Show A Non-Min Phase System Cannot be Controlled via Controllability Matrix?
The state space system will have \$A \in \mathbb{R}^{2 \times 2}\$ and other matrices of appropriate size, so it will be
$$  \dot{x} = Ax + B u$$
The poles of the system will only depend on the eigenvalues of the matrix \$A\$. And since
$$G(s)=\frac{s-z}{(s+p)(s-p)}$$
We have that \$ \det(sI-A) = (s+p)(s-p) = 0\$. Writing that in the controllable canonical form we have
$$  \dot{x} = \begin{bmatrix} 0 &amp; 1 \\ -p^2 &amp; 0\end{bmatrix} x + \begin{bmatrix} 0 \\ 1\end{bmatrix} u$$
The transfer function of that state space equation will be
$$  C(sI-A)^{-1}B$$
with
$$(sI-A)^{-1} = \begin{bmatrix}  -s/(-s^2 + p^2) &amp; s/(s^3 - sp^2) \\
 p^2/(s^2 - p^2) &amp; s/(s^2 - p^2)
  \end{bmatrix} $$
we will have
$$(sI-A)^{-1}B =   \begin{bmatrix} s/(s^3 - sp^2) \\ s/(s^2 - p^2)
 \end{bmatrix} = \begin{bmatrix} 1 \\
   s
 \end{bmatrix} \frac{1}{s^2 - p^2}
$$
Now it should be very simple to determine the \$C\$ matrix
$$C = \begin{bmatrix} -z &amp; 1\end{bmatrix}.$$
All that being said, the non-minimum phase should not be a problem if you assume that you know the state \$x\$ and your model is correct \$(A,B,C)\$. As far as I know, non-minimum phase is a limitation of classical control and PID. While State-space control has different drawbacks (like needing a good model of the plant, fast estimators and dealing with nonlinearities &quot;explicitly&quot;.)