Señales y Sistemas · 2020. 7. 1. · Sistemas lineales realimentados • Realimentación (feedback): usar la salida de un sistema para modiﬁcar o controlar la entrada. • Control:

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


• 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


• 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 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=">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</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=">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</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=">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</latexit>

siQ(j!) =H(j!)

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

K<latexit sha1_base64="8cpHiOuAu6I8TXAfdliTDMX6074=">AAADJ3icdVHLbhMxFHWGVxleKSzZWESVipCimYKADVIlNpW6aRFpK+IocpybxOnYHuw7JJE1H8QHsOcP2CFYsmELn4AniSBJy5UsHZ3rc3yPby/PpMMk+V6Lrly9dv3G1s341u07d+/Vt++fOFNYAS1hMmPPetxBJjW0UGIGZ7kFrnoZnPbOX1f90w9gnTT6Lc5y6Cg+1HIgBcdAdevvjnfHlBkFQ/74FRtYLvzBP6b06ZNDukpQ9kYOR8itNRO6oqXMSQXv6cIiLf1h2a03kmYyL3oRpEvQIMs66m7XPrG+EYUCjSLjzrXTJMeO5xalyKCMWeEg5+KcD6EdoOYKXMfPP6GkO4Hp04Gx4Wikc3ZV4blybqZ64abiOHKbvYq8tDd1Aa2/7qdVzI2BcPCy46XOCwQtFvMMioyiodXH0760IDCbBcCFlSESFSMebDCsJ96hq/ZCVBlcMKHVPMtEmXFgKz+hiphpmAijFNd9z8aTsp10vGegXWGh0vjxYjNlyf4GWJPgZRIcAf5X4mTIVYkYwhSt8nOijMOi0821XgQne830aXPv+Flj//ly5VvkIXlEdklKXpB9ckCOSIsI8pn8JL/I7+hj9CX6Gn1bXI1qS80DslbRjz9p9giT</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=">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</latexit>

Q(s) =Y (s)

X(s)=

H(s)


Q(s) =Y (s)

X(s)=

H(s)


• 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=">AAAC7XicdVFNbxMxEHWWr7J8pXDkYhFV4kK0aRFwQKgSF6ReCiJtpewq8jqziVt/rOxZksjaK/+AG+LKhWv5L/wbvOkKNWkZydLTm3njeTN5KYXDJPnTiW7cvHX7ztbd+N79Bw8fdbcfHzlTWQ5DbqSxJzlzIIWGIQqUcFJaYCqXcJyfvW/yx1/AOmH0Z1yWkCk21aIQnGGgxl3KXhzkbxOafhLTGTJrzZwevEsLy7hntc/rcbeX9JNV0Ktg0IIeaeNwvN05TyeGVwo0csmcGw2SEjPPLAouoY7TykHJ+BmbwihAzRS4zK+s1HQnMBNaGBueRrpiLys8U84tVR4qFcOZ28w15LW5hQto/Xe/aGxuDITFm8wLXVYIml/MU1SSoqHN+uhEWOAolwEwbkWwRPmMhTYYlhzv0MvtOW88uNCENvO0jqRxYJt+XFVxqmHOjVJMT3x6Oq9HSeZ9CtpVFhqNP02Ngimr6/SfgTUJXifBGeB/JU4EX40oRVigVX5F1HE49GDzrFfB0W5/sNff/fiyt/+qPfkWeUqekedkQF6TffKBHJIh4eQr+UXOye/IRN+i79GPi9Ko02qekLWIfv4Fd6jy3g==</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=">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</latexit>

Estabilización de sistemas inestables: caso discretoPolos en lazo abierto

Polos en lazo cerrado

Q(z) =Y (z)

X(z)=

H(z)


Q(z) =Y (z)

X(z)=

H(z)


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.


H(s) =b

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


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>

p�a

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


sin ✓(t) ' ✓(t)


Aproximación lineal en * ≃ + (equilibrio inestable):

aceleración angular


Ld2✓(t)


sin ✓(t) ' ✓(t)



Ld2✓(t)


sin ✓(t) ' ✓(t)


Aproximación lineal en * ≃ + (equilibrio inestable):

aceleración angular


Ld2✓(t)


sin ✓(t) ' ✓(t)



Ld2✓(t)


sin ✓(t) ' ✓(t)


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)


sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1


⇥(s) = H(s)(LX(s)�A(s))

H(s) =1


Ld2✓(t)


sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1


⇥(s) = H(s)(LX(s)�A(s))

H(s) =1


Ld2✓(t)


sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1


⇥(s) = H(s)(LX(s)�A(s))

H(s) =1


INESTABLEESTABLECon polo en semiplano derecho el sistema es inestable.

• Realimentemos para buscar estabilidad

• ¿Cómo elegimos (diseñamos) '(")?

Sistemas lineales realimentados: péndulo invertidoLd2✓(t)


sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1


⇥(s) = H(s)(LX(s)�A(s))

H(s) =1


⇥(s) =LH(s)

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

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

• Realimentación proporcional:


Ld2✓(t)


sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1


⇥(s) = H(s)(LX(s)�A(s))

H(s) =1


⇥(s) =LH(s)

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


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>

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=">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</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:


Ld2✓(t)


sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1


⇥(s) = H(s)(LX(s)�A(s))

H(s) =1


⇥(s) =LH(s)

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


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=">AAADKHicdVHLblMxEHUurxJeKSzZWESVihDRTUDABqkSGyRYFIm0leIQOc7cxK3te7Hn0kSWf4g9H8AfsEPdsmELf4BvGlV5lJEsHZ2ZM54zMyyUdJimZ7XkytVr129s3azfun3n7r3G9v0Dl5dWQFfkKrdHQ+5ASQNdlKjgqLDA9VDB4fDkTZU//ALWydx8xFkBfc3HRmZScIzUoNFzr5+yzHLh3w18JwTfoe8DZYWmzH226JmCDHc3KpiV4wk+/hSJJ8sl4+AvkmHQaKatdB50E7QXoEkWsT/Yrn1jo1yUGgwKxZ3rtdMC+55blEJBqLPSQcHFCR9DL0LDNbi+n28h0J3IjGiW2/gM0jm7rPBcOzfTw1ipOU7ceq4iL81NXUSrv/tpZXdtIMxe9b00RYlgxPk8Wako5rTaPB1JCwLVLAIurIyWqJjw2Abjfeo7dLm9EJUHF5vQap6FI5U7sFU/ocs6M3Aqcq25GXl2fBp6ad97BsaVFiqNP2a5hjEPgV0YWJHgZRKcAP5X4mT0VYkYwhSt9nMi1OOh2+tn3QQHnVb7Wavz4Xlz78Xi5FvkIXlEdkmbvCR75C3ZJ10iyHfym/whf5OvyY/kZ3J2XprUFpoHZCWSX/8Ap60LtQ==</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:


Ld2✓(t)


sin ✓(t) ' ✓(t)

cos ✓(t) ' 1

Ld2✓(t)

dt2 � g✓(t) = Lx(t)� a(t)

⇥(s) =1


⇥(s) = H(s)(LX(s)�A(s))

H(s) =1


⇥(s) =LH(s)

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


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=">AAAC7nicdVFNb9NAEN2Yr2K+UjhyWYgqISFFdqiAU1SJC1I5FIm0lWJjrTdjZ9v9sHbXNNHKZ/4BN8SVC0fgt/BvWKcWatIy0khPb/bNzpvJK86MjaI/veDa9Rs3b23dDu/cvXf/QX/74aFRtaYwoYorfZwTA5xJmFhmORxXGojIORzlp2/a+tEn0IYp+cEuK0gFKSUrGCXWU1n/SVJoQl3cuF38rsH7mRs1H30+L8f7WTwux1HWH0TDaBX4Mog7MEBdHGTbvZ/JTNFagLSUE2OmcVTZ1BFtGeXQhEltoCL0lJQw9VASASZ1Ky8N3vHMDBdK+5QWr9iLCkeEMUuR+5eC2LnZrLXklbWF8Wj9d7dozW8MZIvXqWOyqi1Iej5PUXNsFW73h2dMA7V86QGhmnlLmM6Jb2P9lsMdfLE9pa0H45vgdp7OEVcGdNuPijpMJJxRJQSRM5ecnDXTKHUuAWlqDa3GnSRKQEmaJvlnYE1ir5LYOdj/SgzzvlpRYmFhtXArogn9oePNs14Gh6Nh/GI4er872HvZnXwLPUZP0TMUo1doD71FB2iCKPqMfqBf6HdQBV+Cr8G386dBr9M8QmsRfP8LEZ/yOw==</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?


H(s) =b


Polos en lazo abierto

a < 0


a < 0


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

H(s) =b


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…


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=">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</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%*).


Y (ej✓) =

H(ej✓)

1 +H(ej✓)X(ej✓)

��

H(ej✓)

1 +H(ej✓)D(ej✓)

�

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

https://en.wikipedia.org/wiki/Control_system%23/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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Documents

Señales y Sistemas · 2020. 7. 1. · Sistemas lineales realimentados • Realimentación (feedback): usar la salida de un sistema para modiﬁcar o controlar la entrada. • Control: