controlo de sistemas diagramas polares
TRANSCRIPT
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Controlo de SistemasDiagramas polares
Alexandra Moutinho
Dep. Engenharia Mecรขnica, Instituto Superior Tรฉcnico, Universidade de Lisboa, ([email protected])
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โข Resposta em frequรชncia
โ Diagrama de Bode (revisรตes)
โข Anรกlise de estabilidade
โ Margens de estabilidade (diagrama de Bode)
โ Relaรงรฃo entre margens de estabilidade e LGR
โ Margens de estabilidade de sistemas com atraso
Controlo de Sistemas
Aula anterior
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โข Resposta em frequรชncia
โ Diagrama polar
โข Anรกlise de estabilidade:
โ Margens de estabilidade em diagramas polares
Controlo de Sistemas
Esta aula
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โข O diagrama de Bode representa a resposta em frequรชncia de um sistema, ๐บ(๐๐), atravรฉs
โ Do grรกfico do seu mรณdulo, ๐บ(๐๐) , ou 20 log10 ๐บ(๐๐) , e
โ Do grรกfico da sua fase, arg[๐บ(๐๐)]
โข Exemplo:
๐บ ๐๐ =1
๐๐๐๐
2
+ 2๐๐๐๐๐
+ 1
Controlo de Sistemas
Resposta em fequรชncia:Diagrama de Bode
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โข O diagrama polar representa a resposta em frequรชncia de um sistema, ๐บ(๐๐), representando num mesmo grรกfico o seu mรณdulo, ๐บ(๐๐) , e a sua fase, arg[๐บ(๐๐)]
โข Exemplo:
๐บ ๐๐ =1
๐๐๐๐
2
+ 2๐๐๐๐๐
+ 1
Controlo de Sistemas
Resposta em frequรชncia:Diagrama polar
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๐ฎ ๐๐ =1
๐๐๐๐
2
+ 2๐๐๐๐๐
+ 1
=1
2๐๐๐๐
๐ + 1 โ๐2
๐๐2
= ๐๐ ๐ฎ ๐๐ + ๐๐๐ฆ ๐ฎ ๐๐
= ๐ฎ ๐๐ ๐๐โ ๐ฎ ๐๐
๐บ ๐๐ =เต
1
1 โ๐2
๐๐2
2
+ 2๐๐๐๐
2
โ ๐บ ๐๐ = โarctan2๐
๐๐๐
1 โ๐2
๐๐2
Controlo de Sistemas
Resposta em frequรชncia:Diagrama polar
๐บ(๐๐) Im[๐บ ๐๐ ]
Re[๐บ ๐๐ ]
โ ๐บ ๐๐
๐บ(๐๐)
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Controlo de Sistemas
Resposta em frequรชncia:Diagrama de Bode vs. Diagrama polar
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Controlo de Sistemas
Resposta em frequรชncia:Diagrama polar
โข Vantagens dos diagramas polares:
โ Integraรงรฃo num รบnico grรกfico da evoluรงรฃo do mรณdulo e da fase de ๐บ(๐๐)
โ Mais fรกcil analisar a estabilidade relativa de sistemas de controlo
โ O critรฉrio de estabilidade de Nyquist vai basear-se nesta representaรงรฃo
46
๐บ(๐๐) Im[๐บ ๐๐ ]
Re[๐บ ๐๐ ]
โ ๐บ ๐๐
๐บ(๐๐)
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๐ฎ ๐๐ = ๐๐ ๐ฎ ๐๐ + ๐๐๐ฆ ๐ฎ ๐๐
= ๐ฎ ๐๐ ๐๐โ ๐ฎ ๐๐
๐ฎ ๐๐ = ๐๐ ๐ฎ ๐๐ ๐ + ๐๐ฆ ๐ฎ ๐๐ ๐
๐ฎ โ๐๐ = ๐ฎ ๐๐
โ ๐ฎ ๐๐ = arg ๐ฎ ๐๐ = atan๐๐ฆ ๐ฎ ๐๐
๐๐ ๐ฎ ๐๐
arg ๐ฎ โ๐๐ = โarg ๐ฎ ๐๐
arg๐(๐ )
๐(๐ )= arg ๐ ๐ โ arg[๐(๐ )]
Controlo de Sistemas 47
Funรงรตes complexas
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Controlo de Sistemas
Diagrama polar:Termo integrativo ๐๐ โ1
๐บ ๐๐ =1
๐๐= โ๐
1
๐=1
๐๐โ๐ arctan
๐0 =
1
๐๐โ๐90ยฐ, ๐ โ 0,+โ
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Controlo de Sistemas
Diagrama polar:Termo derivativo ๐๐ +1
๐บ ๐๐ = ๐๐ = ๐๐๐ arctan
๐0 = ๐๐๐90ยฐ, ๐ โ 0,+โ
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Controlo de Sistemas
Diagrama polar:Fator de 1ยช ordem ๐๐๐ + 1 โ1
๐บ ๐๐ =1
๐๐๐ + 1=
1
๐2๐2 + 1๐โ๐ arctan ๐๐ , ๐ โ 0,+โ
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Controlo de Sistemas
Diagrama polar:Fator de 1ยช ordem ๐๐๐ + 1 +1
๐บ ๐๐ = ๐๐๐ + 1 = ๐2๐2 + 1๐๐ arctan ๐๐ , ๐ โ 0,+โ
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Controlo de Sistemas
Diagrama polar:
Fator de 2ยช ordem ๐๐
๐๐
2
+ 2๐๐๐
๐๐+ 1
โ1
๐บ ๐๐ =1
๐๐๐๐
2
+ 2๐๐๐๐๐
+ 1
, ๐ > 0
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Controlo de Sistemas
Diagrama polar:
Fator de 2ยช ordem ๐๐
๐๐
2
+ 2๐๐๐
๐๐+ 1
+1
๐บ ๐๐ = ๐๐
๐๐
2
+ 2๐๐๐
๐๐+ 1, ๐ > 0
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Controlo de Sistemas
Diagrama polar: formas genรฉricas
โข Segundo o tipo do sistema (sistema com ganho positivo)
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Controlo de Sistemas
Diagrama polar: formas genรฉricas
โขPara ๐ โ +โ (sistemas causais sem zeros ou pรณlos no SPD, com ganho positivo)
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Controlo de Sistemas
Diagrama polar: formas genรฉricas
โขPara sistemas com dinรขmica no numerador (zeros)
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โข Desenhar o diagrama polar do sistema:
๐บ ๐ =1
๐ (๐๐ + 1)
๐บ ๐๐ =1
๐๐(๐๐๐ + 1)=
1
โ๐๐2 + ๐๐=โ๐๐2 โ ๐๐
๐2๐4 +๐2
=โ๐๐2
๐2๐4 +๐2โ ๐
๐
๐2๐4 +๐2= โ
๐
๐2๐2 + 1โ ๐
1
๐(๐2๐2 + 1)
lim๐โ0
๐บ ๐๐ = โ๐ โ ๐โ = โ๐โ๐90ยฐ
lim๐โ+โ
๐บ ๐๐ = โ0 โ ๐0 = 0๐โ๐180ยฐ
Controlo de Sistemas
Diagrama polar: exemplo
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โข Desenhar o diagrama polar do sistema:
๐บ ๐ =1
๐ (๐๐ + 1)
lim๐โ0
๐บ ๐๐ = โ๐ โ ๐โ
= โ๐โ๐90ยฐ
lim๐โ+โ
๐บ ๐๐ = โ0 โ ๐0
= 0๐โ๐180ยฐ
Controlo de Sistemas
Diagrama polar: exemplo
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Controlo de Sistemas
Diagrama polar de FT simples
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โข Diagrama polar do termo de atraso:
๐บ ๐๐ = ๐โ๐๐๐ = 1. arg[cos ๐๐ โ ๐ sin(๐๐)] = โ๐๐
โข Mรณdulo unitรกrio
โข Fase varia linearmente com ๐
Controlo de Sistemas
Diagrama polar de sistemas com atraso
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โข Exemplo: desenhar o diagrama polar do sistema:
๐บ ๐๐ =1
๐๐๐ + 1๐โ๐๐๐
๐บ ๐๐ =1
๐๐๐ + 1๐โ๐๐๐ =
1
๐2๐2 + 1
arg ๐บ ๐๐ = arg1
๐๐๐ + 1+ arg ๐โ๐๐๐ = โarctan๐๐ โ ๐๐
Controlo de Sistemas
Diagrama polar de sistemas com atraso
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Controlo de Sistemas
Margens de estabilidade em diagramas de Bode
๐๐๐: arg[๐พ๐บ(๐๐๐๐)] = โ180ยฐ, ๐๐๐: ๐พ๐บ ๐๐๐๐ = 1
MG๐พdB = โ20 log10 ๐พ๐บ ๐๐๐๐ , MF๐พ = 180ยฐ + arg[๐พ๐บ(๐๐๐๐)]
๐๐๐
arg[๐บ ๐๐๐๐ ]
๐๐๐
20 log10 ๐บ ๐๐๐๐
๐พ = 1: ๐พ๐บ ๐๐ = ๐บ ๐๐
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โข ๐พ = 1: ๐พ๐บ ๐๐ = ๐บ ๐๐
โข ๐๐๐: ๐บ ๐๐๐๐ = 1
โข MF๐พ = ๐พ = 180ยฐ + arg[๐บ(๐๐๐๐)]
โข ๐๐๐: arg[๐บ(๐๐๐๐)] = โ180ยฐ
โข MG๐พdB = โ20 log10 ๐บ ๐๐๐๐
= 20 log101
๐บ ๐๐๐๐
โข MG๐พ = ๐พ๐ =1
๐บ ๐๐๐๐
Controlo de Sistemas
Margens de estabilidade em diagramas polares
๐บ ๐๐๐๐
๐บ ๐๐๐๐
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Controlo de Sistemas
Margens de estabilidade em diagramas polares
MG๐พ =1
๐พ๐บ ๐๐๐๐=
1
๐พ๐, MF๐พ = 180ยฐ + arg[๐พ๐บ(๐๐๐๐)]
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โข Determine as margens de ganho MG e de fase MF, e respetivas frequรชncias de cruzamento de fase ๐๐๐ e de ganho ๐๐๐,
para o sistema seguinte, considerando ๐พ = 10 e ๐พ = 100
Controlo de Sistemas
Margens de estabilidade no Matlab
>> num=1;
>> den=conv([1 1 0],[1 5])
>> G=tf(num,den)
>> figure,margin(10*G)
>> figure,margin(100*G)
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Controlo de Sistemas
Margens de estabilidade no Matlab
๐พ = 10 ๐พ = 100
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Diagrama polar e margens de estabilidade
๐บ(๐๐) Im[๐บ ๐๐ ]
Re[๐บ ๐๐ ]
โ ๐บ ๐๐
๐บ(๐๐)
https://create.kahoot.it/details/at9-diagrama-polar-e-margens-estabilidade/4a72e83f-fafe-401c-bcf5-329008ac097c
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โข Resposta em frequรชncia
โ Diagrama de Nyquist
โข Anรกlise de estabilidade
โ Critรฉrio de Nyquist
Controlo de Sistemas
Prรณximas aulas
68
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โ Control Systems Engineering, Norman Nise, John Wiley & Sons (6ยช ediรงรฃo), 2011
โ Controlo de Sistemas, Miguel Ayala Botto, AEIST Press, 2008
โ Modern Control Engineering, K. Ogata, PrenticeโHall International (4ยช ediรงรฃo), 2002
โ Feedback Control of Dynamic Systems, Gene F. Franklin, J. David Powell, Abbas EmamiโNaeini, Pearson (6ยช ediรงรฃo), 2010
Controlo de Sistemas
Referรชncias usadas
69