Step Response: Rise Time, Overshoot, Settling

A quick way to judge any controller is to ask for a sudden change and watch what happens. The step response tells you how fast the system gets there, how far it overshoots, and how long it takes to settle down.

Natural frequency ωn Damping ratio ζ

The second-order model

Many controlled systems behave like a mass, a spring, and a damper, described by a natural frequency $\omega_n$ and a damping ratio $\zeta$:

$$\ddot y + 2\zeta\omega_n\,\dot y + \omega_n^2\, y = \omega_n^2\, u \qquad \text{overshoot} = e^{-\pi\zeta/\sqrt{1 - \zeta^2}} \qquad t_s \approx \frac{4}{\zeta\,\omega_n}$$

What to notice

Rise time is how long the output takes to go from 10 to 90 percent of the step. Overshoot is how far it passes the target, as a percent. Settling time is when it finally stays within 2 percent. Low damping gives a fast rise but big overshoot and ringing. Damping above 1 gives no overshoot but a slow, sluggish rise. A damping ratio near 0.7 is a common compromise. Raising $\omega_n$ speeds everything up without changing the shape.