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

In summary, the conversation discusses the use of observable canonical form and a state transformation to find the L error feedback matrix for a state variable feedback system. It is important for the C matrix to be in observable canonical form in order to use the transformation matrix T.

## 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 ##

Last edited by a moderator:

Hello,

Thank you for your question. It appears that you are on the right track with your solution. To answer your question, yes, the C matrix needs to be in observable canonical form in order to use the transformation matrix T. This can be achieved by multiplying the existing C matrix by the P matrix, which will result in the observable canonical form. So your new C matrix would be:

## C = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\616.8 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \end{bmatrix} ##

Then, you can proceed with your solution using the transformed C matrix. I hope this helps. Let me know if you have any other questions or concerns. Good luck with your work.

## 1. Can a value in the Observable System L Matrix be negative?

Yes, a value in the Observable System L Matrix can be negative. The values in the matrix represent the coefficients of the equations used to describe the system, and these coefficients can be positive or negative depending on the specific system being studied.

## 2. How do negative values in the L Matrix affect the observable system?

The negative values in the L Matrix can affect the observable system in different ways depending on the specific system being studied. In some cases, negative values may indicate a decrease in the observable quantity, while in other cases they may indicate an increase. It is important to carefully analyze the system and understand the role of each coefficient in order to interpret the impact of negative values.

## 3. What does it mean if all values in the L Matrix are negative?

If all values in the L Matrix are negative, it means that the system is unstable. This can indicate that the system is not in a steady state and may exhibit chaotic behavior. It is important to further analyze the system and potentially make adjustments in order to stabilize it.

## 4. Can negative values in the L Matrix be changed to positive values?

Yes, it is possible to change negative values in the L Matrix to positive values. This can be done by adjusting the coefficients in the equations that make up the matrix. However, it is important to carefully consider the implications of changing these values and to ensure that the new values accurately reflect the behavior of the system.

## 5. How do negative values in the L Matrix relate to the eigenvalues of the system?

Negative values in the L Matrix can indicate that the eigenvalues of the system are also negative. This can have implications for the stability and behavior of the system. It is important to analyze the eigenvalues in conjunction with the L Matrix values in order to fully understand the behavior of the system.

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