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cl.IYY YYY YYYY YYY YYY YYYY YYY YYY YYYY YYY YYY YYYY YYY YY 78 t cf(c jml8y jwheeg~tx­higcfq8iwhertqYeKhigigqYeKxl8iwl.AªlB

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

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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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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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8 F i%$6'qYe/rthσmax

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8 F H ~t l8ighe¥rl8eiwl8;&he¥rtqd(qFrthfqKqY(xtfhjwq[Ω(s)/Vref (s)]9x9

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[0 . . . KESP

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*l8fl8e (¢hml8rl8e AªlBqYigrth C Aζmin < 15%

BhA¢?BqYigrth 7Aζmin > 15%

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*l8fl8e (¢hml8rl8e AªlBqYigrth H Aζmin < 15%

BhA¢?BqYigrth H7Aζmin > 15%

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jml8rtq 'yl8~tegcfekjwh*lrthwqYigrthO(¢qYig*l8rtqx­qYihz~l8n21Fhe^rtc+(¢highytcl8cfeAª &/F @ l &F Bhl8fYtigcfpl8e.Aª & F 7h & F !B ekjwh(qzrthfqhekj0$2rthegigcjwqqY*l8cfqYigherthjml8fthe&hK A @ HB qYytrthqY~.jwiwl8etcfx 8jwheghedegcf(xtfc+'pl|rtqYiwl8eKx­qzrthveghidhytqYy jwiwl8rl8e eegcfYytc+'pl8rtqYe (¢fegcfqYerl8e)*8l8igc $ *Yhcfertq(egcfekjwh*lhekjmo8qrthegigcjwqYeKyl ¥ d F @

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 B

E ′′q =1

T ′′d0[E ′q − (X ′

d −X ′′d )Id − E ′′q ]

A F HB

ω =1

2H[Pmec − Pelet] =

1

2H[Pmec − VdId − VqIq −Dω]

A F CB

δ = ω0(ω − 1)A F B

E ′′d = RaId −X ′′q Iq + Vd

A F 7B

E ′′q = RaIq +X ′′d Id + Vq

A F !B

jwiwl8yte(¢qYig*l8npo8qUrl8e(jwhyte 1FheVrhVm

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

*l|jwigc;& rthjwiml|ytewcfegemo|q*rtcfighjml.xq*YhjwqYi/rthhekjml8rtqYe

uq9*YhjwqYi/rthhy jwiwl8rl*rtqegcfekjwh*l(h

ylewl8frlrtqewcegjwh(l.

/~l8ytrtqeghjwiwl|jmlrthegcfekjwh*l8erth Yiwl8ytrthx­qYikjwh8l (¢q8iw&~tl8no8qqYy*YhytcfqYyl8 Aª &?F Brl8e hz~l8n21Fheh'hegxl8nqrthhekjml8rtqYeeghjwqYigylcfyC*Fc $ *Yhxl8iwlighxtigheghy jml8i&l8e rtcfyF8(cpl8ehyF*YqY+*Fcfrl8e/rth5*FcfrtqOl8qxtigqzhegewl8(hy jwqO~tekjwqYegqOrtqYe $8f~tfqYe/xl8iwll8y$8fcfegh8bdhekjmlE(¢qYig*l.$~t*lqY~.jwiwl.(¢qYig&~tl8npo8q(¢qYcxtiwq8x­qYekjml~.jwcffc;&pl8ytrtq| eghl*l|jwigc;& Yl8qYtcl|ylrtqegcfekjwh*l:A ddb

!

@ 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

qYytrthl2*l|jwigc;&(#rtc+(¢highy jwhrl2cfrthy jwcfrl8rthpl8iwl8jwhigc;&ll (¢qYig~tl|npo8qrthegigcjwqYiwl.eghytrtqInl**l|jwigc;&cfrthyzjwcfrl8rthrth&rtcf(hytewo8q

ncfY~l8¥l8q*jml8*l8yttq*rtq*YhjwqYi/rth&hekjml8rtqYe

xJl**l|jwigc;&

Yl8qYtcl8yl.Ba

l*l|jwigc;&/rth hy jwiwl8rll8~t(hyzjml8rlhCa

l*l|jwigc;&/rth ewl8frll8~t(hy jml8rl.jwiwl*Yerlhfcf(cyl8npo8q&rl8e*|l8igc $*YhcfeKl8fYtigcfpl8eA

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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B lhz~l8npo8qjwh(x­qYiwl8 qYigighegx­qYytrth & F C!F9$qY~.jwigqYejwcfx­qYerth(hy jwiwl8rl8erthcfy jwhighegegh8qY(qqcf(xt~tfegq A

U(s) = 1Bhl*iwl8(xl A

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y(t) =n∑

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A F C!B

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|G(jω)| =∣∣∣∣∣

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∠G(jω) = ∠

(n∑

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jω − λi+ d

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rthG(jω)

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σB

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Gζ(jω) =n∑

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Ri

(− ζω√1−ζ2

+ jω)− λi+ d

A F B

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∆Bij(s)

B qYytrthl2ewl8frl∆Pij(s)

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F H. u b Ob/6 !

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h (ªl&hytrtq*l (¢ightytclω*8l8igcl8irth −∞ l

+∞ ^qY(q2qrtcl8Yiwl8*lOxqYl8iegcf(jwigcfqh ighl8npo8ql8q2hc ".qighpl8 qigcjwigcfqrth Fz~tcfekj

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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

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rt~tnpo8qrth&~t ewcyl8 l8rtcfcfqYyl8 hekjml8tcffc;&pl8rtqYirthigc+*|l8rtqrth∆Ω

h~t Yhiwl8rtqYikA¢hegqYftcfrtqt

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B H7

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∆Ωk/∆V kref

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−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

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(1 + saT )/(1 + sT )$qYytrth

a > 1

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φmax = sen −1

(a− 1a+ 1

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ωmax =1

T√a

A F BH!

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l|jwiwl8egqta =

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T =1

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H(s)

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1 + sT

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B

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! 2! k,!dqYeRtfqzqYe K A¢e0Bh6.Aªe0BUyl8F 7&ighxtigheghy jml8Al8e (¢~tytn21zherth/jwiwl8yte(¢higytcl&rtqighY~tl8rtqYi^rth/jwhytewo8qh/rtqegcfyl8l8rtcfcfqYyl8,hekjml8tcffc;&pl8rtqYide(¢~tytn21Fhedrth jwiwl8yte(¢higytcl

F1(s)hF2(s)

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Efd

B qYlOjwhytewo8qjwhig(cfyl8rl$8z~tcfyl AVbB h

qY+l *8l8igc $ *Yh~.jwcffc+&pl8rl2qYqXegcfyl8^rthhy jwiwl8rlrtqXqYy jwigqYl8rtqYi Aω, Pt, θ, R, Pacel

B ighegx­hjwc+*|l8(hy jwh8

8?F 7.bdcl8Yiwl8*lrth tfqFqYerl*l8l&rtq(ighY~tl8rtq8irth ¥hytewo8q(qY#q(hekjml8tcffc;&l8rtqYi

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ωB x­q8jwytcfljwhig(cfyl8 A

PtB .(¢ightytclWylWl8igiwl

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RBqY~Mx­q8jwytcl4l8hfhiwl8y jwh

APacel = Pmec − Pt

B 8 F !(ighxtigheghyzjmlqhegz~th*lYhiwl8rtqhekjml8tcffc;&pl8rtqYixl8iwlH

8?F !Fbdcl8Yiwl8*lrth tfqFqYertq(egjml8tcfc;&pl8rtqYirthFcfekjwh*l8erth x­q8jwytcl Aª. B

$8z~tcfyl8e egfytigqYyl8e 'jwigqdxl8egewl l8c "tlKrth^qYigrthnYl8fY~t*l8e *Yh2&he ythhege%$8igcfqt8jwh

qY(q.(¢~tytnpo8qOl|jwhyz~l8iig~tfrtqYe/rthl8jml(¢ightytcl. 'jwigq ' &l|jwhyz~ll8c "tl8e(¢ightytcl8eyo8qx­hig(cjwcfytrtqz~thqqYyzjwigqYl|rtqYil|jw~th&xl8iwl*8l8igcl8n21Fhe/fhy jml8ertqegcfyl8rth hy jwiwl8rl.

" J """ U g /hekjwhArtcfegx­qYegcjwc+*Yq rthjwhytqYfqYYcl:8FDEl8xtigheghy jml (¢qYikjwhOx­q8jwhytcl8xl8iwlq l8(qYikjwhcfhyzjwqXrthOqYegcfl8n21FhehfhjwigqY(h F8ytcfpl8eheg~ttegegfytigqYyl8e,l8egegcf#qY(q(xhig(cjwh q*l8~t(hyzjwq(rthpl8xl8crl8rthrthjwiwl8yteg(cfegewo8qrtqYed.¥hekjwhjwiwl8l8ftqegqY(hy jwhcfi%$&cfyzjwhighegewl8iKleg~l(l8xtfcfpl8npo8q&xl8iwlql8(qYikjwhcf(hy jwqrth qYegcfl8n51FhehfhjwigqY(h F8ytcfpl8eekjwhqYy jwigqYl8rtqYijwh qY(qhyzjwiwl8rlq Frt~tfq*rtq ~".qrlxq8jwytcll|jwc+*|l*rlfcfytlrthjwiwl8yteg(cfegewo8q A |Pij| Bdh8¥qY(qewl8frl. leg~teghx.j0F8ytclh5(¢hjwc+*8lrlfcfytl:A Bij

BdqY~eg~lighpl|j0F8ytcll8xl8cjwc+*8l A

Xij

B 'ighl8npo8qOl8q. lrtc+(¢hiwhytnplO$8egcfplrtqhegz~th*lO~t'tfqzqOrth(l8rtcfcfqYyl8 xl8iwlewl|jw~tiwl8npo8qrtqegcfyl8rthdewl8frlA 8/F B zz~thdewhKjmiwl|jml rthd~tfcf(c jml8rtqYirtcfyF8(cqt x­qYcfeh5*Fcjmldz~thljwhytewo8qrtq/pl8xl8cjwqYi¥eg~tx­highq/$ ".cf(qrtq/xtigq6-khjwq A

2×Vnominal

rtqt)B

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(4 v ( ^)*!& R, WO) !(

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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||∞)

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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

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@

−1.5 −1 −0.5 00

1

2

3

4

5

6

7

8

9

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IDM Norma Infinita

Eixo real

Eix

o im

agin

ario

polos − modelo completo (62)polos − modelo reduzido (20)

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± K ± K ± K ± K ± K ± K ± K ± K ± K ± K ± K ± K ± K ± K

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G(s) =Y (s)

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A H. @ !B

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c 0

x(sk)

U(sk)

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v(sk)

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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

1

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)

AªlB A¢?B

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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)|

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Limite superior dos erros modais dos modelos G4(s)

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0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

8

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Ordem dos modelos reduzidos (Gr(s))

||G(s

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r(s)|

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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

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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

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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

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r1 = 18r2 = 23

r3 = 18

hr4 = 40

B

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7

10−2

10−1

100

101

−100

−90

−80

−70

−60

−50

−40

Mag

nitu

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Diagrama de Bode (Modulo) − G1(s) para ordem 20

10−2

10−1

100

101

−50

0

50

Frequencia (rad/s)

Ang

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Diagrama de Bode (Fase) − G1(s) para ordem 20

Truncamento BalanceadoIDM norma infinitaIDM valor inicialIDM valor finalModelo Completo (62)

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−25

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−15

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Diagrama de Bode (Modulo) − G2(s) para ordem 27

2 4 6 8 10 12 14

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−150

−100

−50

Frequencia (rad/s)

Ang

ulo

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Diagrama de Bode (Fase) − G2(s) para ordem 27

Truncamento BalanceadoIDM norma infinitaIDM valor inicialIDM valor finalModelo Completo (62)

AªlB A¢?B

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100

101

−30

−20

−10

0

10

20

30

40

Mag

nitu

de (

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Diagrama de Bode (Modulo) − G3(s) para ordem 15

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100

101

−600

−500

−400

−300

−200

−100

0

100

Frequencia (rad/s)

Ang

ulo

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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

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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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BALMRIDMNIIDMVIIDMVFModelo Completo

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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

8 t . igrthyl8npo8qx­qYi busxl8iwll (¢~tytnpo8q(rthjwiwl8yte(¢higytclF1(s)

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

8 t @ . igrthyl8npo8q(xqYiK bu'xl8iwll4(¢~tytnpo8qrthjwiwl8yte(¢higytclF2(s)

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!

20 40 60 80 100 120 140−100

−80

−60

−40

dB

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

8 t @@ igrthyl8npo8q(xqYiK bu'xl8iwll4(¢~tytnpo8qrthjwiwl8yte(¢higytclF3(s)

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0 5 10 15 20 25 3010

−3

10−2

10−1

F1(s) : Erro ||F(jω)−F

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dB

Ordem do modelo reduzido

IDMNIIDMVIIDMVF

0 5 10 15 20 25 3010

−3

10−2

10−1

100

F1 (s) : Gráficos dos Erros e seus Limitantes Superiores

dB

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

AªlB A¢?B

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0 5 10 15 20 25 30

10−3

10−2

F2 (s) : Erro ||F(jω)−F

r(jω)||∞

dB

Ordem do modelo reduzido

IDMNIIDMVIIDMVF

0 5 10 15 20 25 3010

−3

10−2

10−1

100

F2 (s) : Gráficos dos Erros e seus Limitantes Superiores

dB

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

AªlB A¢?B

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0 5 10 15 20 25 30

10−3

10−2

F3 (s) : Erro ||F(jω)−F

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dB

Ordem do modelo reduzido

IDMNIIDMVIIDMVF

0 5 10 15 20 25 30

10−3

10−2

10−1

F3 (s) : Gráficos dos Erros e seus Limitantes Superiores

dB

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

AªlB A¢?B

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10−2

10−1

100

101

−50

−45

−40

−35

−30

−25

Mag

nitu

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Diagrama de Bode para F1(s)

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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0 5 10 15−90

−85

−80

−75

−70

−65

−60

−55

−50

−45

Frequencia (rad/s)

Mag

nitu

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Diagrama de Maginute de Bode para os Erros

Erro para o modelo obtido pelos pólos ordenados pelo IDMNIPontos máximos dos errosErro para o modelo obtido pelos pólos encontrados pelo DPSE

−3 −2 −1 0 1 2 3

x 10−3

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ωo = 0.01rad/s

ωo = 0.01rad/s

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10−2

10−1

100

101

−80

−75

−70

−65

−60

−55

−50

−45

−40

−35

−30

∆ Pij(s)/∆ B

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Frequência (rad/s)

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ALPHABETAGAMMADELTAMODELO COMPLETO

ωd = 0,2397 rad/s

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10−2

10−1

100

101

−75

−70

−65

−60

−55

−50

−45

−40

−35

Freqüência (rad/s)

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ALPHABETAGAMMADELTAModelo Completo

8 t @ . Khegx­qYekjmlytqrtqY(fytcfq&rl4(¢ightytclrtqYeK(qzrthfqYeighrt~&cfrtqYe

^qYvl8eghyl8edegcf&~tl8n21FheKrl8eighegx­qYekjml8edrtqYed(qzrthfqYeKighrt~&cfrtqYe­(qYekjwiwl8rl8eyl8e8t @ Xh t @ .x­qFrth eghOytq8jml8iz~thqYe(qFrthfq8ez~thOx­q8eweg~thLq(qFrtqXrthOcfy jwhiwl8npo8ql8r/* higewlAλ34 = −0, 1151±0, 2397j

B qY.jwc+*Yhiwl8 ighegx­qYekjml8e*l8cfe xti".cf*l8el8q (qFrthfqqYigcfYcfyl8ªYeghytrtqz~th8xhl4¥ d t .tqYeK(qFrthfqYe ¥ hu2uO l|xtiwhewhyzjml8#higigqYeK(hytqYighe

q rtcl8Yiwl8*lrthK*l8Yytcjw~trth^rth ^qzrthA 8 t @ B Yz~l8ytrtql(¢ightytcljwhytrthl*8l8fqYighe

d t .igigqYeKrl8eKighegxqYekjml8eh (¢ightytclrtqYeKz~l|jwigq((qzrthfqYeighrt~&cfrtqYeKh#ighl8npo8q8eKrtq((qzrthfqqY(xtfhjwqt

uOqFrthfqighrt~&cfrtq igigqhy jwighl8eighegx­qYekjml8e @ . ! ×10−4 ¥ !F ! @ ×10−4u2uO ×10−4b/ ¥ @ F ! ×10−4

l8jwqYe qK l8yttqKegh*l8yzjwMxtiwl|jwcfpl8(hyzjwh qYytekjml8y jwh(qYekjwiwl8ytrtqz~thl (¢~tytnpo8qKrthjwiwl8yte(¢higytcfl^tcfxtiYxtigcl.¥ l8iwlhekjwh^jwcfx­qdrth (¢~tytnpo8qt ldighpl8fc;&pl8npo8qhWhegxl8nqrthhekjml8rtqYex­qYegeg~tc.q/hfh(hy jwqrthdjwiwl8yteg(cfegewo8q&rtcfighjmlrtc+(¢highy jwh/rth&higq A

d = 4, 8803× 10−3 .xl8iwl&hekjml(¢~tytnpo8qCB .~6-kq *8l8fq8ix­qzrthKeghiqY.jwcfrtq l|jwiwl * erlighegx­qYekjmlytqjwh(x­qtYxl8iwl

t = 0A 8 t @ B hekjwhKpl8egqt q *|l8fqYi

rlewl8frlrth5*YheghiK&~tjwcxtfcpl8rtq&x­hll|(xtfcjw~trth/rlhy jwiwl8rlrtqjwcfx­qrthYiwl8~Ol8xtfcfpl8rl A @ B

−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

agin

ário

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

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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]

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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)

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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

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t F u b/ /d b $ b

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H

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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

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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

'~.jwcffc+&pl8npo8q rth(qzrthfqYeeg~tx­hi ighrt~&cfrtqYexl8iwl xtigq-khjwqRrthqYy jwigqYl8rtqYighextigqYx­qYekjwqYeythekjmleghnpo8q&l8fpl8ytnqY~igheg~tjml8rtqYeewl|jwcfe(ªl|jYigcqYe ¥qYigO jwiwl|jml eghKegqY(hy jwhKrth~thekjw~trtqxtigh fcfcyl8i eghytrtq&l8cfytrlythhege%$8igcl8e^ytq*|l8e^l8y$8fcfeghexl8iwl (hftqYih#".xtfqYiwl8il cfrtclhhekjml8­hfhhi~t*lO(hjwqFrtqYfqYYcflh&l8ewl8rlh (¢~tytrl8(hy jwqYejwhqYigcfpl8(hy jwh*qYytegcfekjwhy jwhexl8iwlOq2pl8egqrll8­qYigrl8YhG(qFrl8 Aªrtq8(cfyF8ytcl&rth x YfqYeh &higqYe0B

$) ( , WO) W$ -/ K-/p

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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

qYytrthcjihbkiewo8qttighegx­hjwc+*|l8(hyzjwh8tqYe (ªl|jwqYigherthqYteghi*8l8tcffcfrl8rthdhqYy jwigqYl8tcffcrl8rth/l8egegq|

cl8rtqYeKl8qx YfqλiqYytegcfrthiwl8ytrtq*lhy jwiwl8rl

khlewl8frl

jrtqegcfekjwh*l(uOku

@

eewcegjwh(l8e jmhegjwhehekjw~trl8rtqYe/ythekjwhpl8xtjw~tfqewo8q*qYed(heg(qYeKrtqYedpl8xtjw~tfqYeKl8yzjwhigcfqYighex­qYig l8yl|fcfewl8rtqYeeghY~tytrtqRlXx­hiwegxhjwc+*8l&~tjwc+*8l8igc;$ *Yh )q Fcfekjwh*l ¥hekjwhOEA¢dx­ytrtcfh. @ B(¢qYiwl8Ll8xtfcfpl8rl8ert~l8ejwytcfpl8e(rthighrt~tnpo|q rthO(qFrthfqtqXjwig~tytpl8(hyzjwql8l8ythpl8rtqA ^ ¥u B^h qjwig~tytpl8(hyzjwq(qFrl8 A¢ buB $^qY#l8egh ytqYeKqFrthfqYeighrt~&crtqYehApl|rl*l8flrtqYe Yhiwl8rtqYigheDA

∆Ωk/∆V kref

B (¢qYczl8yl8fcewl8rtqqrthegh(x­hyttqdrth~t>hekjml8tcffc;&pl8rtqYi Aª. B&~tjwc+*|l8igc $ *Yhtrtheghyzjwiwl8fc;&pl|rtqtC*Fclrtcl8Yiwl8*l8erthf~t l8igherl8eiwl8;&hexl8iwl ~t*l rthjwhig(cfyl8rl(ªl8c "tl&rth l8yttqtekjwh xtigqFhrtcf(hy jwq rthytqY(cfyl8rtq /'2 '( ytq*­b

l8iwlqFcfekjwh*l ¥hekjwhdgA¢dxytrtcfh . B Fewq8(hyzjwh/l&l8­qYigrl8Yh (qFrl8(¢qYc$l8xtfcfpl8rll8qegcfekjwh*l~tjwc *8l8igc;$ *YhqY(x­qYekjwqx­hfqYe/jwige qYy6-k~ty jwqYe rth~tegcfyl8erthegigcjwqYe ylOFhnpo8q t F @ igheg~tjml8rtqqY.jwcfrtq&ythekjmleghnpo8qeghi%$&jwhekjml8rtqh/qY(xl8iwl8rtqqY qY~.jwiwljwytcfpl&rthqY.jwhytnpo8qrthx YfqYertqY(cfyl8y jwhe&~tjwc+*8l|iwc;$ *Yhcfe¥xtigheghyzjwhytq ­b rthytqY(cfyl|rl ' %,!% & # +%, ' . 2' ! % !/ A ub. B z~th qYytegcfekjwh h#~t*l&h#"Fjwhytewo8q*rl&jwytcfplb/.xl8iwlegcfekjwh*l8euOgu

, 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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