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Thread: mapping from one invariant subspace to another

  1. #1
    Newbie
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    mapping from one invariant subspace to another

    Hello to everybody,

    I am trying to find a constructive proof for the solution of a problem in control engineering, and I got stuck at some point for the last few days. The problem is as follows:

    ------------------
    For a system defined in state space as:

    $\displaystyle
    \dot{x}(t)=Ax(t)+Bu(t)
    $
    $\displaystyle
    y(t)=Cx(t)
    $

    where

    $\displaystyle
    A \in \mathbb{R}^{nxn}
    $
    $\displaystyle
    B \in \mathbb{R}^{nxm}
    $
    $\displaystyle
    C \in \mathbb{R}^{pxn}
    $

    There are defined subspaces $\displaystyle \mathcal{S}$ and $\displaystyle \mathcal{V}
    $ of $\displaystyle \mathbb{R}^n$ that the rest of the problem deals with, with properties:

    $\displaystyle
    \mathcal{S} \subset \mathcal{V} \subset \mathbb{R}^n \\
    $

    $\displaystyle
    \exists \text{ a } G \text{ st. } (A+GC)\mathcal{S} \subset \mathcal{S} \\
    $

    $\displaystyle
    \exists \text{ an } F \text{ st. } (A+BF)\mathcal{V} \subset \mathcal{V} \\
    $

    The problem is to find an $\displaystyle N$ st.

    $\displaystyle
    (A+BNC)\mathcal{S} \subset \mathcal{V}
    $
    ------------------

    Of course, the first thing that comes to mind is to find $\displaystyle N$ st.

    $\displaystyle
    (A+BNC)\mathcal{S} \subset \mathcal{S}
    $

    which satisfies the problem requirement as $\displaystyle \mathcal{S} \subset \mathcal{V}$. However, such a method $\displaystyle N$ appears to be too "tight" (meaning that more freedom on $\displaystyle N$ is needed) for many examples from application.

    Another idea is to find $\displaystyle N$ st.

    $\displaystyle
    (A+BNC)\mathcal{V} \subset \mathcal{V}
    $

    then find a restriction of the resulting $\displaystyle N$ on subspace $\displaystyle \mathcal{S}$. But then my mathematical knowledge is limited on restrictions on linear operators.

    In short, I am stuck at finding an elegant approach for the construction of $\displaystyle N$. Any help and ideas on the topic would be of great help. Thanks in advance.
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  2. #2
    A Plied Mathematician
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    CT, USA
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    Your second idea is definitely better, and it is sufficient, I believe. If you compare the statement

    $\displaystyle \exists\,F\;\text{s.t.}\;(A+BF)\mathcal{V}\subset\ mathcal{V}$ with

    Find an $\displaystyle N$ s.t.

    $\displaystyle (A+BNC)\mathcal{V} \subset \mathcal{V},$

    then why not just solve

    $\displaystyle F=NC$ for $\displaystyle N?$

    Dimensions:

    $\displaystyle A$ is n x n,

    $\displaystyle B$ is n x m,

    $\displaystyle C$ is p x n

    $\displaystyle F$ must be m x n, and so

    $\displaystyle N$ must be m x p.

    As for the restriction, I wouldn't worry about it at all. If an operator takes $\displaystyle \mathcal{V}$ into $\displaystyle \mathcal{V},$ then it must, of necessity, take a subset $\displaystyle \mathcal{S}\subset\mathcal{V}$ into $\displaystyle \mathcal{V}.$
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  3. #3
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    thank you for the answer, ackbeet!
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  4. #4
    A Plied Mathematician
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    You're welcome. Have a good one!
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