The unit-step response for a second-order system (Y(s) / U(s)=omega_{n}^{2} /left(s^{2}+2 zeta omega_{n} s+omega_{n}^{2} ight)) is given by [y(t)=1-mathrm{e}^{-zeta omega_{mathrm{n}} t}left[cos left(omega_{mathrm{d}} t ight)+frac{zeta}{sqrt{1-zeta^{2}}} sin

The unit-step response for a second-order system \(Y(s) / U(s)=\omega_{n}^{2} /\left(s^{2}+2 \zeta \omega_{n} s+\omega_{n}^{2}\right)\) is given by

\[y(t)=1-\mathrm{e}^{-\zeta \omega_{\mathrm{n}} t}\left[\cos \left(\omega_{\mathrm{d}} t\right)+\frac{\zeta}{\sqrt{1-\zeta^{2}}} \sin \left(\omega_{\mathrm{d}} t\right)\right]\]

Prove that the relationship between the overshoot \(M_{p}\) and the damping ratio is

\[M_{p}=\mathrm{e}^{-\pi \zeta / \sqrt{1-\zeta^{2}}}\]