What is a good, non-trivial example of a control system that is unstable without feedback and stable with feedback for certain parameter values? Consider the input/state/output (i/s/o) feedback control system 
$
\left\{
 \begin{array}{lll}
  \frac{d}{dt}x = Ax + Bu  \\
  y = Cx \\
         u = Ky 
 \end{array}
\right.$
Here, $A \in \mathbb{R}^{3 \times 3}$, $B \in \mathbb{R}^{3 \times 1}$, $C \in \mathbb{R}^{1 \times 3}$ and $K \in \mathbb{R}$. I am interested in an example where we first have:
$B = (0,0,0)^{\intercal}$ (so there is no feedback at all) and the system resulting system $\frac{d}{dt}x = Ax$ is unstable for some $(3\times3)$ matrix $A$ (so all eigenvalues of $A$ have positive real part). 
And then (second), with the same matrix $A$ as previously chosen, we choose $B$ and $C$ in such a way that there are certain values for $K$ for which the whole system now becomes stable. If you can give an example with the appropriate $A$, $B$ and $C$, then I can calculate the range of $K$ by means of the Routh-Hurwitz algorithm to actually stabilise the system. 
Do you have an example of this? And does it allow a physical interpretation? Any help would be very much appreciated. 

An inverted pendulum, to give an example that is studied to exhaustion in controls teaching material and demos, and also present in real life - the Segway, or, well, us bipeds.
