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Similarity transformation to get minimal realization

  1. Sep 22, 2009 #1
    I derived a LTI system H described in state space as

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

    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

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

    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

    [tex] T^{-1} \begin{pmatrix} A+BKC & -BLC \\ 0 & A \end{pmatrix} T = \begin{pmatrix} A+BKC & 0 \\ 0 & X \end{pmatrix} [/tex]

    [tex] T^{-1} \begin{pmatrix} B \\ B \end{pmatrix} = \begin{pmatrix} B \\ 0 \end{pmatrix} [/tex]

    [tex] \begin{pmatrix} KC & -LC \end{pmatrix} T = \begin{pmatrix} (K-L)C & 0 \end{pmatrix} [/tex]

    Where X can be anything.

    Someone knows how I can find T?
     
    Last edited: Sep 22, 2009
  2. jcsd
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