# Similarity transformation to get minimal realization

1. Sep 22, 2009

### 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?

Last edited: Sep 22, 2009