SKEDSOFT
Second-order systems settling Time:
Fig: 1Pair of envelope curves for the unit step response curve of the system Fig:2
Here the settling time is obtained in terms of 
. The system is assumed to be underdamped.
Considering the equation,
For an underdamped second-order system, the transient response is obtained as:
The curves 
are the envelope curves of the transient response for a unit-step input. The response curve c(t) always remains within a pair of the envelope curves, as shown in Figure 1. The time constant of these envelope curves is 
. The speed of decay of the transient response depends on the value of the time constant . For a given ωn, the settling time ts is a function of the damping ratio 
From Figure 2, we see that for the same ωn and for a range of between 0 and 1 the settling time ts for a very lightly damped system is larger than that for a properly damped system. For an overdamped system, the settling time ts becomes large because of the sluggish start of the response.
The settling time corresponding to a ±2% or ±5% tolerance band may be measured in terms of the time constant 
from the curves of Figure 2 for different values of 
