state-space reduction

Question

I am confused about state-space reduction. I learned it in the class but am not skilled in it.

If $A,B,C,D$ matrices are given with values, we can
1. find its controllability matrix to see if controllable, if uncontrollable
2. find a transformation matrix $P$
3. reduce the state space to a controllable state space representation

However, if NOT given values, just like the following:

Can I reduce the above state space representation to the following:

Answer 1

I've found Lemma 3.20 in Zhou/Doyle's Robust and Optimal Control on p.72 which might be what you're looking for:

Lemma 3.20 Let $\left[\begin{array}{cc}A & B \\ C & D\end{array}\right]$ be a state space realization of a (not necessary stable) transfer matrix $G(s)$. Suppose there exists a symmetric matrix

$$ P = P^* = \left[\begin{array}{cc}P_1 & 0 \\ 0 & 0\end{array}\right] $$

with $P_1$ nonsingular such that

$$ AP + PA^* + BB^* = 0. $$

Now partition the realization $(A,B,C,D)$ compatibly with $P$ as

$$ \left[\begin{array}{ccc}A_{11} & A_{12} & B_1 \\ A_{21} & A_{22} & B_2 \\ C_1 & C_2 & D \end{array}\right]. $$

Then $$ \left[\begin{array}{cc}A_{11} & B_1 \\ C_1 & D \end{array}\right] $$ is also a realization of $G$. Moreover, $(A_{11},B_1)$ is controllable if $A_{11}$ is stable.

$A^*$ denotes the complex conjugate of $A$ here.