# Thread: mapping from one invariant subspace to another

1. ## 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:

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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.

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

2. 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 $\displaystyle \mathcal{V}.$

3. thank you for the answer, ackbeet!

4. You're welcome. Have a good one!