Espacio De Estados Presentacion
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Analisis de sistemas de control en el espacio de estados Cuevas, Hernández, Valderrama
Indice • Representación de Funciones de transferencia en el Espacio de Estados – Representaciones canónicas en el espacio de estados • • • •
– – – –
Forma Canónica controlable Forma Canónica Observable Forma Canónica Diagonal Forma Canónica de Jordan
Eigenvalores de una matriz A de nxn Diagonalización de una matriz de nxn Invarianza de EigenValores No-unicidad de un conjunto de variables de estado
• Solución de una ecuación de estado invarante en el tiempo – Solución de ecuaciones de estado homogéneas – Matriz Exponencial – Acercamiento por la transformada de Laplace para la solución de una ecuación de estado homogénea – Matriz de transición de estado – Propiedades de la matriz de transición de estado – Solución de ecuaciones de estado no homogéneas – Acercamiento por la transformada de Laplace para la solución de una ecuación de estado no homogénea
Un sistema cualquiera se puede representar asi: QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Forma canónica controlable
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Forma canónica observable QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Forma canónica diagonal
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Forma canónica de Jordan
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Ejemplo de representaciones Sea: QuickTimeª and a None decompressor are needed to see this picture.
Representese en espacio de estados en la forma canónica controlable, observable y de Jordan
Forma canónica controlable
QuickTimeª and a None decompressor are needed to see this picture.
Forma canónica observable
QuickTimeª and a None decompressor are needed to see this picture.
Forma canónica diagonal
QuickTimeª and a None decompressor are needed to see this picture.
Eigenvalores de una matriz A de nxn Los eigenvalores son las raices de la siguiente ecuación característica
QuickTimeª and a None decompressor are needed to see this picture.
Por ejemplo, considerese: QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Los eigenvalores son entonces -1, -2 y -3
Diagonalización de una matriz nxn Sea una matriz con eigenvalores distintos:
QuickTimeª and a None decompressor are needed to see this picture.
La transformación x=Pz, donde P es
QuickTimeª and a None decompressor are needed to see this picture.
Donde λn son distintos eigenvalores de A
Transformará P-1AP en la matriz diagonal:
QuickTimeª and a None decompressor are needed to see this picture.
Ejemplo Considerese la siguiente representacion en espacio de estado de un sistema
Donde: QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
o de forma estándar: QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Los eigenvalores de A son:
QuickTimeª and a None decompressor are needed to see this picture.
Se tienen pues tres eigenvalores distintos. Si se define una variable de estado z mediante la transformacion: QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
donde: QuickTimeª and a None decompressor are needed to see this picture.
entonces al sustituir en la ecuacion de espacios de estados original se tiene: QuickTimeª and a None decompressor are needed to see this picture.
y al multiplicar por P-1 QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
simplificando da QuickTimeª and a None decompressor are needed to see this picture.
La ecuación de salida se modifica asi: QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Invarianza de Eigenvalores Para comprobar que los eigenvalores son identicos aun despues de una transformacion lineal se demostrará que se mantiene la relacion: QuickTimeª and a QuickTimeª and a decompressor None decompressor areNone needed to see this picture. are needed to see this picture.
=
Puesto que la determinante de un producto es el producto de las determinantes, se tiene:
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
La no unicidad de un conjunto de variables de estado Se comprobará que un conjunto de variables de estado no es unico para un sistema dado. Sean x1, x2, …, x3 un conjunto de variables de estado. Entonces se pueden tomar como otro conjunto de variables de estado cualquier conjunto de funciones:
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this corresponda picture.
Siempre que para cada conjunto de valores un conjunto único de valores x1, x2, …, xn, y viceversa. Por lo cual, si x es un vector de estado, QuickTimeª and None decompressor to seea this picture. entonces are ,needed donde QuickTimeª and a decompressor areNone needed to see this picture.
Es también un vector de estado, mientras que P sea no singular. Se puede obtener la misma información sobre el comportamiento de un sistema de diferentes vectores de estado.
Solución de ecuación de estado homogenea QuickTimeª None decompressor are needed toand see athis picture. QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a decompressor areNone needed to see this picture.
Matriz Exponencial QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Y tiene las propiedades QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
Es convergente para cualquier t finita, la cual la hace apropiada para ser calculada por metodos computacionales
Acercamiento a la solución de una ecuación de estado homogénea por medio de la transformada de Laplace QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Matriz de transición de estado QuickTimeª and a decompressor areNone needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Donde
QuickTimeª None decompressor are needed toand see athis picture.
es una solución de
QuickTimeª and a None decompressor are needed to see this picture.
y se verifica de la siguiente manera QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
La matriz de transición contiene toda la información acerca del movimiento libre del sistema que describe
Si los eigenvalores de la matriz de coeficientes A, entonces la matriz de transición contendrá los n exponenciales QuickTimeª and a None decompressor are needed to see this picture.
Si la matriz A es diagonal, se tiene que
QuickTimeª and a None decompressor are needed to see this picture.
Si hay multiplicidad en los eigenvalores, si por ejemplo los eigenvalores de A son: QuickTimeª and a None decompressor are needed to see this picture.
Entonces la matriz de transición contendrá ademas de los exponenciales QuickTimeª and a None decompressor are needed to see this picture.
terminos como
QuickTimeª QuickTimeª athis None decompressor None decompressor are needed toand see picture. needed toand see athis picture. y are
Propiedades de la matriz de transición de estado Para el sistema invariante en el tiempo QuickTimeª and a decompressor areNone needed to see this picture.
para el cual
QuickTimeª and a None decompressor are needed to see this picture.
se tienen las siguientes propiedades
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Ejemplo Sea un sistema descrito por la ecuación de estado descrita por la matriz QuickTimeª and a None decompressor are needed to see this picture.
Obtengase la matriz de transición de estado y su inversa Para este sistema:
QuickTimeª and a None decompressor are needed to see this picture.
La matriz de transición esta dada por: Puesto que
QuickTimeª and a None decompressor are needed to see this picture.
la inversa de sI-A esta dada por
QuickTimeª and a None decompressor are needed to see this picture.
Por lo cual
QuickTimeª and a None decompressor are needed to see this picture.
Puesto que
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Solución de una ecuación de estado no homogénea Considerando el caso escalar
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
Considerando el caso para la ecuación de estado no homogenea descrita por: QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
Empleo de la transformada de Laplace para la solución de una ecuación de estado no homogénea QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
Ejemplo Obtener la respuesta en el tiempo del siguiente sistema QuickTimeª and a None decompressor are needed to see this picture.
Donde u es el escalón unitario que ocurre en t=0
QuickTimeª and a decompressor areNone needed to see this picture.
Para este sistema se tiene que: QuickTimeª and a None decompressor are needed to see this picture.
Se había obtenido previamente la matriz de transición para este sistema: QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Asumiendo la condición inicial x(0) = 0, se tiene:
QuickTimeª and a None decompressor are needed to see this picture.
Indice • Representación de Funciones de transferencia en el Espacio de Estados – Representaciones canónicas en el espacio de estados • • • •
– – – –
Forma Canónica controlable Forma Canónica Observable Forma Canónica Diagonal Forma Canónica de Jordan
Eigenvalores de una matriz A de nxn Diagonalización de una matriz de nxn Invarianza de EigenValores No-unicidad de un conjunto de variables de estado
• Solución de una ecuación de estado invarante en el tiempo – Solución de ecuaciones de estado homogéneas – Matriz Exponencial – Acercamiento por la transformada de Laplace para la solución de una ecuación de estado homogénea – Matriz de transición de estado – Propiedades de la matriz de transición de estado – Solución de ecuaciones de estado no homogéneas – Acercamiento por la transformada de Laplace para la solución de una ecuación de estado no homogénea
Un sistema cualquiera se puede representar asi: QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Forma canónica controlable
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Forma canónica observable QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Forma canónica diagonal
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Forma canónica de Jordan
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Ejemplo de representaciones Sea: QuickTimeª and a None decompressor are needed to see this picture.
Representese en espacio de estados en la forma canónica controlable, observable y de Jordan
Forma canónica controlable
QuickTimeª and a None decompressor are needed to see this picture.
Forma canónica observable
QuickTimeª and a None decompressor are needed to see this picture.
Forma canónica diagonal
QuickTimeª and a None decompressor are needed to see this picture.
Eigenvalores de una matriz A de nxn Los eigenvalores son las raices de la siguiente ecuación característica
QuickTimeª and a None decompressor are needed to see this picture.
Por ejemplo, considerese: QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Los eigenvalores son entonces -1, -2 y -3
Diagonalización de una matriz nxn Sea una matriz con eigenvalores distintos:
QuickTimeª and a None decompressor are needed to see this picture.
La transformación x=Pz, donde P es
QuickTimeª and a None decompressor are needed to see this picture.
Donde λn son distintos eigenvalores de A
Transformará P-1AP en la matriz diagonal:
QuickTimeª and a None decompressor are needed to see this picture.
Ejemplo Considerese la siguiente representacion en espacio de estado de un sistema
Donde: QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
o de forma estándar: QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Los eigenvalores de A son:
QuickTimeª and a None decompressor are needed to see this picture.
Se tienen pues tres eigenvalores distintos. Si se define una variable de estado z mediante la transformacion: QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
donde: QuickTimeª and a None decompressor are needed to see this picture.
entonces al sustituir en la ecuacion de espacios de estados original se tiene: QuickTimeª and a None decompressor are needed to see this picture.
y al multiplicar por P-1 QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
simplificando da QuickTimeª and a None decompressor are needed to see this picture.
La ecuación de salida se modifica asi: QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Invarianza de Eigenvalores Para comprobar que los eigenvalores son identicos aun despues de una transformacion lineal se demostrará que se mantiene la relacion: QuickTimeª and a QuickTimeª and a decompressor None decompressor areNone needed to see this picture. are needed to see this picture.
=
Puesto que la determinante de un producto es el producto de las determinantes, se tiene:
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
La no unicidad de un conjunto de variables de estado Se comprobará que un conjunto de variables de estado no es unico para un sistema dado. Sean x1, x2, …, x3 un conjunto de variables de estado. Entonces se pueden tomar como otro conjunto de variables de estado cualquier conjunto de funciones:
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this corresponda picture.
Siempre que para cada conjunto de valores un conjunto único de valores x1, x2, …, xn, y viceversa. Por lo cual, si x es un vector de estado, QuickTimeª and None decompressor to seea this picture. entonces are ,needed donde QuickTimeª and a decompressor areNone needed to see this picture.
Es también un vector de estado, mientras que P sea no singular. Se puede obtener la misma información sobre el comportamiento de un sistema de diferentes vectores de estado.
Solución de ecuación de estado homogenea QuickTimeª None decompressor are needed toand see athis picture. QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a decompressor areNone needed to see this picture.
Matriz Exponencial QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Y tiene las propiedades QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
Es convergente para cualquier t finita, la cual la hace apropiada para ser calculada por metodos computacionales
Acercamiento a la solución de una ecuación de estado homogénea por medio de la transformada de Laplace QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Matriz de transición de estado QuickTimeª and a decompressor areNone needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Donde
QuickTimeª None decompressor are needed toand see athis picture.
es una solución de
QuickTimeª and a None decompressor are needed to see this picture.
y se verifica de la siguiente manera QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
La matriz de transición contiene toda la información acerca del movimiento libre del sistema que describe
Si los eigenvalores de la matriz de coeficientes A, entonces la matriz de transición contendrá los n exponenciales QuickTimeª and a None decompressor are needed to see this picture.
Si la matriz A es diagonal, se tiene que
QuickTimeª and a None decompressor are needed to see this picture.
Si hay multiplicidad en los eigenvalores, si por ejemplo los eigenvalores de A son: QuickTimeª and a None decompressor are needed to see this picture.
Entonces la matriz de transición contendrá ademas de los exponenciales QuickTimeª and a None decompressor are needed to see this picture.
terminos como
QuickTimeª QuickTimeª athis None decompressor None decompressor are needed toand see picture. needed toand see athis picture. y are
Propiedades de la matriz de transición de estado Para el sistema invariante en el tiempo QuickTimeª and a decompressor areNone needed to see this picture.
para el cual
QuickTimeª and a None decompressor are needed to see this picture.
se tienen las siguientes propiedades
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Ejemplo Sea un sistema descrito por la ecuación de estado descrita por la matriz QuickTimeª and a None decompressor are needed to see this picture.
Obtengase la matriz de transición de estado y su inversa Para este sistema:
QuickTimeª and a None decompressor are needed to see this picture.
La matriz de transición esta dada por: Puesto que
QuickTimeª and a None decompressor are needed to see this picture.
la inversa de sI-A esta dada por
QuickTimeª and a None decompressor are needed to see this picture.
Por lo cual
QuickTimeª and a None decompressor are needed to see this picture.
Puesto que
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Solución de una ecuación de estado no homogénea Considerando el caso escalar
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
Considerando el caso para la ecuación de estado no homogenea descrita por: QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
Empleo de la transformada de Laplace para la solución de una ecuación de estado no homogénea QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture. QuickTimeª and a None decompressor are needed to see this picture.
Ejemplo Obtener la respuesta en el tiempo del siguiente sistema QuickTimeª and a None decompressor are needed to see this picture.
Donde u es el escalón unitario que ocurre en t=0
QuickTimeª and a decompressor areNone needed to see this picture.
Para este sistema se tiene que: QuickTimeª and a None decompressor are needed to see this picture.
Se había obtenido previamente la matriz de transición para este sistema: QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
QuickTimeª and a None decompressor are needed to see this picture.
Asumiendo la condición inicial x(0) = 0, se tiene:
QuickTimeª and a None decompressor are needed to see this picture.
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