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miniradman

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## Homework Statement

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} ##

## Homework Equations

## T = [O_m P]^-1 ##

## L_z = T L_x ##

## O_m = \begin{bmatrix}C \\CA \end{bmatrix} ##

## 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 ##

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