señales y sistemas · 2020. 7. 1. · sistemas lineales realimentados • realimentación...

Instituto de Ingeniería Eléctrica

Señales y SistemasSistemas (Lineales) Realimentados

Sistemas lineales realimentados

• Realimentación (feedback): usar la salida de un sistema para modificar o controlar la entrada. • Control: velocidad, ángulo, altura, temperatura, presión, …

• Sensado, medida: velocímetro, potenciómetro, termómetro, …

• Además de la corrección de variaciones o errores, puede reducir la sensibilidad a los parámetros.

• Permite estabilizar sistemas inestables, y puede desestabilizar un sistema estable.

• Entender los efectos de los cambios de los parámetros de la realimentación en el comportamiento de un sistema es esencial en el diseño de sistemas realimentados.

Sistema en lazo abierto

Sistemas lineales realimentados

• Realimentación (feedback): usar la salida de un sistema para modificar o controlar la entrada. • Control: velocidad, ángulo, altura, temperatura, presión, …

• Sensado, medida: velocímetro, potenciómetro, termómetro, …

• Además de la corrección de variaciones o errores, puede reducir la sensibilidad a los parámetros.

• Permite estabilizar sistemas inestables, y puede desestabilizar un sistema estable.

• Entender los efectos de los cambios de los parámetros de la realimentación en el comportamiento de un sistema es esencial en el diseño de sistemas realimentados.

Sistema en lazo cerrado

Sistemas lineales realimentados

• Algunas aplicaciones: • Diseño de sistemas inversos

• Compensación de elementos no ideales

• Estabilización de sistemas inestables

• Sistemas de seguimiento o rastreo

• Desestabilización de sistemas estables

3

Sistemas lineales realimentados

• Sistemas causales (ROC: un semiplano derecho o el exterior de un círculo).

• Convención: realimentación negativa ( , )

4

Q(s) =Y (s)

X(s)=

H(s)

1 +G(s)H(s)<latexit sha1_base64="Ios6Ky5WTIkAhuTy0dGfi5u6dNk=">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</latexit>

Q(z) =Y (z)

X(z)=

H(z)

1 +G(z)H(z)<latexit sha1_base64="n3gWJRSC42XzUIaysjqt3FzAOZU=">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</latexit>

e(t) = x(t)� r(t)

<latexit sha1_base64="qXk/kzuFJHVeYVzafd0YXW692Iw=">AAACKHicbVDLSsNAFJ1ofcVnddnNYCnowpKIYEUEwY1LBfuAWstkemuHTiZh5qZYQn/AL3CrS79GV+LWL3HSdqHWC/dyOPd1OEEshUHP+3Tm5nMLi0vLK+7q2vrG5lZ+u2aiRHOo8khGuhEwA1IoqKJACY1YAwsDCfWgf5H16wPQRkTqBocxtEJ2r0RXcIaWuoM93D97sOVA29LeKnplbxx0FvhTUDwvPFYOB/h+1c47udtOxJMQFHLJjGn6XoytlGkUXMLIvU0MxIz32T00LVQsBNNKx7JHtGSZDu1G2qZCOmZ/bqQsNGYYBnYyZNgzf3sZ+V+vmWC30kqFihMExSePuomkGNHMA9oRGjjKoQWMa2G1Ut5jmnG0TrmlEv35iPNMnbFXaPZpqlVGBnR2kIeJa33z/7o0C2qHZf+ofHJtDTwlk1gmBbJL9ohPjsk5uSRXpEo40eSJPJMX59V5cz6cz8nonDPd2SG/wvn6BoV7phM=</latexit>

e[n] = x[n]� r[n]

<latexit sha1_base64="/bd/tIQBqfaNYG2WqJ1PjdchBQc=">AAACLnicbVBNSwJBGJ41K7MvtWOXIRW6JLsSVEQgdOlokB+gi8yOrzo4O7vMzIYi/pWudezXBB2iaz+jWd2DaS+8Lw/P+/XweCFnStv2p5XaSm/v7Gb2svsHh0fHuXyhqYJIUmjQgAey7REFnAloaKY5tEMJxPc4tLzxfdxvPYNULBBPehqC65OhYANGiTZUL1colaAj3LuJKRfSlFKplyvaFXsReBM4CSiiJOq9vJXu9gMa+SA05USpjmOH2p0RqRnlMM92IwUhoWMyhI6Bgvig3NlC/ByXDdPHg0CaFBov2NWNGfGVmvqemfSJHqn1Xkz+1+tEenDtzpgIIw2CLh8NIo51gGMncJ9JoJpPDSBUMqMV0xGRhGrjV7ZcxquPKI3VKXMFx58SrTxQIOOD1I+yxjdn3aVN0KxWnMvKzWO1WLtNHMygU3SGzpGDrlANPaA6aiCKJugFvaI36936sL6s7+Voykp2TtCfsH5+AenDpRc=</latexit>

• Consideremos un sistema con transferencia !(") y el siguiente sistema de lazo cerrado

• # debe ser suficientemente grande, y el resultado es independiente de #.

• Amplificadores operacionales son un tipo de dispositivo para este tipo de realimentación.

Diseño de sistemas inversos

5

H(s) = K<latexit sha1_base64="kLkzvwQFsz/rngd6uG5nAtsafJA=">AAAC0XicdVFNa9tAEF2rX6n6lbTHXpaaQHoxUlJoLoVAL4FeUlo7AUuE1Xpkr7MfYnfU2CyC0msvvbb/oz+m/6YrR5TYSQcWHm/mzc6bKSopHCbJn1505+69+w+2HsaPHj95+mx75/nImdpyGHIjjT0rmAMpNAxRoISzygJThYTT4uJ9mz/9AtYJoz/jsoJcsakWpeAMAzU63nOv33043+4ng2QV9CZIO9AnXZyc7/R+ZxPDawUauWTOjdOkwtwzi4JLaOKsdlAxfsGmMA5QMwUu96txG7obmAktjQ1PI12x1xWeKeeWqgiViuHMbeZa8tbcwgW0/rtflJbxjYGwPMy90FWNoPnVPGUtKRrarohOhAWOchkA41YES5TPWGiDYZHxLr3envPWgwtNaDtP50gaB7btx1UdZxouuVGK6YnP5pfNOMm9z0C72kKr8fPMKJiypsn+GViT4G0SnAH+V+JE8NWKMoQFWuVXRBOHQ6ebZ70JRvuD9GCw//FN/+iwO/kWeUlekT2SkrfkiByTEzIknMzJD/KT/Io+Rcvoa/TtqjTqdZoXZC2i738BONjnnw==</latexit>

G(s) = P (s)<latexit sha1_base64="Wyqjy4HwhgqFiokKl1nUKF9u4sQ=">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</latexit>

Q(s) =K

1 +KP (s)

KP (s)�1' 1

P (s)<latexit sha1_base64="N3iidTis8N9cLG8wHxags24I2ws=">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</latexit>

Q(s) =K

1 +KP (s)

KP (s)�1' 1

P (s)<latexit sha1_base64="N3iidTis8N9cLG8wHxags24I2ws=">AAADEnicdVFdaxNBFJ2sX3X9SvXRl8FQqAhht5Xqi1AQQehLCqYtZJYwmdwk087Mbmfu2oRh/4X/wmdffBNfffFR/TPOJkGatF5YOJx7z5l79g4KJR0mya9GdOPmrdt3Nu7G9+4/ePioufn4yOWlFdAVucrtyYA7UNJAFyUqOCkscD1QcDw4e1v3jz+CdTI3H3BWQKb52MiRFBwD1W++O9x2z9+wkeXCH1Q+fXFAO4GpmEMuziwovyAoG49pWnnmpIbzii4UgZhP95utpJ3Mi14F6RK0yLI6/c3GFzbMRanBoFDcuV6aFJh5blEKBVXMSgdF2ICPoReg4Rpc5ud5K7oVmCEd5TZ8BumcvazwXDs304MwqTlO3HqvJq/tTV1Aq6/7aZ10bSEcvc68NEWJYMRin1GpKOa0/sd0KC0IVLMAuLAyRKJiwoMNhkvEW/SyvRB1BhdMaL3PMpHKHdjaT+gyZgYuRK41N0PPTi+qXpJ5z8C40kKt8acs1zDmVcX+BViR4HUSnAD+V+JkyFWLGMIUrfZzoorDodP1s14FRzvtdLe9c/iytb+3PPkGeUqekW2Skldkn7wnHdIlgnwmP8lv8if6FH2NvkXfF6NRY6l5QlYq+vEXARwApg==</latexit>

Compensación de elementos no ideales

• Corregir alguna propiedad no ideal de un sistema en lazo abierto. • Ejemplo: respuesta no constante de un amplificador en una banda de frecuencia.

• Consideremos el siguiente sistema con $(%&) como la transferencia de un amplificador para el cual queremos una respuesta plana en una banda de frecuencias

Compensación de elementos no ideales

Q(j!) =H(j!)

1 +KH(j!)) Q(j!) ' 1

K<latexit sha1_base64="8cpHiOuAu6I8TXAfdliTDMX6074=">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</latexit>

|KH(j!)| � 1<latexit sha1_base64="LraBskKrEgea/0oHnDBHs//NzPQ=">AAAC4nicdVFNb9NAEN2Yr9Z8pSBOXFZElcolslsEHCtxqcSlSKStFFvRej12Nt0Pa3dNE239B7ghrly4wpkf03/DOrFQk5aRVnp6M2923kxWcWZsFF31gjt3791/sLUdPnz0+MnT/s6zE6NqTWFEFVf6LCMGOJMwssxyOKs0EJFxOM3OP7T50y+gDVPys11UkApSSlYwSqynJv0Xlx/x0d4MJ0pASV5f4qQscTzpD6JhtAx8E8QdGKAujic7vT9JrmgtQFrKiTHjOKps6oi2jHJowqQ2UBF6TkoYeyiJAJO65fwN3vVMjgul/ZMWL9nrCkeEMQuR+UpB7NRs5lry1tzceLT+u5sXmtCNgWzxPnVMVrUFSVfzFDXHVuF2ZzhnGqjlCw8I1cxbwnRKfBvrNxvu4uvtKW09GN8Et/N0jrgyoNt+VNRhIuGCKiGIzF0yu2jGUepcAtLUGlqNm62u0TTJPwNrEnubxE7B/ldimPfVihILc6uFWxJN6A8db571JjjZH8YHw/1PbwaHb7uTb6GX6BXaQzF6hw7RETpGI0SRQz/RL/Q7yIOvwbfg+6o06HWa52gtgh9/ASnL7VY=</latexit>

siQ(j!) =H(j!)

1 +KH(j!)) Q(j!) ' 1

K<latexit sha1_base64="8cpHiOuAu6I8TXAfdliTDMX6074=">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</latexit>

• Comentarios • Si '(%&) = # > 1 en el rango de frecuencia deseado, entonces ((%&) atenúa!

• Asumimos que podemos tener un '(%&) con respuesta plana ('(%&) = #) en la misma banda de interés que $(%&), entonces ¿para qué hacer $(%&)?

• Si '(%&) = # < 1 cubrimos ambos comentarios simultáneamente y se puede construir con elementos pasivos. • $(%&) debe tener una ganancia bastante mayor a la deseada para que, dado un # < 1, siga valiendo |KH(j!)| � 1

<latexit sha1_base64="LraBskKrEgea/0oHnDBHs//NzPQ=">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</latexit>

• Sistema de un polo: • Evaluamos la realimentación proporcional:

• Estable si:

Estabilización de sistemas inestables

H(s) =b

s� a<latexit sha1_base64="j3HeZ3w+4fgWkhxjI/IRokzeBMk=">AAAC33icdVFNb9NAEN2Yr2K+0nLksiKqVA5EdkHABakSlx6LRNpKsRWtN+Nk2/2wdsc00cpnbogrF67wB/gx/BvWqYWatIy00tObebPzZopKCodJ8qcX3bp95+69rfvxg4ePHj/pb+8cO1NbDiNupLGnBXMghYYRCpRwWllgqpBwUpx/aPMnn8E6YfQnXFaQKzbTohScYaAm/Z3DPffifVZaxn3RePeSNZP+IBkmq6DXQdqBAeniaLLd+51NDa8VaOSSOTdOkwpzzywKLqGJs9pBxfg5m8E4QM0UuNyvhm/obmCmtDQ2PI10xV5VeKacW6oiVCqGc7eZa8kbcwsX0PrvftEa3RgIy3e5F7qqETS/nKesJUVD24XRqbDAUS4DYNyKYInyOQttMKw13qVX23PeenChCW3n6RxJ48C2/biq40zDBTdKMT312dlFM05y7zPQrrbQavxZZhTMWNNk/wysSfAmCc4B/ytxIvhqRRnCAq3yK6KJw6HTzbNeB8f7w/TVcP/j68HBm+7kW+QZeU72SErekgNySI7IiHCyID/IT/IrYtGX6Gv07bI06nWap2Qtou9/AcpH7UY=</latexit>

G(s) = K<latexit sha1_base64="VwkqnL+8+ScS6v13V2JtRfntPlw=">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</latexit>

Q(s) =H(s)

1 +KH(s)=

b

s� a+Kb<latexit sha1_base64="7pkZDSrTFSjg8ioEQy43T+qOef0=">AAAC/HicdVHLbhMxFHWGVxleKSzZWESRihBhpiBgg1SJTSU2rUTaSplR5HFuEre2Z2TfoYms4S/4A3aIJWzYwkfwN3imA2rSciVLR+f6HN/jmxVSWIyi353gytVr129s3Axv3b5z91538/6BzUvDYchzmZujjFmQQsMQBUo4KgwwlUk4zE7e1v3DD2CsyPV7XBaQKjbTYio4Q0+Nu8/2t+xj+iaZGsbdrseVi5+8ow36S2eVs0+ZZ7OKjru9aBA1RS+CuAU90tbeeLPzLZnkvFSgkUtm7SiOCkwdMyi4hCpMSgsF4ydsBiMPNVNgU9ckq2jfMxM6zY0/GmnDnlc4pqxdKj9YXzGc2/VeTV7aW1iPVl93izrt2kA4fZ06oYsSQfOzeaalpJjT+jfpRBjgKJceMG6Ej0T5nHkb9H8e9ul5e87rDNab0HqeNpHMLZjaj6syTDSc8lwppicuOT6tRlHqXALalgZqjTtOcgUzVlXJvwArErxMgnPA/0qs8LlqUYKwQKNcQ1ShX3S8vtaL4GB7ED8fbO+/6O28bFe+QR6SR2SLxOQV2SG7ZI8MCSefyA/yk/wKPgafgy/B17OrQafVPCArFXz/A2969s0=</latexit>

Q(s) =Y (s)

X(s)=

H(s)

1 +G(s)H(s)<latexit sha1_base64="Ios6Ky5WTIkAhuTy0dGfi5u6dNk=">AAAC/HicdVHLbhMxFHWGVxkeTWHJxiKqVIQUZgoCNkiVWNBlK5E2KDOKPM5N4taPke2hiazhL/gDdoglbNjCR/A32NMRatJyJY/PnOtzfY9vUXJmbJL86UTXrt+4eWvjdnzn7r37m92tB0dGVZrCgCqu9LAgBjiTMLDMchiWGogoOBwXp29D/vgjaMOUfG+XJeSCzCSbMkqsp8bdZ4c75smbbKoJdR88rN0wfFtmv2HSp+/8jpufcbeX9JMm8GWQtqCH2jgYb3W+ZxNFKwHSUk6MGaVJaXNHtGWUQx1nlYGS0FMyg5GHkggwuWuc1XjbMxM8VdovaXHDXlQ4IoxZisKfFMTOzXoukFfmFsaj1dvdInhea8hOX+eOybKyIOl5P9OKY6tweE08YRqo5UsPCNXMW8J0TnwZ69883sYXy1MaPBhfBId+WkdcGdChHhVVnEk4o0oIIicuOzmrR0nuXAbSVBqCxp1kSsCM1HX2z8CKxF4lsXOw/5UY5n0FUWZhYbVwDVHHftDp+lgvg6Pdfvq8v3v4orf3sh35BnqEHqMdlKJXaA/towM0QBR9Rj/RL/Q7+hR9ib5G386PRp1W8xCtRPTjLxw89ws=</latexit>

Q(s) =Y (s)

X(s)=

H(s)

1 +G(s)H(s)<latexit sha1_base64="Ios6Ky5WTIkAhuTy0dGfi5u6dNk=">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</latexit>

• Tal vez la aplicación estrella de la realimentación. !

Polos en lazo abiertoPolos en lazo cerrado

a�Kb < 0 ) K >a

b<latexit sha1_base64="qbB6+lqh/IGXwJ1DL/8ioqQvfAw=">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</latexit>

root-locus

• Sistema de un polo: , inestable si |)|>1

• Evaluamos la realimentación proporcional:

• Estable si:

Q(z) =zb

z(1 +Kb)� a=

z⇣

b1+Kb

⌘

z �⇣

a1+Kb

⌘

<latexit sha1_base64="Ht44a8Np9RPljhBxgM98oci4/iI=">AAADLXicdVFNbxMxEHWWrxK+UjhysYgqJUKNdgsCLkiVuCBxaSXSVspGwevMJm5t78qepUms/U2c+Av8Aw5IiCsXrnDGmwRI0mJppef35s3O8yS5FBbD8EstuHL12vUbWzfrt27fuXuvsX3/yGaF4dDlmczMScIsSKGhiwIlnOQGmEokHCdnryr9+D0YKzL9Fqc59BUbaZEKztBTg8a7w9as/TJODeNuRpPSzVrR4zc0ae+y8g8dS0ixtbj4Cq8nZWzEaIxtX093V3U21+m/gkGjGXbC+aEXQbQETbI8B4Pt2sd4mPFCgUYumbW9KMyx75hBwSWU9biwkDN+xkbQ81AzBbbv5k9R0h3PDGmaGf9ppHN21eGYsnaqEl+pGI7tplaRl2oT69H6392kirwxEKYv+k7ovEDQfDFPWkiKGa2enw6FAY5y6gHjRvhIlI+Zb4N+SfUdutqe8yqD9U1oNc8ykcwsmKofV0U91nDOM6WYHrr49LzshX3nYtC2MFB53GmcKRixsoz/Bliz4GUWHAP+12KFz1WZYoQJGuXmRFn3i44213oRHO11oiedvcOnzf1ny5VvkYfkEWmRiDwn++Q1OSBdwskn8oP8JL+CD8Hn4GvwbVEa1JaeB2TtBN9/A+N3DFs=</latexit>

Estabilización de sistemas inestables: caso discretoPolos en lazo abierto

Polos en lazo cerrado

Q(z) =Y (z)

X(z)=

H(z)

1 +G(z)H(z)<latexit sha1_base64="n3gWJRSC42XzUIaysjqt3FzAOZU=">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</latexit>

Q(z) =Y (z)

X(z)=

H(z)

1 +G(z)H(z)<latexit sha1_base64="n3gWJRSC42XzUIaysjqt3FzAOZU=">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</latexit>

H(z) =b

1� az�1=

zb

z � a<latexit sha1_base64="oTYGFdyLsKbE4/NhskoM3nMDUNk=">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</latexit>

G(z) = K<latexit sha1_base64="4fRv6TwlsURt9vZGcw/QD6WUibI=">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</latexit>

K >|a|� 1

|b|

<latexit sha1_base64="XuJnQkKwJ2P0FTWMTi018Dwxn6k=">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</latexit>

• Sistema de dos polos: • Es inestable.

• Veremos las opciones de realimentación con el siguiente ejemplo.

Estabilización de sistemas inestables

H(s) =b

s2 + a<latexit sha1_base64="1cZyXnVkggWC/IN1dWhHh7LU/7o=">AAAC43icdVFNixNBEO2MX+v4lVXw4qUxLKwIYSaKehEWvOxxBbO7kBlDT6eS9G5/DN01bkI7v8CbePXiVa/+GP+NPdlBNtm1oOHxql51vaqilMJhkvzpRNeu37h5a+t2fOfuvfsPutsPD52pLIchN9LY44I5kELDEAVKOC4tMFVIOCpO3zX5o09gnTD6Ay5LyBWbaTEVnGGgxt3H+7vu2dtsahn3Re3dRz+on7N63O0l/WQV9DJIW9AjbRyMtzu/s4nhlQKNXDLnRmlSYu6ZRcEl1HFWOSgZP2UzGAWomQKX+5WBmu4EZkKnxoanka7YiwrPlHNLVYRKxXDuNnMNeWVu4QJa/90vGrMbA+H0Te6FLisEzc/nmVaSoqHN0uhEWOAolwEwbkWwRPmchTYYVhvv0IvtOW88uNCENvO0jqRxYJt+XFVxpuGMG6WYnvjs5KweJbn3GWhXWWg0/iQzCmasrrN/BtYkeJUE54D/lTgRfDWiDGGBVvkVUcfh0OnmWS+Dw0E/fdEfvH/Z23vVnnyLPCFPyS5JyWuyR/bJARkSTj6TH+Qn+RVB9CX6Gn07L406reYRWYvo+18qLO70</latexit>

• Tal vez la aplicación estrella de la realimentación. !

Polos en lazo abierto

a < 0

a > 0<latexit sha1_base64="UdvnpOQnIWNQCz6ivuDsyD3RiDg=">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</latexit>

a < 0

a > 0<latexit sha1_base64="UdvnpOQnIWNQCz6ivuDsyD3RiDg=">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</latexit>

p�a

<latexit sha1_base64="VrypRYvLi+qrMOjfkAnTIV62+jk=">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</latexit>

p�a

<latexit sha1_base64="VrypRYvLi+qrMOjfkAnTIV62+jk=">AAAC1HicdVFNb9NAEN2Yr2K+2nLksiKqxIXIblHhWIlLj0UiaVFsVevNONl2P9zdMU20+IS4cuEKP4Mfw79hnVqoSctIKz29mTc7b6aopHCYJH960Z279+4/2HgYP3r85Omzza3tkTO15TDkRhp7UjAHUmgYokAJJ5UFpgoJx8X5+zZ//BmsE0Z/xEUFuWJTLUrBGQbqU+YuLPrXrDnd7CeDZBn0Jkg70CddHJ1u9X5nE8NrBRq5ZM6N06TC3DOLgkto4qx2UDF+zqYwDlAzBS73y4kbuhOYCS2NDU8jXbLXFZ4p5xaqCJWK4cyt51ry1tzcBbT6u5+XlvG1gbB8l3uhqxpB86t5ylpSNLTdEp0ICxzlIgDGrQiWKJ+x0AbDLuMder09560HF5rQdp7OkTQObNuPqzrONFxyoxTTE5+dXTbjJPc+A+1qC63Gn2VGwZQ1TfbPwIoEb5PgDPC/EieCr1aUIczRKr8kmjgcOl0/600w2h2ke4PdD2/6B/vdyTfIC/KSvCIpeUsOyCE5IkPCiSI/yE/yKxpFX6Kv0ber0qjXaZ6TlYi+/wW+tunT</latexit>

Sistemas lineales realimentados: péndulo invertido

Ld2✓(t)

dt2= g sin ✓(t) + Lx(t)� a(t) cos ✓(t)

sin ✓(t) ' ✓(t)

cos ✓(t) ' 1<latexit sha1_base64="D1fQXjL7aSXweF0anzkMfqpQB7w=">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</latexit>

Modelo de la ecuación de movimiento:

Ld2✓(t)

dt2= g sin ✓(t) + Lx(t)� a(t) cos ✓(t)

sin ✓(t) ' ✓(t)

cos ✓(t) ' 1<latexit sha1_base64="D1fQXjL7aSXweF0anzkMfqpQB7w=">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</latexit>

Aproximación lineal en * ≃ + (equilibrio inestable):

aceleración angular

Sistemas lineales realimentados: péndulo invertido

Ld2✓(t)

dt2= g sin ✓(t) + Lx(t)� a(t) cos ✓(t)

sin ✓(t) ' ✓(t)

cos ✓(t) ' 1<latexit sha1_base64="D1fQXjL7aSXweF0anzkMfqpQB7w=">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</latexit>

Modelo de la ecuación de movimiento:

Ld2✓(t)

dt2= g sin ✓(t) + Lx(t)� a(t) cos ✓(t)

sin ✓(t) ' ✓(t)

cos ✓(t) ' 1<latexit sha1_base64="D1fQXjL7aSXweF0anzkMfqpQB7w=">AAADb3icdVLbbhMxEPUmXMJyS+GBByRkEbVqhBrtBkT7glSJFx76UCTSVopD5DiTZNu1N9izNJHln+Ef+Bg+gz/Au4lgk5aRLJ09M2d2xsejeZoYjKJfQa1+5+69+40H4cNHj588be48OzNZrgX0RJZm+mLEDaSJgh4mmMLFXAOXoxTOR1cfi/z5d9AmydQXXM5hIPlUJZNEcPTUsPljz544yiaaC2vH7qvt+i+cAfJ9i67tPIkl6z5MKTOJqmbpG2pPFq7EB5avAGUiM9UqxsK9qhLbRSMJ36pEUVPR/auJ6bDZijpRGfQmiNegRdZxOtwJfrJxJnIJCkXKjenH0RwHlmtMRAouZLmBORdXfAp9DxWXYAa2vExHdz0zppNM+6OQlmxVYbk0ZilHvlJynJntXEHemlsYjzb/bhfFxW8NhJOjgU3UPEdQYjXPJE8pZrQwkI4TDQLTpQdc6MSvRMWM+zbobQ53abW9EMUOxjehxTzrjdLMgC76CZmHTMG1yKTkamzZ5bXrRwNrGSiTayg09pJlEqbcOfZ3gQ0J3iYpXfyfxL8FUYoYwgK1tCXhQm90vG3rTXDW7cRvO93P71rH79eWN8hL8prsk5gckmPyiZySHhFBI+gEh8FR7Xf9Rf1Vna5Ka8Fa85xsRL39B5wqHTk=</latexit>

Aproximación lineal en * ≃ + (equilibrio inestable):

aceleración angular

Sistemas lineales realimentados: péndulo invertido

Ld2✓(t)

dt2= g sin ✓(t) + Lx(t)� a(t) cos ✓(t)

sin ✓(t) ' ✓(t)

cos ✓(t) ' 1<latexit sha1_base64="D1fQXjL7aSXweF0anzkMfqpQB7w=">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</latexit>

Modelo de la ecuación de movimiento:

Ld2✓(t)

dt2= g sin ✓(t) + Lx(t)� a(t) cos ✓(t)

sin ✓(t) ' ✓(t)

cos ✓(t) ' 1<latexit sha1_base64="D1fQXjL7aSXweF0anzkMfqpQB7w=">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</latexit>

Ld2✓(t)

dt2 = g sin ✓(t) + Lx(t)� a(t) cos ✓(t)

sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1

Ls2 � g(LX(s)�A(s))

⇥(s) = H(s)(LX(s)�A(s))

H(s) =1

Ls2 � g<latexit sha1_base64="RFL6pMwtmaVNEVyypQlR8toXSa4=">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</latexit>

Aproximación lineal en * ≃ + (equilibrio inestable): Ld2✓(t)

dt2 = g sin ✓(t) + Lx(t)� a(t) cos ✓(t)

sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1

Ls2 � g(LX(s)�A(s))

⇥(s) = H(s)(LX(s)�A(s))

H(s) =1

Ls2 � g<latexit sha1_base64="RFL6pMwtmaVNEVyypQlR8toXSa4=">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</latexit>

Ld2✓(t)

dt2 = g sin ✓(t) + Lx(t)� a(t) cos ✓(t)

sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1

Ls2 � g(LX(s)�A(s))

⇥(s) = H(s)(LX(s)�A(s))

H(s) =1

Ls2 � g<latexit sha1_base64="RFL6pMwtmaVNEVyypQlR8toXSa4=">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</latexit>

Ld2✓(t)

dt2 = g sin ✓(t) + Lx(t)� a(t) cos ✓(t)

sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1

Ls2 � g(LX(s)�A(s))

⇥(s) = H(s)(LX(s)�A(s))

H(s) =1

Ls2 � g<latexit sha1_base64="RFL6pMwtmaVNEVyypQlR8toXSa4=">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</latexit>

INESTABLEESTABLECon polo en semiplano derecho el sistema es inestable.

• Realimentemos para buscar estabilidad

• ¿Cómo elegimos (diseñamos) '(")?

Sistemas lineales realimentados: péndulo invertidoLd2✓(t)

dt2 = g sin ✓(t) + Lx(t)� a(t) cos ✓(t)

sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1

Ls2 � g(LX(s)�A(s))

⇥(s) = H(s)(LX(s)�A(s))

H(s) =1

Ls2 � g<latexit sha1_base64="RFL6pMwtmaVNEVyypQlR8toXSa4=">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</latexit>

⇥(s) =LH(s)

1 +G(s)H(s)X(s)

<latexit sha1_base64="migbJynw2mwncwPAcbVrHOxakco=">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</latexit>

• Realimentación proporcional:

Sistemas lineales realimentados: péndulo invertido

Ld2✓(t)

dt2 = g sin ✓(t) + Lx(t)� a(t) cos ✓(t)

sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1

Ls2 � g(LX(s)�A(s))

⇥(s) = H(s)(LX(s)�A(s))

H(s) =1

Ls2 � g<latexit sha1_base64="RFL6pMwtmaVNEVyypQlR8toXSa4=">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</latexit>

⇥(s) =LH(s)

1 +G(s)H(s)X(s)

<latexit sha1_base64="migbJynw2mwncwPAcbVrHOxakco=">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</latexit>

a(t) = K1✓(t)

G(s) = K1

⇥(s) =1

s2 �⇣

g�K1

L

⌘X(s)

<latexit sha1_base64="SgjHpuphOIt/4Ts3gMVJqIguTLU=">AAADOHicdVJNb9NAEF2br2K+UjhyWRG1Sg+N7IBaLkiVOIAEhyI1baRsiDabcbKN17Z2xzTRyj+LC/+CIzduiCsXrqydCJq0jGTp6b1943k7O8oTaTAMv3n+jZu3bt/Zuhvcu//g4aPG9uNTkxVaQFdkSaZ7I24gkSl0UWICvVwDV6MEzkaz15V+9gm0kVl6goscBopPUhlLwdFRw8Zsl7dwj756N7RRSRlOAWuCMRrsvmmZf1JFsJNar1gWay6cYM1H2yn3WQIxtpbkZL+2lPZ9ybScTHGvpD1nGjaaYTusi14F0Qo0yaqOh9veZzbORKEgRZFwY/pRmOPAco1SJFAGrDCQczHjE+g7mHIFZmDrWynpjmPGNM60+1KkNXvZYbkyZqFG7qTiODWbWkVeq82NQ+t/t/Mq+cZAGL8cWJnmBUIqlvPERUIxo9Um6FhqEJgsHOBCSxeJiil3bdDtK9ihl9sLUWUwrgmt5lklSjIDuuonVBGwFC5EphRPx5adX5T9cGAtg9QUGiqPPWeZggkvS/Y3wJoFr7PUD+J/FiNdrsrEEOaola2JMnCLjjbXehWcdtrR83bnw4vm0cFq5VvkKXlGWiQih+SIvCXHpEsE+Up+e8Tz/C/+d/+H/3N51PdWnidkrfxffwAkRwr4</latexit>

a(t) = K1✓(t)

G(s) = K1

⇥(s) =1

s2 �⇣

g�K1

L

⌘X(s)

<latexit sha1_base64="SgjHpuphOIt/4Ts3gMVJqIguTLU=">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</latexit>

s = ±r

g �K1

L<latexit sha1_base64="wrtxFQ5JCKbfHEcNLMUOrL4lvvA=">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</latexit>

Polos en:

K1 < 0

K1 > 0<latexit sha1_base64="WgZN0cOwijg/Acm96ZHdgeFS6tQ=">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</latexit>

K1 < 0

K1 > 0<latexit sha1_base64="WgZN0cOwijg/Acm96ZHdgeFS6tQ=">AAAC13icdVFNaxRBEO0dv+L4lejRS+MS8LTMREFBkYAXwUsEN4lsD0tPb+1uJ/0xdNeYXZrBm3j14lX/hD/Gf2PPZpDsJhY0/XjVr7peVVkp6THL/vSSa9dv3Ly1dTu9c/fe/QfbOw8Pva2dgKGwyrrjkntQ0sAQJSo4rhxwXSo4Kk/ftvmjz+C8tOYjLisoNJ8ZOZWCY6TY+3H+OmPt9SYbb/ezQbYKehnkHeiTLg7GO73fbGJFrcGgUNz7UZ5VWATuUAoFTcpqDxUXp3wGowgN1+CLsGq6obuRmdCpdfEYpCv2oiJw7f1Sl/Gl5jj3m7mWvDK38BGt/x4WU8fFRkM4fVkEaaoawYjzfqa1omhpOyg6kQ4EqmUEXDgZLVEx57EMxnGmu/RieSFaDz4WoW0/nSNlPbi2ntB1ygycCas1N5PATs6aUVaEwMD42kGrCSfMapjxpmH/DKxJ8CoJzgH/K/Ey+mpFDGGBTocV0aRx0fnmWi+Dw71B/myw9+F5f/9Vt/It8pg8IU9JTl6QffKOHJAhEaQiP8hP8iv5lHxJvibfzp8mvU7ziKxF8v0vRPnpkg==</latexit>

s = ±j

rK1 � g

L(K1 > g)

<latexit sha1_base64="HfIYeK6Vm3SOkUf7mgPtT+AinPI=">AAAC+HicdVFNaxRBEO0dv+L4tdGjl8YlEBGXmSiYixLwIughgpsEtoelp6d2tzfTPZPuGrNrMz/Cf+BNvAriVX+G/8aezSDZTSxoeLyqV12vKi1zaTGK/nSCK1evXb+xcTO8dfvO3XvdzfsHtqiMgIEo8sIcpdxCLjUMUGIOR6UBrtIcDtPj103+8CMYKwv9ARclJIpPtBxLwdFTo+4T+5KVis4osycGHRsbLtzbUfx0Urt3dc1OKp7RbU+8mjwedXtRP1oGvQjiFvRIG/ujzc4PlhWiUqBR5NzaYRyVmDhuUIoc6pBVFkoujvkEhh5qrsAmbumqplueyei4MP5ppEv2vMJxZe1Cpb5ScZza9VxDXpqbW49Wf3fzxvjaQDjeTZzUZYWgxdk84yqnWNBmkzSTBgTmCw+4MNJbomLKfRv0+w636Pn2QjQerG9Cm3laR3lhwTT9hKpCpuFUFEpxnTk2O62HUeIcA20rA43GzVihYML9Uf4ZWJHgZRKcAv5XYqX31YgYwhyNckuiDv2h4/WzXgQHO/34WX/n/fPe3m578g3ykDwi2yQmL8geeUP2yYAI8pn8JL/I7+BT8CX4Gnw7Kw06reYBWYng+1+a5fb6</latexit>

No es posible estabilizarlo con una realimentación proporcional.

⇥(s) =1

s2 �⇣

g�K1

L

⌘X(s)

<latexit sha1_base64="JaT2MmLOIJSi5g2Asg6a6r/haSQ=">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</latexit>

• Realimentación con derivador:

Sistemas lineales realimentados: péndulo invertido

Ld2✓(t)

dt2 = g sin ✓(t) + Lx(t)� a(t) cos ✓(t)

sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1

Ls2 � g(LX(s)�A(s))

⇥(s) = H(s)(LX(s)�A(s))

H(s) =1

Ls2 � g<latexit sha1_base64="RFL6pMwtmaVNEVyypQlR8toXSa4=">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</latexit>

⇥(s) =LH(s)

1 +G(s)H(s)X(s)

<latexit sha1_base64="migbJynw2mwncwPAcbVrHOxakco=">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</latexit>

Polos en:

⇥(s) =1

s2 + s (K2/L)� g/LX(s)

<latexit sha1_base64="7xwT85WE0S189fLFg8jekBcalMs=">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</latexit>

s = �K2

2L±

s✓K2

2L

◆2

+⇣ g

L

⌘

<latexit sha1_base64="ODllB5dY17/YLTXIYMFKCbVlZ0k=">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</latexit>

K2 < 0

K2 > 0<latexit sha1_base64="WQOzZ3Y/inwO3sgQ37ou/aDs388=">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</latexit>

K2 < 0

K2 > 0<latexit sha1_base64="WQOzZ3Y/inwO3sgQ37ou/aDs388=">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</latexit>

s2 = �K2

2L+

s✓K2

2L

◆2

+⇣ g

L

⌘K2!1�����! 0

<latexit sha1_base64="F2aUSkG6v+xp8t9olsZMmJl3P40=">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</latexit>

No es posible estabilizarlo con una realimentación proporcional.

G(s) = K2s<latexit sha1_base64="v/mcGNpaCkAAacvUdLbkxH4OmUM=">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</latexit>

a(t) = K1✓(t)

G(s) = K1

⇥(s) =1

s2 �⇣

g�K1

L

⌘X(s)

<latexit sha1_base64="SgjHpuphOIt/4Ts3gMVJqIguTLU=">AAADOHicdVJNb9NAEF2br2K+UjhyWRG1Sg+N7IBaLkiVOIAEhyI1baRsiDabcbKN17Z2xzTRyj+LC/+CIzduiCsXrqydCJq0jGTp6b1943k7O8oTaTAMv3n+jZu3bt/Zuhvcu//g4aPG9uNTkxVaQFdkSaZ7I24gkSl0UWICvVwDV6MEzkaz15V+9gm0kVl6goscBopPUhlLwdFRw8Zsl7dwj756N7RRSRlOAWuCMRrsvmmZf1 JFsJNar1gWay6cYM1H2yn3WQIxtpbkZL+2lPZ9ybScTHGvpD1nGjaaYTusi14F0Qo0yaqOh9veZzbORKEgRZFwY/pRmOPAco1SJFAGrDCQczHjE+g7mHIFZmDrWynpjmPGNM60+1KkNXvZYbkyZqFG7qTiODWbWkVeq82NQ+t/t/Mq+cZAGL8cWJnmBUIqlvPERUIxo9Um6FhqEJgsHOBCSxeJiil3bdDtK9ihl9sLUWUwrgmt5lklSjIDuuonVBGwFC5EphRPx5adX5T9cGAtg9QUGiqPPWeZggkvS/Y3wJoFr7PUD+J/FiNdrsrEEOaola2JMnCLjjbXehWcdtrR83bnw4vm0cFq5VvkKXlGWiQih+SIvCXHpEsE+Up+e8Tz/C/+d/+H/3N51PdWnidkrfxffwAkRwr4</latexit>

a(t) = K1✓(t) +K2d✓(t)

dt<latexit sha1_base64="ECVg7NB42ZRXWxEHn20YzX55fRA=">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</latexit>

✓K2

2

2L

◆� K1 � g

L< 0 ) K1 >

1

4LK2

2 + g > 0

K2 > 0<latexit sha1_base64="DDUqI6rU5B5gM5EpjVWa0r+2Ims=">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</latexit>

• Realimentación proporcional y con derivador:

Sistemas lineales realimentados: péndulo invertido

Ld2✓(t)

dt2 = g sin ✓(t) + Lx(t)� a(t) cos ✓(t)

sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1

Ls2 � g(LX(s)�A(s))

⇥(s) = H(s)(LX(s)�A(s))

H(s) =1

Ls2 � g<latexit sha1_base64="RFL6pMwtmaVNEVyypQlR8toXSa4=">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</latexit>

⇥(s) =LH(s)

1 +G(s)H(s)X(s)

<latexit sha1_base64="migbJynw2mwncwPAcbVrHOxakco=">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</latexit>

Polos en:

Es posible estabilizarlo con una realimentación proporcional y derivación.

G(s) = K1 +K2s<latexit sha1_base64="jszPXJt+fl+E7AR8XSsYteBq0EM=">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</latexit>

a(t) = K1✓(t) +K2d✓(t)

dt<latexit sha1_base64="ECVg7NB42ZRXWxEHn20YzX55fRA=">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</latexit>

⇥(s) =1

s2 + s (K2/L)� g/L + K1/LX(s)

<latexit sha1_base64="9bjwt4BuwyHdOrVVgIE2IRg/Xl4=">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</latexit>

s =�K2

2L±

s✓K2

2L

◆2

�✓K1 � g

L

◆

<latexit sha1_base64="6qIVRfqL0rahLXCAnN/HFFxyi/o=">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</latexit>

Polos complejos conjugados si:

Polos reales con parte real negativa si:K1 > g > 0

K2 > 0<latexit sha1_base64="ecwhz++kE2y/RaP3NCXAQd+tBKs=">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</latexit>

1

4LK2

2 + g > K1 > g > 0<latexit sha1_base64="Y9cJ+cImFA31p5y/JrzdXlJld0A=">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</latexit> ✓

K22

2L

◆� K1 � g

L< 0 ) K1 >

1

4LK2

2 + g > 0

K2 > 0<latexit sha1_base64="DDUqI6rU5B5gM5EpjVWa0r+2Ims=">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</latexit>

Q(s) =b

s2 + bK2s+ (a+K1b)⇡ !2

n

s2 + 2⇣!ns+ !2n

<latexit sha1_base64="EfqYOWEWkTW9rqYR2yfFRYyHzsM=">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</latexit>

• Sistema de dos polos: • Es inestable.

• No alcanza con una realimentación proporcional ni con una con un derivado, necesitamos usar ambas ¿por qué?

Estabilización de sistemas inestables

H(s) =b

s2 + a<latexit sha1_base64="1cZyXnVkggWC/IN1dWhHh7LU/7o=">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</latexit>

Polos en lazo abierto

a < 0

a > 0<latexit sha1_base64="UdvnpOQnIWNQCz6ivuDsyD3RiDg=">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</latexit>

a < 0

a > 0<latexit sha1_base64="UdvnpOQnIWNQCz6ivuDsyD3RiDg=">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</latexit>

G(s) = K1 +K2s<latexit sha1_base64="opO3aCEUnFXRrsMD9cYiEfmC9sA=">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</latexit>

H(s) =b

s2 + a<latexit sha1_base64="1cZyXnVkggWC/IN1dWhHh7LU/7o=">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</latexit>

Las condiciones de estabilidad de este SLIT vimos que recaen sobre ,>0 (-#2>0) y

garantizar que el término independiente sea positivo ()+#1->0) por lo tanto

precisamos ambos términos (#1 y #2")

Sistemas realimentados para datos muestreados

p[n] = y(nT )<latexit sha1_base64="5CcGuu13iNoH+GCui611dQ3Z7xI=">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</latexit>

z(t) = d[n], nT t < (n+ 1)T<latexit sha1_base64="cvxQzWmLsXqYsZYRN4ljKrqqs0g=">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</latexit>

x(t) = r[n], nT t < (n+ 1)T<latexit sha1_base64="tqYh4kikc8k26YiPrT0u/OSM5o4=">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</latexit>

Si

Sistemas de seguimiento o rastreo

• Sistema donde la salida siga o copie la entrada. • Ejemplo: posicionamiento de un telescopio (*. es el ángulo deseado)

• Asumamos que $/(0) representa la transferencia del sistema (planta) que se busca controlar su salida y $1(0) la transferencia de un controlador a diseñar cuya entrada es la diferencia 2[3] = 4[3] - 5[3], y notemos $(0)= $1(0) $/(0)

• Si queremos un buen seguimiento buscamos minimizar el error en la respuesta frecuencial

• Para las frecuencias donde 6(2%*) no es cero necesitamos que |$(2%*)| sea grande.

• Buen seguimiento requiere altas ganancias. • Pero puede traer malas consecuencias…

Sistemas de seguimiento o rastreo

Y (z) =H(z)

1 +H(z)X(z), Y (z) = H(z)E(z) ) E(z) =

1

1 +H(z)X(z)

<latexit sha1_base64="vPmJym9ymEgsgHBMz3HW3c3abas=">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</latexit>

E(ej✓) =1

1 +H (ej✓)X

�ej✓�' 0

<latexit sha1_base64="Zdb1XCEEjOsjsK0TOOrF0epR92s=">AAADIHicdVHLbhMxFHWGR0t4pbBkYxFVaoUUzbSosEGqhJC6LIK0keIQOc6dxKntGew7NJE1P8M/sOAP2CGWsGELf4EniVCTtleydHSuz/U9PoNcSYdx/LMW3bh56/bG5p363Xv3HzxsbD06cVlhBbRFpjLbGXAHShpoo0QFndwC1wMFp4Oz11X/9BNYJzPzHmc59DQfGZlKwTFQ/ca7NzvwgU1w9xVLLRc+KX3y7IgpSDE0/IQyHAPyklk5GuNuSTvX9ihzUsNHGvcbzbgVz4teBskSNMmyjvtbtS9smIlCg0GhuHPdJM6x57lFKRSUdVY4yLk44yPoBmi4Btfzc/cl3Q7MkKaZDccgnbMXFZ5r52Z6EG5qjmO33qvIK3tTF9Dq635a/dLaQpi+7Hlp8gLBiMU+aaEoZrT6cTqUFgSqWQBcWBksUTHmYQyGXOrb9OJ4ISoPLgyh1T5LRypzYKt5Qhd1ZuBcZFpzM/Rscl524573DIwrLFQaP2GZhhEvS/bfwIoEr5IskrxG4mTwVYkYwhSt9nOirIegk/VYL4OTvVay39p7+7x5eLCMfJM8IU/JDknIC3JIjsgxaRNBvpLf5A/5G32OvkXfox+Lq1FtqXlMVir69Q95JQdN</latexit>

• Al medir 5[3] siempre cometemos un error que llamaremos como 7[3]

• Para disminuir el efecto de 7[3] necesitamos que |$(2%*)| sea bajo. ! • ¿Cómo hacer que |$(2%*)| sea alto y bajo?

• Depende de las características en frecuencia de 6(2%*) y .(2%*).

Sistemas de seguimiento o rastreo

Y (ej✓) =

H(ej✓)

1 +H(ej✓)X(ej✓)

��

H(ej✓)

1 +H(ej✓)D(ej✓)

�

<latexit sha1_base64="eoJxQTdLij9xklmlMGYLR4srUj4=">AAADOXichVFNbxMxEPUuHy3hoykcuVhElYoQ0W5BhQtSJTj0WCTSBsVL5DiziVN7d2XP0kTW/ixO/AtuHLkhrlw44k1XVZMWMZKlpzd+43l+o0JJi1H0PQhv3Lx1e2PzTuvuvfsPttrbD49tXhoBPZGr3PRH3IKSGfRQooJ+YYDrkYKT0enbun/yGYyVefYBFwUkmk8ymUrB0VPDtvq4C5/YDJ++YQpSHLDUcOEOG7Jy8bMLTPsNYkZOppg8/7/i3api2O5E3WhZ9CqIG9AhTR0Nt4MvbJyLUkOGQnFrB3FUYOK4QSkUVC1WWii4OOUTGHiYcQ02cctvqeiOZ8Y0zY0/GdIle1nhuLZ2oUf+puY4teu9mry2N7cerb7u5vU3rC2E6evEyawoETJxvk9aKoo5raOgY2lAoFp4wIWR3hIVU+7HoA+stUMvjxei9mD9EFrv0zhSuQVTzxO6bLEMzkSuNc/Gjs3OqkGUOMcgs6WBWuNmLNcw4VXFLgysSPA6CU4B/ymx0vuqRQxhjka7JVG1fNDxeqxXwfFeN37R3Xv/snOw30S+SR6TJ2SXxOQVOSCH5Ij0iCDfyJ8gCMLwa/gj/Bn+Or8aBo3mEVmp8Pdf8XgOIg==</latexit>

|$(2%*)| alto

|$(2%*)| bajo.

y[n] = y[n] + d[n]<latexit sha1_base64="aGeM2TBp9kX3uPoMt0s2Xc82ODQ=">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</latexit>

(error en la medida)

Modelo paraY (ej✓) =

H(ej✓)

1 +H(ej✓)X(ej✓)

��

H(ej✓)

1 +H(ej✓)D(ej✓)

�

<latexit sha1_base64="eoJxQTdLij9xklmlMGYLR4srUj4=">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</latexit> (uniforme en ) ✓

<latexit sha1_base64="CUiIjly43BqpCw+mxf5eNRp/DZI=">AAACIHicbVC7TgJBFJ1FVMQXqJ3NREJiRXaNiRoLSSykxEQeCRAyO1xgdPaRmbsmSPgHWy39AP/Bzs5YauefOAsUCJ5kkpNzX2eOG0qh0ba/rMRScnllNbWWXt/Y3NrOZHeqOogUhwoPZKDqLtMghQ8VFCihHipgniuh5t5dxvXaPSgtAv8GByG0PNbzRVdwhkaqNrEPyNqZnF2wx6CLxJmS3MXb1evew0+p3M5ayWYn4JEHPnLJtG44doitIVMouIRRuhlpCBm/Yz1oGOozD3RrOLY7onmjdGg3UOb5SMfq7MSQeVoPPNd0egz7er4Wi//VGhF2T1tD4YcRgs8nh7qRpBjQ+O+0IxRwlANDGFfCeKW8zxTjaBJK5/N09hDnsTttttD40tSrDDSoeCH3orTJzZlPaZFUjwrOceHs2s4Vz8kEKbJPDsghccgJKZISKZMK4eSWPJIn8my9WO/Wh/U5aU1Y05ld8gfW9y+LkKTP</latexit>

Desestabilización a causa de la realimentación

Q(s) =K1

1�K1K2e�sT<latexit sha1_base64="UE8lJYdlrRVA8PZXFZPG0obMi+w=">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</latexit>

• #1 > 1 (amplificación), 8 (retardo) y #2 < 1 (atenuación) dados por distancia.

• Inestable si #1#2 > 1

• Para este análisis se precisan herramientas que se verán en cursos posteriores: Sistemas y Control(root-locus, criterios de Nyquist).

Controlador proportional–integral–derivative (PID)

PID controller in a feedback loop, 9(:) is the desired process value or "set point", and 5(:) is the measured process value.

Effects of varying PID parameters (Kp,Ki,Kd) on the step response of a system.

https://en.wikipedia.org/wiki/Control_system#/media/File:PID_Compensation_Animated.gif

Sistemas realimentados: resumen

• La realimentación permite modificar la respuesta de un sistema.

• Podemos lograr diferentes objetivos: • Estabilización

• Respuesta uniforme

• Invertir la respuesta de un sistema

• Desestabilizar

• Tiempo discreto y tiempo continuo.

• Tenemos algunas herramientas para analizar sistemas realimentados. Necesitamos otros métodos y fundamentos para el análisis y diseño de sistemas realimentados, que se verán en otros cursos de la carrera

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