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Are there any values of \$\alpha\$ for which it is possible to find a state-feedback gain for the system by bringing all closed-loop poles to zero? If your answer is affirmative, find a feedback gain yielding the desired closed-loop poles. If you believe there are no such values of α, explain why. Are there values of α for which a state-feedback stabilizing controller can be found?

$$ x(k+1) =\begin{pmatrix}0&1&1\\0&\alpha&0\\1&2&2\end{pmatrix}x(k) + \begin{pmatrix}0\\0\\1\end{pmatrix}u(k) $$

$$ y(k) = \begin{pmatrix} 0 & 0 & 1 \end{pmatrix} x(k) $$

From the first part of the question, it is clear that the question wants to know if it possible to design a deadbeat controller, and if I calculate the controllability matrix of A I get

$$ \begin{pmatrix}0&1&2\\0&0&0\\1&2&5\end{pmatrix} $$

Is this enough to prove that there is no deadbeat controller? If I try and find the feedback vector K = \$\begin{pmatrix} 2 & X & 3\end{pmatrix}\$, were X is a value that can't be extracted from solving the characteristic equation of (A - BK). Is this further proof of there being no $\alpha$ to enable deadbeat control?

In the second part they are looking for a stabilizing controller (i.e. one that places all closed-loop poles in the open unit circle). How would I find this? I'm not sure what the question wants because from my understanding I thought the feedback vector was the one that stabilized the system. Any help appreciated.

Note: Exam revision, Not a homework question.

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Lets rephrase:

When (if ever) is the system stabilizable?

If it is stabilizable, you can perform a controllable decomposition and then apply pole placement.

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