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Second-Order System and Overshoot

A second-order system with natural frequency $\omega_n$ and damping ratio $\zeta$ may oscillate before settling, depending on how lightly damped it is. For $0<\zeta<1$ the system oscillates with an overshoot that shrinks as $\zeta$ grows, while $\zeta\ge1$ gives no oscillation at all.

G(s)=\dfrac{\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2}

second-order standard form

M_p=e^{-\pi\zeta/\sqrt{1-\zeta^2}}

overshoot for 0<\zeta<1

Symbols

$\omega_n$	undamped natural frequency	rad/s
$\zeta$	damping ratio
$M_p$	relative overshoot

Example

$\omega_n=6$ , $\zeta=0.3$ :

$M_p=e^{-\pi\cdot0.3/\sqrt{1-0.09}}\approx0.37$ , i.e. about 37%.

Low

\zeta

gives large overshoot and heavy oscillation;

\zeta\ge1

gives no oscillation at all.

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Part of Control Engineering: Laplace and transfer functions.