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8.4: Ejercicios

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    Ejercicio 8.1

    Supongamos que deseamos realizar una ecuación diferencial de dos entradas de la forma

    \ [\ begin {alineado}
    y^ {(n)} +a_ {n-1} y^ {(n-1)} +\ cdots+a_ {0} y=& b_ {01} u_ {1} +b_ {11}\ punto {u} _ {1} +\ cdots+b_ {n-1,1} u_ {1} ^ {(n-1)}\
    &+b_ {02} u_ {2} +b_ {12}\ punto {u} _ {2} +\ cdots+b_ {n-1,2} u_ {2} ^ {(n-1)}
    \ final {alineado}\ nonumber\]

    Muestre cómo modificaría la observabilidad realización canónica para lograr esto, aún utilizando solo\(n\) integradores.

    Ejercicio 8.2

    ¿Cómo se modificaría la realización canónica de alcanzabilidad si la ecuación diferencial lineal con la que comenzamos variara en el tiempo más que invariante en el tiempo?

    Ejercicio 8.3

    Mostrar cómo modificar la alcanzabilidad realización canónica- pero aún utilizando solo\(n\) integradores- para obtener una realización de un sistema de dos salidas de la forma

    \ [\ begin {array} {ll}
    y_ {1} ^ {(n)} +a_ {n-1} y_ {1} ^ {(n-1)} +\ cdots+a_ {0} y_ {1} & =c_ {10} u+c_ {11}\ punto {u} +\ cdots+c_ {1, n-1} u^ {(n-1)}\\
    y_ {2} ^ {(n)} +a_ {n-1} y_ {2} ^ {(n-1)} +\ cdots+a_ {0} y_ {2} & =c_ {20} u+c_ {21}\ punto {u} +\ cdots+c_ {2, n-1} u^ {(n-1)}
    \ end {array}\ nonumber\]

    Ejercicio 8.4

    Considere el sistema de dos entradas y dos salidas:

    \ [\ begin {array} {l}
    \ punto {y} _ {1} =y_ {1} +\ alpha u_ {1} +u_ {2}\
    \ punto {y} _ {2} =y_ {2} +u_ {1} +u_ {2}
    \ end {array}\ nonumber\]

    a) Encontrar una realización con el número mínimo de estados cuando\(\alpha \neq 1\).

    b) Encontrar una realización con el número mínimo de estados cuando\(\alpha = 1\).


    This page titled 8.4: Ejercicios is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Mohammed Dahleh, Munther A. Dahleh, and George Verghese (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.