 te%e#'/:
T =(# 1 s2+ % 1 )[ r (# 3 s2+ '3 s )[ X r ( # 2 s2+ '3 s + % 1 )+( ( s2+f v s+ % 2 ) Xr+ ()r ]−( ' 3 s)2 X
% 1r (# 3 s2+ '3 s)
]− % 1 X rT ={(# 1 s
2+ % 1 ) s [
r (# 3 s2+ '3 s) [(# 2 s
2+ '3 s+ % 1 )+( ( s2+ f v s+ % 2 )r
2
]−( '3 s)2 r
% 1r (# 3 s2+ '3 s) ]− % 1
1
r } X +(# 1 s
2+ % 1 % 1(# 3 s
2+
F
F& M) F2
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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.0 E" #'t'r ($a/ (ara(ter-/ti(a/ ,e ar ('%tra &e"'(i,a, /e #e/tra e% "a4)ra #e&e "a (ar)a 3e /e &e e% e" ,ia)ra#a* A")%'/ ,e "'/ e%)ra%a@e/
tie%e% i%er(ia* E%(e%tre "a %(i=% ,e tra%/ere%(ia1 G ( s )="2 (s )/ Ea (s ) .
De" )r;4(' te%e#'/:
%
Ra=
T
Ea=5
5=1
# 1
"́1=T m−f 2r1 # 2"́3=f 2 r2−T 2
# 1 s2 "1=T m− ! 2 r1 # 2 s2"3= ! 2 r2−T 2
C'#i%a%,' e(a(i'%e/ te%e#'/:
T 2=
T m−# 1 s2
"1
r1
r2−#
2s2
"3
T2
T#
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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A,e#;/ e% )e%era" /ae#'/ 3e:
r1
r2=
"2
"1=
& 1
& 2
Le)':
T 2=
T m−# 1 s2
"1
& 1
& 2−#
2s2
"3
T 2=
T m−s2
"1
1020−2s2"
3
T 2=2T m−2 s2
"1−2 s2
"3
#
(¿¿2+# 3) "́3=T 2−T 3¿
#
(¿¿2+# 3)s2
"3=T
2−T
3
¿
# 4
"́2=T
4−T '
# 4
"́2=T
4− ' "́
2
# 4
"́2=T
3
20
10− ' "́
2
T T2
T< TD
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8/19/2019 Modelamiento de sistemas eléctricos y mecanicos
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