finding zeta and wn for a 4th order system?
Two possible methods of several:


A 2nd order CLTF model of a higher order system (viz. the \$\small\zeta\$ and \$ \omega _n\$ values), can be derived from the open-loop gain cross-over frequency, and phase margin, thus:


$$\small\left(\frac{\omega _c}{\omega _n}\right)^2 \approx \sqrt{(4\zeta ^4 +1)}\:-2\zeta^2 $$

and

$$\small\zeta \approx 0.01\times PM $$

where \$\small PM\$ is the phase margin in degrees, and \$\small \omega _c\$ is the frequency at which the open loop gain crosses \$\small 0\:dB\$ (i.e. the frequency at which the phase margin is defined).

This assumes that \$ \omega _c\$ (and consequently the phase margin) exists.


Dominant poles: the dominant poles are located at \$\small s=0\$ and \$\small s=-5\$, since the other two poles are at least three times further from the origin than the \$\small (s+5)\$ pole. Hence ignoring the less dominant poles gives the OLTF:


$$\small G_c (s)=\frac{K(s+7)}{s(s+5)} $$

with CLTF:

$$\small G(s)=\frac{K(s+7)}{s(s+5)+K(s+7)}=\frac{K(s+7)}{s^2 +(K+5)s+7K} $$