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I have this state-space system

\begin{align} \dot{x}&=\begin{bmatrix}1 & 0\\3 & -2\end{bmatrix}x+\begin{bmatrix}10\\0 \end{bmatrix}u\\ y&=\begin{bmatrix}1 & 0\end{bmatrix}x \end{align}

And I'm asked to design an observer such that its poles are at $-1+j$ and $-1-j$.

The problem I'm having is that the system described above is not observable, as its observability matrix's rank is 1.

Is there anything I can do in a case like this?

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If the unobservable dynamics is stable, you should be able to separate the unobservable states and the observable states via a similarity transform. You then design an observer for the observable part.

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