# Observable System L Matrix, can a value be negative?

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1. Nov 1, 2015

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. Nov 7, 2015