# Determine output of state space model to determine observability

I have the given input for a system

$\begin{bmatrix}\dot{a}\\\dot{b} \\\dot{c} \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -18 & -27 & -10 \end{bmatrix} \begin{bmatrix} a \\ b \\ c \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}d$

where the sensor measures c. In order to test for observability, I need the output matrices of the system (system is observable if observability matrix $\begin{bmatrix}C \\ CA \\ CA^2 \end{bmatrix}$ is full rank. Is there anyway to do this from the given problem, or is there another method of determining observability without the C matrix?

• Where the sensor measures $c$. What does that imply? Can you figure out $C$ from this statement? – Math Lover Dec 10 '17 at 0:26
• Would C simply be [0 0 1]? – Matt Dec 10 '17 at 0:27
• If $y = c$ then $y = [0~0~1]\cdots$ – Math Lover Dec 10 '17 at 0:28
• Appreciate the help! – Matt Dec 10 '17 at 0:29
• You are given that $C=\begin{bmatrix} 0 & 0 & 1 \end{bmatrix}$. And no, without knowing $C$ there is no way to check the observability of the above system. – copper.hat Dec 10 '17 at 2:13