Overdamped System

Topics: Feedback Control Systems

An overdamped system corresponds to $\zeta>1$ in the standard form. Similarly to the critically damped case, the system exhibits no overshoot nor oscillations, but the step response rises slower.

Step Response

$$y(t) = 1 + \frac{0.5}{\zeta \sqrt{\zeta^2-1} + \zeta^2-1} e^{-\left(\zeta+\sqrt{\zeta^2-1}\right)\omega_nt} + \frac{0.5}{-\zeta \sqrt{\zeta^2-1} + \zeta^2-1} e^{-\left(\zeta-\sqrt{\zeta^2-1}\right)\omega_nt}$$

Poles

An overdamped system has 2 real poles:

• a slow pole, which is closer to the origin and decays slower $s=-\zeta \omega_n + \omega_n \sqrt{\zeta^2-1}$
• a fast pole, which is farther away from the origin and decays quicker $s=-\zeta \omega_n - \omega_n \sqrt{\zeta^2-1}$