Compensator for a specific damping ratio ζ
If you assume the plant has no zeros (big assumption, but we don't have anything else to go on) then the controller below will get you pretty close:

\begin{equation}
K(s) = \frac{2 \zeta s}{s/\omega_{BIG} + 1}
\end{equation}

How close you get depends on where you place the pole in the expression above.  For 10 rad/s, I got pretty close.

Some Scilab code:

s = %s

G = syslin('c', 1/(s^2 + 1))

K = syslin('c', 0.6*s/(s/10 + 1))

T=GK/(1+GK)

plzr(T)