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Observable System L Matrix, can a value be negative?

  1. Nov 1, 2015 #1
    1. The problem statement, all variables and given/known data
    An error matrix is in the form, has a characteristic equation:
    ## CE: s^2 + 120s + 7200 = 0 ##

    A state variable feedback system is described by:
    ## A_F = \begin{bmatrix}0 & 1 \\-616.8 & -40 \end{bmatrix} ##
    ## B = \begin{bmatrix}0 \\ 1 \end{bmatrix} ##
    ## C = \begin{bmatrix}616.8 & 24 \end{bmatrix} ##
    ## D = \begin{bmatrix}0 \end{bmatrix} ##

    Find the L error feedback matrix, using a state transformation to convert to observer canonical form:
    ## L = \begin{bmatrix}l_1 \\ l_2 \end{bmatrix} ##

    2. Relevant equations
    ## T = [O_m P]^-1 ##
    ## L_z = T L_x ##
    ## O_m = \begin{bmatrix}C \\CA \end{bmatrix} ##

    3. The attempt at a solution
    From what I understand, observable canonical form entails that:
    ## P = \begin{bmatrix}1 & 0 \\616.8 & 1 \end{bmatrix} ##
    ## A_e = \begin{bmatrix} -120 & 1 \\ -7200 & 0 \end{bmatrix} ##
    And equating with the existing system:
    ## A_e = \begin{bmatrix} -(40+l_1) & 1 \\ -(616.8+l_2) & 0 \end{bmatrix} = \begin{bmatrix} -120 & 1 \\ -7200 & 0 \end{bmatrix} ##
    This yields the L matrix:
    ## \begin{bmatrix} l_1\\ l_2 \end{bmatrix} = \begin{bmatrix} 80 \\ 6583,2 \end{bmatrix} ##
    However this is form is not in observer canonical form, my question is when I use the Observable Matrix with my Transformation matrix:
    ## O_m = \begin{bmatrix}C \\CA \end{bmatrix} ##
    Does C need to be in observable canonical form?
    ## C = \begin{bmatrix} 1 & 0 \end{bmatrix} ##
    Because my current C matrix is not in that standard form. If so, how do I convert it, or does it fall out in the wash when I apply the P matrix?
    ## T = [O_m P]^-1 ##
     
    Last edited by a moderator: Nov 1, 2015
  2. jcsd
  3. Nov 7, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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