Underdamped Response Characteristics

Topics: Feedback Control Systems

Maximum Overshoot

The maximum overshoot for underdamped systems is a measure of the maximum amount by which the system's output exceeds the steady-state value. It is typically expressed as a percentage of the steady-state value

$$M_p = \frac{1}{y(\infty)} \Big[(\max \, y(t)) - y(\infty) \Big]$$

For a second-order system in standard form, the maximum overshoot can be written as a function of the damping factor

$$M_p = \exp \Bigg[ {\frac{-\pi \zeta}{\sqrt{1-\zeta^2}}} \Bigg]$$

Maximum overshoot is an important design metric because it is desired to limit it in the step response depending on the system characteristics.

Using the above equation, the required damping factor for a desired maximum overshoot is given by

$$\zeta = \frac{| \ln M_p|}{\sqrt{\pi^2 + (\ln M_p)^2}}$$

Time Constant

The time constant governs the rate of decay of the exponential envelop around the step response of the underdamped system. It is defined as

$$\tau=\frac{1}{\zeta \omega_n}$$

Settling Time

The settling time is the total time required for the system's step response to reach and stay within a specific error band (tolerance) around the steady-state value. For an underdamped system, we have the following approximations:

• 5% tolerance band: $t_s \approx 3\tau = \frac{3}{\zeta \omega_n}$
• 2% tolerance band: $t_s \approx 4\tau = \frac{4}{\zeta \omega_n}$
• 1% tolerance band: $t_s \approx 5\tau = \frac{5}{\zeta \omega_n}$

Peaking Time

The peaking time is the time required for the system to reach its first peak, i.e., its maximum overshoot value:

$$t_p = \arg \Big( \max \, y(t) \Big)$$

For an underdamped second-order system:

$$t_p = \frac{\pi}{\omega_d} = \frac{\pi}{\omega_n \sqrt{1-\zeta^2}}$$

Rise Time

The rise time is the time required for the system to reach the steady-state value for the first time

$$t_r = \min \{t \, | \, y(t) = y(\infty)\}$$

The equation of the rise time depends on the phase angle defined as $\theta=\sin^{-1} \zeta$. In other words, $\theta$ is the angle between the imaginary axis and the line connecting the origin and the pole.

$$t_r = t_p \Big( \frac{1}{2} + \frac{\theta}{\pi} \Big) = \frac{\pi}{\omega_n \sqrt{1-\zeta^2}} \Big( \frac{1}{2} + \frac{\sin^{-1} \zeta}{\pi} \Big)$$