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G1(s) A¢?B
G2(s)
A¢BG3(s)
hA¢r?BG4(s)
2Y YYY YYYY YYY YYY YYYY YYY YYYY YYY YYY Y 78 t 7 igigqYeKrth (qFrthfqYexl8iwl.AªlB
G1(s)=A¢?B
G2(s) A¢B
G3(s)hA¢r?B
G4(s) YYY 77
8 t ! bdcl8Yiwl8*l8eKrth ^qFrthxl8iwl AªlBG1(s)
A¢qYigrth B A¢?BG2(s)
A¢qYigrth!B A¢B
G3(s)A¢qYigrth @ B A¢r?B
G4(s)A¢qYigrth HB YYY YYYY YYY YYY Y 7
8 t egxhjwigqrtqYex YfqYe¥rtq/egcfekjwh*lqY(xtfhjwqh^rtqdeg~tqYy6-k~ty jwqrtqYex YfqYehytqYy jwiwl8rtqYexhfq(b/.&YY YYYY YYY YYY YYYY YYY YYY YYYY YYY YY !H
8 t igrthyl8npo8q(xqYiK bu'xl8iwll4(¢~tytnpo8qrthjwiwl8yte(¢higytclF1(s)
vYY YYYY YYY !8 t @ igrthyl8npo8q(xqYiK bu'xl8iwll4(¢~tytnpo8qrthjwiwl8yte(¢higytcl
F2(s)vYY YYYY YYY !
8 t @@ igrthyl8npo8q(xqYiK bu'xl8iwll4(¢~tytnpo8qrthjwiwl8yte(¢higytclF3(s)
vYY YYYY YYY !78 t @ igigqYe/rth&(qFrthfqYedqY.jwcfrtqYedxhfq8e/ bu+xl8iwl
F1(s)AªlBqY(xl8iwl*q8e
jwige bu A¢?BqY(xl8iwlq bu*l8xtfcfpl8rtqUlUrtqYcfe(qYy6-k~tyzjwqYe*rthl8~.jwq*8l8fqYighe YYYY YYY YYY YYYY YYY YYY YYYY YYY YYY YYYY YYY YY !!
8 t @ HCigigqYe/rth&(qFrthfqYedqY.jwcfrtqYedxhfq8e/ bu+xl8iwlF2(s)
AªlBqY(xl8iwl*q8ejwige bu A¢?BqY(xl8iwlq bu*l8xtfcfpl8rtqUlUrtqYcfe(qYy6-k~tyzjwqYe*rthl8~.jwq*8l8fqYighe YYYY YYY YYY YYYY YYY YYY YYYY YYY YYY YYYY YYY YY !!
8 t @ TigigqYe/rth&(qFrthfqYedqY.jwcfrtqYedxhfq8e/ bu+xl8iwlF3(s)
AªlBqY(xl8iwl*q8ejwige bu A¢?BqY(xl8iwlq bu*l8xtfcfpl8rtqUlUrtqYcfe(qYy6-k~tyzjwqYe*rthl8~.jwq*8l8fqYighe YYYY YYY YYY YYYY YYY YYY YYYY YYY YYY YYYY YYY YY !
8 t @ bdcl8Yiwl8*lrth *l8Yytcjw~trthrtq ^qFrth rl (¢~tytnpo8q(rthjwiwl8yte(¢higytclF1(s)
YYY ! 8 t @ 7Cbdcl8Yiwl8*l8e (¢ighthytcl8ce(rtqYe(higigqYerl8e(ighegxqYekjml8e*xl8iwl
F1(s) AªlB
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(ªl|jwqYirthl8(qYikjwhcfhyzjwqrth. CF) YYY YYY YYYY YYY YYY YYYY YYY YY @8 t @ KhegxqYekjmlytqrtqY(fytcfq&rtqjwh(xqrtqYeK(qFrthfqYeighrt~&cfrtqYe4 YYY YYYY YYY 8 t @ KhegxqYekjmlytqrtqY(fytcfq&rl4(¢ightytclrtqYeK(qzrthfqYeighrt~&cfrtqYe YYYY YYY 8 t T^qY(xl8iwl8npo8q4rtqYe &higqYeOrtqYcfyl8yzjwhertq)(qFrthfqRqY(xtfhjwq4qYIqYe
&higqYeKrtqYeK(qFrthfqYeighrt~&cfrtqYeYYY YYY YYY YYYY YYY YYY YYYY YYY YY H8 t @ ~t l8i^rl8e^iwl8;&hexl8iwlq&qFrthfqrth/qYigrth qY(xtfhjmlqY q b4A¢e0B
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qY.jwcfrtqxhfq(b YYY YYYY YYY YYY YYYY YYY YYY YYYY YYY YY 8 t ~t l8igheKrl8eKiwl8;&heKxl8iwlqYeK(qFrthfqYeighrt~&cfrtqYeKqY(xhytewl8rtq8eqY#q
b4A¢e0B AªlB^ =A¢?B ¥ A¢Bu2uOWh9Aªr?Bb/ ¥H YYY YY 8 t H igrthyl8npo8qrtqYex YfqYertq(qFrthfq qY(xtfhjwq rtqFcfekjwh*l hekjwhg xhfq
(jwqFrtq bumY YYY YYY YYYY YYY YYY YYYY YYY YYY YYYY YYY YY !8 t TigigqYehegh~tedfcfc jml8y jmheKeg~txhigcfqYigherth(qFrthfqYeqY.jwcfrtqYeKeghY~tytrtqqYi
rthyl8npo8q&xqYi^ buh rtqYcfehegxhjwigqYejwqFrtqYeqYe^x YfqYertq&(qzrthfqqYigcfYcfyl8h qYeKhytqYy jwiwl8rtqYe*Fclb/.s YYY YYYY YYY YYY YYYY YYY YY
8 t Kheg~tjml8rtqYertq(qFrthfq ighrt~&cfrtqrthqYigrth H ~.jwcffc+&pl8ytrtq buAªlBighegxqYekjmlyl4(¢ightytclhytqjwh(xqt A¢?B^f~t l8irl|eKiwl8;&he YYYY YYY YY
8 t 7Cbdcl8Yiwl8*l8erthtfqzqYe(rtq egcfekjwh*lh *l8l (¢hml8rlU~.jwcffc+&pl8rtqytq %,+ %, +Y YYYY YYY YYY YYYY YYY YYY YYYY YYY YYY YYYY YYY YY
8 t ! bdcl8Yiwl8*l8eh#tfqFqYertqegcfekjwh*lh#*l8fl(¢hml8rl.'YY YYY YYYY YYY 8 t KhegxqYekjmll8qrthYiwl8~Rrth @ yl8e~tegcfyl8e&rthE ~t~tig~tqYq(qzrthfq
ighrt~&cfrtq(rth qYigrth3FvYYY YYYY YYY YYY YYYY YYY YYY YYYY YYY YY 8 t KhegxqYekjmll8qrthYiwl8~Rrth @ yl8e~tegcfyl8e&rthE ~t~tig~tqYq(qzrthfq
ighrt~&cfrtq(rth qYigrth H.vYYY YYYY YYY YYY YYYY YYY YYY YYYY YYY YY 8 t H KhegxqYekjmll8qrthYiwl8~Rrth @ yl8e~tegcfyl8e&rthE ~t~tig~tqYq(qzrthfq
ighrt~&cfrtq(rth qYigrth H tvYYY YYYY YYY YYY YYYY YYY YYY YYYY YYY YY H8 t H @ ^qY(xl8iwl8npo8qrl8eighegxqYekjml8eh *l8fldl8hikjmlrtqYe(qzrthfqYe ighrt~&cfrtqt
eg~txhi ighrt~&cfrtqhqY(xtfhjwqt AªlBjwh(xqYiwl8cfeA¢xhikjw~tl8npo8q*rth @ yleg~teghx.j0F8ytcl(rlfcfytlqYikjwh kF~t BhA¢?BD(¢ighthytcl8cfe YYY YYYY YYY YY
8 t H ~t l8i rl8eiwl8;&he rtqYe(qFrthfqYe.AªlB/ighrt~&cfrtqt¥qY qYigrth 7h A¢?Beg~txhi ighrt~&cfrtqtqY#qYigrth @ . Y YYY YYY YYYY YYY YYY YYYY YYY YY
8 t HH KhegxqYekjml(l8qrthYiwl8~Orth @ yl8eK~tegcfyl8eKrth ¥~t~tig~txl8iwlqYe(qFrthfqYeighrt~&cfrtq*heg~txhi ighrt~&cfrtqhv*l8fl (¢hml8rl.AªlB
KPOD = 900h
A¢?BKPOD = 1200
ZYYY YYY YYY YYYY YYY YYYY YYY YYYY YYY YYY Y 78 t H I^qY(xl8iwl8npo8qKrl8e ighegxqYekjml8e¥rtq(qFrthfqKqYxtfhjwqKqYWl8e rtqYe (qFrthfqYe
eg~txhi ighrt~&cfrtqYe A¢qYigrth @ ^ ¥u WhqYigrth @ &x YfqYeh&higqYertqY(cfyl8y jwhe0B AªlBbdcl8Yiwl8*l rth ^qFrthdh A¢?B KhegxqYekjmll8qrthYiwl8~rth@ Y YYY YYYY YYY YYY YYYY YYY YYY YYYY YYY YYY YYYY YYY YY !
8 t H ~t l8irl8el8;&hexl8iwlq((qzrthfqeg~txhi ighrt~&cfrtq(rth qYigrth @ F YYY YYY !8 t H7 KhegxqYekjml(l8qrthYiwl8~Orth @ yl8eK~tegcfyl8eKrth ¥~t~tig~txl8iwlqYe(qFrthfqYe
ighrt~&cfrtq(h eg~txhi ighrt~&cfrtq(hA(l8fl(¢hml8rlxl8iwlKPOD = 900
sYYY Y @@
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[0 . . . KESP
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*l8fl8e (¢hml8rl8e AªlBqYigrth C Aζmin < 15%
BhA¢?BqYigrth 7Aζmin > 15%
B (YYY YYYY YYY YYY YYYY YYY YYY YYYY YYY YYY YYYY Y @ 8 F ~t l8irl8eOiwl8;&heOxl8iwl4qYeO(qFrthfqYeighrt~&cfrtqYe ^ ¥u 'qYIeg~l8e
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BhA¢?BqYigrth H7Aζmin > 15%
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rlUl8igiwlUqYythjml8rl l8q Yhiwl8rtqYi*hLjwhyte 1Fheigh5(¢highytcl8cfecfy jwhigyl8ertqigq8jwqYiAª &F B hljwiwl8yte(¢qYig*l8npo8qiwhl|jwc+*8l 8eqYigighy jwherl8el|iwiwl8eAª & F B^ewo8q(l8qYigrl8rl8ej l|#ytqhz~l8cfqYyl8(hy jwqrtq((qFrthfqt
6H
¥ d F @ Khl8npo8qrl8e)*8l8igc $ *Yhcfertq(egcfekjwh*lF E′
q E′′d E′′
q
! #"%$"'&)(+* #"%, &)-(),.*"/"%01&)(+* #"%, &)-(),.*"3245"3!, 645"32, ()7&)4828950#*2(+*&)0(+*T ′
d0 T ′′q0 T ′′
d0
: 4 #";&+* 5&)"<2=&)!>@?4A&+(+* "%, &)-(),.*88"%01&)(+* #"%, &)-(),.*"'()!B.*C7, 4 #*2*"*45"3!, 645"<2,.()D&+4@28950#*2(+*&)0(+*Xd X ′
d X ′′d Xq X ′′
q
E<*&+F #C7,.*"<"%G #C7()4 #* &)(+* #"%, &)-(),.*H"%01&)(+* #"%, &)-(),.*"245"3!, 645"<2, ()7&)4@82950#*2(+*&)0(+*Ra
E<"%,.";&)I! #C7,.*@2*J*()>J*20(+*H24@";&+*&+4(
Id Iq
: 4()()! 5&)"/, KL7&+*2*"3 #*"31#*()(+*"/()!B.*C7, 4 #*2*"/*45"/!, 645"<2, ()7&)4828950#*2(+*&)0(+*Vd Vq
! #"%$"/, 5&)!() #*"<2*A>JM950, #*A()!B.*C7, 4 #*2*"/*45"3!, 645"32, ()7&)4828950#*2(+*&)0(+*Vr Vm N *(%&)=()*BO, >J*P, #M(),.* ()"%?QCD&), R*>@! 5&) 2*A&)! #")S4H&)!()>@, #*BT24@P!(+*24( Vb
H: 4 #";&+* 5&)2, U!(+C7,.*
D: 4 #";&+* 5&)28*>@4(%&)C7, >@! &)4
Pmec N 4&)I! #C7,.*@>@!C!F ,.C!*@*?B ,.C!*2*@*4A!, 64J2*A>JM950, #*
Efd
! #")S4@28C!*>@?4J2*A>JM950, #*
ω V !B 4C7,.2*2* P0B.*(<24@()4&)4(Wω0 X R!B 4C7,.2*2* P0B.*(<"%G #C7()4 #*
= 2πf0 Yδ N 45"%,.Z!S4@* P0B.*(<24J()4&)4(340[2"%B 4C!*>@! 5&)4@* P0B.*(<2*H&)! #")S4276C7, &+*Z!S4A824A&)!()>@, #*B\2*A>JM950, #*
[Vd
Vq
]=
[sen δ − cos δcos δ sen δ
][Vr
Vm
]A F B
[Id
Iq
]=
[sen δ − cos δcos δ sen δ
][Ir
Im
]A F B
dcfytrlXythekjmlU(qzrthl8YhO^lXrtcfyF8(cfplrtq KhY~tl8rtqYid~.jwqY$|jwcfqXrth ¥hytewo8qA K>BxqzrthOeghicfytqYigxqYiwl8rtqtrth(qFrtqz~thO*l8cfe~t*lhz~l8npo8qrtc+(¢highytcl8rth @ (qYigrth^]&xqFrthxl8egewl8il (ªl&hiKxl8ikjwh rl (¢qYig&~tl8npo8qt
Efd = −1
Ta1Efd +
Ka1
Ta1(Vref + Vesp − |Vb|)
A F @ B
qYytrth |Vb| =√V 2r + V 2m
Vref = 1
x, ~,l*jwhytewo8qrthigh5(¢higytcl.Vesp
qegcfyl8rth&jwhytewo8qrth&ewl8frl(rtqekjml8tcffc;&pl8rtqYiKrthFcfekjwh*l(rth¥q8jwytcl Aª. Bh
Vbl(jwhytewo8qytql8igiwl8(hyzjwq
jwhig(cfyl8$rtq(8hiwl8rtqYi
_D` !?Q! #2! #24J2*J*?B ,.C!*Z!S4 4J>@42!B 4J24JEbacd?Q42"%!(b2>J*, 4(34(+2!>[e
F @ FkKuO K /b . xq8eweg~tcK~t*l(qzrthl8Yh pl8iwl8jwhigfekjwcfplqYytrthOyo8qXegqY(hyzjwhOl8ertcfyF8(cfpl8e&rl8e
$8z~tcfyl8e&ewo8qXfh5*8l8rl8eh+qYytegcfrthiwl8npo8qt*l|e&jml8&+l8ecf(xhrF8ytcl8e&rl8ecfyzjwhigfc l8n21Fhehy jwighqYe(rtc+*YhigegqYe*l8igiwl8(hy jwqYe(qYytrthOhekjmo8q qYythjml8rl8el8e(pl8ig l8e(hl8e(~tewcyl8e*Yhiwl8rtqYiwl8eqYy/'Y~tiwl8npo8q rlighrth2rthOjwiwl8yteg(cfegewo8qUhekj0$Ul8eghpl8rlUyl8e*cf(xhrF8ytcl8erl8efcfytl|ep^rtqYejwiwl8yte(¢qYig*l8rtqYigheh rl8eKpl8ig l8e
qYy6-k~ty jwqpl8ig l Yhiwl8npo8q qYy/'Y~tiwl~txqYy jwqrthhz~tcffftigcq z~thdpl8iwl8jwhigc;&pl8rtqxhl8ejwhyte 1Fheh2qYigighy jwheyl8el8igiwl8eh2xqFrthCeghipl8f~tl8rl8el|jwiwl *Yerth(jwqzrtqYeyz~t(igcfqYeA¢rthjwhig(cfyl8npo8qrtq ~".qrthpl8ig lB rthjml8fl8rtqYehWu k A @ HB$h Ku 9A @ HB
*l|jwigc;&&l8rt(cj0F8ytcl(ytqFrl8A & F @@ B(¢qYig&~tl8rl*h / A @ 7B ,ighxtigh eghy jmllKighl8npo8q*l|jwigcfcl|8hy jwigh^l8eqYigighyzjwhe¥rthcfy6-khnpo8qhl8ejwhyte 1Fhe¥rl8e l8igiwl8e rthqY(xqYekjml8exhl8eighegxhjwc+*|l8exl8ikjwheKighpl8h cf*l8Ycy$8igcflF
Im1
Ir1
Im2
Ir2Imn
Irn
=
B11 G11 −B12 −G12 . . . −B1n −G1nG11 −B11 −G12 B12 . . . −G1n B1n
−B21 −G21 B22 G22 . . . −B2n −G2n−G21 B21 G22 −B22 . . . −G2n B2n −Bn1 −Gn1 −Bn2 −Gn2 . . . Bnn Gnn
−Gn1 Bn1 −Gn2 Bn2 . . . Gnn −Bnn
︸ ︷︷ ︸
Vr1
Vm1
Vr2
Vm2Vrn
Vmn
A F @@ B
/~l8ytrtql8el8igiwl8eKyo8q*xqYegeg~th Yhiml|rtqYiqYythjml8rtqtl8eqYigighyzjwheIrhImewo8q*cfY~l8cfeKl
&higqt^qYighl8npo8q *l|jwigc;&X qYetfqFqYerl rtcl8YqYyl8txtigcfytcfxl8tewo8qcfY~l8cfel8qegqY*l|jYigcfqrthjwqFrl8el8rtcj0F8ytcl8el8qYxtl8rl8e igh5(¢higcfrll8igiwl.tcfytf~tegc+*Yheg~teghx.j0F8ytpcl|e ' thq8eKrth*l8cfehfh(hy jwqYe/hekjmo8qrthl8qYigrtqqY l8e/cf(xhrF8ytcl8edrl8e/qYyth#"/1Fhe h#".cfekjwhy jwhe hy jwighl8e/l8igiwl8edcfytrl.,qYe/hfh(hy jwqYe
Bij
hGij
ewo8qt,ighegxhjwc+*|l8(hyzjwh8 l8e/eg~teghx.j#F|ytpcl8ehl8e/qYytrt~.j0F8ytcl8ehy jwighl|eKl8igiwl8e
ihj
F @ H ^ u db u b/ dehz~l8n21zhe rthegigcjml8eyl8e Fhn21FheF @ @ h9F @ (qY(xtighhytrth ~t*l*xl8ikjwh&cf(xqYikjml8y jwh
rl (¢qYig~tl|npo8qyo8q|fcythpl8irtqegcfekjwh*lrth xq8jwytcl.l8iwlz~thll8y$8fcfegh/rtqewcegjwh(lh (ªl8hlxhz~thyl8e^xhikjw~tiwl|n21Fheegh-gl (¢hcjml.zl8fY~t*l8e eg~ttekjwcjw~tcfn21FhehKegcf(xtfc 'pl8n21Fheewo8q ythhege0$8igcl8erth qFrtqz~th egh qY.jwhytl~t#hz~l8cfq8yl8(hyzjwqz~th hytYfqYl*8l8igc $ *Yhcferthhekjml8rtq(h*8l8igc $ *Yhcfel8fYtigcfpl8e
F~ttekjwcjw~tcfytrtq|egh&l8e*8l8igc $ *YhcfeVdhVqh3(¢~tytnpo8qrth
VrhVm,q8y/(¢qYig(hl & F *yl8e
& F t?F 7hF !F.jwhegh8
E ′q =1
T ′d0[Efd − (Xd −X ′
d)Id − E ′q]A F @ B
E ′′d =1
T ′′q0[(Xq −X ′′
q )Iq − E ′′d ]A F @ HB
E ′′q =1
T ′′d0[E ′q − (X ′
d −X ′′d )Id − E ′′q ]
A F @ CB
ω =1
2H[Pmec − (Vrsen δ − Vm cos δ)Id − (Vr cos δ + Vmsen δ)Iq −Dω]
A F @ B
δ = ω0(ω − 1)A F @ 7B
0 = −E ′′d +RaId −X ′′q Iq + (Vrsen δ − Vm cos δ)
A F @ !B
0 = −E ′′q +RaIq +X ′′d Id + (Vr cos δ + Vmsen δ)
A F @ B
¥qFrth egh^ytq8jml8i¥z~thVrVmIdhIqewo8q*8l8igc $ *Yhcfel8fYtigcfpl8e hhekjmo8q/l8qYxtl8rl8el8qighekjml8yzjwh
rtqegcfekjwh*ll|jwiwl*YeKrl*l|jwigc;&/rthl8rt(cj0F8ytclytqFrl8 Aª & F @@ B
F @ dK $ b u b/ b FkKuO b/6 /^l8iwlegh6hytqYyzjwiwl8iR~t*l (¢qYig~tl|npo8q>z~thHxqYegewlWighxtigheghy jml8i4qegcfekjwh*ll|jwiwl*YeRrth
hz~l8n21Fheh hegxl8nq rth2hekjml8rtqYeythhege%$8igcl lUfcfythpl8igc;&l8npo8qrl8ehz~l8n21zhel8xtigheghy jml8rl8eyl(Fhnpo8q.F @ HyF~t#rthjwhig(cfyl8rtqxqYy jwq(rth qYxhiwl8npo8qtekjwh xqYy jwq( rth5'ytcfrtqxhfqYe *8l8fqYigherl8ejwhyte 1FhehdqYigighy jmherl8e^l8igiwl8e^rtq&egcfekjwh*l.FylqYy/'Y~tiwl8npo8q&rth/hz~tcffftigcqhyzjwigh/lYhiwl npo8qhlrth*l8ytrl*rthhythigYcl A Fhnpo8q F @ B $igheg~tjml8ytrtqh *|l8fqYighe/qYytekjml8y jwhexl8iwl
δVr
VmIdhIqtrl8rtqYexqYi
δ0Vr0Vm0
Id0hIq0h#".xl8ytewo8q(rl&egigcfhrth l FfqYirth @ KqYigrth
rl8e & F @ 9F @ igheg~tjml8GhO
∆E ′q = − 1
T ′d0∆E ′q −
Xd −X ′d
T ′d0∆Id +
1
T ′d0∆Efd
A F @ B
∆E ′′d = − 1
T ′′q0∆E ′′d +
Xq −X ′′q
T ′′q0∆Iq
A F B
∆E ′′q =1
T ′′d0∆E ′q −
1
T ′′d0∆E ′′q −
X ′d −X ′′
d
T ′′d0∆Id
A F @ B
∆ω = −(−Vr0Id0 cos δ0 − Vm0Id0sen δ0 + Vr0Iq0sen δ0 − Vm0Iq0 cos δ02H
)∆δ +
+(−Vr0sen δ0 + Vm0 cos δ0
2H)∆Id + (
−Vr0 cos δ0 − Vm0sen δ02H
)∆Iq +1
2H∆Pmec
+(−Id0sen δ0 − Iq0 cos δ0
2H)∆Vr + (
Id0 cos δ0 − Iq0sen δ02H
)∆Vm −D∆ωA F B
67
∆δ = ω0∆ωA F HB
0 = −∆E ′′d + (Vr0 cos δ0 + Vm0sen δ0)∆δ +Ra∆Id −
−X ′′q∆Iq + sen δ0∆Vr − cos δ0∆Vm
A F CB
0 = −∆E ′′q + (−Vr0sen δ0 + Vm0 cos δ0)∆δ +Ra∆Iq +
+X ′′d∆Id + cos δ0∆Vr + sen δ0∆Vm
A F B
e=*8l8fqYigherthδ0Ir0hIm0
ewo8q/qY.jwcfrtqYel|jwiwl * erl/cfytcfcfl8c+&pl8npo8qrl8e qYigighy jwhe rtqYe Yhiwl8rtqYighe
δ0 = arctan(Eqm
Eqr
)A F 7B
qYytrthEqr
hEqm
ewo8qrl8rtqYeKxqYi
Eqr + jEqm = (Vr0 + jVm0) + (Ra + jXq)(Ir0 + jIm0)
Eqr + jEqm = (Vr0 +RaIr0 −XqIm0) + j(Vm0 +RaIm0 +XqIr0)A F !B
h
Ir0 + jIm0 =P + jQ
Vr0 + jVm0
Ir0 + jIm0 = (PVr0 +QVm0V 2r0 + V 2m0
) + j(PVm0 −QVr0V 2r0 + V 2m0
)A F B
h eg~ttekjwcjw~tcfytrtq|eghqYe)*8l8fqYigherthIr0hIm0
yl & F igheg~tjml8#hId0hIq0
F @ 8 du /^ 4 8 du /u¥/bKdll8y$8fcfeghrth*hekjml8tcffcfrl|rth(lOxhz~thyl8e&xhikjw~tigl8n21zhel (¢qYig~tl|npo8qqYyC*YhytcfqYyl8rtq
(qFrthfq fcfythpl8igc+&pl8rtq qYytegcfekjwh8Fxl8iwl~tegcfekjwh*luOku zyl8ehz~l8n21Fhehhegxl8nqrthdhekjml8rtqYeA uO / @ B
x = Ax+Bu
y = Cx+Du
A F B
qYytrthAl**l|jwigc;& rtcfyF8(cfplrtqegcfekjwh*l.
Bq*YhjwqYi/rth&hy jwiwl8rl.
Cq* hjwqYidrth&ewl8frl.
Dl
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ylewl8frlrtqewcegjwh(l.
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!
@ CB .z~th eghl8eghclhGhz~l8n21Fhel8~t(hy jml8rl8exl8iwl&rthegigh5* hidl&rtcfyF8(cfplrtqegcfekjwh*lFde & F @ l9F xqFrth#eghiKhegigcjml8eKrth (¢qYig*l&qY(xl8jml.
x = f(x, r, u)
0 = g(x, r, u)
y = h(x, r, u)
A F HB
qYytrthxq.*YhjwqYirth&hekjml8rtqYe
rq.*YhjwqYirth *|l8igc $*Yhcfe/l8fYtigcfpl8e
uq.*YhjwqYirthhy jwiwl8rl8ertq
egcfekjwh*lhyq *YhjwqYiKrth ewl8frl8ertq(egcfekjwh*l.
l8iwl l8y$8fcfeghlUxhz~thyl8exhikjw~tigl8n21zhextighegeg~tx 1zh eghlfcfythpl8igc;&pl8npo8qrl & F HUhjwqYigytqUrth~t rl8rtqUxqYy jwqUrthqYxhiwl8npo8q A
x0r0hu0B eghY~tytrtqRl8e*rthigc+*8l8rl8e(xl8igcl8cferl8e
(¢~tytn21Fhefghhh#ighl8npo8qlpl8rl *8l8igc $ *Yh
∆x
0
∆y
=
∂f
∂x
∣∣∣x0,r0,u0
∂f
∂r
∣∣∣x0,r0,u0
∂f
∂u
∣∣∣x0,r0,u0
∂g
∂x
∣∣∣x0,r0,u0
∂g
∂r
∣∣∣x0,r0,u0
∂g
∂u
∣∣∣x0,r0,u0
∂h∂x
∣∣∣x0,r0,u0
∂h∂r
∣∣∣x0,r0,u0
∂h∂u
∣∣∣x0,r0,u0
∆x
∆r
∆u
A F H @ B
& F H @ xqzrth eghiighhegigcjmlqY(qtE︷ ︸︸ ︷
In 0
0 0
x
r
=
J︷ ︸︸ ︷ J1 J2
J3 J4
x
r
+
Ba︷ ︸︸ ︷ Bx
Br
u
y =[Cx Cr
]
︸ ︷︷ ︸Ca
x
r
+ Du
A F HB
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ncfY~l8¥l8q*jml8*l8yttq*rtq*YhjwqYi/rth&hekjml8rtqYe
xJl**l|jwigc;&
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rB l8e*l|jwigc+&heKrth hekjml8rtq(rtq(egcfekjwh*l
ewo8q(q8.jmcrl8exqYi
A = (J1 − J2J−14 J3)
A F HHB
B = Bx − J2J−14 Br
A F H CB
C = Cx − CrJ−14 J3
A F HB
D = −CrJ−14 Br
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6
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Gζ(jω) =n∑
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Ri
(− ζω√1−ζ2
+ jω)− λi+ d
A F B
8?F FF~txhi(¢fcfhrth |G(s)| txl8iwl Re(s) ∈ [−8; 0] h Im(s) ∈ [0; 12]
dl 8CF F cff~tekjwiwl8rtq~t h#".h(xtfq rlighegxqYekjml hG(¢ightytcl rl8rlxhl*l8Yytcjw~trthrth
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l &higq2h*lO~ti*8lh *Yhig(hftq A |Gζ(jω)|dBB(ªl|jwqYirthl8(qYikjwhcfhyzjwqcfY~l8l F CFxl8iwl
~t*lrthjwhig(cfyl8rl(ªl8c "tlrth (¢ightytclrthl @ &iwl8r?<|eH
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WigqYt~tekjwh2&rtq(qYy jwigqYl8rtqYihekj0$ighl8cfqYyl8rlqYAleg~ll8tcffcrl8rthh#*l8yzjwhiKlhekjml tcffcrl8rthqY~qOrthegh(xhyttq2xtighrthjwhig(cfyl8rtqrtqOegcfekjwh*l. hytz~l8y jwqYiwl8ytrthxl8ikjwh(rtqYe egh~texl8i%F8(hjwigqYe*|l8igcl8qYy jwcfyz~l8(hy jmh| qYrthegh(xhyttqhekj0$rtcfighjml8(hyzjwhl8egegqFcl8rl pl8xl8cfrl8rthrtq qYy jwigqYl8rtqYi(h *l8yzjwhi*lUrtcfyF8(cplXrtq egcfekjwh*lUh qYytrtcfn21Fhe*eghY~tiwl8ehqYy/'?$ * hcferth ~.jwcffc+&pl8npo8qt.~t(xtigcfytrtq~t#qY/-khjwc+*Yqhegxh+'qt
&qYiwlq&(qzrthfqrth/~tG.4xqYegeg~tc,~t(lYiwl8ytrthdz~l8yzjwcfrl8rthdrth/hy jwiwl8rl8ehdewl8frl8e.qxtigq6-khjwqrthqYyzjwigqYl8rtqYighexqFrtheghiighekjwigcjwq l/~t*l(¢~tytnpo8q rth^jwiwl8yte(¢higytclhegpl8l8i qY~l/~t*l*l|jwigc;&rthjwiwl8yte(¢higytclrthKxqY~tpl8ehy jwiml|rl8ehewl8frl8ezrthK(qzrtq z~thKq qY(xqYikjml8(hy jwq'z~thh'ighxtigheghyzjml8rtq2xl8iwlOl8y$8fcfeghrth(~t'xtigqYtfh*lhegxh+'qt 'h#".h(xtfq*lE(¢~tytnpo8qrthjwiwl8yte(¢higytcl(~.jwcffc+&pl8rlxl8iwl(l8yl8fcfewl8iq*(qzrtq(rthqYegcfl8npo8q(l8egegqFcl|rtq cfy jwhigfcf l|npo8qytqYikjwh eg~ttiwl8egcffhcfiwl A ∆Pij(s)
∆Bij(s)
B qYytrthl2ewl8frl∆Pij(s)
q ~".qXrthxq8jwytcll|jwc+*|l2z~thxl8egewlytqjwighmtq(rlcfy jwhigfcfYl8npo8qh lhy jwiwl8rl
∆Bij(s)leg~teghx.j0F8ytclh5(¢hjwc+*8lrthekjml(heg*lfcfytl.
F H. @ ^KDK kKk^b qYyzjwigqYfhrth. xqYegeg~tcdpl8iwl8jwhigfekjwcfpl8e($8egcfpl8e(ighl8cfqYyl8rl8e 8e*ighekjwigcfn21FherthqYi
ighy jwhedrl8edrtcfekj0F8ytcl8eKhyzjwighqYedegh~tedhfh(hy jwqYeqY(xqYythy jwhedhrl *Fcl8tcffcrl8rth rth~.jwcffc;&pl8iKlighpl8fcfhyzjml8npo8qrth/egcfyl8cfe^rthdeml|frlxl8iwl~t*lrl8rl*l8fl. ¥qYipl|~temlrthekjml8epl8iwl8jwhigfekjwcfpl8e*cfyC*Fc $ *Yh^l~.jwcffc+&pl8npo8qOrtq2qYy jwigqYfh*hy jwiwl8fc;&pl8rtqteghytrtq~.jwcffc+&pl8rlOlighpl8fcfhyzjml8npo|qrth9*8l igc $ * hcfefqFpl8cfertcfegxqYyt+*Yhcfexl8iwl(hrtcfnpo8qtekjmlighekjwigcfnpo8qylhekjwig~.jw~tiwl(rtqqYy jwigqYl8rtqYih#".cf8h~t*ll8y$8fcewhigcjwhigcfqYewl(xl8iwl*rthjwhig(cfyl8npo8q*rl8e/*l8fl8erth&qYy jwigqYfh*l8cfecfytrtcfpl8rl8edxl8iwl*ql8(qYikjwhcf(hy jwqrtqYeK(qzrtqYerth qYegcfl8npo8qxtigqYtfh$|jwcfqYe
F H. u b Ob/6 !
igcjwigcfqrth(hekjml8tcffcfrl8rthrth( Fz~tcfekj(ll8egh(rl*l8cfqYigcflrtqYe (jwqFrtqYe rthighegxqYekjmlOh(¢ightytcl)~.jwcffc+&pl8rtqYeyl4jwhqYigclRrthUqYyzjwigqYfhX $8egegcfq)xl8iwl)l)l8y$8fcfeghrl)hekjml8tcffcfrl8rthrthegcfekjwh*l8eekjmleghnpo8ql8xtigheghy jmll8f8~t*l8e/qYytegcfrthiwl8n21FhelighegxhcjwqOrtheg~ll8xtfcfpl8npo8qxl8iwlxtigq6-khjwq4hl6-k~tekjwhrthXegcfyl8cfel8rtcfcfq8yl8cfehekjml8tcffc;&pl8rtqYigherthXegcfekjwh*l8ehfjwigcfqYerthXxq8jwytclA uO / @ B
H
^qYytegcfrthiwl8ytrtq*q(egcfekjwh*lighpl8fcfhyzjml8rtql8xtigheghyzjml8rtq*yl98?F H.l4(¢~tytnpo8q*rth jwiwl8yte(¢h igytclrth**l8fll8hikjmlrthekjwhegcfekjwh*lO
G(s)H(s)Aªl8hikjwq2ytq2xqYy jwq @ B hl (¢~tytnpo8q2rth
jwiwl8yte(¢higytcl rthK*l8fl(¢h l8rl G(s)/(1+G(s)H(s))
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igcjwigcfq*rthekjml8tcffcfrl8rthrth Fz~tcegjhekjml8hfhhz~thqegcfekjwh*l*h *l8fl (¢h l8rleghi%$he j0$ * hewh|h(egqY(hy jwh(egh8¥qOrtcl8Yiwl8*lxqYl8i9Aªbdcl8Yiwl8*lrth* Fz~tcfekj#Brth
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@ CB
8?F H.bdcl8Yiwl8*lrth^fqFqh#*l8fl(¢hml8rl.
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jωyl:(¢~tytnpo8q rth
jwiwl8yte(¢higytclrth *l8fl&l8hikjmlG(s)H(s)
h (ªl&hytrtq*l (¢ightytclω*8l8igcl8irth −∞ l
+∞ ^qY(q2qrtcl8Yiwl8*lOxqYl8iegcf(jwigcfqh ighl8npo8ql8q2hc ".qighpl8 qigcjwigcfqrth Fz~tcfekj
xqzrtheghiegcf(xtfc 'pl8rtqxl8iwlp/2
*YqYjml8e(ytq eghy jwcfrtq l8yzjwc tqYi%$8igcqXhLjwqYigytqUrtq xqYyzjwq −1 qYytegcfrthiwl8ytrtqOl8xhyl8el8e *|l8igcl8n21FhexqYegcjwc+*|l8erth4(¢ightytcl.egqYf~tnpo8qrlhz~l8npo8qOpl8iwl8 jwhigfekjwcfpl A
det|A − λI| = 0B (¢qYigythhegh(xtighl8~.jwq *|l8fqYigheighpl8cfeqY~ qY(xtfh#".qYeqYy6-k~t l8rtqYe
ekjwqqFqYigighrth5*FcfrtqUl8q (ªl|jwqXrthOlX*l|jwigc;&Aeghi(ighpl8 A
aij ∈ Rxl8iwljwqFrtq
ijB )degegcfO^l
egcf(hjwigclrtq*bdcl8Yiwl8*lrth Fz~tcfekjxqFrth eghiKrth5*Fcfrl8(hy jwh qYytegcfrthiwl8rl. qYYqt8lKxl8ikjwcfirtqqYytthcf(hy jwqdrtqyCt(higqrthx YfqYecfytekj0$*Yhcfe hrlKighegxqYekjmlytqrtqY(fytcfq
rl(¢ightytclrth&~t*l(¢~tytnpo|qrthjwiwl8yte(¢higytclG(s)
hegxh+'pl.qigcjwigcfq*rthhekjml8tcffcfrl8rthrth Fz~tcfekj^xhiwcjwhKl rthjwhig(cfyl8npo8q rtqYexl8i%F8(hjwigqYerth
H(s)xl8iwl z~thKqegcfekjwh*lrthK*l8fl
(¢hml8rl*egh-gl*hekj0$ *Yh xtigqzhrtcf(hyzjwq($8egcfq*xl8iwl(eghxtigq-khjml8ihl6-k~tekjml8iq*hekjml8tcffc;&pl8rtqYilrthjwhig(cfyl8npo8qrlqY(xhytewl8npo8qrth(ªl8eghdh l8yttqxl8iwlz~thdqrtcl8Yiwl8*lxqYl8irl (¢~tytnpo8qrthjwiwl8yte(¢higytclrth *l8fl&l8hikjml
G(s)H(s)hyC*YqY+*8lqxqYy jwq @ rth (¢qYig*ll8rthz~l8rl.
F~txqYytrtqz~th ~t#rl8rtqegcfekjwh*l&hAl8y$8fcfeghdxqYegeg~l(l8xhyl8e~t#xl8iqY(xtfh#".qrth l8~.jwq|*8l8fqYighe cfytekj0$*Yhcfe
λc = +σc±jωc hKz~thKegh-glxqYegeg+*Yhhekjml8tcffc;&pl8iqewcegjwh(l l|jwiwl *Yerlcfy jwigq|
rt~tnpo8qrth&~t ewcyl8 l8rtcfcfqYyl8 hekjml8tcffc;&pl8rtqYirthigc+*|l8rtqrth∆Ω
h~t Yhiwl8rtqYikA¢hegqYftcfrtqt
xqYih#".h(xtfqt$l|jwiwl*Ye/rl*l8y$8fcfeghrtqYedighegfrt~tqYe/rl (¢~tytnpo8qrthjwiwl8yte(¢higytcl*rth∆Ω/∆Vref
B H7
lqY(xhytewl8npo8q(rth (ªl|ewhh l8yttqythhege%$8igclxqzrth eghiKqY.jwcfrll|jwiwl * ertqrtcl8Yiwl8*lxqYl8irl(¢~tytnpo8qXrthjwiwl8yte(¢higytcl
∆Ωk/∆V kref
R38 F l8xtigheghy jmlXqYertcl8Yiwl8*l8e&xqYl8ighexl8iwl(hftqYih#".h(xtfc+'pl8iqxtigqFhrtcf(hyzjwqt
−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
α0 = −2.4491oA Z
(a) − Diagrama de Nyquist ∆Ω(s)/∆Vref
(s) − Paulo Afonso IV
Eix
o Im
agin
ario
−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.5
0
0.5
ωc = 1,12rad/s
A
Z
(b) − Diagrama de Nyquist − malha aberta com controlador
Eix
o Im
agin
ario
Eixo Real
ωc = 1,12rad/s
8?F tbdcl8Yiwl8*l|erth Fz~tcegjxl8iwlqxtigq-khjwqrth qYy jwigqYl8rtqYi
l8iwlqY(xhytewl8iql|jwiwl8egqα0
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(1 + saT )/(1 + sT )$qYytrth
a > 1
xl8iwl/l *8l8ytnq/qY~a < 1
8xl8iwl/l|jwiwl8egqt (¢ightytcl8e *l8cfel8jml8e8q/xl8i%F8(hjwigq/ldighxtigheghy jmlq l8yttqrtcfyF8(cqrthegegh tfqFqtFeghytrtqz~thxl8iwl*|l8fqYighehfh5*8l8rtqYetxtigqYtfh*l8erthl8(xtfc 'pl8npo8qrthig~tfrtqYeqY(hnpl8+leg~tigYcfi4bdhE(¢qYig*l2Yhiwl8 eghqXhekjml8tcffc;&pl8rtqYixqYegeg~tcKrtq8cfetfqFqYerthl *8l8ytnqt ighqY(hytrl egh*z~thq *8l8fqYi rtq2 l8yttqrtcfyF8(cfqOhspl8egpl|jml A
a2Begh-glytq2$ ".cf(q
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φmax = sen −1
(a− 1a+ 1
)A F @ B
ωmax =1
T√a
A F BH!
8l&hytrtq|eghφmax α0
hωmax = ωc
qY.jweghq8e xl8i%F8(hjwigqYeahTrtqtfqFqOl *8l8ytnqOqY~
l|jwiwl8egqta =
1 + sen α0
1− sen α0A F HB
T =1
ωc√a
A F CBl6-k~tekjwh&qYy*YhytcfhyzjwhrtqqYyzjwigqYl8rtqYirth5*YheghiqY.jwcfrtql|jwiwl*Ye/rthrth jmhiwcfyl8npo8qrtqYe
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l8yttqrtqhekjml8tcffc;&pl8rtqYirth5*Yheghi l -k~tekjml8rtqrth(¢qYig*ll*eghqY.jwhi/~tl8(qYikjwhcf(hy jwql8rthz~l8rtqOxl8iwljwqzrtqYe qYe (qFrtqYerthqYegcfl8npo8qrtqY(cfyl8y jwhe(bdh5*Yh(l * hi ~tcfrl8rtqxl8iwlz~thyo8qqFqYigiwl Yiwl8ytrthKrthjwhigcfqYiwl8npo8qytql8(qYikjwhcf(hy jwq rthdl8fY~tyte(qFrtqYeh (ªl * qYirl (hftqYigclrth qY~.jwigqYe
! " "ekjwhigcjwigcfq/~.jwcffc+&plq ~t l8i rl8e l8;&he rtqegcfekjwh*l/l8egegqFcl8rtq 8e cfy/(¢qYig*l8n21Fhe rtqighegfrt~tqrtq(qFrtqigjwcfqt F8ytY~tfqrthxl8ikjwcfrlrtq*iwl8(qrtq(f~t l8irl8eiwl8;&herthekjwh(qFrtqqYigighegxqYytrth (ªl8egh(rtqegh~ xtiYxtigcfqOighegfrt~tqtOFhytrtq2l8egegcfOl8e &>F Hh.F eghi*Yhrth(l8egh*xl8iwlOq $8f~tfql8yl8fjwcqrtqqYy jwigqYl8rtqYi
H(s)
H(s) =
(1 + sTa
1 + sT
)n
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ahTewo8q qY.jwcfrtqYe(h (¢~tytnpo8q
rlxl8ikjwh(cf*l8Ycy$8igcflrtqO(qFrtqigjwcfq Aωc = Im(λc)
B/h(rl (ªl|ewhrtqOewh~ ighegfrt~tql8egegqFcl8rtqAα0 = ∠Rc
B
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8?F Fbdcl8Yiwl8*lrth tfqFqYertq9KhY~tl8rtqYid~.jwqY$|jwcfq&rth ¥hytewo8q A K B
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F1(s)hF2(s)
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B ighegxhjwc+*|l8(hy jwh8
8?F 7.bdcl8Yiwl8*lrth tfqFqYerl*l8l&rtq(ighY~tl8rtq8irth ¥hytewo8q(qY#q(hekjml8tcffc;&l8rtqYi
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APacel = Pmec − Pt
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BdqY~eg~lighpl|j0F8ytcll8xl8cjwc+*8l A
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(4 v ( ^)*!& R, WO) !(
hekjwhOpl8xtjw~tfqXewo8qrthegigcjml8e(l8fY~t*l8e&jwytcfpl8erthOighrt~tnpo8qrthOqYigrth+rthO(qFrthfqrthegcfekjwh*l8e(rtcfyF8(cqYerthO(qFrtqz~thegh-gl8 hy/(ªl|jwc;&pl8rl8e(l8fY~t*l8exh~tfcl8igcfrl8rthertqYe*(jwq|rtqYe*z~th~.jwcffc+&pl8LlrtqY(cfyF8ytcl(qzrl8Kh qY(xl8iwl8npo8qqY lXjwytcfplUrthjwig~tytpl8(hyzjwql8l8ythpl8rtq A @ B ¥l8(xtfl8(hy jwh~.jwcffc+&pl8rlxl8iwlhekjwh 'Odf8~t*l8eqY(xl8iwl8n21zhehy jwighdqYe^(jwqzrtqYerthjwig~tpl8(hy jwql8l8ythpl8rtqEA ^ ¥u zrtqcfytYfe 42 0 1 2' % Bh*rtqY(cfyF8ytcl(qFrl8 jwsegcfrtqighpl8fc;&pl8rl8e¥yo8qegqY(hyzjwh(xl8iwlOegcfekjwh*l8erth*xq8jwytclA F// d t^ @ 7B |*l8ejml8&xl8iwlKegcfekjwh*l8e rthqY~.jwiwl8e $8ighpl8e rthxhegz~tcfewlqY(qqYeegcfekjwh*l8e(h F8ytcfqYe Aªbd @ m @ !Bh/hGhekjml8tcffcfrl8rthdrth/cf*l8Yhyte^rthjwhfheg Yxtcfq A t 8>8^ @ B AjwqFrl|edl8edl8xtfcfpl8n21FheKrthekjwhpl8xtjw~tfqeg~tx 1Fh eghz~thqhegxhjwigqOqY(xtfhjwqrtqYex YfqYe/rtqewcegjwh(lrthjwhig(cfyl8rtqxtigh5*Fcl8(hyzjwhl|jwiwl *Ye rl8eigq8jwcfyl8e RqY~ h#".hjwq yldl8xtfcfpl8npo8qrlFhnpo8q H. Hdz~thjwiwl|jmlrtq/(jwqFrtqdxl8iwl $8f~tfqdrtq/hegxhjwigqrth x YfqYertqY(cfyl8yzjwhe
dert~l8el8q8iwrl|Yhyte fh5*|l8WhWqYytegcfrthiwl8npo8q/~tWhigigq/rthl8xtigq ".cf*l8npo8qhy jmigh^qYe(qFrthfqYeqY(xtfhjwqh ighrt~&cfrtq(rth5'ytcfrtqqY(q
e = ‖G(s)−Gr(s)‖∞ytz~l8y jwq*ljwytcfpl^ ¥u
(cfytcfc+&pl~t fcfc jml8y jwheg~txhigcfqYiqjwig~tytpl8(hy jwqU(qFrl8 Yhiwl8f(hy jwh8^ l8iwl8yzjwhOl8xhyl8elh#".cfekjwytclrth~t fcfcjwl8yzjwheg~txhigcfqYirthegegh(higigqtegegqeghi%$(qYekjwiwl8rtqythekjwhpl8xtjw~tfqt,~t*l*Yh2& z~thl8Y~tyteKl8~.jwqYighetqY(q&xqYih#".hxtfq4) A @ B .cjml8 z~th~t*l&rthe*8l8y jml8Yhrtq(jwqFrtqrthjwig~tytpl8(hy jwq((qzrl8, -k~tekjml8(hy jwhl (ªl8jml&rthekjml l8iwl8y jwcl.
efcf(c jml8y jwheeg~txhigcfqYigherthekjwh/higigqYexl8iwlq jwig~tytpl8(hyzjwql8l8ythpl8rtqxqYegeg~th *8l8fqYighe(hytqYighez~thqYerl/l8qYigrl8YhW(qFrl8 A @ B 8z~th~.jwcffc+&plKqdegqY*l|jYigcfqrl8eytqYig*l8ecfy/'ytcjml8e rtqYe(qFrtqYerlrtc+(¢highytnplOhy jmigh*qYe(qzrthfqYe.A Fhnpo8q H. F HB Oekjml2l8qYigrl8Yh'jwhqY(qqY-khjwc+* qxtigqFrt~&cfi~t (qzrthfqighrt~&crtqt.~6-kqYe^x YfqYeh/ighegfrt~tqYel8egegqFcl8rtq8eegh-gl8 ~teg~tqYy6-k~ty jwqXrtqqYy6-k~ty jwqrthx YfqYehighegfrt~tqYertq(qzrthfq2qY(xtfhjwqtekjwqyo8qqzqYigighytqpl8egq(rl&jwytcfplrthjwig~tytpl8(hy jwql8l8ythpl8rtqttz~thyo8q*l8y jwAl(heg*ll8egh (qFrl8ª
a "%, >81Q4B 4P,.* ‖G(s)‖∞ 2! 4&+* *b 4()>J* H∞
2G(s) 950/950, R*B /*4 "%0?()!>@4 24 >J*, 4(R*B 4( "%, P0B.*(2
G(jω) ?#*(+* ω ∈ Re
@
l8iwlh#".h(xtfc+'pl8iqYeKxtigqFhrtcf(hy jwqYeKrth pl8rl(jwqFrtqt(¢qYc,~.jwcffc+&pl8rtq~tA(qFrthfqrth qYi rth 7Fzz~thdighxtigheghy jml~t . jwiwl|jml8rtqh A @ !B qYytthcfrtqqY(q ! 4 A¢dxytrtcfh . @ B (¢~tytnpo8q*rthjwiwl8yte(¢higytcl*hegqYtcfrl(
∆Ω(s)/∆Pmec(s) z~thqYyte jwcjw~tc l(*l8fl&rthqYy jwigqYfh rtq*ighY~tl8rtqYiKrth *YhfqFcfrl8rth h&l8(q8igjwhhq*(qzrtq(hfhjwigqY(h F8ytcfql8egegqFcl8rtq(l8q(Yhiwl8rtqYiKrll8igiwl9H7rtq*egcfekjwh*l. (qFrthfqrtq(egcfekjwh*l jwhekjwh (¢qYc,qY.jwcfrtq(xhfqb eghytrtq(z~thxl8iwll8eegcf~tl8n21zheq*uO 6j l| (¢qYc,~.jwcffc+&pl8rtqt
H. @ b b/ b/u b/ u b/ d/^u ^ //b A ^ ¥u B
ekjwh&(jwqFrtq*&l8(xtl8hyzjwh ~.jwcffc;&pl8rtq(xhl*qY&~tytcfrl8rthrthegcfekjwh*l8edrthqYyzjwigqYfhxl8iwlighrt~tnpo8q)rthXqYigrthTrthX(qFrthfq8erthXegcfekjwh*l8erthX(rtcfqRxqYikjwh8l|jwiwl *YerthX~tIl8fYqYigcjwqegcfekjwh$|jwcfqh *l|jwh*l|jwcpl8hyzjwh/igcfYqYigqYegqt
ekjmlXeghnpo8qUrthegigh5*Yhtigh5*Yh(hyzjwhl(hjwqFrtqYfqYYcflrthOighrt~tno8qUrth(qFrthqYefcfythpl8igherthegcfekjwh*l8ertcfyF8(cqYe&~tjwc+*8l8igc;$ *Yhcfe~.jwcffc+&pl8ytrtq/jmig~tytpl|(hyzjwql8l8ythpl8rtqA uOd^ m@ @ @ K B *l8iwlOh#".xtfccj0$ q(l8xtigheghyzjml8rtqq¥hqYigh*l9H. @ ~6-glxtigq*8legh hytqYy jwimlh A @ Bh z~theghi*Fcfi%$rth l8egh jwh Yigcfplxl8iwlqYeKhekjw~trtqYextigqYxqYekjwqYe
! G(s) ∈ RH+∞
G(s) =
A B
C D
=
A11 A12 B1
A21 A22 B2
C1 C2 D
) %/ 42 2 +% / 2 ' 4 %,% !/! %,% % !-%&
Σ = diag(Σ1,Σ2) / + -/ !.
AΣ + ΣAt +BBt = 0
AtΣ + ΣA+ CtC = 0,
'
Σ1 = diag(σ1Is1, σ2Is2, . . . , σrIsr)
Σ2 = diag(σr+1Isr+1 , σr+2Isr+2 , . . . , σNIsN)
∆Ω R!B 4C7,.2*2=* P0B.*(<24@()4&)4(3 ∆Pmec ?4&)I! #C7,.*A>@C!F ,.C!*A*?B , C7*2*@*4@!, 64@2*@>JM950, #*e
2 σ1 > σ2 > . . . > σr+1 > σr+2 > . . . > σN
σi
'( + ' % . % 2% sii =
1, 2, . . . , Ns1 + s2 + . . .+ sN = n
3 ' +%&'(' ! 2
Gr(s) =
A11 B1
C1 D
42 %,'(' % 2 '( ' 43 %& / % !-4 % * ' %& %,'
‖G(s)−Gr(s)‖∞ ≤ 2(σr+1 + σr+2 + . . .+ σN)
¤
qY-khjwc+* qrtq((jwqzrtq( pl8f~tl8i~tA(qzrthfqGr(s)
rth qYigrthAighrt~&cfrlrlxl8ikjwcfirth
G(s)A¢rth*qYigrth
n | n > rB rth*jml8 (¢qYig*lOz~thlytqYig*l
H∞rlrtc+(¢highytnplhy jwighhekjwhe
rtqYcfe(qzrthfqYe Aª & HF @ BxqYegeg~lXfcfc jml8y jwheg~txhigcfqYih+rth5'ytcfrtq Aª & H. B qYy/(¢qYig(hq¥hqYigh*l H. @
‖G(s)‖∞ := supRe(s)>0
σ[G(s)] = supω∈R
σ[G(jω)] = supu
∞∫0
y(t)Ty(t)dt
∞∫0
u(t)Tu(t)dt
A H. @ B
‖G(s)‖∞ = ‖G(s)−Gr(s)‖∞ ≤ e = 2n∑
i=r+1
σiA H. B
qYytrthσ[G(s)]
dq(l8cfqYi *|l8fqYi^egcfyt8~tl8irthG(s)
σiewo8q&qYe
n− r (hytqYighe *8l8fqYighe^egcfytY~tl8igherth)dl8yYhtrtq(qzrthfq/qY(xtfh jmq
G(s)y(t)
l/ighegxqYekjmlrthG(s)
xl8iwl/~t*ldhy jwiwl8rl/fcf(c jml|rlu(t)
jwiwl*Ye rlqYigrthyl8npo8qKrthigheghy jwhrtqYe *|l8fqYigheegcfytY~tl8ighe,rth dl8yYh?A
σkB h#".hxtfc+'pl8rl
ylE8H. @ l.,xqFrth eghrth5'ytcfi~t*l*qYigrthrxl8iwlq(qzrthfqighrt~&cfrtqjml8z~th
σr À σr+1egh-gl2ewl|jwcfe(¢hcjmlqY~Rz~thl2iwl&po8qσr/σr+1
l8xtigheghy jwh~t*lqYigrth rthYiwl8ytrth2&plε ≈ 10−1
qY~8r dq h#".h(xtfqUxtigqYxqYekjwqA Fcfekjwh*l !4 B lqYigrth ighrt~&cfrlUz~thxqFrtheghihegqYftcfrl.l8rtq8jml8ytrtq|eghhekjwhigcjwigcfqt.
r = 20
de'Y~tiwl8ez~th2hekjmo8q4l8xtigheghy jml8rl8eythekjwhpl8xtjw~tfqtz~th2(qYekjwiwl8 qYeigheg~tjml8rtqYe*rl8eqYigrthyl8n21FhertqYeD*8l|fqYigheegcfytY~tl8igherth dl8y8h A¢qY~fytrtcfhe^rth/rtqY(cfyF8ytcl (qzrl8 Bh/qYe^higigqYertqYe(qFrthfqYeqYTegh~teOfcf(c jml8y jwhejwIl4hegpl8l4fq8 l8igjwcplA¢r )Bxl8iwl (ªl8cffc jml8il4eg~l*Fcfeg~l8fc;&pl8npo8qhl|y$8fcfegh8
$8f~tfq&rtqYeD*8l|fqYigheegcfytY~tl8igherth dl8y8h,xl8iwl&hekjwhegcfekjwh*l jwhekjwh/yo8q&h#".cfYcf~Yiwl8ytrthhe(¢qYignqRq8(xt~.jml8cfqYyl8 rth5*Fcrtq4lUeg~lUxhz~thyl rtcf(hytewo8qt>q hy jml8y jwqtKxl8iwlq pl8egqRrtqegcfekjwh*ldrthxq8jwytcl/tiwl8egcffhcfigqdrth @ 4A l8xtjw~tfq CB¥~6-gl/qYigrth ~tjwiwl8xl8egewld(cfFheghcfeghy jwqYe
H
0 10 20 30 40 50 60−350
−300
−250
−200
−150
−100
−50
0
dB
Valores Singulares de Hankel (Ω / Pmec
)
Ordem dos modelos reduzidos (Gr(s)) 0 10 20 30 40 50
−400
−350
−300
−250
−200
−150
−100
−50
0
Ordem dos modelos reduzidos (Gr(s))
(dB
)
Truncamento Balanceado (Ω(s)/Pmec
(s))
Erro do modelo (||G−Gr||∞)
Limite superior do erro (Σ σi)
AªlB A¢?B
8H. @ AªlB igrthyl8no8qrtqYeD*8l8fqYigheegcfytY~tl8ighe^rth dl8y8h,xl8iwll(¢~tytnpo8qrth/jwiwl8yte(¢higytcflΩ(s)/Pmec(s)
hA¢?Bigigq(rtq(qFrthfqh fcfc jwh/eg~txhigcfqYirtq(higigqh#pl8rlqYigrthO
hekjml8rtqYe.qhe(¢qYignq&qY(xt~.jml8cfqYyl8cfyC*Fcl8tcfc+&qY~(lighpl8fc;&l8npo8qrthekjwh/ $8f~tfqtz~.jwcffc+&pl8ytrtq|eghK~txtigqFhegewl8rtqYiKrthF &h (h Yigcl Ku @ zjwhe
ehigigqYertqYe(qFrthfqYe&ighrt~&cfrtqYe(qYekjwiwl8rtqYeyl 8>H. @ ,xqYegeg~th *8l|fqYighe&~tcjwqxti ".cf(qYertqYe/egh~tefcfc jml8y jwhedeg~txhigcfqYighel|jw&l*qYigrth F/qhyzjml8y jwqt$xl8iwl*qYigrthyte/eg~txh igcfqYighe l Fhegjwhe higigqYe pl8h xl8iwl*8l8fqYighe&~tcjwqxti ".cf(qYe/rth &higq A
10−16B ¥l8~t(hy jml8ytrtq
q~tekjwq qY(xt~.jml8cfqYyl8rtqUegh~E $8f~tfqUlX(hrtcfrlz~th2lq8iwrth eghl8xtigq ".cf*lrlqYigcfYcfyl|An = 62
B txqFrthytrtq(cfytrt~&cfilrtcfekjwqYign21zheKyz~t(igcfpl8e
−1.5 −1 −0.5 00
1
2
3
4
5
6
7
8
9
10
0.2
0.15 0.1 0.05
Eixo real
Eix
o Im
agin
ario
Mapa de polos
Polos do modelo completo (62)Polos do modelo reduzido (20)
8H. F qFpl8fc;&pl8npo8qrtqYeKx YfqYertqY(cfyl8y jwhertqYeK(qFrthfqYeqYigcfYcfyl8h ighrt~&cfrtqt
ekjwhjwcfxqrthjwytcfpl.Yl|frthh#".cfYcfi xtig qYytrtcfcqYyl8(hyzjwqYez~l8ytrtq/eghjwiwl|jmlKrth^egcfekjwh*l8ecfytekj0$ * hcfe|hytqYy jwiwl~t*lKegqYf~tnpo8qhWz~thyo8qd$K~tWqY(xtigqY(h jmc(hy jwqKqYWlxqYegcfnpo8qh#"tl|jmlrtqYex YfqYe¥qYigcfYcfyl8ce qY(qd(qYekjwiwll86H. F8xhigrthytrtq/cfy/(¢qYig*l8n21zhe¥ighfh5*|l8yzjwhexl8iwldl8y$8fcfegh(qFrl8th egcfekjwh*l8erthKxq8jwytcl.¥qYigO A @ B(qYekjwiwlz~thqYehigigqYertqYe(qFrthfqYeighrt~&cfrtqYeYhiwl8rtqYe&xhfqjwig~tytpl8(hy jwq2l8l8ythpl8rtqewo8q2(hytqYighez~thl8z~thfhel8xtigheghy jml8rtqYexhfqYe(jwqFrtqYe(qFrl8cferthegigcjwqYeKylxti".c*l&ewhnpo8qt
H. b b/ b/u b/ u b/ K , #b b/ b/ dbdkb/;b uO //^ u bK A¢ bu B
xtigcfytfxtcfqrtqYe(jwqFrtqYe rthegigcjwqYeythegjmlOeghnpo8qqYytegcfekjwh*l8egcfl8(hyzjwhhsl8xtfcfpl8i*8l fqYigheKz~thz~l8y jwc+'pl8AlrtqY(cfyF8ytcl&rth pl8rl(qFrtqttrthytqY(cfyl8rtqYeytrtcfherthbdqY(cfyF8ytcluOqFrl8A¢ bu B l8egegc+'pl8ytrtq|qYeh*xtigheghi*8l8ytrtqXytq2(qzrthfqighrt~&cfrtql8z~thfhe
rx 8fqYe*l8cfe
rtqY(cfyl8y jwheh egh~teighegfrt~tqYel8egegqzcl8rtqYeekjwhefytrtcfhe (¢qYiwl8 xtigh5*Fcl8(hyzjwh(l8qYigrl8rtqYe xqYi A @ HB z~th(xtigqYx e ~t
(jwqFrtqxl8iwlOqYigrthyl8npo8qOrth(x YfqYe xqYi rtqY(cfyF8ytcl(qFrl8rthy jwigqrth(~t'(qFrthfqOrthegigcjwqyl (¢qYig*lrthighpl8fc;&pl8npo8q2h hegxl8nqrthhekjml8rtqYeqY~Ryl (¢qYig*lrth.(¢~tytnpo8qrthjwiwl8yte(¢higytcl. ~.jwiwl8el8jwhigyl|jwc+*|l8erth buGewo8qhytqYy jwiwl8rl8e h A / 6 u @ 7 m @ ! uO/b/ d d @ Kbd @ t 8>8^ @ B $4qY~.jwiwl8el8qYigrl8Yhytez~thrthekjml8pl8'lqY&tcfyl8npo8qrthjwytcfpl8e(qFrl8cfe/qYqY~.jwigqYe(jwqFrtqYerthighrt~tnpo8qqY'l'yl8fcfrl8rthrth^l8(hytc;&pl8i¥l8e ighekjwigcfn21Fhe¥rtqd(jwqzrtq(qFrl8 qY(qdlKeg~txqYekjml(ªl8jmlrth l8iwl8y jwclrthfcfc jml8y jwhexl8iwlhigigqYerth/l|xtiwq ".cf*l8npo8q rth(qFrthfqYehpl8igytcl rthdxtigqzhrtcf(hyzjwqYel8rthz~l8rtqYerthl8y$8fcfegh/rth rtqY(cfyF8ytcl(qFrl8 A ) @ B
l8iwlxqFrthiKighxtigheghy jml8iK~t#(qzrthfqrthqYigrth#ighrt~&cfrll jmiwl *YeKrll|y$8fcfegh/rtqYe(qFrtqYerth^jwqFrtq/egcfekjwh*ldyz~t rl8rtq/xqYy jwqrthqYxhiwl8npo8qtYeg~/'cfhy jwhz~theg~l(¢~tytnpo8q/rthjwiwl8yte(¢higytcflegh-glh#".xtighegeml2h (¢iwl8n21Fhexl8igcl8cferth(qzrtq2l (¢qYig*l8i~tsqYy6-k~tyzjwq2rth*xl8ighl8e&leghighl8yl8fcfewl8rl8ecfytrthxhytrthy jwh(hy jwh84hekjwhpl8egqteg~tx 1Fh eghOll8~tegytclrthx YfqYe& tjwcxtfq8e&ytqegcfekjwh*l.ekjmldl8qYigrl8Yh~.jwcffc+&pll8e &6H. HlH. z~theghxl8iwl8Wl8e¥xl8ighl8e qYigighegxqYytrthy jwhel8qYeKqFrtqYeighpl8cfeh qY(xtfh#".qYeqYy6-k~t l8rtqYeleghighAhekjw~trl8rtqYe
G(s) =nr∑
i=1
Gri (s) +
nc∑
i=1
Gci(s)
A H. HB
G(s) =nr∑
i=1
Ri
s− λi+
nc∑
i=1
2Re(Ri)s− (Re(Ri)Re(λi) + Im(Ri)Im(λi))
s2 − 2Re(λi)s+Re(λi)2 + Im(λi)2A H. CB
G(s) =nr∑
i=1
Ri
s− λi+
nc∑
i=1
αs+ β
s2 − 2Re(λi)s+ |λi|2A H. B
qYytrthα = 2Re(Ri)
β = −(Re(Ri)Re(λi) + Im(Ri)Im(λi))
nrq&yt(higq&rthx Yfq8eighpl8cfe
hnc q(yCt(higqrth (qFrtqYeqY(xtfh#".qYeh#".xtighegemo8q(xl8iwlq(qFrthfq&rth qYigrthGighrt~&cfrl(¢hcjml&rth *l8ythcfiwlhz~tc+*8l8fhy jwh8eghytrtq
z~thl(qYigrthv(hytqYi(r ¿ n | n = 2nc + nr)
hegqY(hy jwh&l8exl8ighl8edrtqYed(qzrtqYe*l8cfertqY(cfyl8y jwhehekjmo8q(ytqYeKegqY*l|jYigcfq8e
ekjwhe(jwqFrtqYedxqYegeg~th~t igcjwigcfq*h qY~tvrthegigcjwqyl &H. @ $z~th&ighxtigheghy jmlqhigigq($ ".cf(q&rlighegxqYekjmlytq(rtqY(fytcqrl (¢ightytclhyzjwigh qYeKrtqYcfe(qFrthfqYe dexl8ighl8eigheg~tjml8y jwherlrtc (¢highytnpl
G(s)−Gr(s) = G(s)ewo8q-k~tekjml8(hy jwh/l8e (¢iml|n21Fhe^xl8igcl8cfeighl|jwc+*|l8e
l8qYen− r
(qzrtqYe(hytqYeKrtqY(cfyl8y jwhehekjmleghnpo8qOeghiwo8qrthegigcjwqYejwige (jwqFrtqYe~.jwcffc+&pl8rtqYe/xl8iwlhytqYy jwiwl8i ~tqYy6-k~tyzjwqrth
x YfqYe h*egh~te&ighegfrt~tqYe&l8egegqFcl8rtqYez~thighxtigheghy jwh ~ts(qzrthfqOrth*qYigrthsighrt~&cfrlh*z~thighxtigqFrt~&pl8 l8eighegxqYekjml8ertq (qFrthfqqY(xtfhjwq/rthK(l8ythcfiwl/ewl|jwcfe(ªl|jYigclFbdhy jwighKqYe(jwqFrtqYexhegz~tcfewl8rtqYez~th2~.jwcffc;&pl8 l8qYigrl8Yhyterth2rtqY(cfyF8ytcl(qzrl8 Kl|xhyl8eq8ertqYcfextigcf(hcfigqYeA Fhn21FheH. F @ h H. F BqYyzjwigcft~tfiwl8xqYegcjwc+*8l8(hy jwhdxl8iwl&l8xtfcfpl8n21Fhe^hGegcfekjwh*l8erth/Yiwl8ytrthxqYikjwh8eghi*Fcfytrtqrthl8eghxl8iwlqY(xl8iwl8npo8q(qYAq*(jwqFrtq(xtigqYxqYekjwq*ythekjwhjwiwl8l8ftq A Fhnpo8qH. F HB z~th&rth5'yth~tvfytrtcfhrth&rtqY(cyF8ytcll8eghpl8rtqyl*ytqYig*l(cfy/'ytcjml(rl9(¢iwl8npo8q*xl8igcl8qYigighegxqYytrthy jwhlpl8rl(qFrtqt
ertqYcfextigcf(hcfigq8eigcjwigcfqYel8yl8fcfewl8 lrtqY(cfyF8ytcl(qzrl8l|jwiwl*Ye^rthhy jwiwl8rl8e^rtqjwcfxqcf(xt~tfegq A l8fqYiytcfcl8 B&hrtqUjwcfxqUrthYiwl8~ A l8fqYi98cfyl8 B Ml8iwlUl8qYe*qYe*(jwqzrtqYe*eghi%$~.jwcffc+&pl8rll (¢qYig~tfl8npo8qleghY~tcfi
Y (s) = G(s)U(s) =
[n∑
i=1
Gi(s)
]U(s)
A H. 7B
qYytrthU(s)
hY (s)
ighxtigheghy jml8O.ighegxhjwc+*8l8(hy jwh8.l8ejwiwl8yte(¢qYig*l8rl8erth l8xtl8hKrlhy jwiwl8rlh rlewl8frl.th ∑n
i=1Gi(s)qYigighegxqYytrth h#".xl8ytewo8q*h (¢iwl8n21zheKxl8igcl8cferth
G(s)
eKfc(c jml8y jwheeg~txhigcfqYigheKrtqYehigigqYeKrth(qFrthfqYe AeB^rth5'ytcfrtqYexl8iwl&jwqFrtqYeqYe(jwqFrtqYeKl
eghighArthegigcjwqYeKqYhrthhvqigcjwigcfql8qYigrl8rtqyl(Fhnpo8qH. F H.tqYy/(¢qYig(h l &?H. !F
e ≤ e =n∑
i=r+1
‖Gi(s)‖∞ =n−r∑
i=1
‖Gi(s)‖∞A H. !B
qYytrthGi(s)
ewo8ql8e (¢iwl8n21Fhexl8igcl8cfe(qzrl8cferthG(s)
qYigrthyl8rl8eeghY~tytrtqqYe *8l8fqYigherl8eeg~l8eytqYig*l8ecfy/'ytcjml8e
l8h/ighegewl8jml8iz~th8tqYy/(¢qYig(h A @ B .lytqYig*l L∞ A¢ytqYig*lcfy/'ytcjmlB /rth5'ytcfrlxl8iwlq2hegxl8nqXrl8e(¢~tytn21zheiwl8cfqYyl8cfextiYxtigcl8eh A¢hegewhytpcl8f(hyzjwhB fcf(c jml|rl8eewq8tiwhq2hc ".qcf*l8Ycy$8igcq Aª & H. B ,rtc+(¢hiwhyzjwh&rlytqYig*l H∞ ,z~thrth5'ytcfrlegqY(hyzjwh&xl8iwlqYeegcfekjwh*l8ehekj0$ * hcfeAª &?H. @ B
7
‖G(s)‖∞ :=hewesupω∈R
σ[G(jω)]A H. B
H. F @ u , b b \ k^ A¢ bu /Bekjwh^(jwqFrtqdq8ytewcegjwhh l8xtfcfpl8i~t*lxhikjw~tigl8npo8qdrtqjwcfxqcf(xt~tfegqKyF~t*lKrl8ehy jwiwl8rl8e
rtq2egcfekjwh*l. pl8f~tl8ytrtq|egh*q *|l8fqYicfytcfcfl8At −→ 0
B rl2ighegxqYekjml.eghY~tytrtqq ¥hqYigh*lrtql8fqYiytcfcl8
limt→0
y(t) = lims→∞
n∑
i=1
sGi(s) =
nr∑i
Ri,eghλi ∈ R
nc∑i
αi,eghλi ∈ C
A H. B
qYytrthRi
ighegfrt~tqhαihekj0$rth5'ytcfrtqyl( &?H. Fl8&qYeighl|jwc+*YqYel
λi
e*8l8fqYigherthegigcjwqYeyl & H. xl8iwlpl8rl(qFrtqt,egh-glighpl8q8~XqY(xtfh#".qtrth5*Yh'eghiqYigrthyl8rtqYe(rthO*l8ythcfiwlrthigheghy jwhYighegewl8jml8ytrtqqYe(rthO*l8cfq8iighfh5*6F8ytclXxl8iwlXq(qFrthfqt 8 H. H l*h0".h(xtfc+'pl*hekjmll8egegc+'pl8npo8qhl.8 H. H 2(qYekjwiwl.$xl8iwl*pl8rlqYigrth rtqYe(qFrthfqYedighrt~&crtqYeqYe*8l8fqYighedrtqYefcfc jml8y jwhedeg~txhigcfqYighe/rtqYehigigqYe/rth(qzrthfqqY(xl8iwl8rtqqY#qYeKxtiYxtigcfqYehigigqYe
0 10 20 30 40 50 60−220
−200
−180
−160
−140
−120
−100
−80
−60
−40
−20
IDM Valor Inicial (Ω(s)/Pmec
(s))
Ordem do modelo
dB
0 10 20 30 40 50 60−250
−200
−150
−100
−50
0
Ordem dos modelos reduzidos (Gr(s))
(dB
)
IDM Valor Inicial (Ω(s)/Pmec
(s))
Erro do modelo (||G−Gr||∞)
Limite superior do erro (Σ||Gi||∞)
AªlB A¢?B
8H. H.AªlB igrthyl8npo8q(rtq8eK bu dxl8iwll4(¢~tytnpo8qrthjwiwl8yte(¢higytclΩ(s)/Pmec(s)
hA¢?Bigigq(rtq(qFrthfqh fcfc jwh/eg~txhigcfqYirtq(higigqh#pl8rlqYigrthO
higigq(rtqYe(qFrthfqYeighrt~&cfrtqYeA 8H. H ?Byo8q (qYytq8jwqYytcfpl8hyzjwh/rthigheghyzjwhqYq&l8~t(hy jwqrlqYigrthOzytq&hy jml8y jwqt.qfcfc jml8y jwhKeg~txhigcfqYirtq&higigqtFz~thdhz~tc+*8l8fhdl8qegqY*l|jYigcfq
limt→0
h(t) = lims→∞
sH(s)e
!
−1.5 −1 −0.5 00
1
2
3
4
5
6
7
8
9
100.15 0.1 0.05
IDM Valor Inicial
Eixo real
Eix
o im
agin
ario
polos − modelo completo (62)polos − modelo reduzido (20)
8H. t^qY(xl8iwl8npo8qrtqYeKx YfqYertqYeK(qFrthfqYe l8fqYiytcfcl8
rl8e ytqYig*l8ecfy/'ytcjml8e(qzrl8cfertqYe x YfqYeyo8q|rtqY(cfyl8y jwheegh(xtigh(qYytq8jwqYytcfpl8(hy jwhrth igheghy jwhYekjml8epl8iwl8jwhigfekjwcfpl8eeghiwo8q jml8 qYteghi*8l8rl8e^xl8iwl q bu l8fqYi 8cfyl8 A Fhnpo8qH. F B^hqYig*l&y/'ytcjml A Fhnpo8qH. F HB
dl8H. t.l8xhyl8e^~t (qFrtqhfhjwigqY(h F8ytcfqqY (ªl|jwqYirth/l8(qYikjwhcfhyzjwq hjwqYigytqrth . yo|q (¢qYcKcfytf~tfrtqhy jwigh2qYeEx YfqYe(*l8cfe(rtqYcfyl8yzjwhe(qYigrthyl8rtqYe*eghY~tytrtqRqU bKu /m¥qYigOrtc+(¢highyzjwh(hy jwhrtq ^ ¥u lxqYegcfnpo8qXrtqYe(x YqYertq(qzrthfqXqYigcfYcyl8ewo8qxtigheghi*8l8rtqYe
H. F u , b b 8 A¢ bu 8 Bekjwh(jwqFrtq qYytegcfekjwhKh l8xtfcfpl8i ~t*lxhikjm~tigl8no8qrtqjwcfxq rthYiwl8~(yz~t*l rl|ehy jwiwl8rl8e
rtqegcfekjwh*lhpl8f~tl8idq.*8l8fqYi'yl8 rlighegxqYekjml At −→ ∞ B eghY~tytrtqqE¥hqYigh*lrtqEl8fqYi
8cfyl8
limt→∞
y(t) = lims→0
n∑
i=1
Gi(s) =
nr∑i
Ri
λi,eghλi ∈ R
nc∑i
βi
|λi|2 ,eghλi ∈ C
A H. @ B
qYytrthRi
qighegfrt~tq(hβihekj0$rth5'ytcfrtq(yl &?H. Fl8&qYeKl8egegqFcl8rtqYeKl
λi
8=H. h#".h(xtfc+'plOq(jwqFrtqrlO(heg*l (¢qYig*lz~thq buTl8y jwhigcfqYirthl8qYigrtqqY#qYe)*|l8fqYigheKrthewiwc jmq8eyl &?H. @ .
eigheg~tjml8rtqYehytq8yzjwiwl8rtqYe(qYekjwiwl8#z~thqYeKhigigqYertqYe(qFrthfqYeighrt~&cfrtqYedl8xtigheghy jml8qY (¢ighz~tytcl*l8xhyl8ed~t*lxhz~thyl*ighrt~tnpo8q*xl8iwlqYigrthyteeg~thegewc+*|l8eAqYxhytewl8npo8qt
lim
t→∞h(t) = lim
s→0sH(s)
e
0 10 20 30 40 50 60−260
−240
−220
−200
−180
−160
−140
−120
−100
−80
−60
IDM Valor Final (Ω/Pmec
)
Ordem do modelo
dB
0 10 20 30 40 50 60−250
−200
−150
−100
−50
0
Ordem dos modelos reduzidos (Gr(s))
(dB
)
IDM Valor Final (Ω(s)/Pmec
(s))
Erro do modelo (||G−Gr||∞)
Limite superior do erro (Σ||Gi||∞)
AªlB A¢?B
8¥ H. FA lB igrthyl8npo8qrtqYeK bu 8)xl|imll (¢~tytnpo8qrthjmiwl8yte(¢higytclΩ(s)/Pmec(s)
hA¢?Bigigq(rtq(qFrthfqh fcfc jwh/eg~txhigcfqYirtq(higigqh#pl8rlqYigrthO
−1.5 −1 −0.5 00
1
2
3
4
5
6
7
8
9
100.15 0.1 0.05
IDM Valor Final
Eixo real
Eix
o im
agin
ario
polos − modelo completo (62)polos − modelo reduzido (20)
8H. 7.^qY(xl8iwl8npo8qrtqYeKx YfqYertqYeK(qFrthfqYe l8fqYi)8cfyl8
l(cfyt~temo|q(rthrthjwhig(cfyl8rtqYeK(qFrtqYextigq *YqFpl(~t*lighrt~tnpo8q*egcfYytc+'pl|jwc+*8l.tqYy/(¢qYig(h (qYekjwiwll 8H. ,
^qYy/(¢qYig(hl 8?H. 7.xqFrth eghytq8jml8iz~thrtqYcfeK(qzrtqYeKhfhjwigqY(h F8ytcfqYeqY (ªl|jwqYigheKrthl8(qYikjwhcf(hy jwqhjwqYigytqrth yo8qE(¢qYiwl8l8egegc+'pl8rtqYexhfq bu 8HqY(qrtqY(cfyl8y jwherthy jwighqYex YfqYerthekjwh(qzrthfqighrt~&cfrtqt
H. F H u , b b duOW 8 ¥ A¢ buBekjwh(jwqFrtqhytqYy jwiwl~t>fytrtcfhh&l8ewl8rtqylKxtigqYxtigcfhrl8rthrlKrthegcfY~l8frl8rthjwigcl8ytY~tl8i
rl8e*ytqYig*l8epxhlz~l8qUegqY*l|jYigcfqXrl8e*ytqYig*l8ecfy/'ytcjml8e(rthpl8rlxl8ighlighekjml8y jwh2ylrtc+(¢highytnpl
G(s)−Gr(s)rth5'yth~t#fcfc jml8y jwh/eg~txhigcfqYiA
eBxl8iwlqhigigq(rth (qzrthfq A
eB
e = ‖G(s)−Gr(s)‖∞ = ‖G(s)‖∞ = ‖n−r∑
i=1
Gi(s)‖∞ ≤n−r∑
i=1
‖Gi(s)‖∞ = eA H. @@ B
q hy jml8y jwqtqYy/(¢qYig(hl &DH. @ z$~t*lrtc+(¢highytnplUhy jwighl|e*ytqYig*l8e(rl8e9(¢iwl8n21Fhexl8igcl8cfertqYex YfqYe^ighpl8cfe^hdrtqYex YfqYeqY(xtfh#".qYe^ylh#".xtighegemo8q
G(s).~6-gl8eD(¢qYig*l8e^l8yl8fjwcpl8e
ewo8qrth5'ytcfrl8e&yl2 &>H. @ Hhyl2 &>H. @ t ighegxhjwc+*8l8(hy jwh8xl8iwlqYen − r
x Yfq8e(hytqYertqY(cfyl8y jwhe
n−r∑
i=1
‖Gi(s)‖∞ =nr∑
i=1
‖Gri (s)‖∞ +
nc∑
i=nr+1
‖Gci(s)‖∞
A H. @ B
‖Gri‖∞ =
∥∥∥∥Ri
s− λi
∥∥∥∥∞
=
∣∣∣∣Ri
s− λi
∣∣∣∣s=0
=
∣∣∣∣Ri
λi
∣∣∣∣A H. @ HB
‖Gci‖∞ =
∣∣∣∣Ri
s− λi+
R∗is− λ∗i
∣∣∣∣s=jωd
A H. @ CB
qYytrthn − r = 2nc + nr
nchnrigh5(¢higheghl
GωdOl (¢ightytclXrtqXx Yfq2qY(xtfh#".q
h ‖Gci‖∞
h ‖Gri‖∞
qYigighegxqYytrthOighegxhjwc+*8l8(hy jwh8 l8qYe burtqYe(qFrtqYe qY(xtfh#".qYe highpl8cfe
fytrtcfhrthegigcjwq&h A @ Bd~.jwcffc+&pl8rtq jml8yzjwqxl8iwlq&(qzrtqighpl8z~l8yzjwq&xl8iwlpl8rl x Yfq rtq(qzrtq qY(xtfh#".qqYy6-k~t l8rtqeghxl8iwl8rl8(hy jwhKhKrl8rtqxqYi |Ri/Re(λi)|
¥qYiglytqYig*lcfy/'ytcjmlOrtqxl8iqY(xtfh#".q2qYy6-k~t l8rtq2xqYegeg~tc~t *8l8fq8itytcfq2xl8iwl2q2(qFrtqt z~thqYhrthhOl2rthegcfY~l8frl8rthEH. @ hz~thjwhqYigcfpl8(hy jwh(hytqYi&z~thqfcf(c jml|yzjwh*eg~txhigcfqYixqYihfhrth5'ytcfrtqt ekjwhE(ªl|jwqXhekj0$cff~tekjwiwl8rtq2yl:8 H. !z~th(qYekjwiwl2qXYi%$6'qXrtqYefcf(c jml|yzjwheeg~txhigcfqYigheKrtqYeKhigigqYeKrth (qzrthfqrtqFcfekjwh*l ¥hekjwhmxl8iwlpl8rl4*|l8fqYirth
r
‖Gci‖∞ =
∣∣∣∣Ri
s− λi+
R∗is− λ∗i
∣∣∣∣s=jωd︸ ︷︷ ︸
IDMNI
≤∣∣∣∣
Ri
s− λi
∣∣∣∣s=jωd
+
∣∣∣∣R∗i
s− λ∗i
∣∣∣∣s=jωd
A H. @ B
*l8e∣∣∣∣
Ri
s− λi
∣∣∣∣s=jωd
+
∣∣∣∣R∗i
s− λ∗i
∣∣∣∣s=jωd
=
∣∣∣∣Ri
Re(λi)
∣∣∣∣+∣∣∣∣
R∗i2jωd −Re(λ∗i )
∣∣∣∣ <∣∣∣∣
Ri
Re(λi)
∣∣∣∣+∣∣∣∣
R∗iRe(λ∗i )
∣∣∣∣︸ ︷︷ ︸
GREEN A H. @ 7B
‖a+ b‖ ≤ ‖a‖+ ‖b‖ e
6
0 10 20 30 40 50 6010
−12
10−10
10−8
10−6
10−4
10−2
100
Ordem dos modelos reduzidos (Gr(s))
dB
Limitantes Superiores dos Erros − IDMNI x GREEN (Ω(s)/Pmec
(s))
GREEN (Norma Infinita do pólo = Σ |Ri/Re(λ
i)|)
IDMNI (Norma Infinita do modo = Σ||Gi||∞)
8H. !F^qY(xl8iwl8npo8qrtqYeKfcf(c jml|yzjwheeg~txhigcfqYigheKrtqYeKhigigqYeK bu " tjwcf*l rthegcfY~l|frl8rthdrl& &/H. @ 7e&eghi%$&hekjwigcjmlytq&pl8egq&rthl8~.jwq*8l8fqYigheqY(xtfh#".qYe^qYxl8ikjwhighpl8¥rtc+(¢highy jwhrth &higqt$qz~th&~t*l(xtigh(cfegewl(ythekjwhjwiwl8l8ftqt- $*z~th&eghqYytegcfrthiwl8(qFrthfqYe
G(s) ∈ RL∞
0 10 20 30 40 50 60−250
−200
−150
−100
−50
0
IDM Norma Infinita (Ω(s)/Pmec
(s))
Ordem do modelo
dB
5 10 15 20 25 30 35 40 45 50 55 60−240
−220
−200
−180
−160
−140
−120
−100
−80
−60
−40
Ordem dos modelos reduzidos (Gr(s))
dB
IDM Norma Infinita (Ω(s) / Pmec
(s))
Erro do modelo (||G−Gr||
∞)
Limite superior do erro (Σ||Gi||
∞)
AªlB A¢?B
8H. .AªlB igrthyl8npo8q(rtq8eK bu2Kxl8iwll4(¢~tytnpo8qrthjwiwl8yte(¢higytclΩ(s)/Pmec(s)
hA¢?Bigigq(rtq(qFrthfqh fcfc jwh/eg~txhigcfqYirtq(higigqh#pl8rlqYigrthO
^qYy/(¢qYig(hl.8 H. .q* bul8xtigheghyzjwqY~,$xl8iwljwqFrl(qYigrthvrth&(qzrthfqtfcf(c jml|yzjwheeg~txhigcfqYighe (hytqYighe z~thqYe rth*l8cfe¥ buYrth5*Fcfrtql8q/egqY*l|jYigcfqdrtqYextiYxtigcfqYefytrtcfhe)A¢ytqYig*l8ecfy/'ytcjml(rtqYed(qFrtqYed(hytqYe/rtqY(cfyl8y jwhe0B eghi*Fcfytrtqrth&igh5(¢hiwytclxl8iwl*qYedhigigqYe/rtqYed bu /h bu 8^
@
−1.5 −1 −0.5 00
1
2
3
4
5
6
7
8
9
100.15 0.1 0.05
IDM Norma Infinita
Eixo real
Eix
o im
agin
ario
polos − modelo completo (62)polos − modelo reduzido (20)
8H. .^qY(xl8iwl8npo8qrtqYeKx YfqYertqYeK(qzrthfqYe^^qYig*l&y/'ytcjml.
¥ d?H. @ l8hlqY(xl8iwl|jwc+*|l&rtqYeK busxl8iwlqYigrth . 424 ` b4()>J*L # , &+* ` V *B 4(L ,.C7,.*B ` V *B 4(, #*B (+2!> V *B 4( (+2!> V *B 4( (+2!> V *B 4(
± K ± K ± K ± K ± K ± K ± K ± K ± K ± K ± K ± K ± K ± K
8=H. (qYekjwiwlOz~th| ythekjwhpl8egqxl8ikjwcf~tl8i¥lOfqzpl8fc;&l8npo8qOrtq8ex YfqYe (¢qYcl(heg*lz~thlrtq bu /migheg~tjml8ytrtqytq(heg(qhigigqrth (qFrthfqtFxqYijwhigh#q(heg(qqYy6-k~tyzjwqrthx YfqYe
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G(s) =∆Pij(s)
∆Bij(s)
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Resolver equação (autovetor à direita):
Resolver equação (autovetor à esquerda):
( ) ( )
( )
0( )
10 ( )
k ki i
ki
s I A b x s
c U s
− −
=
( ) ( )
( )
0( )
10 ( )
k t t ki i
t ki
s I A c v s
b U s
− −
=
Fatorização LDU:
( )
0
kis I A b
c
− −
Calcula os m novos autovalores estimados:( ) ( ) ( )
( ) ( ) ( )
( ) ( ) 1 ( )
( )
( )
( )
t
t
k k kmxm
k k kmxm
k k kmxm
kmxm
G V X
H V AX
B G H
B
−
=
=
=
Λ
( ) ( )1
( ) ( ) ( )
, 1,...,
( )i
k ki B
k k ki i i
s i m
Ax x tol
λ
λ+ = =
− ≤
k = k+1
i = 1, ... , m
k = 1Valores estimados iniciais :
( )1 , 1,...,k
is i m+ =
Resultado Final
8¥ H. @ .dfYqYigcjwqxl|imlq $8f~tfqrtqhegxhjwigq*rth x YfqYertqYcfyl8yzjwhe67
−14 −12 −10 −8 −6 −4 −2 0
−10
−5
0
5
10
0.9
0.45 0.150.10.05
0.9
0.45 0.150.10.05
Eixo real
Eix
o Im
agin
ario
Mapa de estimativas iniciais
−14 −12 −10 −8 −6 −4 −2 0
−10
−5
0
5
10
0.9
0.45 0.150.10.05
0.9
0.45 0.150.10.05
Eixo realE
ixo
Imag
inar
io
Mapa de pólos convergidos
AªlB A¢?B
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0 5 10 15 20 25 30 35 40−2
−1.5
−1
−0.5
0
0.5
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1.5
2x 10
−5
∆ P
ij (pu
)
G(s)=Bij(s)/P
ij(s)
modelo completomodelo reduzido 1 (ordem 24)modelo reduzido 2 (ordem 22)
10−1
100
101
−65
−60
−55
−50
−45
−40
−35
Frequência (rad/s)
Mag
nitu
de (
dB)
G(s)=Bij(s)/P
ij(s)
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B/qY~cfytekj0$ * hcfeA∆Ω(s)/∆Vref (s)
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=
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−Im(λ1) Re(λ1) 0 0 0 −Im(R1)√2
0 0 λ2 0 0 R2
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7
0 10 20 30 40 50 60−250
−200
−150
−100
−50
0
50IDM Norma Infinita
Ordem do modelo
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G1(s)
G2(s)
G3(s)
G4(s)
8 t @ buxl8iwll8eKz~l|jwigq4(¢~tytn21FheKrth jwiwl8yte(¢higytclF
0 10 20 30 40 50 60−250
−200
−150
−100
−50
0
50IDM Valor Inicial
Ordem do modelo
dB
G1(s)
G2(s)
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8 t F bu /xl8iwll8eKz~l|jwigq4(¢~tytn21FheKrth jwiwl8yte(¢higytclF
7 @
0 10 20 30 40 50 60−250
−200
−150
−100
−50
0
50IDM Valor Final
Ordem do modelo
dB
G1(s)
G2(s)
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8 t H. bu 84xl8iwll8eKz~l|jwigq4(¢~tytn21Fherthjwiwl8yte(¢higytcl.
0 10 20 30 40 50 60−250
−200
−150
−100
−50
0
50Valores Singulares de Hankel
Ordem do modelo
dB
G1(s)
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rtq z~th2hG1(s)
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rhekjml8rtqYe
A 'jwfcfqYe0B*l8cfe&cf(xqYikjml8y jwhe&xl8iwlXlq l2iwhqYytekjwiw~tno8qUrtqXqY(xqYikjml8(hyzjwq2hyzjwiwl8rl ewl8frll8xl8ighh+ytqqFrthfqighrt~&cfrtqt4¥qYicfegegqt^lh5*YqYf~tnpo8qXrtqYe4*8l8fqYigheegcfytY~tl8ighe&rth dl8yYh qYAql8~t(hy jwq*rth
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l8xtigheghy jml8ytrtqUqYe(higigqYeighl|jwc+*YqYe A ‖G(s)‖∞‖G(s)‖∞
B ^z~thOighxtigheghy jml8 lXighl8npo8qXhy jwighOqYe(hiwigqYerth(qFrthfq A ‖G(s)‖∞
B*hl ytqYig*lUcfy/'ytcjmlUrtqR(qFrthfqUqYigcfYcfyl8A ‖G(s)‖∞ B hekjwhpl8egqtdl8eytqYig*l8e&cfy/'ytcjml8e&rtqYe(qzrthfqYe&qYigcfYcfyl8ceewo8qt ‖G1(s)‖∞ = 0, 0132 ‖G2(s)‖∞ = 1, 0539
‖G3(s)‖∞ = 266, 92
h ‖G4(s)‖∞ = 196, 97
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uOjwqFrtqG1(s)
Ar = 20
BG2(s)
Ar = 27
BG3(s)
Ar = 15
BG4(s)
Ar = 35
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G3(s)Ar = 15
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• efcfc jml8y jwhe/eg~txhigcfqYigherth5'ytcfrtqYe xhfq bu/l8xtigheghyzjml8iwl8 *8l8fqYighe(hytqYighe z~thqYe* bu /h bu 8^xl8iwlz~l8fz~thi*qYigrth rtqYe*(qFrthfqYeighrt~&cfrtqYe*hxl8iwlXjwqzrl8el8e(¢~tytn21Fhe&rth(jwiwl8yte(¢higytcl.2ekjwhigheg~tjml8rtq2hiwlhegxhiwl8rtqt~t*l * h2&z~thq buqYigrthylqYe(qFrtqYeeghY~tytrtq(eg~l8eytqYig*l8ecfy/'ytcjml8e Aª &?H. !B
7
0 5 10 15 20 25 30 35 400
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Ordem dos modelos reduzidos (Gr(s))
Σ||G
i|| ∞
Limite superior dos erros modais dos modelos G1(s)
IDM Valor FinalIDM Valor InicialIDM Norma InfinitaTruncamento Balanceado
Ordem Escolhida
0 5 10 15 20 25 30 35 400
0.5
1
1.5
2
2.5
3
Ordem dos modelos reduzidos (Gr(s))
Σ||G
i(s)|
| ∞
Limite superior dos erros modais dos modelos G2(s)
Ordem Escolhida
IDM Valor FinalIDM Valor InicialIDM Norma InfinitaTruncamento Balanceado
AªlB A¢?B
0 5 10 15 20 25 30 35 400
10
20
30
40
50
60
70
80
90
100
Ordem dos modelos reduzidos (Gr(s))
Σ||G
i(s)|
| ∞
Limite superior dos erros modais dos modelos G3(s)
Ordem Escolhida
IDM Valor FinalIDM Valor InicialIDM Norma InfinitaTruncamento Balanceado
0 5 10 15 20 25 30 35 400
50
100
150
200
250
300
350
400
450
Ordem dos modelos reduzidos (Gr(s))
Σ||G
i(s)|
| ∞
Limite superior dos erros modais dos modelos G4(s)
Ordem Escolhida
IDM Valor FinalIDM Valor InicialIDM Norma InfinitaTruncamento Balanceado
A¢B A¢r?B
8 t F cf(c jml8y jwheeg~txhigcfqYigheKrtqYeKhigigqYeKxl8iwl.AªlBG1(s)
A¢?BG2(s)
A¢B
G3(s)hA¢r?B
G4(s)
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• /~l8ytrtq4qY(xl8iwl8rtqRqYCqYe(jwqFrtqYe b uKlUjwytcfpl ^ ¥u xqYegeg~tcfcf(c jml|yzjwheeg~txhigcfqYighe^(hytqYighexl8iwlqYe^higigqYehy jwigh/qYe(qzrthfqYeqY(xtfhjwqh/ighrt~&cfrtqYe.xqYcfe^q&egh~
7
0 5 10 15 20 25 30 35 400
1
2
3
4
5
6
7
8
9x 10
−3
Ordem dos modelos reduzidos (Gr(s))
||G(s
)−G
r(s)|
| ∞
Erros dos Modelos G1(s)
IDM Valor FinalIDM Valor InicialIDM Norma InfinitaTruncamento Balanceado
0 5 10 15 20 25 30 35 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ordem dos modelos reduzidos (Gr(s))
||G(s
)−G
r(s)|
| ∞
Erros dos Modelos G2(s)
IDM Valor FinalIDM Valor InicialIDM Norma InfinitaTruncamento Balanceado
AªlB A¢?B
0 5 10 15 20 25 30 35 400
5
10
15
20
25
30
35
40
45
50
55
Ordem dos modelos reduzidos (Gr(s))
||G(s
)−G
r(s)|
| ∞
Erros dos Modelos G3(s)
IDM Valor FinalIDM Valor InicialIDM Norma InfinitaTruncamento Balanceado
0 5 10 15 20 25 30 35 400
20
40
60
80
100
120
140
160
180
200
Ordem dos modelos reduzidos (Gr(s))
||G(s
)−G
r(s)|
| ∞
Erros dos Modelos G4(s)
IDM Valor FinalIDM Valor InicialIDM Norma InfinitaTruncamento Balanceado
A¢B A¢r?B
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=A¢?BG2(s)
A¢BG3(s)
hA¢r?BG4(s)
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e =r∑i=1
‖Gi(s)‖∞ < ε ×n∑i=1
‖Gi(s)‖∞B qpl8egqrthekjwh
egcfekjwh*l jwhekjwh8eg~tYhigh eghε = 1%
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r1 = 18r2 = 23
r3 = 18
hr4 = 40
B
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∥∥∥∞
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7
10−2
10−1
100
101
−100
−90
−80
−70
−60
−50
−40
Mag
nitu
de (
dB)
Diagrama de Bode (Modulo) − G1(s) para ordem 20
10−2
10−1
100
101
−50
0
50
Frequencia (rad/s)
Ang
ulo
(o )
Diagrama de Bode (Fase) − G1(s) para ordem 20
Truncamento BalanceadoIDM norma infinitaIDM valor inicialIDM valor finalModelo Completo (62)
2 4 6 8 10 12 14−35
−30
−25
−20
−15
−10
−5
0
Mag
nitu
de (
dB)
Diagrama de Bode (Modulo) − G2(s) para ordem 27
2 4 6 8 10 12 14
−200
−150
−100
−50
Frequencia (rad/s)
Ang
ulo
(o )
Diagrama de Bode (Fase) − G2(s) para ordem 27
Truncamento BalanceadoIDM norma infinitaIDM valor inicialIDM valor finalModelo Completo (62)
AªlB A¢?B
10−1
100
101
−30
−20
−10
0
10
20
30
40
Mag
nitu
de (
dB)
Diagrama de Bode (Modulo) − G3(s) para ordem 15
10−1
100
101
−600
−500
−400
−300
−200
−100
0
100
Frequencia (rad/s)
Ang
ulo
(o )
Diagrama de Bode (Fase) − G3(s) para ordem 15
Truncamento BalanceadoIDM norma infinitaIDM valor inicialIDM valor finalModelo Completo (62)
2 4 6 8 10 12 14
10
20
30
40M
agni
tude
(dB
)
Diagrama de Bode (Modulo) − G4(s) para ordem 35
2 4 6 8 10 12 14
−500
−400
−300
−200
−100
Frequencia (rad/s)
Ang
ulo
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Diagrama de Bode (Fase) − G4(s) para ordem 35
BALMRIDMNIIDMVIIDMVFModelo Completo
A¢B A¢r?B
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A¢qYigrth B=A¢?BG2(s)
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A¢qYigrth @ B A¢r?BG4(s)
A¢qYigrth HB
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pólos do modelo completopólos encontrados pelo DPSE
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20 40 60 80 100 120 140
−80
−60
−40
dB
Ω/Vref
(P. Afonso IV)
Modos ordenados segundo o IDMNIModos encontrados pelo DPSE
20 40 60 80 100 120 140
−80
−60
−40
−20
dB
Modos ordenados segundo o IDMVIModos encontrados pelo DPSE
20 40 60 80 100 120 140
−80
−60
−40
Ordem do modelo completo
dB
Modos ordenados segundo o IDMVFModos encontrados pelo DPSE
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20 40 60 80 100 120 140
−80
−60
−40
dB
Ω/Vref
(Xingo)
Modos ordenados segundo o IDMNIModos encontrados pelo DPSE
20 40 60 80 100 120 140
−80
−60
−40
dB
Modos ordenados segundo o IDMVIModos encontrados pelo DPSE
20 40 60 80 100 120 140
−80
−60
−40
Ordem do modelo completo
dB
Modos ordenados segundo o IDMVFModos encontrados pelo DPSE
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20 40 60 80 100 120 140−100
−80
−60
−40
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Ω/Vref
(Itaparica)
Modos ordenados segundo o IDMNIModos encontrados pelo DPSE
20 40 60 80 100 120 140−100
−80
−60
−40
dB
Modos ordenados segundo o IDMVIModos encontrados pelo DPSE
20 40 60 80 100 120 140−100
−80
−60
−40
Ordem do modelo completo
dB
Modos ordenados segundo o IDMVFModos encontrados pelo DPSE
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IDMNIIDMVIIDMVF
0 5 10 15 20 25 3010
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10−2
10−1
100
F1 (s) : Gráficos dos Erros e seus Limitantes Superiores
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Ordem do modelo reduzido
Erros de modelos − espectro dos pólos do modelo completoLimitantes Superiores dos Erros − espectro dos pólos do modelo completoErros de modelo − espectro dos pólos encontrados pelo DPSELimitantes Superiores dos Erros − espectro dos pólos encontrados pelo DPSE
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Ordem do modelo reduzido
IDMNIIDMVIIDMVF
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10−1
100
F2 (s) : Gráficos dos Erros e seus Limitantes Superiores
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Ordem do modelo reduzido
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0 5 10 15 20 25 30
10−3
10−2
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F3 (s) : Gráficos dos Erros e seus Limitantes Superiores
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Ordem do modelo reduzido
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10−2
10−1
100
101
−50
−45
−40
−35
−30
−25
Mag
nitu
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10−2
10−1
100
101
−150
−100
−50
0
50
100
150
Frequencia (rad/s)
Fas
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Modelo reduzido − ordem 32 (pólos encontrados pelo DPSE)Modelo reduzido − ordem 32 (pólos ordenados pelo IDMNI)Modelo completo (ordem 1662)
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−85
−80
−75
−70
−65
−60
−55
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Frequencia (rad/s)
Mag
nitu
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Diagrama de Maginute de Bode para os Erros
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ωo = 0.01rad/s
ωo = 0.01rad/s
ωc = 4.15rad/s
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10−1
100
101
−80
−75
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−55
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Frequência (rad/s)
Mag
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ALPHABETAGAMMADELTAMODELO COMPLETO
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−60
−55
−50
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Freqüência (rad/s)
Mag
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ALPHABETAGAMMADELTAModelo Completo
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B qY.jwc+*Yhiwl8 ighegxqYekjml8e*l8cfe xti".cf*l8el8q (qFrthfqqYigcfYcfyl8ªYeghytrtqz~th8xhl4¥ d t .tqYeK(qFrthfqYe ¥ hu2uO l|xtiwhewhyzjml8#higigqYeK(hytqYighe
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−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.60
0.5
1
1.50.2 0.15 0.1 0.05
Mapa de pólos e zeros
Eixo Real
Eix
o Im
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Pólos do modelo completoZeros do modelo ALPHAZeros do modelo BETAZeros do modelo GAMMAZeros do modelo DELTAZeros do modelo completo
λ12
(Modo Norte−Sul)
λ34
(Modo de Interação Adversa)
8 t .^qY(xl8iwl8npo8qrtqYe&higqYeKrtqY(cfyl8y jwhertq((qzrthfqqY(xtfhjwqqY#qYe&higqYertq8e(qFrthfqYeighrt~&crtqYe
^qYy/(¢qYig(hl 8 t . q&higqrtq(qFrthfqqY(xtfhjwq xti ".cf(ql8q (qFrtqλ34
l8xl8ighh8FytqYe(qFrthfqYe ighrt~&cfrtqYez~th*qxqYegeg~thOhs*l8cfe xti ".cf(qOrthekjwh*(qFrtqtekjwh9(ªl|jwql8jwhiwlqYeiwl8(qYe/rtqf~t l8idrl8eiwl8;&hehegjw~trl8rtq8eA 8 t @ h t Bh8,qYyteghthpy jwh(hyzjwh8 cfy ~thytclytqxtigq-khjwqrth qYy jwigqYl8rtqYighe
H
qYyzjwigqYfh~.jwcffc;&pl8rtqKylcfyzjwhigfc l8npo8qKytqYikjwh eg~tA t)B 8rthegigcjwqdh uO /EA BxqYegeg~tc$qY(q(¢~tytno8qxtigcfytcfxl8ql8(qYikjwhcf(hy jwqrth/qYegcfl8n21Fhe^hfhjwigqY(h F8ytcpl8e A 2%,,4' % #. %, ! b B 8qYy/(¢qYig(hl & t 7. ekjwhqYy jwigqYl8rtqYiKighegxqYyte%$*Yhxhfql8(qYi jwhcf(hy jwq&rtq&(qzrtq&qYikjwh kF~t>A
λ12 = −0, 0335± 1, 0787jBrth(¢qYig*l z~thl8fpl8ythd~t (ªl|jwqYi
rthl|(qYikjwhcf(hy jwq*l8cfqYi^z~th @ l8iwlcfegegqtthfh/~.jwcffc;&plqY(xhytewl8npo8q(rth(ªl8egh8l|jwiwl *YeKrthtfqFqYerthl|jwiwl8egqh#egigcfh qY#~t#tfqzq '
POD(s) = K5s
1 + 5s
(1 + 0, 3s
1 + 2, 2s
)2 A t 7B
f~t l8i&rl8eiwl8;&hertqX(qFrthfq2rthqYigrth+qY(xtfhjml (¢qYc^qY.jwcfrtqXxhfqb qYytrthl8xhyl8e(rtqYcfe(x YfqYe(igjwcfqYe A
λ12hλ34
B qYxt~.j l|rtqYe(xhfqUb/. (¢qYiwl8Ll8yl8fcfewl8rtqYe eiwl8(qYeqY.jwcfrtqYeqYytegcfrthiwl8iwl8 *8l8fqYighertcfegighjwqYexl8iwlqYe l8yttqYertqU bqYy/(¢qYig(hl8 t @ q8jwh*z~th8xl8iwl
K = 896qO(qFrtqqYikjwh kF~tA
λ12B l8pl8ytnpl @ rth (ªl|jwqYi rth
l8(qYikjwhcf(hy jwqt$hytz~l8y jwqOz~thq(qzrtqrthcfyzjwhiwl8npo8ql8r/* higewl Aλ34
B*8l8c xl8iwlcfytekjml8tcffcrl8rthz~l8ytrtq(q l8yttqrtq* bW~tjwiwl8xl8egewl
K = 2.965
ef~t l8igherl8eiwl8;&hertqYez~l|jwigq(qFrthfqYeighrt~&cfrtqYehekjmo8q(qYekjwiwl8rtqYeyl48 t F.highegewl8jml8'lcf(xqYikj0F8ytclrthpl8rlqYtcfyl8npo8qrth(x YfqYe xl8iwllqY.jwhytnpo8qOrth(hz~tc+*8l8hyzjwhel8rthz~l8rtqYedl8qxtigq6-khjwqrth qYyzjwigqYl8rtqYighe
8 t @ ~t l8irl8eKiwl8;&hexl8iwlq(qFrthqrth qYigrth#q8(xtfhjmlqY#q( b4A¢e0BqY.jwcfrtqxhfq(db
'";&)b1B 4C74U<()"%?4 #")M R!B?4( *&)! 0#*(c1#*, 6*" ()9I! #C7,.*" #S4=?Q!()>@, &), #248950<48C74 &)()4B.*24(c*&)0<?#*(+*R*(),.*Z7$"/B ! &+*"b24J"%, #*B\2! &)(+*2* W∆Pij Y e
Lugar das Raizes
Eixo Real
Eix
o Im
agin
ario
−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
20.2 0.15 0.1 0.05
K=838
K=6220
Lugar das Raizes
Eixo Real
Eix
o Im
agin
ario
−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
20.2 0.15 0.1 0.05
K=863
K=3740
AªlB A¢?BLugar das Raizes
Eixo Real
Eix
o Im
agin
ario
−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
20.2 0.15 0.1 0.05
K = 897
K = 2810
Lugar das Raizes
Eixo Real
Eix
o Im
agin
ario
−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
20.2 0.15 0.1 0.05
K=877
K=2930
A¢B A¢r?B
8 t F ~t l8igheKrl8eKiwl8;&hexl8iwlqYeK(qFrthfqYeighrt~&cfrtqYeKqY(xhytewl8rtqYeKqY#q( b4A¢e0B AªlB^ =A¢?B ¥=A¢Bu2uOWhA¢r?Bb/ ¥
! ! ^qYigighegxqYytrth*l~t (qFrthfqrthqYigrth z~th8¥l8xhewl8irthjwhil&eghY~tytrl*l8cfqYiqYigrth rthyzjwighqYel8xtigheghy jml8rtqYeyo8qqY.jwh5*YhighegxqYekjml8eytq&jwh(xqhyl(¢ightytcl ewl|jwcfe(ªl|jYigcl|eYrth5*Fcfrtq l8~tegytclrtq (qzrtqrthcfyzjwhiwl8npo|q l8r/* higewl.^qYighl8npo8ql8qOf~t l8irl8eiwl8;&he.A 8 t lB q(qzrthfqjwhytrthlOeg~ttekjwcjw~tcfi qOiwl8(qOrtq(qFrtqOrth(cjwhiwl8npo8ql8r/* higewlxqYiqY~.jwigqiwl8(qt ~6-glqYigcfYh (~tsx YqOiwhl8 ¥qYigsq l8yttq
K = 6220z~thqYytrt~& (cfytekjml8tcfcfrl|rth l8xtigq ".cf*l8rl8(hy jwh rt~l8e* h2&he/*l|cfqYi
z~thq*rtq*(qFrthfq(qYigcfYcyl8ªq*z~thxqFrthqYytrt~&cfidq*egcfekjwh*lighpl8¥hv*l8l4(¢hml8rl cfytekjml8tcffcrl8rth8.rthxhytrthyrtq*rtq l8yttqt
! %& , rthqYigrth 7hegh~tex YfqYe>(¢qYiwl8 hegqYftcfrtqYerth)(¢qYig*lz~th|
l8igheghy jml8ytrtqq(qzrtq rthKcfy jwhiwl8npo8ql8r/* hiwewl.Fl8eighegxqYekjml8eytq rtqY(fytcfq/rtq jwh(xqhKytqrl (¢ightytcleghKjwqYigyl8(l8cfexti".cf*l8e 8eighegxqYekjml8e^rtqqFrthfq rthqYigrthqY(xtfhjml.xqYegeg~tcfytrtq~t#higigq(xhz~thytq*hyzjwighl8eKighegxq8egjml8edxhewl8irthOxqYegeg~tcfi~t+higigq(hytqYihighxtigheghy jml8i*qiwl8(qrtq(qFrtql8r/*Yhigegq A 8t ?B qf~t l8idrl8e/iwl8;&hedyo8qigh hjwh~t*l**l8igYhvrthhekj l|tcffcfrl8rthhz~tc+*8l8fhy jwhl8q(qFrthqq8iwcYcfyl8ªFjwhytrtq(x YfqYeegqYtigh q(hc ".qcf*l8Ycy$8igcq&xl8iwl~t# l8yttq
K = 3740
! 4 W W 2ekjwh (qFrthfq)jwh qYigrth )h (hftqYiOighxtigheghyzjwqY~ lrtcfyF8(cplrtqegcfekjwh*lhsighl8npo8ql8qYe&l8y jwhigcfqYighe¥qYigOeghsq(qFrtqOcfy jwhi xtl8yzjmlrl $8ighpl ^hf¥~t~tig~t A
λ20 = −1, 4463± 1, 4565jB^rthegigcjwqyl ¥ d t .ll8y$8fcfegh
rl8eKighegxqYekjml8eKxl8iwl~t*lxhikjw~tl8npo8q(yl8eK~tegcfyl8erth ¥~t~tig~t eghjwqYigylcfxtiwhcfewl.¥qFrth egh^ytq8jml8i|yl8 t @ Kz~th8Yxl8iwlqYe¥(qFrthfqYe )h ¥|qYe¥egh~te ighegxhjwc *YqYe>&higqYeighxtigheghy jml8rtqYexhfqYe(fytcfqYe¥fqFpl8cfe¥rl8e ~ti*|l8eAªl8xtigq ".cf*l8rl8(hy jwhDH/iwl8r?< e#BxqYegeg~thv~tAl6(ªl8ekjml8(hyzjwqhAighl8npo8q 4(¢ightytcl(rtq9&higq*rtq(egcfekjwh*lqY(xtfhjwqt8 t qYy/'ig*lhekjwh rthegfqFpl8(hy jwqt f~t l8irl8e^iwl8;&he^xl|imlhekjwh/hz~tc+*|l8fhy jwhd(qFrl8 A 8 t Bighxtigheghy jwqY~,trth(¢qYig*l*l8cfel8rthz~l8rl.qf~t l8irl8eiwl8;&hertq(qFrthfq&qYigcfYcfyl8ª l8y jwqq l8yttqrtq( bWz~thqYytrt~&q(qzrtqqYikjwh kF~tFxl8iwldq(ªl|jwqYi¥rthl8(qYikjwhcf(hy jwqdrth @ A
K = 896B Yz~l8y jwq
q l8yttqz~th&cfytekjml8tcffc+&plqegcfekjwh*l AK = 2810
Bhekjmo8qxti ".cf(qYedrtqYedqYigcfYcfyl8ceeghytrtql*l8igYhGrth hekjml8tcffcfrl8rth/*l8cfeqYyteghi*8l8rtqYiwl.
! $^ 2ekjwh(qFrthfqOrth*qYigrth H 2*qY(xqYekjwq2xqYijwqFrtqYeqYex |fqYertqY(cfyl8y jwhextigh5*Fcl8(hy jwh rthjwhig(cfyl8rtqYexhfq(b/. , cf(xq8igjml|yzjwh/ytq8jml8iz~thhfhxqYegeg~tcqYe(qFrtqYe
λ20hλ34
ythhege0$8igcfq8exl8iwl ~t*lqY(xtfh j l/ighxtigheghy jml8npo8qrtqegcfekjwh*l.ekjwh(qFrthfqjml8&l8xtigheghy jmlrtc+(¢highytnpl8e*l8cfqYighehy jwighdl8eighegxqYekjml8e.xqYigO qY(qeghi%$4*Fcfekjwqyl(l8y$8fcegh/jwiwl8ytegcjYigclEA Fhnpo8q t F HB thekjml8eKrtc+(¢highytnpl8eKyo8q(ewo8qighfh5*8l8y jwhe&qYiwlqX(qFrthfq ¥vxqYegeg~lXqYe(qFrtqYertqY(cfyl8y jwhecf(xqYikjml8y jwhexl8iwlXl8y$8fcfeghqY(xtfhjmlrtq*egcfekjwh*lhl8cfytrlhigigq*(hytqYilrtc+(¢highytnpl($8egcfplhy jwighq((qFrthfq ¥hqb/ ¥ hekj0$2ytq2f~t l8irl8e&iwl8;&he A 8 t r?B xqYcfe&l2*l|iw8hsrth l8yttqrtq(qFrthq(b/ C¥ (¢qYc,egcf(cffl8il8q(qFrthfqqYigcfYcfyl8ª
ekjwheh#".h(xtfqYedxhig(cjwhAqYytf~tcfiz~thqYed(hytqYighe/higigqYedrth&(qFrthfq(yth egh(xtigh&pl iwl8jwhigc;&pl8A~tAhz~tc+*|l8fhyzjwh(qzrl8ewl|jwcfe(ªl|jYigcfqz~thighxtigheghy jwhl8e*l8igYhyterthhekjml8tcffcfrl8rthrtq egcfekjwh*lxl8iwlU~t xtigq-khjwqRl8rthz~l8rtq rth2qYy jwigqYl8rtqYighe>ekjwh2~t cfytrtfcfqrth2z~thlxtigheghi*8l8npo8qrtqYe &higqYertqY(cfyl8y jwhe xqFrthjml8 rthegh(xhytl8i~t xl8xhighfh5*|l8yzjwhxl8iwl
7
l q lUiwhxtiwhewhyzjml8npo8q4rl8ertcfyF8(cpl8e*cf(xqYikjml8y jwhe*rtqR(qFrthfq q8(xtfhjwqt ekjwhigheg~tjml8rtqxqzrth/eghiqYteghi*8l8rtql|jwiwl * ertqYe^(qFrthfqYeighrt~&cfrtqYe ¥>hb/ ¥ (qzrthfq4u2uOl8xtigheghy jwqY~Ujml8y jwqOq(hytqYi higigqz~l8y jwq2l*l8igYh rth( l8yttqqY(xl|jw+*YhcfeqY'qYe *8l8fqYigheighegxhjwc+*YqYertq(qzrthfqqYigcfYcfyl8ª^qYy jw~trtqt~t*lOl8y$8fcfeghrl8eiwhewxqYekjml8ejwiwl8ytegcjYigcl8eegh9(ªl&ythhege%$8igclxl8iwlqYy/'ig*l8ieghhekjwh q(hftqYihz~tc+*8l8fhy jwh (qFrl8$rthyzjwigh qYeKhekjw~trl8rtqYe
bdh(qFrtqll *8l8fcl8idqigheg~tjml8rtqrll8xtfcfpl8npo8qrtq bu/l8qqYy6-k~ty jwqOrthx YfqYertqY(c yl8y jwhehytqYy jwiwl8rtqYe *Fclb/.l 8 t H(qYekjwiwl2l2qYigrthyl8npo8qrthjwqzrtqqhegxhjwigqXrthx YfqYertq(qFrthqqYigcfYcfyl8ªrthekjml8pl8ytrtqlfqzpl8fc;&pl8npo8qrtqYe(qFrtqYe rtqY(cfyl8y jwhe rthegigcjwqYeyl¥ d t .qzrth eghxhighhidz~thl8fY~tytex YfqYeewo8qqYytegcfrthiwl8rtqYeKrtqY(cfyl8yzjwhexhfq bumxqYig#yo8q4(¢qYiwl8Gpl8x.jw~tiwl8rtqYeKytq(db
8 t H(qYekjwiwl&l8xhyl8eqYe @ x YfqYe^*l8cfertqY(cfyl8y jwhe^rtq(qzrthfq&qY(xthjwq&qYigrth yl8rtqYeeghY~tytrtqq buA * hig(hftqCBh l&xqYegcfnpo8q&rthpl8rlx Yfq&qY.jwcfrtq&xhfqb/. AªxtighjwqCB qYy/(¢qYig(h&hekjml(heg*l*qYigrthyl8npo8qt (qzrtqrthcfy jwhiwl8npo8ql8r/*Yhigewl A
λ34Bxqzrtheghi*Fcfekjmqyl
qYigrthyl8npo8qrtq bum xqYigO eghigcl/ythhege%$8igcfq ighjwhil8xtigq ".cf*l8rl8(hy jwh)/x YfqYeYpl8egq (¢qYegeghhegqYftcfrl~t*lqYigrthsfytcf(lythhege%$8igclxl8iwlqY.jwhytnpo8qrth*~t'(qFrthfqxqYi bu z~thxqYegeg~lhekjwh(qzrtqtbdhy jwighqYe H x YfqYedqY.jwcfrtqYedxhfqb/.lxqYegcfnpo8qeghY~tytrtqq burthekjwh(qFrtqdKljwigcfYegcf*l.|eghytrtqt hy jmo8qtYythhege0$8igcfqhegqYfthi ~t (qFrthfqdighrt~&cfrtqrthqYigrthH.txl8iwlxtigheghi*8l8iq(qFrtq
λ34
10 20 30 40 50 60 70 80 90 100 110 120
−101.5
−101.6
−101.7
−101.8
−101.9
dB
∆ Pij/∆ B
ij
λ2 = −2.9445+4.8214iλ
4 = −3.1928+9.2818iλ
6 = −2.60331+10.7219i
λ8 = −6.42313+8.69489iλ
10 = −4.51541+11.7469i
λ12
= −0.033534+1.0787i
λ16
= −5.5632+7.751i
λ18
= −0.75843+4.9367iλ
22 = −0.55674+3.6097i
λ32
= −0.61201+0.35873i
λ34
= −0.11506+0.23972i
λ26
= −1.2936+1.4028i
λ20
= −1.4463+1.4565i
λ28
= −4.5467+5.7598i
λ24
= −1.2786+7.2546i
λ30
= −1.2891+8.5414i
λ14
= −4.9526+7.1481i
Pólos ordenados segundo IDMNI Modos encontrados pelo DPSE
8 t H. igrthyl8npo8q(rtqYeKx YfqYertq(qFrthfqqY(xtfhjwqrtqFcfekjwh*l ¥hekjwhg^xhfq(jwqFrtq bum!
xl8ikjwcfirtqYe(qzrthfqYe rth(qYigrthyteighrt~&cfrl8ertcfekjwcfy jml8e qY.jwcfrtqYe qY'l8egh*ytqYex YfqYe rtqhegxhjwigqqY(xtfhjwqqYigrthyl8rtqYe*Fcl( buhqYe(qzrthfqYeKYhiwl8rtqYexhfqYeKx YfqYeKrl9 d t .qYigrthyl8rtqYedeghY~tytrtqq*(heg(q* buqYedhigigqYedhy jwigh&l8edighegxqYekjml8e/yl (¢ightytcl*hegh~te/fc(c jml8y jwheeg~txhigcfqYighe (¢qYiwl8 pl8f~tl8rtqYexl8iwl pl8rlqYigrth *|l8igcl8ytrtqrthlH tzqYy/(¢q8iwhK(qYekjwiwll8t t ¥qFrth eghytq8jml8iz~th8 xl8iwldqYigrthyte rth(qFrthfqYe¥hyzjwigh)dhF8qYe higigqYe rtqYe (qFrthfqYeYhiwl8rtqYedl(xl8ikjwcfiKrtq*hegxhjwigqxl8igcl8DAªb/. BKewo8q*(hytqYigheKegegq(qYekjwiwlz~th&l8fY~tytex YfqYehytqYy jwiwl8rtqYexhfq&b/.tl8xl8ighyzjwh(hy jwhd(hytqYe^rtqY(cfyl8y jwheeghY~tytrtq&q&igcjwigcfq bum.eghqY&tcfyl8rtqYel8rthz~l8rl8(hy jwh8l8fpl8ytnpl8 igheg~tjml8rtqYel8cfytrl(hftqYighey jwighjml8yzjwqt¥qYefc(c jml8y jwheeg~txhigcfqYigherthegjwhehigigqYeewo8q&*l8cfqYighertq&z~th/qYe^rtqYe(qzrthfqYeiwhrt~&cfrtqYeqY x YfqYe^rtqhegxhjwigqjmq|j l|q8iwrthyl8rtqYe*Fcl& bu2K
0 5 10 15 20 25 30
10−3
10−2
10−1
Gráficos dos Erros e seus Limitantes
dB
Ordem do modelo reduzido
Erros de modelo − espectro dos pólos encontrados pelo DPSE Limitantes Superiores dos Erros − espectro dos pólos encontrados pelo DPSE Erros de modelos − espectro dos pólos do modelo completo Limitantes Superiores dos Erros − espectro dos pólos do modelo completo
8 t tigigqYehegh~teKfcf(c jml8y jwheeg~txhigcfqYigheKrth(qzrthfqYeqY.jwcfrtqYeeghY~tytrtqqYigrthyl8npo8qxqYi buh rtqYcfehegxhjwigqYe jwqFrtqYeqYex YfqYertq(qFrthfq qYigcfYcfyl8thqYehytqYy jwiwl8rtqYe *Fclb/.
(qFrthfqrthOqYigrth HXqY.jwcfrtqXxhlqYigrthyl8npo8qrtqYe(x YfqYerl: dt *Fcl buqY.jwh5*Yhl8eighegxqYekjml8eKytqjwh(xqhyl(¢iwhtytclrth l8qYigrtqqY#l48 t p l&hqf~t l8i^rl8eiwl8;&he eghY~tytrtql 8 t ,*l8igYhrth(hegjml8tcfcfrl8rth(l*l8cfe/qYyteghi*8l8rtqYiwlrthyzjwighqYe rth*l8cfe/(qFrthfqYe/ighrt~&cfrtqYeA¢ ¥=u2uh(b/ ¥ rthegigcjwqYe ylE¥ dt B txqYig#xq8eweg~tcq((l8cfqYihigigqhyzjwighl8eKighegxqYekjml8e(¢ighthytcl8cfeA
eω = 13, 162× 10−4B
0 5 10 15 20 25 30 35 40−2
−1
0
1
2
3
4
x 10−5
Tempo (s)
∆ ω
(pu
)
Resposta ao degrau − Modelo DPSE+IDMNI − ( 30 )
10−2
10−1
100
101
−70
−65
−60
−55
−50
−45
−40
−35
Freqüência (rad/s)
Mag
nitu
de (
dB)
Diagrama de Bode − Modelo DPSE+IDMNI − ( 30 )
modelo completomodelo reduzido
Lugar das Raízes − DPSE+IDMNI (30)
Eixo Real
Eix
o Im
agin
ário
−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
K=881
K=2730
AªlB A¢?B
8 t F Kheg~tjml8rtqYertq((qzrthfqighrt~&cfrtq(rth qYigrth H~.jwcffc+&pl8ytrtq bu AªlB^ighegxqYekjmlyl4(¢ightytclhytqjwh(xqt A¢?B^f~t l8irl8eKiwl8;&he
t F H / k. K//F K uuO G8 D/bK uOk ekjmleghnpo8q*l8xtigheghyzjml~t*ll8y$8fcfegh/rl8eKighegxqYekjml8ejwiwl8ytegcjYigcl8ertqYeK(qFrthfqYeqY(xtfhjwqh
ighrt~&cfrtqYehG*l8fl (¢h l8rl&qY qqYyzjwigqYl8rtqYirthegigcjwqyl & t 7. 8 t 7(qYekjwiwlqrtcl8Yiwl8(l %,+ %, ~.jwcffc+&pl8rtqxl8iwl2egcf&~tl8n21zherthighegxqYekjml8el2~t*lxhikjw~tl8npo8qXrth~txqYihy jwq(ylxq8jwytcfl(h F8ytcfpl A
∆Pmec
BytqYeKhc ".qYertqYeKYhiwl8rtqYigheKrl8eK~tegcfyl8eKrth ¥~t~tig~t
8 t 7.bdcl|Yiwl8*l8erth tfqFqYertq(egcfekjwh*l&hA*l8fl(¢hml8rl~.jwcffc;&pl8rtq&ytq %,+ %,
(¢~tytnpo8qrth jwiwl8yte(¢higytclKrthegigcjmlyl)¥ dt @ D(¢qYcY~.jwcffc;&pl8rl^qY\qcyzjw~tcjwqrthl *8l8fcl8i$qrthegh(xhyttqrtqK(qzrthfqxl8iwlxhikjw~tigl8n21zhe¥h#" YYhyl8e *l8fl^rthhekjml8tcffc;&pl8npo8qtpqY(xl8iwl8ytrtq
eg~l8eKighegxqYekjml8eKjwh(xqYiwl8cfeqYAl8z~thl8ertq((qzrthfqqY(xtfhjwqt¥hl ¥ d t @@ q (qzrtq rl $8ighplrth ^hf¥~t~tig~t A −1, 4463±1, 4565j BxqYegeg~tcq
eghY~tytrtq(*l8cfqYi *8l8fqYil8tegqYf~.jwqrtqegh~Oighegfrt~tqttrthekjml8pl8ytrtqleg~lcf(xqYikj0F8ytclxl8iwl (¢~tytnpo|qrthjwiwl8yte(¢higytclhekjw~trl8rl.
¥ d t @ . 8~tytnpo8qrthjwiwl8yte(¢higytclhegpl8l8i^∆Pij/∆Pmec
¥~t~tig~t
*G.2* W 3*()(+*" Y \4C!*B , *Z!S4 &)(+*2* W <*()(+*" Y \4C!*B , *Z!S4Pij
W
Y L>@?!(+*&)(), : 4B , #*" Pmec
W
Y T0#C70()0G
¥ige eg~tqYy6-k~ty jwqYeRrth;x YfqYeUqY.jwcfrtqYeUl|jwiwl*Ye rtql8fYqYigc jwq\b. A¢qYigrth FHh H CB (¢qYiwl8 ~.jmcfc;&pl8rtqYexl8iwl\ighxtigqFrt~&cfi (qzrthfqYeRuOk F~l8eRighpl8fc;&pl8n21zheUh hegxl8nqrthXhekjml8rtqYe~.jwcffc;&pl8 qYeighegfrt~tqYerl8eE(¢~tytn21Fherthjwiwl8yte(¢higytcl
H1(s) = ∆Pij/∆Pmec
hH2(s) = ∆Pij/∆Bij
qY(q*l|jwigc;&/rthhy jwiwl8rl.W*l|jwigc;&/rthewl8frl cfY~l8l~t * hjwqYiKqYyFjwhytrtql8xhyl8e/hfh(hy jwqYedcfY~l8cfedl*~t h&l**l|jwigc;&rthjwiwl8yteg(cfegewo8q*rtcfighjml*
D = [d1 d2] =
[0 4, 803× 10−3]
8 t !Fbdcl8Yiwl8*l8eh#tfqzqYeKrtqegcfekjwh*lh#*l8fl(¢hml8rl.
8 t !Orthegigh5*Yhq2egcfekjwh*l2uOk hekjw~trl8rtqXl|jwiwl *Ye&rth*rt~l8e(¢~tytn21zhe&rth*jwiwl8yte (¢higytcl8e hegpl8l8igherth(*l8ythcfiwleghxl8iwl8rl A
H1(s)hH2(s)
B *ekjwh(jwcfxqOrthrtcl8Yiwl8*lxqYegegc tcffc jml*q $8f~tfql8yl8fjgcfq*rl8e h#".xtighege 1Fherl8e(¢~tytn21Fhe rth&jwiwl8yte(¢higytclh*l8fl9(¢hml8rlAHmf1 = ∆Pij/∆Pmec
hHmf2 = ∆Bij/∆Pmec
B ¥de¥ & t !ht ighxtigheghyzjml8hekjml8e (¢~tytn21zhe
Hmf1 =
∆Pij∆Pmec
=H1
I +H2.POD
A t !B
Hmf2 =
∆Bij
∆Pmec
=H1.POD
I +H2.POD
A t B
¥ d t @@ ^qYy6-k~ty jwq(rtqYeK(qFrtqYertqY(cfyl8y jwherth pl8rl(qFrthfqighrt~&cfrtq∆Pij/∆Pmec
b0> ab0&)4 R*B 4()" E<"%G.2045" 42!B 4>@424 ()*B , >J*P, #M(), 4 >@-20B 4 *"%Γ ∆
± # K ∓
± K ±
± K ∓ ± K ∓ # # ± K ± ± K ∓ # ± K ∓ # ± K ± ± # K ∓
± K ±
± K ∓ #
± K ∓
± K ∓
# ± K ∓ ± K ± ± K ± ± K ±
rtcl8Yiwl8*lh tfqFqYe(qYekjwiwl8rtq2yl 8 t 7 (¢qYc^~.jwcffc+&pl8rtqxhfq %,+ %, xl8iwl2l8eegcf&~tl8n21Fhe*rl8eighegxqYekjml8ertqYe(qFrthfqYe*qY(xtfhjwq highrt~&cfrtqYe A 8 t . t Uh t HB jwh(xq2rth*xtiwqzhegewl8(hy jwqXxl8iwlegcf&~tl8npo8qOrtq2(qFrthfqqY(xtfh jmqh*l8l (¢hml8rl (¢qYchfh5*8l|rtq Aªl8xtigq ".cf*l8rl8(hyzjwh*~t*l2tq8imlBrthqYigighy jwhrl2rtcf(hytewo8q2rtqXegcfekjwh*lrthegigcjwqXyl(¢qYig*l&rth ighpl8fc;&pl8npo8qh#hegxl8nq(rth hekjml8rtqYe A @ 77!*8l8igc $ *Yhcferth hekjml8rtqCB
3(¢~tytnpo8qUrthjwiwl8yte(¢higytclHmf2
l8yl8fcfewl8rlyl8e.8 t . t Xh t HxqYegeg~tcewl8frl∆Bij
rtq b4A¢e0B&hhy jwiwl8rl∆Pmec
rth ¥~t~tig~t qYytrthE(¢qYcl8xtfcfpl8rl~t*lxhikjw~tigl8npo8qrtqjwcfxqrthYiwl8~*rth @ yl8e~tegcfyl8erth¥~t~tig~t deighegxqYekjml8ejwh(xqYiwl8cfeqYigighegxqYytrth l8q*8l8fq8iyth l|jwc+*Yqrth
∆Bij
A¢rth5*Fcfrtq ighpl8fcfhyzjml8npo8qyth l|jwc+*8lB z~l8ytrtqOq l8yttqrtq b4A¢e0B*8l8igclrthl8qYigrtqqY#qYe)*8l8qYigheKrtqYeKhfh(hy jwqYertq *YhjwqYi
KPOD = [200 400 600 900 1200]
8 t (qYekjwiwlz~thdq&(qzrthfqrthdqYigrth F.l8xhewl8irthdxqYegeg~tcfi^(hytqYihigigq&rthd(q|rthfqhd*l8igYh rthdhekjml8tcffcfrl|rthKhz~tc+*8l8fhy jwhdl8q(qzrthfqqYigcfYcyl8=A Fhnpo8q t F B zyo8q&l8xtigheghyFjwqY~ighegxq8egjml8eewl|jwcfe(ªl|jYigcl|e8xtigcfytcfxl8fhyzjwh88xl8iwld l8yttqYe rtq b6*l8cfe¥hfh5*8l8rtqYe tewhi *|l eghz~th/qYeqY~.jwigqYertqYce(qFrthqYe^ighrt~&cfrtqYerth/qYigrth H&h H A 8 t h t HBxtigqFrt~&cfiwl8ighegxqYekjml8eytqjwh(xq*l8cfexti ".cf*l8e 8z~thl8e^rtq&(qFrthfqqY(xtfhjwqtzcfytrthxhytrthy jwhrtq& l8yttqt
q&egcfekjwh*l iwhl8 FqYe^xtcfqYerl8e^ighegxqYekjml8e^jwh(xqYiwl8cfez~l8ytrtq&ewo8qhfh5*8l8rtqYe.xqFrth egq(¢ighi~t*lOewl|jw~tiwl8npo8q2h (¢~tytnpo8q2rl8efcfc jml8n21Fhe (¢fegcfpl8ertqYehz~tcfxl8(hy jwqYe2¥qYicfegegqtq(qFrthfqighrt~&cfrtqrth5*Yh ighxtigqFrt~&cfil8rthz~l8rl8(hyzjwhqYextcfqYejwiwl8ytegcjYigcfqYe^yl*l8lrth hekjml8tcffc;&pl|npo8qrtcl8y jwhrth/xhikjw~tiwl|n21Fheh#" YYhyl8erth/l8jwqcf(xl8jwqt xl8iwleghi*Fcfi^qY(q~t*ll8eghdqYytegcfekjwhy jmh
@
0 5 10 15 20 25 30 35−30
−25
−20
−15
−10
−5
0
5
10
Tempo(s)
Sus
cept
ânci
a de
saí
da d
o T
CS
C
Resposta Transitória (ordem 22)
Modelo completo (K = 200)Modelo completo (K = 400)Modelo completo (K = 600)Modelo completo (K = 900)Modelo completo (K = 1200)Modelo reduzido (K = 200)Modelo reduzido (K = 400)Modelo reduzido (K = 600)Modelo reduzido (K = 900)Modelo reduzido (K = 1200)
8 t . KhegxqYekjml(l8qrthYiwl8~Orth @ yl8eK~tegcfyl8eKrth ¥~t~tig~tqY#q((qzrthfqighrt~&cfrtq(rthqYigrth3F
0 5 10 15 20 25 30 35−35
−30
−25
−20
−15
−10
−5
0
5
10
Tempo(s)
Sus
cept
ânci
a de
saí
da d
o T
CS
C
Resposta Transitória (ordem 30)
Modelo completo (K = 200)Modelo completo (K = 400)Modelo completo (K = 600)Modelo completo (K = 900)Modelo completo (K = 1200)Modelo reduzido (K = 200)Modelo reduzido (K = 400)Modelo reduzido (K = 600)Modelo reduzido (K = 900)Modelo reduzido (K = 1200)
8 t . KhegxqYekjml(l8qrthYiwl8~Orth @ yl8eK~tegcfyl8eKrth ¥~t~tig~tqY#q((qzrthfqighrt~&cfrtq(rthqYigrth H.
xl8iwlqxtigq-khjwqrth qYy jwigqYl8rtqYighe*lqY(xl8iwl8npo8qhy jwigh/qYe^(qFrthfqYeighrt~&cfrtqYe^rthdqYigrth ./Hh H rth5*Yheghdl8eghpl8iyl
pl8xl8cfrl8rthrth ighxtigqzrt~&cfirth(¢qYig*l&*l8cfel8tiwl8ytYhyzjwh8tqrthegh(xhyttq(rtqegcfekjwh*lqYigcfYcyl8ªdxhewl8irthxqYegeg~tfigh *l8cfqYihigigqrthqFrthfqtqYehz~tc *8l8fhy jwhe(qzrl8cfe&rthqYigrth H2h H l8fpl8ytnpl8iwl8 igheg~tjml8rtqYeewl|jwcfe(ªl|jYigcfqYehGighl8npo8ql8q&rthq8iwrth F.z~l8ytrtqewo8qqYytegcfrthiwl8rl8el8eewhY~tcfyzjwhe ighekjwigcfn21Fhehigigqrth(qFrthfqt,*l8igYhrthhekjml8tcffcfrl8rth&h(l8(xtfc jw~trthe9$ ".cf*l8erl8eKighegxqYekjml8ejwiwl8ytegcjYigcl8erl*l8flrth hekjml8tcffc;&pl8npo8qxl8iwlxhikjw~tigl8n21zheh#" YYhyl8e
rtc+(¢highytnplO$8egcfplOhy jwigh*qO(qFrthfqrth*qYigrth HOh(qrth(qYigrth H Ohekj0$yl (¢qYig*lrth
0 5 10 15 20 25 30 35−35
−30
−25
−20
−15
−10
−5
0
5
10
Tempo (s)
Sus
cept
ânci
a de
saí
da d
o T
CS
C
Resposta Transitória (ordem 34)
Modelo completo (K = 200)Modelo completo (K = 400)Modelo completo (K = 600)Modelo completo (K = 900)Modelo completo (K = 1200)Modelo reduzido (K = 200)Modelo reduzido (K = 400)Modelo reduzido (K = 600)Modelo reduzido (K = 900)Modelo reduzido (K = 1200)
8 t H. KhegxqYekjml(l8qrthYiwl8~Orth @ yl8eK~tegcfyl8eKrth ¥~t~tig~tqY#q((qzrthfqighrt~&cfrtq(rthqYigrth H t
qY.jwhytnpo8qt^xqYcfexqYegeg+*YhKrtcf(cfyz~tcilXqYigrth+rtq(qFrthfq(heg(qXz~l8ytrtqUhekjwhOxl8ighhOeghiewl|jwcfe(ªl|jYigcfqt.l|jwiwl*YeKrtq( bul8xtfcfpl8rtqlhekjwhqYy6-k~ty jwq(rth x YfqYe
t F u b/ /d b $ b
xtigq-khjwqrthqYy jwiwq8l8rtqYighexqzrtheghi (¢hcjwq~.jwcffc;&pl8ytrtq| egh&(qFrthfqYe/rthqYigrth ighrt~&crlqY.jwcfrtqYe&xhl8e&jwytcfpl8erthjwig~tytpl8(hy jwqX(qzrl8 ^qYy/(¢qYig(hqYeigheg~tjml8rtqYerthekjwhpl8xtjw~tfqtqYe higigqYe rth(qzrthfqh *l8fll8hikjmlyo8qewo8qeg~/'cfhyzjwhexl8iwll *8l8fcl8ideghqYe hz~tc+*8l8hyzjwhe(qFrl8cfe^ighxtigheghy jml8#qegcfekjwh*ll|rthz~l8rl8(hyzjwh8¥qYikjml8y jwqttl(l8igYh rthhekjml8tcffcfrl8rthdhl8eighegxqYekjml8ejmiwl8ytegcjYigcl|eh#*l8fl(¢hml8rljml8#rth#* hAeghiKfh5*8l8rl8eh#qYytegcfrthiwl8npo8qt
/cfekjwq)z~thl8eOqYigrthyteOrtq8eO(qFrthqYeighrt~&cfrtqYel8cfytrlRewo8q)qYytegcfrthiwl8rl8eYiwl8ytrtheOxl8iwlhekjw~trtqYe/*l8cfel *|l8ytnpl8rtqYedhvegcfekjwh*l8edrthqYyzjwigqYfh8~t*leghY~tytrlighrt~tnpo8ql8xtfcfpl8rl(l(hekjwhehz~tc+*8l8hyzjwhe(qFrl8cfe xqFrthKeghiighpl8fc;&pl8rl/qYteghi*|l8ytrtq&l8fY~tytertqYeigcjwigcfqYe z~thKewo8qighfh5*|l8yzjwhexl8iwlOqxtigq6-khjwqrth*hekjml8tcffc;&pl8rtqYighe¥qY(qthigigqrth*(qFrthfqOh(*l8igYh'rth*hekjml8tcffcrl8rth8 jwig~tytpl8(hy jwql8l8ythpl8rtqtF(jwqFrtql8(xtl8hy jwh/l8hcjwq&xhlqY&~tytcfrl8rthjwytcfq|cfhyzjw+'pl.(¢qYchegqYftcfrtqxtigcf(hcfiwl8hy jwhKxl8iwlighpl8fc;&pl8i~t*l ighrt~tnpo8ql8rtcfcfqYyl8trthdqYigrth ythekjmleghY~tytrlhjml8xl. (qzrthfq4xtigqFrt~&cfrtq)egh8~tytrtq;hekjml)l8qYigrl8YhTrthegcfYyl8rtq)ythekjwhXjwiwl8l8ftq4rth0&#/. ( %
l8iwlhekjmlXl8y$8fcfegh8 qXhz~tc+*|l8fhy jwh(qFrl8^xtigqYxqYekjwq (¢qYc^qY.jwcfrtq2qY l8eghytqYe 72x YfqYertqY(cfyl8y jwheKhytqYyzjwigqYexhfq*b/.l|jwiml *Yerth ~tAyCt(higq(rthhekjwcf*l|jwc *8l8e*l8cfqYirtq(z~thqrth5'ytcfrtq*xl8iwlqYed(qzrthfqYeK ¥u2uO hb/ ¥ higigq*rth(qFrthfq*l8egegcf
H
qY(qRl *l8igYh rth2hekjml8tcffcfrl8rthxl8iwl qR(qzrthfq rth2qYigrth 7 ewo8qRhz~tc+*|l8fhyzjwhel8q8ertqegcfekjwh*lqYigcfYcfyl8ª qz~th/qYy jwigcft~tcxqYegcjwc+*8l8(hy jwhxl8iwlqxtigq-khjwqrtq b qFrthfqeg~txhi ighrt~&cfrtqxqYegew~tc¥qYigrth @ (highxtigheghy jmll8rthz~l8rl8(hyzjwh&l*rtcfyF8(cplrtqegcfekjwh*l(xl8iwl(hekjmhjwcfxqrth xtigqYtfh*l.
8 t H @ (qYekjwiwl/qigheg~tjml8rtq/qY(xl8iwl|jwc+*Yq/rl8eighegxqYekjml8erl(¢~tytnpo8qrthjwiwl8yte(¢higytcfll8egegqFcl8rll*l8fl&rtq( b A
∆Pij(s)/∆Bij(s)B
0 5 10 15 20 25 30 35 40−3
−2
−1
0
1
2
3
4
5x 10
−5Resposta ao degrau (∆ B
ij(s))
Tempo (s)
Var
iaca
o do
flux
o de
pot
ênca
ativ
a (
∆ P
ij)
modelo de ordem 60modelo super−reduzido (10)modelo completo
10−1
100
101
−70
−65
−60
−55
−50
−45
−40
−35
−30Diagrama de Bode
Frequencia (rad/s)
Mag
nitu
de (
dB)
modelo de ordem 60modelo super−reduzido (10)modelo completo
AªlB A¢?B
8 t H @ ^qY(xl8iwl8npo8qrl8eKighegxqYekjml8eh#*l8fl&l8hikjmlrtqYeK(qFrthfqYeighrt~&cfrtqteg~txhi ighrt~&cfrtq(h qY(xtfhjwqtAªlBjwh(xqYiwl8cfe A¢xhikjw~tl8npo8q(rth @ yleg~teghx.j0F8ytclrlfcfytl
qYikjwh kF~t B^hA¢?B (¢ighthytcl8cfe
e&higigqYe&qY.jwcfrtqYeyl8e&ighegxqYekjml8eytq2rtqY(fytcfqOrl (¢ightytcl A 8¥ t H @ ?B rtq2(qFrthfqighrt~&cfrtqOh'*l8fll8hikjml A¢qYigrth 7Bdh(rtqeg~txhi ighrt~&cfrtq A¢qYigrth @ B/ewo8qt¥ighegxhjwc+*|l (hy jwh8
1, 5533×10−4 h 1, 5617×10−4 Y(hytqYigheYxqYikjml8y jwqt8z~thqYehytqYyzjwiwl8rtqYeytqYe (qFrthfqYerthegigcjwqYeKyl(Fhnpo8q t F F
ef~t l8ighe^rl8eiwl8;&hexl8iwlqYe(qzrthfqYe^ighrt~&cfrtqYehekjmo8q&iwhxtiwhewhyzjml8rtqYeyl48 t Hz (qFrtqrthcfy jwhiwl8npo8ql8r/*Yhigewlhekj0$xtigheghy jwh(ytq(qFrthfqrthqYigrth 7hyo8qhekj0$ytq(qFrthfqeg~txhi ighrt~&cfrtqt ¥qYigO|q(qFrthfqKeg~txhi ighrt~&cfrtq/l8xtigheghy jwqY~&~t*l*l8igYh>rthhekjml8tcffcfrl8rth(hftqYiz~th/lrtq(qFrthfq A 8 t lB z~thj l|yo8qxqYegeg~tchekjwh/(qzrtqt dqYeqYe(qzrthfqYeighrt~&cfrtqYeighz~thigh l8yttqYertq bMl8cf*ldrthH. xl8iwldjwqYigyl8iq egcfekjwh*lcfytekj0$ * h Fhytz~l8yzjwqz~th xl8iwlq(qFrthfqqY(xtfhjwqtFq l8yttqigjwcfqrth F 7. *l&qYigighjmll *8l8fcl8npo8qrlO*l8igYh'rth(hekjml8tcffcfrl8rth*cf(xqYikjml8yzjwhxl8iwlOqxtigq-khjwqrth(qYyzjwigqYl|rtqYighe *Fcl(qFrthfqighrt~&cfrtqt.xqYcfe^*l8igYhyte^(hytqYe^qYyteghi*8l8rtqYiwl8ez~thl8ertq&egcfekjwh*lqYigcfYcyl8igheg~tjml8h#qYy jwigqYl8rtqYighe(hytqYeKigqYt~tekjwqYe
Lugar das Raízes
Eixo Real
Eix
o Im
agin
ário
−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
20.15 0.1 0.05
K=887
K=3200
Root Locus
Real Axis
Imag
Axi
s
−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
20.15 0.1 0.05
K=892
K=3480
AªlB A¢?B
8 t HF ~t l8irl8eKiwl8;&heKrtqYeK(qzrthfqYeAªlB^ighrt~&cfrtqttqY#qYigrth 7hA¢?Beg~txhi ighrt~&cfrtqtqY#qYigrth @ .
*lOl8y$8fcfeghqY(xtfh(hy jml8i ((qYekjwiwl8rlyl 8¥ t HHqYsqqY-khjwc+*Yqrth * higc+'pl8ieghqYe^xtcfqYe^rl8e^ighegxqYekjml8e^jwiwl8ytegcjYigcl8ertqYe(qzrthfqYeighrt~&cfrtqYe^ewo8q&hz~tc+*8l8fhy jwhel8qYe^rtq(qFrthfqqY(xtfhjwqt l8frthl *8l8fcfl8iq2qY(xqYikjml8(hy jwqqYegcfl|jYigcfqOrl8e&ighegxqYekjml8eekjml8eighegxqYekjml8e(¢qYiwl8GqY.jwcfrl8exhikjw~tigl8ytrtq|eghlxq8jwytcl(h F8ytcfpl&ytqYeKhc ".qYertqYeKYhiwl8rtqYigheKrth ¥~t~tig~th(qYytcjwqYiwl8ytrtq| egh leg~teghx.j0F8ytclh5(¢hjwc+*8lrlfcfytl&rthjwiwl8yteg(cfegewo8q A
∆Bij
B ¥qFrth egh ytq8jml8iz~th q8extcfqYerth l8(xtfc jw~trth/$ ".cf(lrtqYe(qFrthqYeighrt~&cfrtqYetjml8y jwqxl8iwl
KPOD = 900z~l8yzjwq
KPOD = 1200l8qY(xl8ytl8iwl8+qYertq(qFrthfqXqY(xtfhjwqt)qYigO
z~l8ytrtqegh l8~t(hy jmlq l8yttqt.q&(qFrthfqeg~txhi ighrt~&cfrtqqY(hnpll(qYekjwiwl8i^~t*lpl8iwl8jwhigfe jwcfpl(hytqYe^l8(qYikjwhcfrl.Fl8~t(hy jml8ytrtqq&egh8~tytrtqxtcfqrth/l8(xtfc jw~trthK$ ".cf*l. ekjmlz~thekjmo8q(¢qYcl8qYigrl8rl*ylFhnpo8q t F H.qYytrth(¢qYc¥hy/(ªl|jwc;&l8rll(cf(xqYikj0F8ytclrl*z~l8fcfrl8rthrtq*(qFrthfqighrt~&cfrtq(xl8iwllq lighxtigheghy jml8npo8q*rl8eKighegxqYekjml8ejwiwl8ytegcjYigcl8eh#*l8fl(¢hml8rl.
*lRqY~.jwiwl)l8jwhigyl|jwc+*|l (¢qYccfy*Yhekjwcf l8rlRxl8iwl4qY.jwhytnpo8q)rthX(qFrthfqYeeg~txhi ighrt~&cfrtqYe*Fcll8qYigrl8Yh (qFrl8 .l|jwiwl * erlxtigheghi*|l8npo8q(rth l8fY~tytex YfqYeh/jml8& rthl8fY~tyte &higqYertqY(cfyl8y jwheKqhy jml8y jwqt|qYe ighegfrt~tqYe¥l8egegqzcl8rtqYe¥l8qYex YfqYeegq(¢ighl8jwhiwl8n21Fhe hqK l8yttqKxl8iwll(¢~tytnpo8qrth^jwiwl8yte(¢higytcldrthekjwh(qFrthfqdeg~txhi ighrt~&cfrtqrth5* heghipl8f~tl8rtqdxqYi qY(xl8iwl8npo8qrtcfighjmlrl8eKighegxqYekjml8e(¢ighthytcl8cfeyz~t*lxtigcf(hcfiwl&l8qYigrl8YhO
*lrl8e&*l8ythcfiwl8ertheghxtigheghi*|l8iqYe&x YfqYeh.&higqYe&*l8cfecf(xqYikjml8y jwhel|jwiwl *Yerlfcfc jml8npo8qOrth(~t*lOighYco8qOytq2xtl8ytqOqY(xtfh#".qtrthjml8 (¢qYig*lz~th*~tsqYy6-k~ty jwq2rth*x YqYeh&higqYexqYegeg~l Frt~tfq(hytqYi/z~th~trthjwhig(cfyl8rtq*8l8fqYi4A¢ythekjwhpl8egq
ωn = 8rad/sB ekjwh
igcjwigcfqxq8eweg~tc ~t*lighekjwigcfnpo8qt,xqYcfel8fY~t*l8e*Yh2&he¥eghq*l8xlrthx YfqYe/h4&higqYe(¢qYi/&~tcjwq
0 5 10 15 20 25 30 35 40−25
−20
−15
−10
−5
0
5
Resposta ao degrau (∆ Pmec
(s))
Tempo (s)
Sus
cept
anci
a de
sai
da d
o T
CS
C (
pu)
modelo de ordem 60modelo super−reduzido (10)modelo completo
0 5 10 15 20 25 30 35−30
−25
−20
−15
−10
−5
0
5
10
modelo de ordem 60modelo super−reduzido (10)modelo completo
Sus
cept
anci
a de
sai
da d
o T
CS
C (
pu)
Resposta ao degrau (∆ Pmec
(s))
Tempo (s)AªlB A¢?B
8 . HH. KhegxqYekjml(l|qrthYiwl8~rth @ yl8e~tegcfyl8eKrth ¥~t~tig~t xl8iwl&qYeKqFrthfqYeighrt~&cfrtqheg~txhi ighrt~&cfrtq(h#*l8fl(¢hml8rl.AªlB
KPOD = 900hA¢?B
KPOD = 1200
rthytegqtlz~l8y jwcfrl8rthrth &higqYe*xtigheghi*8l8rtqYexqFrtheghi**l8cfqYiz~th2lz~l8yzjwcfrl8rthOrthx YfqYefh5*8l8ytrtq&l ~t*l (¢~tytnpo8qiwl8cfqYyl8cf(xtiYxtigcfl. (qFrthfqighrt~&cfrtqhegqYftcfrtqxl8iwll8xtfcfpl8ihekjml(hjwqFrtqYfqYYcfl(¢qYc,q(qFrthfq9u2uOWrth qYigrth3 A ¥ d t B
~.jwigql8egxhjwqOleghi l8qYigrl8rtqt,rtc;&&ighegxhcjwql8q l8yttqrl(¢~tytnpo8qrthjwiwl8yte(¢higytclrtq(qFrthfq*eg~txhi ighrt~&cfrtqt,z~thxqzrtheghi/qY.jwcfrtqh (¢~tytnpo|qrlrtc+(¢highytnplhy jwighqYe/ l8yttqYe/rth*l8Yytcjw~trth&rth^qFrthxl8iwll8c "tl8e(¢ightytcl8e & t xtigqYx?1Fh~t*l.(¢qYig*l*rtheghqY.jwhiq l8yttq
Kxl8iwlq(qFrthfq&eg~txhi ighrt~&cfrtqttythekjwh pl8egqtl (¢ightytcl
ωmin
~.jwcffc+&pl8rlcfY~l8. @ iwl8r?<|e
K = 10(|Gr(jωmin)|dB−|Gsr(jωmin)|dB
20).
A t BqYytrth
Gsr(s) =
12∏i=1
(s− zi)
12∏i=1
(s− λi)
,
hλihziewo8q(qYeKx YfqYeh &higqYeKrtqY(cfyl8y jwhertq((qzrthfqighrt~&cfrtq
Gr(s)A¢qYigrth3B
8¥ t H (qYekjwiwl ~t*lqYxl8iwl8npo8qhy jwigh/l8eighegxqYekjml8e^jwh(xqYiwl8cfeh(¢ighthytcl8cfertqYe(qFrthfqYeeg~txhi ighrt~&cfrtqYe/qY.jwcfrtqYe*Fcl ^ ¥u h*Fcl(rtqY(cfyF8ytclrthx YfqYedh &higqYe,qY q(qFrthfqqY(xtfhjwqt e&higigqYe&rtqYe(qzrthfqYeeg~txhi ighrt~&cfrtqYerthqYigrth @ hqYigrth @ ewo8qighegxhjwc+*|l8(hy jmh|
1, 424× 10−3 h 2, 466× 10−3 7
10−2
10−1
100
101
−70
−65
−60
−55
−50
−45
−40
−35
−30Diagrama de Bode
Frequencia (rad/s)
Mag
nitu
de (
dB)
modelo super−reduzido ordem 12modelo super−reduzido ordem 10modelo completo
0 5 10 15 20 25 30 35 40−2
−1
0
1
2
3
4
x 10−5
Tempo (s)
∆ P
ij (pu
)
Resposta ao degrau
modelo super−reduzido de ordem 12modelo super−reduzido de ordem 10modelo completo
AªlB A¢?B
8 t H t^qY(xl8iwl8npo8qrl8eKighegxqYekjml8eKrtq((qFrthqqY(xtfhjwqqYAl8ertqYeK(qFrthfqYeeg~txhi ighrt~&cfrtqYe A¢qYigrth @ ^ ¥u >h qYigrth @ x YfqYeh &higqYeKrtqY(cfyl8yzjwhe0B AªlB
bdcl8Yiwl8*lrth^qFrth hA¢?B KhegxqYekjml(l8qrthYiwl8~Orth @
e f~t l8ighe/rl8e iwl8;&hexl8iwlq(qzrthfqeg~txhi ighrt~&cfrtqOrthqYigrth @ hekj0$(qYekjwiwl8rtqyl8 t HF¥qFrth eghqYteghi*|l8iz~th/eg~l&*l8igYh rth/hekjml8tcffcfrl8rtheghl8xtigq".cf(qY~**l8cferl8z~thlrtq(qzrthfq(qYigcfYcfyl8rtq*z~th&q*eg~txhi ighrt~&cfrtqqY.jwcfrtq*xhfq ^ ¥u ,FqY(hyzjwhxl8iwl( l8yttqYertq( b *l8cfq8ighez~thH. @ z~th q(egcfekjwh*leghjwqYigylcfytekj0$ * h
Lugar das Raizes
Eixo Real
Eix
o Im
agin
ario
−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
20.15 0.1 0.05
K=968
K=3010
8 t HF ~t l8irl8el8;&hexl8iwlq(qFrthfqeg~txhi ighrt~&cfrtq(rth qYigrth @ F
l8iwlq(egcfekjmh*l*uOk qYe&higqYerl(*l8fl&rthxhikjw~tigl8npo8q A∆Pij(s)/∆Pmec(s)
BD(¢qYiwl8xtigheghi*8l8rtqYeOytq)(qzrthfq4eg~txhi ighrt~&cfrtqEqYy/(¢qYig(hxtigqFhrtcf(hy jwq4~.jwcffc+&pl8rtqRyl4*l8fl rtq b A
∆Pij(s)/∆Bij(s)B degegcfOzqYy/(¢qYig(hl & t .Fl8e^ighegxqYekjml8e^jwiwl8ytegcjYigcl|eh*l8fl
!
(¢hml8rlxl8iwl~t*l xhikjw~tigl8npo8qh#" YYhylrth¥~t~tiw~txt~trthiwl8 eghi^egcf&~tl8rl8ehqY(xl8iwl8rl8eqYy/(¢qYig(h(qYekjwiwll 8 t H7.txl8iwl
KPOD = 900
0 5 10 15 20 25 30 35 40−25
−20
−15
−10
−5
0
5
Tempo (s)
Sus
cept
ânci
a de
Saí
da d
o P
OD
(∆
Bij)
Resposta ao degrau dos modelos em malha fechada
modelo super−reduzido ordem 12modelo reduzido ordem 22modelo super−reduzido ordem10modelo completo ordem 1667
8 . H7. KhegxqYekjml(l|qrthYiwl8~rth @ yl8e~tegcfyl8eKrth ¥~t~tig~t xl8iwl&qYeKqFrthfqYeighrt~&cfrtqheg~txhi ighrt~&cfrtq(h#*l8fl(¢hml8rlxl8iwl
KPOD = 900
^qYl8eghytqYe&igheg~tjml8rtqYe&qY.jwcfrtqYe&qYqYe&(qFrthfq8eeg~txhi ighrt~&cfrtqYel8fY~t*l8eqYteghi *8l8n21zherth5*YhAeghiighegewl8jml8rl8e
• (qzrthfqeg~txhi ighrt~&cfrtqq8.jmcrtq*Fcl ^ ¥u Ml8fpl8ytnqY~higigq&rth/(qzrthfqewl|jwcfe(ªl|jYigcfq.xqYigLl*l8igYh rthhekjml8tcfcfrl8rth(¢qYc*l8cfqYi&z~thOlrtqqY(xtfhjwqthy/(ªl jmc+&pl8ytrtqX~t*lighekjwigcfnpo8qcf(xqYikjml8y jwh/xl8iwlqxtigq-khjwqrth hekjml8tcffc;&pl8rtqYighe
• (qFrthfqdeg~txhi ighrt~&cfrtqqY.jwcfrtqdxhldrtqY(cfyF8ytclKrtqYe x YfqYeh &higqYe rth~t (qFrthfq - $ighrt~&cfrtq*Fcll8qYigrl8Yh (qFrl8xqYegeg~tc$higigq&xhz~thytqttxqYig *l8cfqYiz~thdq&rtq(qFrthfqeg~txhi ighrt~&cfrtq *Fcl ^ ¥u A*l8igYhsrthhekjml8tcffcfrl8rth 'qY~4*l8cfexti ".cf*l2l2rtqegcfekjwh*lqYigcfYcfyl88highxtigqFrt~&cf~qK(qFrtqKrthcjwhiwl8npo8ql8r/*Yhigewl.8cf(xqYikjml8y jwh ytq(¢hml8(hyzjwqrl*l8fl&xl8iwll8jwqYe l|yttqYeKrtq* b
• dedighegxqYekjml8ejwiwl8ytegcjYigcl8eKrtq*egcfekjwh*l(hv*l8fl (¢h l8rl(xl8iwl(~t*l(xhikjw~tl8npo8qh#" Yhyl A 8t H7B,(qYekjwiwl8>z~th^qYe¥(qFrthfqYeeg~txhi ighrt~&cfrtqYexqYegeg~thWqY(xqYikjml8(hy jwqegh(hfl8y jwh/l8qYeegh~te(qFrthfqYeighrt~&cfrtqYe A¢qYigrth 7xl8iwlq ^ ¥u HhdqYigrth xl8iwllrtqY(cfyF8ytclrth x YfqYeh &higqYe0B
• igrthyte¥rtqYe (qFrthfqYeeg~txhi ighrt~&cfrtqYe¥(hytqYighe¥z~th^l8e xtigqYxqYekjml8e yo8qxtigheghi*8l8O|ythekjwhpl8egqttjwqFrtqYeqYeK(qFrtq8eKrth qYegcfl|npo8qcf(xqYikjml8y jwhexl8iwlxtigq6-khjwq(rth qYy jwigqYl8rtqYighe
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igcf(hcfiwl8hy jwh88l8frtqYe¥x YfqYe rtqY(cfyl8yzjwhe rth5'ytcfrtqYexhl8e jwytcfpl8erthegigcjml8eytql8xt jw~tfq H..xl8iwl&lqY.jwhytnpo8q&rtqhz~tc+*|l8fhyzjwhd(qFrl8rth/egcfekjwh*l8euOku ythhegegcjml eghrtqYeD(ªl|jwqYigherth/qYy jwigqYl8tcfcrl8rthhdqYteghi*8l8tcfcfrl8rthdl8q&cfy*YertqYeighegfrt~tqYe^rthdpl8rl(¢~tytnpo8q&rthjwiwl8yte(¢higyFcl.qYyzjw~trtqtighz~thigcfrl(l(heg*ljwiwl8yte(¢qYig*l8npo8qrth egcf(cffl|igcfrl|rth/~.jwcffc;&pl8rlh#egcfekjwh*l8eFk Aª & t B *FcfekjwqXz~thOlighpl8fc;&pl8npo8qqY.jwcfrl2xhl8e&jwytcfpl8erthjwig~tytpl8(hy jwqX(qFrl8^qYigcfYcfyl8hy jwhrtcfl8YqYyl8,qY(xtfh#"tl.
de( & F @ DF ighxtigheghyzjml8Oighegxhjwc+*8l8(hy jwh8l8eighpl8fc;&pl8n21zhertcl88qYyl8^qY(xtfh#"tl2htfqFq|rtcl|YqYyl88ighpl8zrth~t(qFrthfqtcfxq8jwjwcfqKrthz~tcfy jmlKqYigrthOekjwhjwcfxqKrthjwiwl8yte(¢qYig*l8npo8q(¢qYc,~.jwcffc+&pl8rl&ythekjwhpl8xtjw~tfqxl8iwllqY.jwhytnpo8q(rth hz~tc+*8l|fhyzjwheK(qFrl8cfe
Gr(s) =
[A1 B1
C1 D
]=
λ1 0 0 0 0 b11 b21
0 λ∗1 0 0 0 b∗11 b∗21
0 0 λ2 0 0 b12 b22
0 0 0 λ3 0 b13 b23
0 0 0 0 λ∗3 b∗13 b∗23
c11 c∗11 c12 c13 c∗13 d11 d12
c21 c∗21 c22 c23 c∗23 d21 d22
We Y
Gr(s) =
[A2 B2
C2 D
]=
[T−1A1T T−1B1
C1T D
]=
Re(λ1) Im(λ1) 0 0 0 Re(b11)√2 Re(b21)
√2
−Im(λ1) Re(λ1) 0 0 0 −Im(b11)√2 −Im(b21)
√2
0 0 λ2 0 0 b12 b22
0 0 0 Re(λ3) Im(λ3) Re(b13)√2 Re(b23)
√2
0 0 0 −Im(λ3) Re(λ3) −Im(b13)√2 −Im(b23)
√2
Re(c11)√2 Im(c11)
√2 c12 Re(c13)
√2 Im(c13)
√2 d11 d12
Re(c21)√2 Im(c21)
√2 c22 Re(c23)
√2 Im(c23)
√2 d21 d22
We Y
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, cf(xqYikjml8yzjwhighegewl8jml8iz~th8xl8iwlegcfekjwh*l8euOku qYe(qFrtqYeEAλiB (¢qYiwl8sqYigrthyl8rtqYe
eghY~tytrtqq *|l8fqYi^$ ".cf(qrthy jwighqYe bu^hytqYy jwiwl8rtqYexl8iwl&l8e (¢~tytn21zherth/jwiwl8yte(¢higytcl8ehegpl8l8ighe*z~th2qY(x 1zh lU*l|jwigc;&rthjwiwl8yte(¢higytclFKqYy/(¢q8iwh2lU &)F H.\ekjwhjwcfxqUrthhy/(¢qFz~th4(¢qYc xtigh5*Fcl8(hyzjwh~.jwcffc+&pl8rtqh / 6 u A @ 7BuO/b/ d d A @ B t 8>8 A @ B th#egcfekjwh*l8erth 4tjwcxtl|ehyzjwiwl8rl8eKheml|frl8e
IDM(λi) = maxk,j[IDM(λi)kj]
A F HB
qYytrthk q(fytrtcfh/rl8eKhyzjwiwl8rl8eKh
jqrl8eKewl8frl8e
F @ FkKuO KKE
eigheg~tjml8rtqYerthekjml(eghnpo8qt(qYekjwiwl8Alpl8xl8cfrl8rthrljwytcfpl(rth jwig~tytpl8(hy jwq*(qzrl8xl8iwl(ighrt~tnpo8qrthqYigrth rth(qFrthfqYeK~tjwc *8l8igc;$ *Yhcfeh&l*qY(xl8iwl(q8 q*(jwqFrtq ^ ¥u G*l|jwigc;&rth*jwiwl8yte(¢higytcl~.jwcffc+&pl8rl
[∆Ω(s)/∆Vref (s)]9×9 z~thqYigighegxqYytrth 8e*l8fl8e
rtqYeKhekjml8tcffc;&pl8rtqYighe Aª. Brl8eKytq *Yh ~tegcfyl8eKYhiwl8rtqYiwl8ertq(egcfekjwh*l. *l8xlOqY(xl|iml jmc * qrth*x YfqYeh9&higqYertqYe(qFrthfqYe ighrt~&cfrtqYeh(qY(xtfhjwqO(qYekjwiwll
qYy jwigcft~tcfnpo8qrtqYe&rtqYcfe(jwqzrtqYeEA ^ ¥u h buBxl8iwl2lqYigrth ! A 8>F @ B 2ekjmlqYigrth (¢qYchegqYftcfrl2rthE(¢qYig*l2z~thjwqzrtqYeqYe(qzrtqYehfhjwigqY(h F8ytcfqYe A (ªl|jwqYi&rthOl8(qYikjwhc (hy jwq(hytqYi/z~th . @ BKhekjmc * hegegh xtigheghyzjwhe ytqYe(qzrthfqYedighrt~&cfrtqYe&¥qFrth eghytq8jml8i/z~thq ^ ¥u vqY.jwh5*Yh2~t xqYegcfcfqYyl8(hy jwqUl8xtigq ".cf*l8ytrtqrthekjwhe(qFrtqYe(h2l8cfytrlXcfytf~tcf~)qY~FjwigqYex Yfq8eighpl8cfe/ytq(qzrthfqiwhrt~&cfrtqtl8iwll8&qYeqYe(jwqzrtqYeqYe &higqYe rtqY(cfyl8y jwhe/yo8q(¢qYiwl8#xtigheghi*|l8rtqYe$eghytrtq*z~thq((jwqzrtq((qFrl8l8igheghy jwqY~ &higqYeqY(xtfh#".qYeKcfytf~tegc+*Yh ytq
@ @
egh(cfxtl8ytq|srlrtcfighc j lF
−2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 00
1
2
3
4
5
6
7
8
9
100.2 0.15 0.1 0.05
Eixo Real
Eix
o Im
agin
ário
Mapa de pólos − ordem 27
Modelo CompletoModelo Reduzido (BALMR)Modelo Reduzido (IDMNI)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.50
0.5
1
1.50.20.150.10.05
Mapa de zeros − modelos reduzidos de ordem 27
Eixo Real
Eix
o Im
agin
ario
Modelo CompletoModelo Reduzido (BALMR)Modelo Reduzido (IDMNI)
AªlB A¢?B
8?F @ YfqYeh &higqYertq(qFrthfqqY(xtfhjwqh rtqYeK(qzrthfqYeighrt~&cfrtqYeKrth qYigrth3!FAªlB*l8xl&rth x YfqYehA¢?B^*l8xlrth &higqYe
GxtigcfytfxtcfqtFhegqYfthytrtq|eghl&qYigrth @ xl8iwll8&qYeqYe(qFrthfqYe^ighrt~&cfrtqYeA ^ ¥u h buB jwqzrtqYeqYe (qzrtqYehfhjwigqY(h F8ytcfqYel8egegqFcl8rtqYel8qYeYhiwl8rtqYigherth5* higcl8 hekjml8ixtigheghy jwheOrl8rl egh~teOl8jwqYe(ªl jmq8iwherthqYy jwigqYl8tcfcrl8rthhqYteghi*8l8tcffcfrl|rth8 qFrl *Fcl cfekjwqyo8ql8qYy jwhh8&qpl8egqrtq ^ ¥u $qqYy6-k~ty jmqrthx YfqYe/rtqY(cfyl8y jwhe'plpl8rl* h2&*l8cfel6(ªl8ekjml8rtq&rtqqYy6-k~ty jwqrth/x YfqYe^rtq(qzrthfq&qY(xthjwqt.l&(hrtcfrlz~thegh/rtcf(cfyz~tcl&qYigrth rtq(qFrthfqt 8l|jwqz~thyo8qqzqYigigh qY#q(qFrthfqqY.jwcfrtqxhfq bum
l8iwl&cff~tekjwiwl8i^l&rtc+(¢highytnplhyzjwigh/qYe(qFrthfqYe^rth/qYigrthyte @ h !F(¢qYcpl8f~tl8rtqq$ ".cf(q l8yttq*rth ^qFrthxl8iwl*l8rl9(¢~tytnpo8q*rthjwiwl8yte(¢higytcl*hegpl8l8irthy jwigqrl (ªl8c "tlrth(¢ightytclrthcfyzjwhighegegh8z~thqYigighegxqYytrth&l8q*$ ".cf(q4*|l8fqYiKegcfytY~tl8iKrl((l|jwiwc+& rth jwiwl8yte(¢higytcl
σmax
A 8?F B rtc+(¢highytnplrl8e~ti*|l8erth
σmax
A¢~ti*8l8e qYy jwfyF~l8e0BdhyzjwighqYe rtqYcfe(qFrthfqYeighrt~&cfrtqYeA ^ ¥u #h buBhq(qzrthfqXqY(xthjwqXyo8qegcfYytc+'pl|jwc+*8l.Rvrtc+(¢highytnplrl8e(~ti*|l8erth
σmin
A¢~ti*8l8ejwiwl8h-gl8rl8e0B KqYy jw~trtqtewo8qRl8ekjml8y jwhh5*Fcfrthy jwhe .l8h eghz~thσmax
hekj0$ighl8cfqYyl8rtql8qYex YfqYeh
σmin
l8qYe)&higqYeqz~thqYy/'ig*lqYeigheg~tjml8rtqYeKrl 8?F @ ¥qYiO*l8cfez~thUeghUqYytegcfYl)qY.jwhiO~tI(qFrthfq4ighrt~&cfrtqEz~thUxqYegeg~l)jwqzrtqYeOqYeO(qFrtqYe
hfhjwigqY(h F8ytcfqYe A¢qYigrth !B lXxqYegcfnpo8qXrtqYe &higqYe*l8cfytrl2~t (ªl|jwqYicf(xqYikjml8y jwhz~thOcfyF~thytclrtcfighjml8(hy jwhytqOxtigq6-khjwqOrthqYy jwigqYl8rtqYighedf rtcfegegqt ql8xl8ighcfhyzjwqrth &higqYeighpl8cfeqY~OqY(xtfh#".qYeytq(egh(c xtl8ytqrlrtcfighcjml&xti".cf(qYertq(hc ".qcf*l8Ycy$8igcqt.z~thyo8qhekjmo8qxtigheghy jwhedytq(qFrthfqqY(xtfhjwqttl6(¢hjmlegcfYytc+'pl|jwc+*8l8hyzjwh/qf~t l8irl8eiwl8;&he
@
100
101
−120
−100
−80
−60
−40
−20
Gráficos do σmax
e σmin
− ordem 27
σmax
− Modelo BALMRσ
min − Modelo BALMR
σmax
− Modelo IDMNIσ
min − Modelo IDMNI
σmax
− Modelo Completoσ
max − Modelo Completo
Frequência (rad/s)
Mag
nitu
de (
dB)
100
101
−120
−100
−80
−60
−40
−20
Gráfico do σmax
e σmin
Frequência (rad/s)
Mag
nitu
de (
dB) σ
max − Modelo BALMR
σmin
− Modelo BALMRσ
min − Modelo IDMNI
σmax
− Modelo IDMNIσ
max − Modelo Completo
σmax
− Modelo Completo
AªlB A¢?B
8?F F i%$6'qYeKrthσmax
hσmin
rtq(qFrthfqqY(xtfhjwqqY(xl8iwl8rtqYeKl8qYeKrtqYeAªlB(qzrthfqYeighrt~&cfrtqYeKrth qYigrth3!&hA¢?B(qzrthfqYeighrt~&cfrtqYeKrth qYigrth @ .
l8iwl*eghl8yl8fcfewl8iKqqY(xqYikjml8(hy jwq*rtqYed(qFrthfqYehv*l8fl4(¢h l8rl. (¢qYc¥~.jwcffc+&pl8rtq(~tqYy6-k~ty jwqrthO. z~thqY(xhytewl8 jwqFrtqYeqYe(qFrtqYe&(hytqYe(l8(qYikjwhcfrtqYe deegcf&~tl8n21Fhel8xtigheghy jml8rl8ehekjmo8q l8eghpl8rl8e*ytqYel -k~tekjwhe*qFqYigrthyl8rtqYe*rtqYe*hekjml8tcffc;&pl8rtqYigherthegigcjwqYe*h8 K GA B ^~6-kqqY-khjwc+* qXOqY.jwhi~t*lXegqYf~tnpo8qqYytrthythytF~tL(qFrtqXrthOqYegcfl8npo8qxqYegeg~l9(ªl|jwqYiKrthl8(qYikjwhcf(hy jwq(hytqYiz~th @ ehekjml8tcffc;&pl8rtqYigheKxqYegeg~th l (¢qYig*lrl & F tqY qYe*|l8fqYighedrtqYe/ l8yttqYe/hrl8e/qYytekjml8y jwherthjwh(xq*rtqYe/tfqFqYedrthl *|l8ytnqrth(ªl8egh rthegigcjwqYeKyl4¥ d F @
ESP (s) = KESP
5s
1 + 5s
(1 + Tns
1 + Tds
)3 A F CB
l8iwlOq(qzrthfqqY(xtfhjwqt (¢q8cqYytekjwig~tfrtqqOYi%$6'qrtqYef~t l8ighe rl8eiwl8;&he.A 8=F HBl|jwigcft~tcfytrtq|egh*8l8fqYighertcfegighjwqYeKl8qYeegh~teK l8yttqYe cfyzjwhi*|l8fq&z~th qYy jw#hekjwhe)*|l8fqYighe *|l8crth &higq*l|jwq( l8yttq$ ".cf(q&rth5'ytcfrtqylqYf~tyl
KESP
rl4¥ d F @ l8highegewl8jml8i^z~thqYex YfqYe^rtqYeK.)hytqYy jwiwl8egh/yl&ighYco8q&qY xl8ikjwh/ighpl8$rthd&l
.8hz~th$d~t hegxl8fl8(hy jwq/rtqYe iwl8qYerthekjwhex YfqYe¥;hewxhc+'pl8npo8qrthl8(qYikjwhcf(hy jwqrtqYe (qzrtqYehfhjwigqY(h F8ytcfqYe(¢q8c l8fpl8ytnpl8rlxl8iwlqK l8yttqK$ ".cf(q A
KESP
B qY>~t*l*l8igYhrth l8yttqOighl|jwc+*|l8(hy jwhl8jml. 8qYc qYytekjml|jml8rtqOz~th(egqY(hy jmhxl8iwl l8yttqYe.*Yh2&he*l8cfqYighertqz~thqYeK$ ".cf(q8ehegxhc+'pl8rtqYe A
KESP
B tqewcegjwh(legh jwqYigylcfytekj0$ *Yh e(heg(qYe. (¢qYiwl8 ~.jwcffc+&pl8rtqYe xl8iwl qY(xhytewl8iqYe(qzrthfqYeighrt~&cfrtqYe deD8CF
h.F (qYekjwiwl8 qYef~t l8ighe rl8eiwl8;&hexl8iwlhekjwhepl8egqYeqYsl(heg*lE*8l8igcl8npo8qrth( l8yttqYel8xtfcfpl8rl&l8q(qFrthfq&qY(xtfhjwqt teghi*8l egh z~thh#".cfekjwh ~tGfcf(cfl|i^xl8iwl&q4*8l8fqYi^rth
rA¢qYigrth
@ H
¥ d F @ l8fqYigherl8eKqYytekjml8yzjwheKrthjwh(xqrtqYeKtfqFqYerthl *8l8ytnq(rth (ªl8egh rtqYeKytq*Yh.¥
hiwl8rtqYiTn
A¢e0BTd
A¢e0BKESPH . H . H.
H @ . H . ! H. H . ! @ . . HH . ! . 7 H. H . 7 . 7. H . H . H. H7 . . F HH! . !! . HH H. H . C . H @ H t
−60 −50 −40 −30 −20 −10 0−30
−20
−10
0
10
20
30
30 20 10
0.150.10.05
0.150.10.05
Eixo Real
Eixo
Imag
inar
io
Lugar das Raízes [∆ Ω/∆ Vref
]9x9
Pólos do modelo não compensadoPólos do modelo compensadoZeros do modelo
Pólos eletromecânicos com ζ < 15%
Pólos dos controladores
8?F H. ~t l8igheKrl8eKiwl8;&heKrtq(qFrthfqqY(xtfhjwq[Ω(s)/Vref (s)]9x9
txl8iwlK = [0 . . . KESP
7Xxl8iwlljwytcfplX buh H7Xxl8iwlXq ^ ¥u BlXxl8ikjwcfirtqz~l8l8ehegxhc+'pl8n21Fhe*rthl8(qYikjwhcf(hy jwqyo8qewo8q*l8cfel|jwhytrtcfrl8exl8iwlhekjwhe (qFrthfqYe/ighrt~&cfrtqYeekjwh4(ªl|jwqrthytq8jmlz~thXlU*l8igYh rth2hekjml8tcffcfrl8rthOxqzrth2eghil|egjml8y jwhl6(¢hjml8rlUrthxhytrthytrtq4rl qYigrth h2rtq(jwqFrtqtt(heg(qz~thqhigigqrth (qFrthfqegh-gl&~tcjwqxhz~thytq A 8?F B
q8jml egh8 hyzjmo8qt z~thlyo8q xtigheghi*|l8npo8qrtqYe &higqYertqY(cfyl8y jwheytq (qzrthfqighrt~&cfrtq xl8iwlegcfekjwh*l8euOgu xqzrthl8pl8igighjml8iKrtcfekjwqYign21zhel8cfytrl*l8cfeYiwl*YheKyl*l8igYhGrth hekjml8tcffcfrl8rth
@
−60 −40 −20 0 20 40 60−60
−40
−20
0
20
40
60
60 40 20
0.150.10.05
0.150.10.05
Eixo Real
Eix
o Im
agin
ario
Lugar das Raízes [∆ Ω/∆ Vref
]9x9
− IDMNI (45)
Pólos do modelo não compensadoPólos do modelo compensadoZeros do modelo Modo do controlador
com ζ < 15%.
Modos Eletromecânicoscom ζ > 15%.
−60 −40 −20 0 20 40 60−60
−40
−20
0
20
40
60
60 40 20
0.150.10.05
0.150.10.05
Eixo Real
Eix
o Im
agin
ario
Lugar das Raízes [∆ Ω/∆ Vref
]9x9
− IDMNI (46)
Pólos do modelo não compensadoPólos do modelo compensadoZeros do modelo
Modos Eletromecânicoscom ζ > 15%.
AªlB A¢?B
8F t ~t l8irl8eiwl8;&hexl8iwlqYe(qFrthfqYeighrt~&cfrtqYe buqYGeg~l8eK*l8fl8e (¢h l8rl8eAªlB^qYigrth C A
ζmin < 15%B^hA¢?BqYigrth 7 A
ζmin > 15%B
−60 −40 −20 0 20 40 60−60
−40
−20
0
20
40
60
60 40 20
0.150.10.05
0.150.10.05
Eixo Real
Eix
o Im
agin
ario
Lugar das Raízes [∆ Ω/∆ Vref
]9x9
− BALMR (35)
Pólos do modelo não compensadoPólos do modelo compensadoZeros do modelo
Modo do controladorcom ζ < 15%.
Modos Eletromecânicoscom ζ > 15%.
−60 −40 −20 0 20 40 60−60
−40
−20
0
20
40
60
60 40 20
0.150.10.05
0.150.10.05
Eixo Real
Eix
o Im
agin
ario
Lugar das Raízes [∆ Ω/∆ Vref
]9x9
− BALMR (36)
Pólos do modelo não compensadoPólos do modelo compensadoZeros do modelo
Modos Eletromecânicoscom ζ > 15%.
Modo do controladorcom ζ > 15%.
AªlB A¢?B
8?F F ~t l8irl8eKiwl8;&hexl8iwlqYeK(qFrthfqYeighrt~&cfrtqYe ^ ¥u >qY#eg~l8eK*l8fl8e(¢h l8rl8eAªlB^qYigrth H A
ζmin < 15%Bh9A¢?BqYigrth H7 A
ζmin > 15%B
h *l8cfqYighertc+'~tfrl8rtheKytqxtigq6-khjwq(rth qYy jwigqYl8rtqYigheKl8rthz~l8rtqYede*jwytcfpl8erthighrt~tnpo8qRrth2qYigrth ewo8qR*l8cfe(h5'pl&hexl8iwl egcfekjwh*l8e*qY l8c ".qYe(ªl
jwqYighe&rthqYy jwigqYl8tcfcrl8rth(hqYteghi*|l8tcffcfrl8rth8 Fcfekjwh*l hekjmh*uOku yo8q2xq8eweg~tc^hekjmlpl8iwl8jwhigfekjwcfplh9(¢qYc~.jwcffc;&pl8rtqythekjmleghnpo8q2xl8iwlOh5*Fcfrthytcl8iz~thljwytcfplrthjwig~tytpl8(hyzjwql8l8ythpl8rtqxqFrth*yo8qeghijmo8q *|l8yzjml -kqYewlOh'jwhig(qYerth(jml "tlOrth*ighrt~tnpo8q2hsqY(xl8iwl8npo8q
@
qY#qjwig~tytpl8(hy jwq((qzrl8,ytq(pl|ewq~tjwc *8l8igc;$ *Yh
F FkKuO KKEFkKuO K /b K/F
Gl8y$8fcfegh&~tjwc *8l8igc;$ * hl8xtigheghyzjml8rlOythekjmleghnpo8qegh*igh5(¢high O*l|jwigc;&rth(jwiwl8yte(¢higytcflqY#hyzjwiwl8rl8eKh ewl8frl8eKrth5'ytcfrl8eKyl l8hlt tekjwhegcfekjwh*l ighegxqYyte%$ * h¥xhlqY(xhyFewl8npo8qrtq(qFrtqcfytekj0$ * h ytqYikjwh eg~t l|jwiwl *Ye rl8e/jwige*l8fl8e/rthqYy jwigqYfh
∆Ω(s)/∆Vref (s)
bd~l8eD(¢qYig*l8erth/q8.jmhytnpo8q&rtqYex YfqYertqY(cfyl8yzjwhe (¢qYiwl8 ~.jwcffc+&pl8rl8e lxtigcf(hcfiwl. egh/igh5(¢high/l8qqYy6-k~ty jwq~tytco8q*rtqYedx YfqYehytqYyzjwiwl8rtqYedxhfqb/.,qYy/(¢qYig(hlFhnpo8q t F @ $h&l*eghY~tytrl.jwiwl|jmlrth ~t#qYy6-k~ty jwq(rth x YfqYertqY(cfyl8y jwhexl8iwl&jwqFrl*l|jwigc;&/rthjwiwl8yte(¢higytcl.
l8fYqYigc jwq/rthKx YfqYe rtqY(cfyl8yzjwheb/. A uO / @ !B¥igcl/qYytrtcfn21Fhe jml8y jwq xl8iwll(ighrt~tnpo8qrth(qFrthfq(rth(¢~tytn21Fhedrthjwiwl8yte(¢higytcl*hegpl8l8ighe4A Fk BKz~l8y jwqxl8iwl(egcfekjwh*l8erthhy jwiwl8rl4tytcfpl A Fku B^qY~ewl8frl tytcfpl A uOk B eigheg~tjml8rtqYertq(l8xtjw~tfq rthekjml8pl|l h5'cytcl rll8qYigrl8Yh (qFrl8txl8iwlqxtigq6-khjwqrthKqYy jwigqYfhhegcfekjwh*l8erthYiwl8ytrthKxqYikjwh8
qhy jml8yzjwqtzz~l8ytrtq&egh/l8yl8fcfewl~tegcfekjwh*lrth tjwcfxtfl8ehyzjwiwl8rl8ehewl8frl8eA uOku B FqqYy6-k~ty jwq(rth x YfqYe~tjwc *8l8igc;$ *YhcfertqY(cfyl8yzjwheYhiwl8f(hy jmh/xqYegeg~tc~t#qYy6-k~tyzjwqegh(hfl8yzjwhl8qYex YfqYe rtqYehz~tc+*8l8fhy jwhe(qFrl8cfehegpl8l8igherlOrtcl8YqYyl8 rlO*l|jwigc;&rthjwiwl8yte(¢higytcl. l8fYqYigc jw(q*rthegigcjwqh'uO / A HBKqYigighegxqYytrth(l8q.(¢~tytrl8(hy jwq$8egcfqxl8iwlq $8 ~tfqrtqYex 8fqYe rtqY(cfyl8y jwherth(~t egcfekjwh*l&~tjwc *8l8igc;$ * hªxqzrthytrtqqYy* higYcfihekjwcf*l|jwc+*8l8ertcfekjwcfy jml8exl8iwlO~t*lO(heg*lOegqYf~tnpo8qtO YfqYertqY(cfyl8y jwhe&uOku rtcfekjwcfyzjwqYe xl8iwlpl8rlOhekjwc *l|jwc+*8lcfytcfcl8,ewo8qqY.jwcfrtqYeKytqb l|jwiwl * edrl (¢higiwl8(hy jmlub.$~6-gl(egqYf~tnpo8q*hekj0$l8eghpl8rlytq $8f~tfqrth
σmax
xl8iwljwqFrl8e l8e(¢~tytn21Fhe rth&jwiwl8yte(¢higytclhegpl8l8ighe rtq(qFrthfq&~tjwc+*|l8igc $ *Yh .qYy/(¢qYig(hhekj0$rthegigcjwq(yl & F F
‖G(s)‖ = σmax = sup
(‖G(s).u‖‖u‖
)A F B
qYytrthG(s)
ld*l|jwigc;&rth^jwiwl8yte(¢higytcl/z~l8rtiwl8rl Am×m B xl8iwld~t rthjwhig(cfyl8rtq*|l8fqYirth s
u/~t *YhjwqYirthhy jwiwl8rl&qY(xtfh#".q&rth/rtcf(hytewo8q
m× 1 σ2max
hz~tc+*8l8fh/l8q*l8cqYi^l8~.jwq*8l8fqYirth
G(s)HG(s)hσmax
q*l8cfqYiD*|l8fqYiegcfytY~tl8irthG(s)
8 F 7 l(qYekjwiwll8fY~t*l|ertc+(¢highytnpl8e(hyzjwighOqYe(qYy6-k~ty jwqYe(rthOx YfqYeqY.jwcfrtqYexhfq
b/.GA *x YfqYe0BKhub.GA 7*x YfqYe0B hytz~l8yzjwql.8 F 7 2qY(xl8iwl(qYe&higqYe/rtqYedrtqYcfe(qFrthfqYeyz~t*lighYco8qxti ".cf*lrlqYigcfYhOqzrth eghytq8jml8iKz~thl*l8cfqYigcfl&rtqYex YqYeKrth (¢igh tytcll8cf*lrth7 iwl8r?<|e (¢qYiwl8 xtigheghi*8l8rtqYe^xhfqb/.zxqYigl8fY~tyterthl8(qYikjwhcf(hy jwq*l8cfqYiegqY(hyzjwh/hekjmo8qxtigheghyzjwheKytqqYy6-k~ty jwqrthx YfqYeqY.jwcfrtqYexhfq(ub.dl48F 7 ,qYe&higqYed*l8igpl8rtqYeKxqYieghjml8edl8xl8ighhvegqY(hy jwhhApl8rl(~tArtqYed(qzrthfqYez~thl8xl8ighyzjwh (hy jwhxqFrth cfy jwhi(¢higcfiyl*qY(xhytewl8npo8qrth(ªl8egh8 *l(*l8ythcfiwl(rth&l*|l8fcl8iKeghl*xtigheghytnpl
@ 7
−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 00
1
2
3
4
5
6
7
8
9
100.2 0.15 0.1 0.05
Mapa de polos
Eixo Real
Eix
o Im
agin
ario
Pólos do modelo de ordem 49Pólos do modelo de ordem 60
−1 −0.8 −0.6 −0.4 −0.2 00
0.5
1
1.5
2
2.5
3Mapa de zeros
Eixo Real
Eix
o Im
agin
ario
Zeros do modelo de ordem 49Zeros do modelo de ordem 60
AªlB A¢?B
8F 7.AªlBul8xlqY(xl8iwl|jwc+*YqrtqYex YfqYeqY.jwcfrtqYexhfq(b/. A¢qYigrth Bhxhfq(ub.A¢qYigrth 7B A¢?B*l|xlqY(xl8iwl|jwc+*YqrtqYe&higqYeKrtqYeK(qFrthfqYeighrt~&cfrtqYe)(¢qYig*l8rtqYexqYihekjwhe
x YfqYe
rthekjwhe)&higqYeKl8jwhiwl8 qYe^iwl8(qYe^rtqYef~t l8ighe^rl8eiwl8;&hel|jwiwl *Yertq&rtcl8Yiwl8*l rtq ~t l8i^rl8el8;&hexl8iwl l8yttqYe)*8l8igcl8ytrtqrthyzjwigqrth ~t*l (ªl8c "tlighekjwigcjml Aªl|jwx, ~, B
xtigq6-khjwqrtqYe&hekjml8tcffc;&pl8rtqYighez~thl8fpl8ytnpl8s~t*lqY(xhytewl8npo8q(fytcf(lOrth @ rtq(qFrtq ytqYikjwh eg~t=A
ωd = 1, 2iwl8r?<|e0B (¢qYcxtigqYxqYekjwqxqYi^uO / A @ B ekjwhe^qYy jwigqYl8rtqYighe
xqYegeg~thGjwigetfqzqYerth/qY(xhytewl8npo8qrth(ªl8egh/h~t 'jwigq ' th/qYe l8yttqYextigqYxqYekjwqYeewo8q(cY~l8cfel
K(k)ESP
xl8iwlqYejwige.¥qYy/(¢qYig(hrthegigh5* hl( & z 7.
ESP3×3(s) = diag(ESP (1)(s), ESP (2)(s), ESP (3)(s)
)3×3
A F 7B
xl8iwl
ESP (k)(s) = K(k)ESP
3s
1 + 3s
(1 + 0, 12s
1 + 0, 012s
)2(1 + 0, 25s
1 + 0, 75s
), k = 1, 2, 3.
8 F !(qYekjwiwlqf~t l8irl8e&iwl8;&hertq8e(qFrthfqYeighrt~&cfrtqYe&qY(xhytewl8rtqYe A¢ig~&he*Yhig(hfl8e0BqY(xl8iwl8rtqYel8q(qzrthfqOqY(xthjwqOqY(xhytewl8rtq Aªl&~t B e l8yttqYehegqYftcfrtqYe*8l8igcl8srth Xl|jw @ 7. H. t 72h @ . .ighegxhjwc+*|l8(hyzjwh8xl|iml2qYe. rthl8~tfq(¢qYytegq &/cfyt h jml8xl8igcfplFqYytrthOq l8ekjwhigcfegqUhz~tc+*8l8fhl8qYe(x YfqYertqYe*(qzrthfqYeegh qY(xhytewl8npo8qAKESP = 0
B rtcl8Yiwl8*lrtqO(qFrthfqqY(xtfhjwqE(¢qYcqY.jwcfrtqxhfqb ¥~.jmcfc;&pl8ytrtq|eghl(heg*lE(ªl8c "tlrth( l8yttqYexl8iwlOl8xhyl8e !(qFrtqYe rtqY(cfyl8y jwhe¥qFrth egh*ytq|j l|iz~th*l8&qYeqYed(qFrthfqYeighrt~&cfrtqYe,xtiwl|jwcfpl8(hy jwh8yo8ql8xtigheghy jml8iwl8 rtc+(¢highytnpl8e/ytqYediwl8(qYedrtqYed(qFrtqYeqYy jwigqY $*YhcfeYl8egegcf qY(q/yl/xqYegcfnpo8q'yl8.rtqYex YfqYeekjwq(qYekjwiwl.Yh xtigcfytfxtcfqt|z~thKl8&qYeqYeK(qzrthfqYeighrt~&cfrtqYeKewo8q(l8rthz~l8rtqYexl8iwlqxtigq-khjwqrth qYy jwigqYl8rtqYighe
@ !
−2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 00
1
2
3
4
5
6
7
8
9
100.15 0.1 0.05
Eixo Real
Eix
o Im
agin
ario
Root Contour [∆ Ω/∆ Vref
]3x3
− ordem 49
−2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 00
1
2
3
4
5
6
7
8
9
100.15 0.1 0.05
Eixo Real
Eix
o Im
agin
ario
Root Contour [∆ Ω/∆ Vref
]3x3
− ordem 60
AªlB A¢?B
8F !zbdcl8Yiwl8*l8ertqYef~t l8ighe^rl8eiwl8;&hertqYe(qFrthfqYe^ighrt~&cfrtqYeAªlBrthqYigrth h9A¢?Brth qYigrth 7.tqY(xl8iwl8rtqYeKl8q(qFrthfqqY(xtfhjwq Aªl&~t B
de8?F hz (qYekjwiwl8#l8eKighegxqYekjml8eytq(rtqY(fytcqrtqjwh(xqrtqYeK(qFrthfqYeqY(xhyFewl8rtqYe/qY .Hrth& l8yttqYedcfY~l8cfelE.$rl8rl*~t*l(xhikjw~tigl8npo8qrtq*jwcfxq*rthYiwl8~2rth @ yljwhytewo8qrthigh5(¢higytcl2rthpl8rl2~tegcfyl A
∆VrefB X¥qFrth eghytq8jml8i*Fcfeg~l8f(hyzjwh8z~thq2(qFrthfq
rthqYigrth37(qYigighegxqYytrth&l(~t*l*l8xtigq ".cf*l8npo8q*l8cfqYiyl8edighegxqYekjml8edxl8iwl(*l8cfqYixl8ikjwhrl8e(¢~tytn21Fhe(rthjwiwl8yte(¢higytclXhegpl8l8ighe l8fYqYigcjwqub.G(¢qYc*l8cfe&h5'cfhyzjwhylXqY.jwhytnpo8qrthx YfqYe/~tjwc *8l8igc;$ *Yhcfe/rtqY(cfyl8y jwhertqOz~thqOb/.l8xtfcfpl8rtqh pl8rl~t*lrl8e (¢~tytn21zherth jwiwl8yte(¢higytcl*l8egegqFcl8rl8e 8e*l8fl8erthq8yzjwigqYfh8~t*l4* h2&z~thq8e7x YfqYeKcf(xqYikjml8yzjwhe(¢qYiwl8GqY.jwcfrtqYeegcf&~tjml8ythpl8hyzjwh8
^qYq qYy6-k~tyzjwqrthhekjwcf*l jmc *8l8ecfytcfcfl8cehegqYftcfrl8eythekjml eghnpo8qxl8iwl qY.jwhytnpo8qrthx YfqYertqY(cfyl8y jwhe*Fcl*l8fYqYigc jw(q*ub.$q*jwh(xq*rth&xtigqzhegewl8(hy jmq(¢qYc rth&l8xtigq".cf*l8rl8hy jwhH(cfyz~.jwqYeytz~l8yzjwqOz~th8~.jwcffc+&pl8ytrtqql|fYqYigcjwqb/.>xl8iwlpl8rlE(¢~tytnpo8qOrthjwiwl8yte(¢higyFcl*hegpl8l8i$qxtigqFhrtcf(hy jwqighz~thi~tvjwh(xq*l8cfqYirth*Yhigc+'pl8npo8qyo8q|egcfekjwh$|jwcpl(xl8iwlqY(xl8iwl8npo8q rthKighegxqYekjml8ejwh(xqYiwl8cfeh)(¢ighthytcl8cfezrth)(¢qYig*ll qY.jwhi~t qYy6-k~ty jwq(fytcfqewl|jwcfe(ªl|jYigcfq rthdx Yfq8ertqY(cfyl8y jwhexl8iwlpl8rlhfh(hy jwqrl*l|jwigc;&Krthjwiwl8yte(¢higytcl.F*l8ez~thxqzrthAeghiKcfyteg~/'cfhy jwhexl|imlighxtigheghy jml8iqqY(xqYikjml8(hy jwqYfqYl8uOku
@
0 5 10 15
−15
−10
−5
0
x 10−5 ∆ ω
1
Vre
f1
0 5 10 15
−15
−10
−5
0
x 10−5 ∆ ω
2
0 5 10 15
−15
−10
−5
0
x 10−5 ∆ ω
3
0 5 10 15
−15
−10
−5
0
x 10−5
Vre
f2
0 5 10 15
−15
−10
−5
0
x 10−5
0 5 10 15
−15
−10
−5
0
x 10−5
0 5 10 15
−15
−10
−5
0
x 10−5
Vre
f3
Time (s)0 5 10 15
−15
−10
−5
0
x 10−5
Time (s)0 5 10 15
−15
−10
−5
0
x 10−5
Time (s)
completoreduzido
8?F . KhegxqYekjmlytqjwh(xqrl*l|jwigc;&/rthjwiwl8yte(¢higytcl[Ω(s)/Vref (s)]3x3
xl8iwlqYe(qFrthfqYeqY(xtfhjwqh ighrt~&cfrtq(rth qYigrth .
0 5 10 15
−15
−10
−5
0
x 10−5 ∆ ω
1
Vre
f1
0 5 10 15
−15
−10
−5
0
x 10−5 ∆ ω
2
0 5 10 15
−15
−10
−5
0
x 10−5 ∆ ω
3
0 5 10 15
−15
−10
−5
0
x 10−5
Vre
f2
0 5 10 15
−15
−10
−5
0
x 10−5
0 5 10 15
−15
−10
−5
0
x 10−5
0 5 10 15
−15
−10
−5
0
x 10−5
Vre
f3
Time (s)0 5 10 15
−15
−10
−5
0
x 10−5
Time (s)0 5 10 15
−15
−10
−5
0
x 10−5
Time (s)
completoreduzido
8?F . KhegxqYekjmlytqjwh(xqrl*l|jwigc;&/rthjwiwl8yte(¢higytcl[Ω(s)/Vref (s)]3x3
xl8iwlqYe(qFrthfqYeqY(xtfhjwqh ighrt~&cfrtq(rth qYigrth 7.
@
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