Complex pole locations arise in conjugate pairs, $\sigma\pm j\omega$. We may analogize the product of a complex conjugate pair of poles to a second order transfer function denominator.

$$ s^2+2\sigma s+\sigma^2+\omega_d^2\approx s^2+2\zeta\omega_ns+\omega_n^2\Rarr\sigma=\zeta\omega_d,~\omega_d=\omega_n\sqrt{1-\zeta^2} $$

For second order systems (and systems which we may approximate as second order) there are various constraints we can apply to the pole locations in order to prescribe certain performance in the time-domain response.

For a second order system with no zeros, the rise time, $t_r$, can be roughly prescribed. Peak time, $t_p$, and overshoot, $M_p$, and settling time, $t_s$, to a certain error percentage, $\delta$, can be prescribed more precisely.

$$ t_r\approx\frac{1.8}{\omega_n},~~~~ t_p=\frac{\pi}{\omega_d},~~~~ M_p=e^{-\pi\zeta/\sqrt{1-\zeta^2}},~~~~ t_s=\frac{-\ln\delta}{\zeta\omega_n} $$

However, the presence of even a single zero may prohibit the above design relationships from holding with accuracy. Considering the below transfer function, a second order with a zero, it is required that $\alpha\gg 1$ in order for standard second order behavior to be preserved.

$$ G(s)=\frac{s/(\alpha\zeta\omega_n)+1}{(s/\omega_n)^2+2\zeta (s/\omega_n)+1} $$

The presence of an additional pole can also change the dynamics, as shown in the following figure. This additional pole will increase the rise time significantly unless it is more than four fold the real component of the complex poles, in which case it is negligible.

$$ G(s)=\frac 1{(s/(\alpha\zeta\omega_n)+1)((s/\omega_n)^2+2\zeta(s/\omega_n)+1)} $$