Time Response

For a standard underdamped second-order system ($0 < \zeta < 1$):

$M_p = e^{-\zeta\pi/\sqrt{1-\zeta^2}} \qquad t_p = \frac{\pi}{\omega_n\sqrt{1-\zeta^2}} \qquad t_s \approx \frac{4}{\zeta\omega_n}\ (2\%)$

$\zeta = 0.7$ gives about 4.6 % overshoot — a common design target.

Steady-state error depends on system type (number of integrators in the loop) and the static error constants: for a step input $e_{ss} = \dfrac{1}{1 + K_p}$, for a ramp $e_{ss} = \dfrac{1}{K_v}$. Adding an integrator removes the steady-state error to a step.