Fundamentos de Controle
SEL611
Prof. Dr. Valdir Grassi Jr
Escola de Engenharia de São Carlos - EESC/USP
Estabilidade Neutra
Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
2
Estabilidade Malha Fechada
K=2
I Considere o seguinte sistema de malha fechada
R
+
!
−
K
1
s(s + 1)2
K
Y
−2
−1
K=2
I A equação caracterı́stica do sistema:
(a) = 0
1 + KG(s)
I Os pontos nos ramos do lugar das raı́zes de G(s) representam
Suponha
que
a função
de transferência
os polos de malha
fechada
para
diferentes
valores de K:em malha fec
demos determinar a estabilidade de um sistema simplesm
o
180os
ou
= 180o as raízes
|KG(s)|
= 1; fatorada
KG(s)(já=que
G(s)
na forma
fatores
fornecerem
se as partes reais são positivas ou negativas. No entanto
3
fechada normalmente não é conhecida; na verdade, todo
Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
Estabilidade Malha Fechada
I Quando polos de malha fechada estão sobre o eixo imaginário,
o sistema está na condição de estabilidade neutra
(marginalmente estável).
I Isso acontece quando a raı́z possui apenas componente
imaginário, ou seja, s = jwc
I Se esse ponto, que cruza o eixo imaginário, está no lugar das
raı́zes:
|KG(jwc )| = 1
;
KG(jwc ) = 180o
Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
4
Figura
de esta
K=2
finição
K>2
lugarBa
da
1
Capítulo 6 O Método de Projeto
Im(s)
Estabilidade Malha Fechada
R
+
!
−
+
K
1
s(s + 1)2
1
2
K<2
Y
−2
−1
2
K = 2−1
K=2
1
K −2
<2
Im(s)Re(s)
K>2
Figu
de e
finiç
luga
IR Para o! sistema
K G(s) do exemplo,
oY
2
(a) s(s + 1)
Re(s)
−2
−1 (b)
lugar−das raı́zes cruza o eixo
−1
imaginário na posição s = j1, ou
Suponha
de um sistema é conh
K=2
seja,
wc =que
1 a função de transferência em malha fechada
demos determinar a estabilidade de um sistema simplesmente−2inspecionando seu de
I Isso acontece quando K = 2, ou
na forma fatorada (já que os fatores fornecerem as raízes do sistema diretamente) pa
parareais
> 2são
o (a)
sistema
instável,
seseja,
as partes
positivaséou
negativas. No entanto, a função
(b) de transferênci
para
K
<
2
o
sistema
é
estável.
fechada normalmente não é conhecida; na verdade, todo o propósito por trás da co
da técnica do lugar das raízes é ser capaz de encontrar os fatores do denominador da
Suponhaem
quemalha
a função
de transferência
em malha
fechada
de um sistema
éc
transferência
fechada
conhecendo apenas
a função
de transferência
5 em ma
demos determinar a estabilidade de um sistema simplesmente inspecionando seu
Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
Estabilidade Malha Fechada
I Se o lugar das raı́zes cruza o eixo imaginário, a raı́z s = jw
pertence ao lugar das raı́zes, e existe um valor de w = wc e
K = Kc para o qual:
|KG(jw)| = 1
;
KG(jw) = 180o
(1)
I O diagrama de bode mostra a resposta de |KG(jw)| e fase
KG(jw) = 180o do sistema KG(s) em função do valor w.
I Portanto, se existe um valor de ganho K e uma frequencia w
para os quais a resposta em frequencia é dada pelas condições
da Eq. (1), o sistema é marginalmente estável, ou seja, está
na condição de estabilidade neutra.
Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
6
Estabilidade Malha Fechada
R
+
!
K
−
1
s(s + 1)2
K=2
K<2
Y
−2
−1
K=2
G(s) =
1
s(s
(a) + 1)2
−
−
I O gráfico de bode de G(s) pode ser construı́do a partir dos
componentes:Suponha que a função de transferência em malha fecha
1
1 simplesmen
1 determinar a estabilidade
demos
de um sistema
; G3 (s) =
;
G1 (s) = ; G2 (s) =
na forma
os fatores
fornecerem
s fatorada (já(sque
+ 1)
(s + 1) as raízes do
partes
reais sãoposteriormente.
positivas ou negativas. No entanto, a
I O ganho se
K as
será
considerado
fechada normalmente não é conhecida; na verdade, todo o
7
da técnica do lugar das raízes é ser capaz de encontrar os fa
Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
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Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
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Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
1000
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Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
1000
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1000
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Exemplo - Fase
Semi-log Paper for Bode Plots
<latexit sha1_base64="8I4XQycTS5qjTzq7FcE5/BJh2HI=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK0CCUpgt0IBRcKbirYB7SxTKaTdtrJJMxMLCVk48ZfceNCEbf+gzv/xmmbhbYeuHA4517uvccNGZXKsr6NpeWV1bX1zEZ2c2t7Z9fc26/LIBKY1HDAAtF0kSSMclJTVDHSDAVBvstIwx1eTvzGAxGSBvxOjUPi+KjHqUcxUlrqmEdX+cGoAC9g2xMIxzdJPBhp5dQu3JeSjpmzitYUcJHYKcmBFNWO+dXuBjjyCVeYISlbthUqJ0ZCUcxIkm1HkoQID1GPtDTlyCfSiadfJPBEK13oBUIXV3Cq/p6IkS/l2Hd1p49UX857E/E/rxUpr+zElIeRIhzPFnkRgyqAk0hglwqCFRtrgrCg+laI+0jHoXRwWR2CPf/yIqmXirZVtG/PcpVyGkcGHIJjkAc2OAcVcA2qoAYweATP4BW8GU/Gi/FufMxal4x05gD8gfH5A+d6luE=</latexit>
<latexit sha1_base64="ORL7zEiCROitdw9H7l6udrIngfo=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBahRShJEexGKLjQZQX7gDaWyXTSTjuZhJmJpYRs3Pgrblwo4tZ/cOffOG2z0NYDFw7n3Mu997gho1JZ1reRWVldW9/Ibua2tnd298z9g4YMIoFJHQcsEC0XScIoJ3VFFSOtUBDku4w03dHV1G8+ECFpwO/UJCSOj/qcehQjpaWueXxdGI6L8BJ2PIFwbCfxcKyVM7t4X066Zt4qWTPAZWKnJA9S1LrmV6cX4MgnXGGGpGzbVqicGAlFMSNJrhNJEiI8Qn3S1pQjn0gnnn2RwFOt9KAXCF1cwZn6eyJGvpQT39WdPlIDuehNxf+8dqS8ihNTHkaKcDxf5EUMqgBOI4E9KghWbKIJwoLqWyEeIB2H0sHldAj24svLpFEu2VbJvj3PVytpHFlwBE5AAdjgAlTBDaiBOsDgETyDV/BmPBkvxrvxMW/NGOnMIfgD4/MHvsCWxw==</latexit>
90
45
0
-90
-180
0.01
0.1
1
10
100
Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
1000
18
Exemplo - Ganho e Fase
Semi-log Paper for Bode Plots
20
10
0
-20
0.01
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0.1
1
10
100
1000
0.1
1
10
100
1000
-90
-180
0.01
Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
19
Exemplo - Ganho e Fase
Semi-log Paper for Bode Plots
20
10
0
Sistema instável
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-20
0.01
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0.1
1
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10
100
1000
10
100
1000
Sistema estável
-90
-180
0.01
0.1
1
Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
20
Exemplo - Ganho e Fase
osta em
40
100
K = 10
20
K=2
0
1
−20
0,1
K = 0,1
db
Magnitude, |KG(jω)|
10
−40
0,01
0,001
0,1
1
10
−60
100
I O sistema é estável quando
|KG(jw)| < 1 no ponto
onde G(jw) = −180o
Fase, ∠G(jω)
ω (rad/s)
−90°
+80°
−180°
−270°
−35°
0,1
1
ω (rad/s)
10
100
Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
21
Margens de Estabilidade
Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
22
Exemplo - Margem de Ganho
Semi-log Paper for Bode Plots
20
10
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0
Sistema instável
<latexit sha1_base64="1/8c3E9VJLwmwXdbKv2v6IrEsWI=">AAAB6nicdVDLSgMxFM3UV62vqks3wSK4GpKho7OSghvBTUX7gHYomTTThmYyQ5IRSuknuHGhiFu/yJ1/Y/oQVPTAhcM593LvPVEmuDYIfTiFldW19Y3iZmlre2d3r7x/0NRprihr0FSkqh0RzQSXrGG4EaydKUaSSLBWNLqc+a17pjRP5Z0ZZyxMyEDymFNirHR7feH1yhXkojO/igOIXB/hAPuWeD5GyIPYRXNUwBL1Xvm9209pnjBpqCBadzDKTDghynAq2LTUzTXLCB2RAetYKknCdDiZnzqFJ1bpwzhVtqSBc/X7xIQkWo+TyHYmxAz1b28m/uV1chMH4YTLLDdM0sWiOBfQpHD2N+xzxagRY0sIVdzeCumQKEKNTadkQ/j6FP5Pmp6LkYtvqpVasIyjCI7AMTgFGJyDGrgCddAAFAzAA3gCz45wHp0X53XRWnCWM4fgB5y3T+IMjX8=</latexit>
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-20
0.01
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0.1
1
<latexit sha1_base64="2ZhP2hxBMOiN7FEOezgRT1PhUGk=">AAAB6nicdVDJSgNBEK1xjXGLevTSGARPQ88YxwQ8BLwIXiKaBZIh9HR6kiY9C909Qgj5BC8eFPHqF3nzb+wsgoo+KHi8V0VVvSAVXGmMP6yl5ZXVtfXcRn5za3tnt7C331BJJimr00QkshUQxQSPWV1zLVgrlYxEgWDNYHg59Zv3TCqexHd6lDI/Iv2Yh5wSbaTb6wu3Wyhiu1LBJcdD2D7D2PUqhuBTt+x5yLHxDEVYoNYtvHd6Cc0iFmsqiFJtB6faHxOpORVsku9kiqWEDkmftQ2NScSUP56dOkHHRumhMJGmYo1m6veJMYmUGkWB6YyIHqjf3lT8y2tnOiz7Yx6nmWYxnS8KM4F0gqZ/ox6XjGoxMoRQyc2tiA6IJFSbdPImhK9P0f+k4doOtp2bUrFaXsSRg0M4ghNw4ByqcAU1qAOFPjzAEzxbwnq0XqzXeeuStZg5gB+w3j4B/m+Nkw==</latexit>
10
100
1000
10
100
1000
Sistema estável
-90
-180
0.01
0.1
1
Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
23
Exemplo - Margem de Fase
Semi-log Paper for Bode Plots
20
10
0
Sistema instável
<latexit sha1_base64="1/8c3E9VJLwmwXdbKv2v6IrEsWI=">AAAB6nicdVDLSgMxFM3UV62vqks3wSK4GpKho7OSghvBTUX7gHYomTTThmYyQ5IRSuknuHGhiFu/yJ1/Y/oQVPTAhcM593LvPVEmuDYIfTiFldW19Y3iZmlre2d3r7x/0NRprihr0FSkqh0RzQSXrGG4EaydKUaSSLBWNLqc+a17pjRP5Z0ZZyxMyEDymFNirHR7feH1yhXkojO/igOIXB/hAPuWeD5GyIPYRXNUwBL1Xvm9209pnjBpqCBadzDKTDghynAq2LTUzTXLCB2RAetYKknCdDiZnzqFJ1bpwzhVtqSBc/X7xIQkWo+TyHYmxAz1b28m/uV1chMH4YTLLDdM0sWiOBfQpHD2N+xzxagRY0sIVdzeCumQKEKNTadkQ/j6FP5Pmp6LkYtvqpVasIyjCI7AMTgFGJyDGrgCddAAFAzAA3gCz45wHp0X53XRWnCWM4fgB5y3T+IMjX8=</latexit>
-20
<latexit sha1_base64="KXtCCGtG+V8VLHv86fas98GsEMw=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR7EUoeBG8VLQf0Iay2W7apZtN2J0IJfQnePGgiFd/kTf/jds2B219MPB4b4aZeUEihUHX/XbW1jc2t7YLO8Xdvf2Dw9LRccvEqWa8yWIZ605ADZdC8SYKlLyTaE6jQPJ2ML6Z+e0nro2I1SNOEu5HdKhEKBhFKz3cXVf7pbJbcecgq8TLSRlyNPqlr94gZmnEFTJJjel6boJ+RjUKJvm02EsNTygb0yHvWqpoxI2fzU+dknOrDEgYa1sKyVz9PZHRyJhJFNjOiOLILHsz8T+vm2JY8zOhkhS5YotFYSoJxmT2NxkIzRnKiSWUaWFvJWxENWVo0ynaELzll1dJq1rx3Ip3f1mu1/I4CnAKZ3ABHlxBHW6hAU1gMIRneIU3RzovzrvzsWhdc/KZE/gD5/MHkeuNSA==</latexit>
0.01
0.1
1
<latexit sha1_base64="2ZhP2hxBMOiN7FEOezgRT1PhUGk=">AAAB6nicdVDJSgNBEK1xjXGLevTSGARPQ88YxwQ8BLwIXiKaBZIh9HR6kiY9C909Qgj5BC8eFPHqF3nzb+wsgoo+KHi8V0VVvSAVXGmMP6yl5ZXVtfXcRn5za3tnt7C331BJJimr00QkshUQxQSPWV1zLVgrlYxEgWDNYHg59Zv3TCqexHd6lDI/Iv2Yh5wSbaTb6wu3Wyhiu1LBJcdD2D7D2PUqhuBTt+x5yLHxDEVYoNYtvHd6Cc0iFmsqiFJtB6faHxOpORVsku9kiqWEDkmftQ2NScSUP56dOkHHRumhMJGmYo1m6veJMYmUGkWB6YyIHqjf3lT8y2tnOiz7Yx6nmWYxnS8KM4F0gqZ/ox6XjGoxMoRQyc2tiA6IJFSbdPImhK9P0f+k4doOtp2bUrFaXsSRg0M4ghNw4ByqcAU1qAOFPjzAEzxbwnq0XqzXeeuStZg5gB+w3j4B/m+Nkw==</latexit>
10
100
1000
10
100
1000
Sistema estável
-90
-180
<latexit sha1_base64="0xOuDYA5cSu7Y8T4QwQ8ynexA2c=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GjK1ju1GCoK4ESrYB7RDyaRpG5tJhiQjlKH/4MaFIm79H3f+jelDUNEDFw7n3Mu994QxZ9og9OFklpZXVtey67mNza3tnfzuXkPLRBFaJ5JL1QqxppwJWjfMcNqKFcVRyGkzHF1M/eY9VZpJcWvGMQ0iPBCszwg2VmpcX8JziLr5AnIrFVTyfIjcU4SKfsUSdFIs+z70XDRDASxQ6+bfOz1JkogKQzjWuu2h2AQpVoYRTie5TqJpjMkID2jbUoEjqoN0du0EHlmlB/tS2RIGztTvEymOtB5Hoe2MsBnq395U/MtrJ6ZfDlIm4sRQQeaL+gmHRsLp67DHFCWGjy3BRDF7KyRDrDAxNqCcDeHrU/g/aRRdD7neTalQLS/iyIIDcAiOgQfOQBVcgRqoAwLuwAN4As+OdB6dF+d13ppxFjP74Aect08+Lo45</latexit>
<latexit sha1_base64="8TBY7hYNbzkV2gS8uWzo/O15dAI=">AAAB7XicdVDLSgMxFM34rPVVdekmWARXQ1I6OgsXBUHcCBXsA9qhZNJMG5vJDElGKEP/wY0LRdz6P+78G9OHoKIHLhzOuZd77wlTwbVB6MNZWl5ZXVsvbBQ3t7Z3dkt7+02dZIqyBk1Eotoh0UxwyRqGG8HaqWIkDgVrhaOLqd+6Z0rzRN6accqCmAwkjzglxkrN60t4DlGvVEYuOvWq2IfI9RD2sWdJxcMIVSB20QxlsEC9V3rv9hOaxUwaKojWHYxSE+REGU4FmxS7mWYpoSMyYB1LJYmZDvLZtRN4bJU+jBJlSxo4U79P5CTWehyHtjMmZqh/e1PxL6+TmcgPci7TzDBJ54uiTECTwOnrsM8Vo0aMLSFUcXsrpEOiCDU2oKIN4etT+D9pVlyMXHxTLdf8RRwFcAiOwAnA4AzUwBWogwag4A48gCfw7CTOo/PivM5bl5zFzAH4AeftExu1jiE=</latexit>
0.01
0.1
1
Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo
24