INGENIERÍA INGENIERÍA ELECTROMECÁNICA ELECTROMECÁNICA ASIGNATURA INGENIERÍA INGENIERÍA DE CONTROL CLASICO CLA SICO “PROYECTO FINAL (MOTOR CD CONTROLADO
POR CORRIENTE DE ARMADURA)” MAESTRO ING. JOSE JOSE ALFONSO CAMILO SANTANA ALUMNOS RODRIGO RIVERA ZAVALA, No. CONTROL: CONTROL: 11090364 OCTAVIO OCTAVIO GOMEZ MONTERO, M ONTERO, No. CONTROL: CONTROL: 11090298 JOHNATAN JAHIR AGUAYO JUAREZ, JUAREZ, No. CONTROL: CONTROL: 11090233 NEIL BONY CÁRDENAS NÁJERA, No. CONTROL: 10091429 JOSÚE DAVID MUSITO ALCANTARA, No. CONTROL: CONTROL: 11090333
Ingeniería de Control Clásico
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1. RESUMEN En este trabajo se presenta el modelo matemático de un motor de corriente continua. Se ha desarrollado el modelo matemático usando ecuaciones diferenciales dife renciales y a su vez ve z también se se desarrolló d esarrolló el análi análisis sis respectivo respe ctivo en términos té rminos de las variables variables físicas que se tiene en e l motor, moto r, como en la carga mecánica mec ánica acoplada acop lada al mismo. Adicionalmente se ha simulado simulado y se ha o btenido bte nido la respuesta de control de la velocidad la cual cual puede ser en función de la corriente de armadura armadura y para para esto se utilizó SIMULINK, una herramienta del programa MATLAB. Los motores de corriente continua son los más comunes y económicos, y se pueden encontrar en la mayoría de los juguetes a pilas, constituidos, por lo general, por dos imanes permanentes fijados en la carcasa y una serie de bobina bob inados dos de cobre ubicados en el e je del mo tor, que q ue habitualmente habitualmente suelen ser tres y a su vez son ampliamente usados a nivel industrial. Los motores de corriente cont co ntinu inuaa permiten un amplio amplio rango de velocidad ve locidad y puede n proporcionar propo rcionar un alto par-motor con control más sencill sencillo o y económico eco nómico que cualquier cualquier motor mo tor de corriente alterna. alterna. En En la actualidad actualidad los métodos métod os de control co ntrol de velocidad ve locidad se han ido desarrollan desarrollando do cons co nsidera iderableme blemente nte y los los más co munes son el control de velocidad por corriente de campo campo y el control de velocidad por corriente de armadura, armadura, que son técnicas de co ntrol no lineal. lineal. Para poder analizar estos métodos se requiere del conocimiento físico del sistema, unidades de las constantes que aparecen en el modelo, selección adecuada de las las variables variables de estado y conocimi conoc imiento entoss de desar d esarrollo rollo de ecuaciones diferenciales dife renciales utilizando utilizando la transformada de Laplace y a su ve z para poder pod er obser ob serva varr el comportamiento un simulador el cual para objeto de estudio se utiliza SIMULINK SIMULINK una herramienta del de l programa pro grama MATLAB. La selección selecc ión de d e variables variables no es evidente, sino más bien resulta de la experiencia en el modelado de sistemas eléctricos y mecáni me cánicos, cos, y así así como de la apropia aprop iada da selección de constantes constantes físicas físicas como de fricción, inercia y torque eléctrico. En esta propuesta, propuesta, se desarrolla desarrolla el modelo mode lo matemático de d e un método méto do de control de velocidad veloc idad el cual cual es: e s: control de velocidad por corriente de armadura. armadura. Para P ara esto
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análisis matemático, pues se emplea la teoría de control para la selección de las variables de estado. estado . Dentro del de l trabajo se presenta una simulación simulación y se determina de termina el comportamiento de la velocidad del motor con respecto a la corriente de armadura con condiciones iniciales establecidas. Finalmente Finalmente se concluye analizan analizando do los resultado resultadoss obte ob tenidos nidos de d e la simulación simulación con co n el modelo matemático determinado.
2. INDICE
3 Marco Teórico Teó rico
4-8
4 Mode Mo delo lo Matemátic Matemát ico o
9-10
5 Diagrama Diagrama de bloq ues
11-13
6 Espacio Espacio de modelos mo delos de esta e stados dos
14-16
7 Respu Resp uesta Dinámica
16-18
8 Error en esta e stado do estable con escalón unitar unitario io
19-22
9 Contro C ontrolado ladores res P, PI, PID, P ID, Zieglers-Nichols Ziegle rs-Nichols
23-27
10 Estabilidad Estabilidad y LGR
28-32
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3. MAR MARCO CO TEORI T EORICO CO
Proceso de modelado. El proceso proc eso de modelado analí analítico tico se se divide en tres grandes etapas et apas.. La primera de de ellas consiste en la delimitación del modelo en función de los fenómenos que resultan relevantes de acuerdo al problema que se quiere resolver. Esta es una etapa que no puede sistematizarse fácilmente y que requiere por ende de una cierta dosis de intuición intuición y por po r sob sobre re todo tod o de una vasta vasta ex experi perienci enciaa en relación relación con el sistema a modelar. Una vez delimitados los fenómenos que se consideraron relevantes para la construcción del modelo, se pasa a la siguiente etapa en la que se deben formali fo rmalizar zar las las relaciones re laciones constitutivas y estructurales e structurales asociadas respe ctivamente a los fenómenos considerados y a la forma en que estos se disponen dentro del sistema. En los sistemas físicos, estas relaciones constitutivas y estructurales encuentran su expresión formal (matemática) en las leyes fundamentales de los dominios de la física física asociados asociados a los fenóme f enómenos nos mencionados. Por Po r este mo tivo, el e l modelado mode lado analítico analítico de un sis sistema tema físico no e s posible sin sin un conocimiento de las leyes físicas elementales asociadas a los fenómenos en cuestión.
Sistemas Me cánicos. cánicos. La mecánica me cánica clásica clásica (newtoniana) (ne wtoniana) se ocupa ocup a de describir de scribir fenóme fe nómenos nos asociados asociado s con co n el movimiento de los cuerpos. Por este motivo, en los sistemas mecánicos tendremos te ndremos habitualmente habitualmente como co mo variables descriptivas de scriptivas las posi po sicione ciones, s, velocid ve locidades ades y aceleraciones, con sus relaciones constitutivas básicas:
̇ ̇ ̇ ̇
=0 =0 =0 =0
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Que establece que la d erivada de la cantidad de mo vimiento vimiento de un cuerpo es igual igual a la fuerza neta aplicada. Esta ecuación constituye c onstituye una ley dinámica que será una relación relación constitutiva constitutiva fundamental asociada asociada a cada cuerpo en el que q ue se cons co nsidere idere la presencia de masa. Recordemos también que definíamos a la cantidad de movimiento como el producto de la masa por la velocidad:
Otras leyes importantes son las asociadas a los fenómenos de fricción y de la elasticidad. En el caso de la fricción, una hipótesis habitual es representar la misma misma como c omo una fuerza que se opo ne al al movimi mo vimiento ento cuya magnitud se relaciona relaciona con la velocidad.
Fricción, Fricción
)=0
En el caso particular de la fricción viscosa la relación se considera lineal, caracterizada por un coeficiente de rozamiento.
Fricción
-
Fricción
=0
El fenómeno de elasticidad, por su parte, vincula la fuerza ejercida sobre un cuerpo cuerp o con la deformación def ormación sufrida por po r el mismo. Una aproximación aprox imación habitual para p ara representar esta relación en un resorte, en función de una constante de elasticidad es la siguiente:
Resorte
El resto de las relaciones que podrán aparecer en sistemas mecánicos serán en general relaciones estructurales.
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Fig. 1 Sistema Mecánico (masa-resorte)
Sistema Mecánico Rotacionales. Todas las leyes constitutivas vistas para los sistemas mecánicos traslacionales tienen tiene n su equivalente eq uivalente rotacional, donde do nde las fuerzas fuerz as son son re emplazad emp lazadas as por po r torques to rques en tanto que las posiciones, velocidades y aceleraciones traslacionales son reemplazadas por sus versiones angulares. Encontramos también elementos roto-traslacionales que vinculan variables de ambos dominios. Un caso típico es una polea, en la cual se verifica:
Siendo r el e l radio radio de la misma misma..
Del sistema físico real al idealizado. Cuando nos encontramos con un sistema real a modelizar. Estos esquemas, que corresponden a los llamaremos sistema físico idealizado, son producto de simplificaciones que se realizan acorde al problema a estudiar. Es muy importante no perder de vista que los modelos obtenidos resultarán adecuados sólo para resolver resolver dete d etermin rminados ados problemas prob lemas y dentro d e un rango de operaciones dado.
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Es decir, de cir, el sistema sistema físico idealizado dep enderá enderá no sólo sólo del de l sistema sistema real en sí, sino también del problema a resolver y del interval intervalo o de validez validez que se pretenda prete nda tener te ner para el modelo resultan resultante. te. Lamentablemente, no hay una metodología que nos permita realizar estas simplifica simplificaciones ciones de forma fo rma sistemática sistemática.. Esta Esta etapa del modela mode lado do (que es quizás la más importante en virtud virtud de que todo to do el resto depe nderá nderá de lo que se haga aquí) se resuelve resuelve en gran medida a partir partir de consideraciones consideraciones suje sujetas tas a la e xperienci xpe rienciaa y al conocimiento del proceso real. Sin embargo los sistemas complejos pueden habitualmente dividirse en subsistemas más simples de los cuales se encuentran modelos en base a simplificaciones ya probadas en problemas similares. Por eso es fundamental antes de comenzar a realizar las primeras simplificaciones de un sistema real, buscar en la literatura mode los de siste sistemas mas similares similares en e n los lo s cuales se manifiesten los mismos mismos fenómenos. fe nómenos. Es muy importante importante tener en cuenta cuenta siempre siempre que debido a que la obtención ob tención de un modelo mod elo se se basa en la aplicación aplicació n de hipótesis hipó tesis simplificado simplificadoras, ras, los modelos mod elos tendrán te ndrán validez validez siempre siempre que se respeten respete n las mencionadas mencionadas hipóte hipótesi sis. s.
Del sistema idealizado ide alizado al sistema sistema matemático. Una vez realizadas las simplificaciones y obte ob teniendo niendo esquemas esque mas decimos de cimos que tenemos un sistema físico idealizado. Aunque estos sistemas en general contienen toda la información información que se necesita para la la constru c onstrucción cción de un modelo mo delo
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De esta forma, una una vez que se tiene el sistema sistema físico físico idealizado idealizado el siguiente siguiente paso hacia la obtención de un modelo matemático será el de reunir las relaciones constitutivas y estructurales involucradas. En muchos casos encontraremos conjuntos de relaciones matemáticamente equivalentes, por lo que no serán necesarias todas las relaciones para la construcción del modelo. Una vez que tengamos todas las relaciones matemáticas necesarias, aún no tendremos un modelo matemático muy útil. Si bien tendremos una especie de sistemas de ecuaciones (en realidad una mezcla de ecuaciones diferenciales y algebraicas algeb raicas), ), este sistema sistema así planteado no tendrá te ndrá una estructura que nos permita p ermita estudiar estudiar y resolver re solver problemas pro blemas.. Por esto, el paso siguiente tras reunir las relaciones constitutivas y estructurales es el obtener un sistema de ecuaciones que tenga una estructura que permita aplicar alguna teoría matemática adecuada ya establecida. Dado que trataremos con sistemas físicos bajo hipótesis de parámetros concentrados, buscaremos arribar a los modelos con que trata la teoría de ecuaciones diferenciales. diferenciales. En general ge neral,, utilizaremos utilizaremos dos do s tipos de ex expresiones: presiones: Las ecuaciones Diferenciales Ordinarias (EDO) y los sistemas de Ecuaciones de Estado y de Salida (EE/ES). Para llegar a este tipo de expresiones deberemos manipular las ecuaciones dadas por las relaciones constitutivas y estructurales siguiendo siguiendo diferentes d iferentes técnicas té cnicas..
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4. MODELO MATEMÁTI MATEMÁTICO. CO.
Cons Co nsiderando iderando perturbaci perturbación ón por el torque nulo. nulo.
Datos:
La= 0.5 J= 0.08 Km = 0.01 Va = 5 v Ra=1 b=0.2 Kb=0.01 Tp=0 (perturbaci (perturb ación). ón).
Ecuaciones:
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Torque del de l motor
Carga expresada ex presada en términ té rminos os de Velocidad angula angular. r.
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5. DIAGR DIAGRAMA AMA DE BLOQUES. BLOQ UES.
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Reducción de bloque s con Matlab Matlab
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Con lo anterior obtenemos:
Con ayuda de Matlab Matlab realizamos real izamos el feedbac fe edback k
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6. ESPACIO DE MODELOS DE ESTADO. Del diagrama diagrama de bloques: b loques:
Se procede a obtener la función de transferencia y a despejar de la siguiente manera:
Despejando la derivada de mayor orden:
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[ ]
)
Para B1:
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[ ] =
[ ] +
7. RESPUESTA DINÁMICA.
LAPLACE
Fracciones Fraccione s parciales parciales
[ ]
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Gráfica: Caracterización con escalón unitario. unitario.
Respuesta ante una señal señal de entrada escalón unitario unitario y un voltaje de entrada 5 v.
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FT = G(s) =
Por lo tanto: tanto:
2λWn=4.5 Wn=4.5 por tanto λ=
Tp= Tp =
= √ √
Wn =
√
Tp=indeterminado
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8. ERROR EN ESTADO ESTABLE CON CO N ESCALON ESCALON UNITARIO. UNITARIO.
V a ( s)
W(s)
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Sustituir
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Grafica en Simulink.
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Grafica Grafica PI de d e la función:
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9. CO CONTROLADORES NTROLADORES P, PI, PID Y ZIEGLERS-NICHOLS. ZIEGLERS-NICHO LS.
APLICACIÓN DE CONTROLADORES AL MOTOR DE CD
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Controlador
Kp
P
45
Ti
Ki
Td
Kd
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1. Gráfica dando los siguientes valores: Proporcional: 45 Integral: 0
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3. Grafica dando los siguientes siguie ntes valores: valores:
Proporcional: 54
Integral: Inte gral: 38.4341 Derivati Deri vati
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