Designing a root-locus compensator to attain design requirements
I don't have access to matlab, but I have played with this in R (see code below) and can confirm that if you're trying to transform your $G(s)$ into:
$$
H(s) = \frac{K G(s)} {1 + KG(s)}
$$
using only a scalar $K$, then you cannot get to the target pole locations (based on the values of $\xi$ and $\omega_n$).

In the diagram:


The black circles are the starting root (pole) positions.
The green circles are the target pole positions.
The red circles are the root locus for varying values of $K$.
The large black circle is the unit circle.






R Code Below

#30341

library("signal")

xi &lt;- 0.5
omega_n &lt;- 0.3

target_den &lt;- c(1, 2*xi*omega_n, omega_n*omega_n)

target_roots &lt;- roots(target_den)

num &lt;- c(0, 0.5, 0.5)
den &lt;- c(1, -2, 1)

start_roots &lt;- roots(den)

par(pty="s")
plot(Re(start_roots),Im(start_roots),xlim=c(-5,5),ylim=c(-5,5), col="black", lwd=5)
points(Re(target_roots), Im(target_roots), col="green", lwd = 3)
for (k in seq(-100.0, 100.0, 0.1))
{
  rts &lt;- roots(k*num + den)
  points(Re(rts), Im(rts), col="red")
}

theta &lt;- seq(-1,1,.01)*pi
lines(sin(theta), cos(theta), col="black")