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 Exemplo - Ganho Semi-log Paper for Bode Plots <latexit sha1_base64="Z05y4iL5FsQqq6pqliBvq3K2h8g=">AAAB/3icbVDLSgMxFM3UV62vUcGNm2AR6qZMRLAboeBClxXsA9qhZNJMmzaTGZKMpYyz8FfcuFDErb/hzr8xbWehrQcuHM65l3vv8SLOlHacbyu3srq2vpHfLGxt7+zu2fsHDRXGktA6CXkoWx5WlDNB65ppTluRpDjwOG16o+up33ygUrFQ3OtJRN0A9wXzGcHaSF376KaLSsPxGbyCHV9ikqA0GY7Trl10ys4McJmgjBRBhlrX/ur0QhIHVGjCsVJt5ETaTbDUjHCaFjqxohEmI9ynbUMFDqhyk9n9KTw1Sg/6oTQlNJypvycSHCg1CTzTGWA9UIveVPzPa8far7gJE1GsqSDzRX7MoQ7hNAzYY5ISzSeGYCKZuRWSATYxaBNZwYSAFl9eJo3zMnLK6O6iWK1kceTBMTgBJYDAJaiCW1ADdUDAI3gGr+DNerJerHfrY96as7KZQ/AH1ucPUkCU/Q==</latexit> 40 20 10 0 -20 -40 0.01 0.1 1 10 100 Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo 1000 8 Exemplo - Ganho Semi-log Paper for Bode Plots <latexit sha1_base64="wkfS0j5smqVCncoUTXE79/gpiRk=">AAACA3icbVDLSgMxFM3UV62vUXe6CRahRSiTItiNUHChywr2Ae0wZNJMG5vJDElGKcOAG3/FjQtF3PoT7vwb08dCqwcunJxzL7n3+DFnSjvOl5VbWl5ZXcuvFzY2t7Z37N29looSSWiTRDySHR8rypmgTc00p51YUhz6nLb90cXEb99RqVgkbvQ4pm6IB4IFjGBtJM8+uPSqpdv7MjyHvUBikqIsNe8TVM48u+hUnCngX4LmpAjmaHj2Z68fkSSkQhOOleoiJ9ZuiqVmhNOs0EsUjTEZ4QHtGipwSJWbTm/I4LFR+jCIpCmh4VT9OZHiUKlx6JvOEOuhWvQm4n9eN9FBzU2ZiBNNBZl9FCQc6ghOAoF9JinRfGwIJpKZXSEZYhOFNrEVTAho8eS/pFWtIKeCrk+L9do8jjw4BEegBBA4A3VwBRqgCQh4AE/gBbxaj9az9Wa9z1pz1nxmH/yC9fENDBSV0w==</latexit> 40 20 10 0 -20 -40 0.01 0.1 1 10 100 Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo 1000 9 Exemplo - Ganho Semi-log Paper for Bode Plots <latexit sha1_base64="4Q7JyMd1JhjTRA4SVqY9MRh1SLE=">AAACA3icbVDLSgMxFM3UV62vUXe6CRahRSgTFexGKLjQZQX7gHYYMmmmjc1khiSjlGHAjb/ixoUibv0Jd/6N6WOh1QMXTs65l9x7/JgzpR3ny8otLC4tr+RXC2vrG5tb9vZOU0WJJLRBIh7Jto8V5UzQhmaa03YsKQ59Tlv+8GLst+6oVCwSN3oUUzfEfcECRrA2kmfvXXonpdv7MjyH3UBikqIsNe8jVM48u+hUnAngX4JmpAhmqHv2Z7cXkSSkQhOOleogJ9ZuiqVmhNOs0E0UjTEZ4j7tGCpwSJWbTm7I4KFRejCIpCmh4UT9OZHiUKlR6JvOEOuBmvfG4n9eJ9FB1U2ZiBNNBZl+FCQc6giOA4E9JinRfGQIJpKZXSEZYBOFNrEVTAho/uS/pHlcQU4FXZ8Wa9VZHHmwDw5ACSBwBmrgCtRBAxDwAJ7AC3i1Hq1n6816n7bmrNnMLvgF6+MbDa+V1A==</latexit> 40 20 10 0 -20 -40 0.01 0.1 1 10 100 Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo 1000 10 Exemplo - Ganho Semi-log Paper for Bode Plots <latexit sha1_base64="ORL7zEiCROitdw9H7l6udrIngfo=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBahRShJEexGKLjQZQX7gDaWyXTSTjuZhJmJpYRs3Pgrblwo4tZ/cOffOG2z0NYDFw7n3Mu997gho1JZ1reRWVldW9/Ibua2tnd298z9g4YMIoFJHQcsEC0XScIoJ3VFFSOtUBDku4w03dHV1G8+ECFpwO/UJCSOj/qcehQjpaWueXxdGI6L8BJ2PIFwbCfxcKyVM7t4X066Zt4qWTPAZWKnJA9S1LrmV6cX4MgnXGGGpGzbVqicGAlFMSNJrhNJEiI8Qn3S1pQjn0gnnn2RwFOt9KAXCF1cwZn6eyJGvpQT39WdPlIDuehNxf+8dqS8ihNTHkaKcDxf5EUMqgBOI4E9KghWbKIJwoLqWyEeIB2H0sHldAj24svLpFEu2VbJvj3PVytpHFlwBE5AAdjgAlTBDaiBOsDgETyDV/BmPBkvxrvxMW/NGOnMIfgD4/MHvsCWxw==</latexit> 40 20 10 0 -20 -40 0.01 0.1 1 10 100 Prof. Dr. Valdir Grassi Jr / São Carlos School of Engineering / University of São Paulo 1000 11 Exemplo - 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Fase Semi-log Paper for Bode Plots <latexit sha1_base64="4Q7JyMd1JhjTRA4SVqY9MRh1SLE=">AAACA3icbVDLSgMxFM3UV62vUXe6CRahRSgTFexGKLjQZQX7gHYYMmmmjc1khiSjlGHAjb/ixoUibv0Jd/6N6WOh1QMXTs65l9x7/JgzpR3ny8otLC4tr+RXC2vrG5tb9vZOU0WJJLRBIh7Jto8V5UzQhmaa03YsKQ59Tlv+8GLst+6oVCwSN3oUUzfEfcECRrA2kmfvXXonpdv7MjyH3UBikqIsNe8jVM48u+hUnAngX4JmpAhmqHv2Z7cXkSSkQhOOleogJ9ZuiqVmhNOs0E0UjTEZ4j7tGCpwSJWbTm7I4KFRejCIpCmh4UT9OZHiUKlR6JvOEOuBmvfG4n9eJ9FB1U2ZiBNNBZl+FCQc6giOA4E9JinRfGQIJpKZXSEZYBOFNrEVTAho/uS/pHlcQU4FXZ8Wa9VZHHmwDw5ACSBwBmrgCtRBAxDwAJ7AC3i1Hq1n6816n7bmrNnMLvgF6+MbDa+V1A==</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 16 Exemplo - Fase Semi-log Paper for Bode Plots <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 17 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 <latexit sha1_base64="KXtCCGtG+V8VLHv86fas98GsEMw=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR7EUoeBG8VLQf0Iay2W7apZtN2J0IJfQnePGgiFd/kTf/jds2B219MPB4b4aZeUEihUHX/XbW1jc2t7YLO8Xdvf2Dw9LRccvEqWa8yWIZ605ADZdC8SYKlLyTaE6jQPJ2ML6Z+e0nro2I1SNOEu5HdKhEKBhFKz3cXVf7pbJbcecgq8TLSRlyNPqlr94gZmnEFTJJjel6boJ+RjUKJvm02EsNTygb0yHvWqpoxI2fzU+dknOrDEgYa1sKyVz9PZHRyJhJFNjOiOLILHsz8T+vm2JY8zOhkhS5YotFYSoJxmT2NxkIzRnKiSWUaWFvJWxENWVo0ynaELzll1dJq1rx3Ip3f1mu1/I4CnAKZ3ABHlxBHW6hAU1gMIRneIU3RzovzrvzsWhdc/KZE/gD5/MHkeuNSA==</latexit> 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 <latexit sha1_base64="1/8c3E9VJLwmwXdbKv2v6IrEsWI=">AAAB6nicdVDLSgMxFM3UV62vqks3wSK4GpKho7OSghvBTUX7gHYomTTThmYyQ5IRSuknuHGhiFu/yJ1/Y/oQVPTAhcM593LvPVEmuDYIfTiFldW19Y3iZmlre2d3r7x/0NRprihr0FSkqh0RzQSXrGG4EaydKUaSSLBWNLqc+a17pjRP5Z0ZZyxMyEDymFNirHR7feH1yhXkojO/igOIXB/hAPuWeD5GyIPYRXNUwBL1Xvm9209pnjBpqCBadzDKTDghynAq2LTUzTXLCB2RAetYKknCdDiZnzqFJ1bpwzhVtqSBc/X7xIQkWo+TyHYmxAz1b28m/uV1chMH4YTLLDdM0sWiOBfQpHD2N+xzxagRY0sIVdzeCumQKEKNTadkQ/j6FP5Pmp6LkYtvqpVasIyjCI7AMTgFGJyDGrgCddAAFAzAA3gCz45wHp0X53XRWnCWM4fgB5y3T+IMjX8=</latexit> -20 0.01 <latexit sha1_base64="KXtCCGtG+V8VLHv86fas98GsEMw=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR7EUoeBG8VLQf0Iay2W7apZtN2J0IJfQnePGgiFd/kTf/jds2B219MPB4b4aZeUEihUHX/XbW1jc2t7YLO8Xdvf2Dw9LRccvEqWa8yWIZ605ADZdC8SYKlLyTaE6jQPJ2ML6Z+e0nro2I1SNOEu5HdKhEKBhFKz3cXVf7pbJbcecgq8TLSRlyNPqlr94gZmnEFTJJjel6boJ+RjUKJvm02EsNTygb0yHvWqpoxI2fzU+dknOrDEgYa1sKyVz9PZHRyJhJFNjOiOLILHsz8T+vm2JY8zOhkhS5YotFYSoJxmT2NxkIzRnKiSWUaWFvJWxENWVo0ynaELzll1dJq1rx3Ip3f1mu1/I4CnAKZ3ABHlxBHW6hAU1gMIRneIU3RzovzrvzsWhdc/KZE/gD5/MHkeuNSA==</latexit> 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 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 <latexit sha1_base64="dvnE+QQjr3/7alRGEl6tWKzVJzc=">AAAB7XicdVDLSgMxFM34rPVVdekmWARXQ1I6OgsXBRe6ESrYB7RDyaSZNjaTGZKMUIb+gxsXirj1f9z5N6YPQUUPXDiccy/33hOmgmuD0IeztLyyurZe2Chubm3v7Jb29ps6yRRlDZqIRLVDopngkjUMN4K1U8VIHArWCkcXU791z5Tmibw145QFMRlIHnFKjJWa15fwHKJeqYxcdOpVsQ+R6yHsY8+SiocRqkDsohnKYIF6r/Te7Sc0i5k0VBCtOxilJsiJMpwKNil2M81SQkdkwDqWShIzHeSzayfw2Cp9GCXKljRwpn6fyEms9TgObWdMzFD/9qbiX14nM5Ef5FymmWGSzhdFmYAmgdPXYZ8rRo0YW0Ko4vZWSIdEEWpsQEUbwten8H/SrLgYufimWq75izgK4BAcgROAwRmogStQBw1AwR14AE/g2UmcR+fFeZ23LjmLmQPwA87bJx09jiI=</latexit> 0 Sistema instável <latexit sha1_base64="1/8c3E9VJLwmwXdbKv2v6IrEsWI=">AAAB6nicdVDLSgMxFM3UV62vqks3wSK4GpKho7OSghvBTUX7gHYomTTThmYyQ5IRSuknuHGhiFu/yJ1/Y/oQVPTAhcM593LvPVEmuDYIfTiFldW19Y3iZmlre2d3r7x/0NRprihr0FSkqh0RzQSXrGG4EaydKUaSSLBWNLqc+a17pjRP5Z0ZZyxMyEDymFNirHR7feH1yhXkojO/igOIXB/hAPuWeD5GyIPYRXNUwBL1Xvm9209pnjBpqCBadzDKTDghynAq2LTUzTXLCB2RAetYKknCdDiZnzqFJ1bpwzhVtqSBc/X7xIQkWo+TyHYmxAz1b28m/uV1chMH4YTLLDdM0sWiOBfQpHD2N+xzxagRY0sIVdzeCumQKEKNTadkQ/j6FP5Pmp6LkYtvqpVasIyjCI7AMTgFGJyDGrgCddAAFAzAA3gCz45wHp0X53XRWnCWM4fgB5y3T+IMjX8=</latexit> <latexit sha1_base64="nBnNnAea71AymYF+uu4PSRUjlIM=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GjK1ju1GCi50I1SwD2iHkknTNjaTDElGKEP/wY0LRdz6P+78G9OHoKIHLhzOuZd77wljzrRB6MPJLC2vrK5l13Mbm1vbO/ndvYaWiSK0TiSXqhViTTkTtG6Y4bQVK4qjkNNmOLqY+s17qjST4taMYxpEeCBYnxFsrNS4voTnEHXzBeRWKqjk+RC5pwgV/Yol6KRY9n3ouWiGAlig1s2/d3qSJBEVhnCsddtDsQlSrAwjnE5ynUTTGJMRHtC2pQJHVAfp7NoJPLJKD/alsiUMnKnfJ1IcaT2OQtsZYTPUv72p+JfXTky/HKRMxImhgswX9RMOjYTT12GPKUoMH1uCiWL2VkiGWGFibEA5G8LXp/B/0ii6HnK9m1KhWl7EkQUH4BAcAw+cgSq4AjVQBwTcgQfwBJ4d6Tw6L87rvDXjLGb2wQ84b58/to46</latexit> -20 0.01 <latexit sha1_base64="KXtCCGtG+V8VLHv86fas98GsEMw=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR7EUoeBG8VLQf0Iay2W7apZtN2J0IJfQnePGgiFd/kTf/jds2B219MPB4b4aZeUEihUHX/XbW1jc2t7YLO8Xdvf2Dw9LRccvEqWa8yWIZ605ADZdC8SYKlLyTaE6jQPJ2ML6Z+e0nro2I1SNOEu5HdKhEKBhFKz3cXVf7pbJbcecgq8TLSRlyNPqlr94gZmnEFTJJjel6boJ+RjUKJvm02EsNTygb0yHvWqpoxI2fzU+dknOrDEgYa1sKyVz9PZHRyJhJFNjOiOLILHsz8T+vm2JY8zOhkhS5YotFYSoJxmT2NxkIzRnKiSWUaWFvJWxENWVo0ynaELzll1dJq1rx3Ip3f1mu1/I4CnAKZ3ABHlxBHW6hAU1gMIRneIU3RzovzrvzsWhdc/KZE/gD5/MHkeuNSA==</latexit> 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
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