7. Response of Second Order System to Unit Step Input - Underdamped System

\(0<\zeta < 1\):

\[Y(t) = K_p\left[1-\frac{1}{\sqrt{1-\zeta^2}}e^{-\zeta t/\tau}\sin(\omega t + \phi) \right]\] where \[\omega = \frac{\sqrt{1-\zeta^2}}{\tau} \qquad \text{ and } \qquad \phi=\tan^{-1}\left[\frac{\sqrt{1-\zeta^2}}{\zeta} \right]\]

Although the response is initially faster and reaches its ultimate value quickly, it does not stay there, but it starts oscillating with progressively decreasing amplitude.

The oscillatory behavior becomes more pronounced with smaller values of the damping factor, \(\zeta\).

Almost all the underdamped responses in a chemical plant are caused by the interaction of the controllers, with the process unit they control.

If \(\zeta = 0\), then such a second-order system is marginally stable in that the response is of constant amplitude in time. This is the undamped case.

If \(\zeta < 0\), then such a second-order system is unstable and the response grows in time without bound.