Examen de septiembre

8QLYHUVLGDGGH1DYDUUD
1DIDUURDNR8QLEHUWVLWDWHD
(VFXHOD6XSHULRUGH,QJHQLHURV
,QJHQLDULHQ*RL0DLODNR(VNROD
$6,*1$785$*$,$
,QJHQLHUtDGH&RQWURO,
120%5(,=(1$
&8562.85762$
)(&+$'$7$
&8(67,21(6
/DILJXUDPXHVWUDODUHVSXHVWDIUHFXHQFLDOGHFLHUWDSODQWDGLDJUDPDGH
%RGHHQOD]RDELHUWR6HSUHWHQGHFRQWURODUODSODQWDFRQXQUHJXODGRU
SXUDPHQWHSURSRUFLRQDO6HSLGH
D'HWHUPLQDUXQDJDQDQFLDDGHFXDGDSDUDHVWHUHJXODGRU
E'LEXMDUGHIRUPDDSUR[LPDGDODUHVSXHVWDIUHFXHQFLDOGLDJUDPDGH%RGHHQ
OD]RFHUUDGRGHODSODQWDFRQODJDQDQFLDHOHJLGDGLEXMDUORVXSHUSXHVWRDOGH
ODSODQWDHQOD]RDELHUWR
F &DOFXODUGHIRUPDDSUR[LPDGDHODQFKRGHEDQGDGHOFRQWURODGRUHOHJLGR
Bode Diagram
20
0
Magnitude (dB)
-20
-40
-60
-80
-100
-120
-140
0
-45
-90
Phase (deg)
-135
-180
-225
-270
10
-1
10
0
10
Frequency (rad/sec)
1
10
2
5HVSXHVWD
•
•
•
Bode Diagram
20
0
-20
Magnitude (dB)
•
8QDSODQWDFRQWURODGDGHEHWHQHUDGHFXDGRVPiUJHQHVGHIDVH\GH
JDQDQFLD(QHVWHFDVRVHYDDHOHJLUODJDQDQFLDGHIRUPDTXHHOPDUJHQGH
IDVHVHDGHXQRVJUDGRV\VHFRPSUREDUDTXHHOPDUJHQGHJDQDQFLDHV
UD]RQDEOHSWRV
/DJDQDQFLDFRQODTXHODSODQWDWLHQHXQPDUJHQGHIDVHGHJUDGRVVH
REWLHQHYLHQGRODJDQDQFLDGHODSODQWDHQHOSXQWRGHFRUWHGHODFXUYDGHIDVHV
FRQ±JUDGRVG%/DJDQDQFLDGHOFRQWURODGRUHVHVWDPLVPDFRQVLJQR
FRQWUDULRHVGHFLUG%!. &RQHVWDJDQDQFLDHQHOFRQWURODGRUHO
PDUJHQGHJDQDQFLDHVGHXQRVG%(VWHYDORUHVUD]RQDEOHPD\RUGH
G%SWRV
(OGLDJUDPDGH%RGHGHODSODQWDFRQHVDJDQDQFLDHQOD]RDELHUWRVHUiHO
PLVPRSHURFRQODFXUYDGHJDQDQFLDVGHVSOD]DGDG%KDFLDDUULED
(OGLDJUDPDGH%RGHHQOD]RFHUUDGRVHUiVLPLODUDOGLDJUDPDGH%RGHDQWHULRU
GRQGHODVJDQDQFLDVVRQSHTXHxDVHQHVWHFDVR\FRPRHVKDELWXDOHQODDOWD
IUHFXHQFLD\SUiFWLFDPHQWHODXQLGDGSDUDEDMDVIUHFXHQFLDVSWRV
(ODQFKRGHEDQGDHVDSUR[LPDGDPHQWHHOSXQWRGRQGHHOGLDJUDPDGH%RGHHQ
OD]RDELHUWRFRUWDDG%(QHVWHFDVRUDGV(OYDORUUHDOHVGHUDGV
SWRV
-40
-60
-80
-100
-120
-140
0
-45
-90
Phase (deg)
•
-135
-180
-225
-270
10
-1
10
0
10
Frequency (rad/sec)
1
10
2
8QLYHUVLGDGGH1DYDUUD
1DIDUURDNR8QLEHUWVLWDWHD
(VFXHOD6XSHULRUGH,QJHQLHURV
,QJHQLDULHQ*RL0DLODNR(VNROD
$6,*1$785$*$,$
,QJHQLHUtDGH&RQWURO,
120%5(,=(1$
&8562.85762$
)(&+$'$7$
&8(67,21(6
(VFRJHUUD]RQDGDPHQWHXQDJDQDQFLDDGHFXDGDSDUDFDGDSODQWDXWLOL]DQGR
ODLQIRUPDFLyQGHOOXJDUGHODVUDtFHV6LWXDUGHIRUPDDSUR[LPDGDODSRVLFLyQ
GHORVSRORVGHODSODQWDSDUDHVDJDQDQFLD5HVSRQGHUHQHVWDPLVPDKRMD
1RWDHQHOOXJDUGHODVUDtFHVXQLGDGHVXQFHQWtPHWUR
Planta A)
* (V ) =
V+5
(V + 1)(V + 3)(V + 6)
Root Locus
6
4
Imag Axis
2
0
-2
-4
-6
-10
-5
0
Real Axis
5
Planta B)
* (V ) =
1
(V + 1)(V + 3)(V + 4)(V + 6)
Root Locus
6
4
2
0
-2
-4
-6
-10
-5
0
Real Axis
5HVSXHVWD
3ODQWD$
• (OFULWHULRJHQHUDOHVHOHJLUORVSRORVGHODSODQWDDOHMDGRVGHOHMHLPDJLQDULR\
FRQXQDPRUWLJXDPLHQWRUD]RQDEOHHVGHFLUHQWUH\XQR
• (VWDSODQWDWLHQHSRORV\XQFHUR6HJ~QDXPHQWDODJDQDQFLDODSRVLFLyQGHO
WHUFHUSRORHQOD]RFHUUDGRWLHQGHDDFHUFDUVHDODSRVLFLyQGHOFHURFRQORTXH
ODLQIOXHQFLDGHDPERVSUiFWLFDPHQWHVHFDQFHOD'HFXDOTXLHUIRUPDGDGRTXH
HOFHURHVWiPiVFHUFDTXHHOSRORHOVREUHLPSXOVRGHODSODQWDVHUiDOJRPD\RU
TXHVLQRVHFRQVLGHUDHOHIHFWRGHOSDUSRORFHUR
• /RVYDORUHVUD]RQDEOHVGHJDQDQFLDVRQSRUWDQWRDTXHOORVTXHSRQHQORVSRORV
HQOD]RFHUUDGRHQWUHODVSRVLFLRQHVTXHVHKDPDUFDGRHQODILJXUD
Root Locus
6
4
2
0
-2
-4
-6
-10
•
•
-8
-6
-4
-2
0
Real Axis
2
SWRV
/RVYDORUHVGHODJDQDFLDVRQVLORVSRORVHVWiQHQHOSXQWRGHGLVSHUVLyQ\
FRQORVSRORVFRPSOHMRVFRQMXJDGRVDJUDGRV(VGHFLUGHIRUPD
DSUR[LPDGD
1.33 < . < 20
SWRV
5HVSXHVWD
3ODQWD%
• (VWDSODQWDWLHQHSRORV6HJ~QDXPHQWDODJDQDQFLDORVSRORVOOHJDQDORV
SXQWRVGHGLVSHUVLyQ\WLHQGHQKDFLDODVDVtQWRWDV/RVSRORVVLWXDGRVDOD
L]TXLHUGDGHOSODQRFRPSOHMRWLHQHQSRFDLQIOXHQFLDHQODUHVSXHVWDWUDQVLWRULD
(QWRGRFDVRHOVLVWHPDVHUiDOJRPiVOHQWR\WHQGUiPHQRVVREUHLPSXOVRTXH
VLQRHVWXYLHUDQORVSRORV
• 3RUWDQWRHVQHFHVDULRTXHORVSRORVVLWXDGRVDODGHUHFKDHVWpQORPiV
DOHMDGRVTXHVHDSRVLEOHGHOHMHLPDJLQDULR(VWRVHFRQVLJXHHQHOSXQWRGH
GLVSHUVLyQ'HFXDOTXLHUIRUPDVLODJDQDQFLDHVDOJRPD\RUODFRPSRQHQWHUHDO
GHORVSRORVQRVHDFHUFDGHPDVLDGRDFHUR\HOWLHPSRGHHVWDEOHFLPLHQWR
SRVLEOHPHQWHGLVPLQX\DDOJR1RHVUD]RQDEOHXWLOL]DUDPRUWLJXDPLHQWRV
PD\RUHVGHSKLGHXQRVJUDGRV/RVSRORVVHVLW~DQSRUWDQWRFRPR
PDUFDODILJXUD
Root Locus
6
4
2
0
-2
-4
-6
-10
-8
-6
-4
-2
0
Real Axis
•
2
SWRV
(OYDORUGHODJDQDQFLDHVDSUR[SDUDHOSXQWRGHGLVSHUVLyQ\DSUR[
SDUDORVSRORVVLWXDGRVDJUDGRV
9 < . < 40
SWRV
8QLYHUVLGDGGH1DYDUUD
1DIDUURDNR8QLEHUWVLWDWHD
(VFXHOD6XSHULRUGH,QJHQLHURV
,QJHQLDULHQ*RL0DLODNR(VNROD
$6,*1$785$*$,$
,QJHQLHUtDGH&RQWURO,
120%5(,=(1$
&8562.85762$
)(&+$'$7$
&8(67,21(6
(QODILJXUDVHPXHVWUDODUHVSXHVWDWUDQVLWRULDGHGRVSODQWDVGLIHUHQWHV
FRQWURODGDVFRQXQ3,'HWHUPLQDUUD]RQDGDPHQWHTXHSDUiPHWURRSDUiPHWURV
GHO3,HVQHFHVDULRPRGLILFDUSDUDPHMRUDUODUHVSXHVWDWUDQVLWRULDGHFDGDXQD
GHODVSODQWDV
127$
• (QDOJXQRVFDVRVODUHVSXHVWDVHSXHGHPHMRUDUGHGLVWLQWDVIRUPDV
5D]RQDUODUHVSXHVWDTXHVHGp
• /DHFXDFLyQGHO3,XWLOL]DGRHV


1
X (W ) = .  E\ (W ) − \ (W ) + ∫ H(W )GW 
7 0


W
VS
L
H(W ) = \
VS
(W )− \ (W )
• La posición de referencia es 1.
3ODQWD$
.6
.4
.2
1
.8
.6
.4
.2
0
0
5
10
15
20
25
30
5
10
15
20
25
30
3ODQWD%
3
2.5
2
1.5
1
0.5
0
0
5HVSXHVWD
3ODQWD$
•
•
3ODQWD%
•
•
'LVPLQXLUODJDQDQFLD.HOVLVWHPDHVPX\RVFLODWRULRSWRV
'LVPLQXLUODFRQVWDQWHGHWLHPSRGHLQWHJUDFLyQ7LHOVLVWHPDWDUGD
PXFKRHQOOHJDUDOUHJLPHQSHUPDQHQWHSWRV
'LVPLQXLUODFRQVWDQWHGHWLHPSRGHLQWHJUDFLyQHOVLVWHPDWDUGD
PXFKRHQOOHJDUDOUHJLPHQSHUPDQHQWHSWRV
'LVPLQXLUHOYDORUGHESDUDHYLWDUHOSULPHUVREUHLPSXOVRSWRV
8QLYHUVLGDGGH1DYDUUD
1DIDUURDNR8QLEHUWVLWDWHD
(VFXHOD6XSHULRUGH,QJHQLHURV
,QJHQLDULHQ*RL0DLODNR(VNROD
$6,*1$785$*$,$
,QJHQLHUtDGH&RQWURO,
120%5(,=(1$
&8562.85762$
)(&+$'$7$
352%/(0$
•
•
•
•
8Q3//3KDVH/RFNHG/RRSHVXQFLUFXLWR
HOHFWUyQLFRXWLOL]DGRSDUDVLQFURQL]DUOD
IUHFXHQFLD\ODIDVHGHORVUHORMHVGHGRV
VLVWHPDVTXHGHVHDQLQWHUFDPELDULQIRUPDFLyQ
(VQHFHVDULRXWLOL]DUORHQVLVWHPDVGLJLWDOHVTXH
HQYtHQODRQGDPRGXODGD6XXWLOL]DFLyQSRU
WDQWRHVDPSOtVLPDWHOpIRQRVPyYLOHVDFFHVR
HQEDQGDDQFKDHWF
7pFQLFDPHQWHHVXQVLVWHPDQROLQHDODXQTXH
EDMRGHWHUPLQDGDVKLSyWHVLVVHSXHGH
FRQVLGHUDUTXHHVXQVLVWHPDOLQHDOFRPRHOTXH
PXHVWUDHOGLDJUDPDGHEORTXHV
(QSULPHUOXJDUVHHVWLPDODGLIHUHQFLDHQWUHHOiQJXORGHODUHIHUHQFLD
(QHOFiOFXORGHHVDGLIHUHQFLDVHLQWURGXFHWpUPLQRVGHDOWDV
IUHFXHQFLDVTXHVHILOWUDQFRQXQILOWURSDVDEDMD
/DVDOLGDGHOILOWURSDVDEDMDVHXWLOL]DFRPRHQWUDGDGHOFRQWURODGRUTXH
VHSLGHGLVHxDU
/DVDOLGDGHHVWHFRQWURODGRUHVODHQWUDGDGHXQRVFLODGRUFRQWURODGR
SRUYROWDMHHVGHFLUXQRVFLODGRUTXHJHQHUDXQDVHxDOGHIUHFXHQFLD
SURSRUFLRQDODOYROWDMHTXHVHDSOLFDVREUHpO$TXtVHKDVXSXHVWRXQD
JDQDQFLDXQLWDULD
(OiQJXORGHODVLQXVRLGHHVGLUHFWDPHQWHODLQWHJUDOGHODIUHFXHQFLD
•
6HSLGHGLVHxDUXQFRPSHQVDGRUTXHFXPSODODVVLJXLHQWHVSURSLHGDGHV
(UURUQXORDQWHXQDHQWUDGDUDPSDHVWRHVHTXLYDOHQWHDSRU
HMHPSORVLQFURQL]DUORVUHORMHVGHXQWHOpIRQRPyYLOHQXQFRFKH\OD
HVWDFLyQEDVH
6REUHLPSXOVRPHQRUGHO
$QFKRGHEDQGDGHXQRVUDGV1RWDORVYDORUHVKDELWXDOHVVRQ
HQWUH\+]SHURVHKDSXHVWRHVWHSDUDIDFLOLWDUORVFiOFXORV
900
Theta ref
s2 +42s+900
Filtro Pasa Baja
Controlador
1
Voltaje
1
Omega
1
s
Theta
VCO
(Oscilador controlado
por voltaje)
127$DXQTXHHOGLDJUDPDGH%RGHTXHVHGDSXHGHVHU~WLOQRHVHVWULFWDPHQWH
QHFHVDULRSDUDODUHVROXFLyQGHHVWHUREOHPD
5HVSXHVWD
•
•
'DGRTXHVHH[LJHTXHHOVLVWHPDWHQJDHUURUQXORDQWHXQDHQWUDGDUDPSDHO
VLVWHPDFRQWURODGRGHEHVHUGHWLSR,,'DGRTXHHOVLVWHPDVLQFRQWURODUHVGH
WLSR,HOILOWURSDVDEDMD\HO9&2HOUHJXODGRUGHEHDSRUWDURWURSRORHQHO
RULJHQSWRV
/DFRQGLFLyQGHVREUHLPSXOVRHQHOGRPLQLRGHODIUHFXHQFLDVHFRQYLHUWHHQ
PDUJHQGHIDVH
0S =H
•
•
•

− ζ

1−ζ 2


π


=H
−
π
tan ϕ
$SOLFDQGRHVWDIRUPXODUHVXOWDTXHHODPRUWLJXDPLHQWRHVDSUR[LPDGDPHQWH
\SRUWDQWRHOPDUJHQGHIDVHGHXQRVJUDGRV(VGHFLUHVQHFHVDULR
HQFRQWUDUXQFRQWURODGRUTXHDSRUWHXQDQFKRGHEDQGDGHUDGVFX\R
PDUJHQGHIDVHVHDGHJUDGRVSWRV
'DGRTXHKD\FRQGLFLRQHVGHUpJLPHQSHUPDQHQWHSRORHQHORULJHQ\GH
UpJLPHQWUDQVLWRULRPDUJHQGHIDVH\DQFKRGHEDQGDVHYDDSUREDUXQ
FRPSHQVDGRUGHDGHODQWRUHWUDVR
6HYDDDMXVWDUSULPHURHOFRPSHQVDGRUGH$GHODQWR\VHVLWXDUDHOSRORGHO
FRPSHQVDGRUGHUHWUDVRXQDGpFDGDSRUGHEDMRSDUDHYLWDUVXLQIOXHQFLDHQHO
UpJLPHQWUDQVLWRULR
(OFRPSHQVDGRUHOHJLGRWLHQHFRPRIXQFLyQGHWUDQVIHUHQFLD
1 + 7V 1 + 7L V
SWR
*F = * DGHODQWR * UHWUDVR = .
1 + α7V 7L V
• /DIDVHGHODSODQWDHQOD]RDELHUWRSDUDUDGVHVGH±JUDGRV(O
FRPSHQVDGRUGHEHDSRUWDUDOPHQRVJUDGRV(QHVWHFDVRVHDxDGLUiQ
JUDGRVGHELGRDOHIHFWRGHODSDUWHGHUHWDVR
sin Φ P =
1−α
1+α
3RUWDQWR
1−α
⇒ α = 0.218
1+α
(VWHDYDQFHVHGHEHDSRUWDUMXVWRDUDGV(OPi[LPRDYDQFHVHFRQVLJXHHQOD
IUHFXHQFLDTXHHVPHGLDJHRPpWULFDHQWUHHOSROR\HOFHURSRUWDQWR
1
7=
= 0.107
20 α
α7 = 0.107α = 0.0233 SWRV
El cero del compensador de retraso se puede colocar con una constante de tiempo 10 veces
mayor que el cero del compensador de adelanto, por tanto,
0.643 =
7 = 107 = 1.07 SWRV
L
'HEHDMXVWDUVHODJDQDQFLDSDUDTXHODIUHFXHQFLDGHFUXFHGHJDQDQFLDVHDGH
UDGV3RUWDQWR
900
1
1 + 7V 1 + 7 V
.
= 1 ⇒ . = 10.3 SWR
2
1 + α7V 7 V V + 42 V + 900 V =20
L
L
V
M
•
/DHFXDFLyQGHOFRQWURODGRUVHUiSRUWDQWR
*F = 10.3
•
1 + 0.107 V 1 + 1.07 V
1 + 0.107 V 1 + 1.07 V
= 9.62
1 + 0.0233 V 1.07 V
1 + 0.0233 V
V
Aunque no se pedía, se muestra el diagrama de Bode del PLL en lazo abierto, y con el
controlador en lazo abierto y en lazo cerrado.
Bode Diagram
40
Planta controlada en lazo cerrado
Planta controlada en lazo abierto
Planta sin controlar en lazo abierto
30
20
10
)
B
d(
e
d
ut
i
n
g
a
M
0
-10
-20
-30
-40
-50
-60
0
-45
-90
)
g
e
d(
e
s
a
h
P
-135
-180
Planta controlada en lazo cerrado
Planta controlada en lazo abierto
Planta sin controlar en lazo abierto
-225
-270
10
-1
10
0
10
1
10
2
Frequency (rad/sec)