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)