Similarity transformation to get minimal realization

[azizz]

I derived a LTI system H described in state space as

$$H=\left( \begin{array}{cc|c} A+BKC & -BLC & B \\ 0 & A & B \\ \hline KC & -LC & I \end{array} \right)$$

system H is not of minimal realization, hence I should be able to find a similarity transformation T such that I get the minimal realization of H. MATLAB cofirms this for me when I just fill in some values for A, B, C, L, K and call the function minreal. I also found that the minimal realization of the system should be

$$H=\left( \begin{array}{cc} A+BKC & B \\ (K-L)C &I \end{array} \right)$$

However, I cannot find the transformation T (matlab just gives me some numbers which I can not link to the original variables). So I need to find a T which satisfies

$$T^{-1} \begin{pmatrix} A+BKC & -BLC \\ 0 & A \end{pmatrix} T = \begin{pmatrix} A+BKC & 0 \\ 0 & X \end{pmatrix}$$

$$T^{-1} \begin{pmatrix} B \\ B \end{pmatrix} = \begin{pmatrix} B \\ 0 \end{pmatrix}$$

$$\begin{pmatrix} KC & -LC \end{pmatrix} T = \begin{pmatrix} (K-L)C & 0 \end{pmatrix}$$

Where X can be anything.

Someone knows how I can find T?

 

[Navin A S]

You can find the transformation T by solving a set of linear equations. Consider the matrix equation in your first equation. You can rewrite it as A+BKC+T11=A+BKC and BLC+T12=0. This gives you two linear equations in two unknowns (T11 and T12). Similarly, you can consider the other two equations. Solving these four equations will give you the transformation matrix T.