Given only a system transfer function, after changing it using state feedback, can we know if the resulting system is controllable? Observable? Practice test question
Given
$$ G(s) = \frac{(s-2)(s-5)}{(s+1)(s-3)(s+4)} $$
Use state feedback to change it to
$$ \frac{(s-5)}{(s+1)(s+4)}$$
Is the resulting system controllable? Observable? If no, explain why not
My answer
a.) Using state feedback, change the transfer function
By putting the function into controllable canonical realization, we get
$$ \dot{x}_c = \begin{bmatrix}0&1&0\\0&0&1\\12&11&-1\end{bmatrix}x_c + \begin{bmatrix}0\\0\\1\end{bmatrix}u, \quad y_c = \begin{bmatrix}10&-7&1\end{bmatrix}x_c $$
Using state feedback (omitting the steps here) we change the transfer function to
$$ G(s) = \frac{(s-2)(s-5)}{s^3+(2+k_{c3})s^2+(-11+k_{c2})s + (-12+k_{c1})} $$
Setting the denominator equal to $(s-2)(s+1)(s+4)$ by letting $k_c = \begin{bmatrix}4&5&1\end{bmatrix}$ changes the transfer function to the desired value.
b.) Is the resulting system controllable? Observable?
It can't be both controllable and observable, as we have canceled out a term in the transfer function. However, without knowledge of the original $\mathbf{A}$ and $\mathbf{b}$ matrices, we cannot derive the transform matrix $\mathbf{T}$ to find $k = k_c\mathbf{T}$
I'm pretty confident about the first part of the question, but I'm not sure if there's more I could be doing on the second question. Is this really the best answer?
 A: The controllability of a LTI state space model does not change under state feedback. This can be shown by using the Hautus test to check for controllability
$$
\begin{bmatrix}
\lambda\,I - A + B\,K & B
\end{bmatrix} =
\begin{bmatrix}
\lambda\,I - A & B
\end{bmatrix}
\begin{bmatrix}
I & 0 \\ K & I
\end{bmatrix},
$$
since the right most matrix is square and full rank and thus the rank of the remaining matrices have to be equal to each other. So just by looking at the transfer function and noticing that there is a pole-zero cancelation one could already conclude that it would not be observable (using the Hautus test for observability one can conclude it would lose one rank).
It can also be noted that all state space models (of the same size), that have the same corresponding transfer function, are equivalent to each other. So the results regarding controllability and observability of your model should also hold for other equivalent state space realizations. So your choice for $A$ and $B$ would allow you to say something about it (assuming one has to use a minimal realization).
I do have a remark regarding the last part of the question, regarding controllability and observability. Namely, one can always start with a non-minimal state space realization, which is not controllable and not observable. Such system will always remain that way.
