Figure 10-4. (cont'd) the continuous curve was calculated by using the exact
equation for the response of a second-order system to a step command (Ref. 27). The example case shown is for a sec- ond-order system with an undamped natural frequency wn = 20 rad/s and a damping ratio = 0.5. In Fig. 10-4(A) the time step T= 0.02s; this gives a sampling rate of about 16 times the damped natural frequency. At this sampling rate the root-matching difference equation gives good results.
When the sampling rate is reduced to about six times the damped natural frequency, the aliasing effect begins to introduce errors as shown in Fig. 10-4(B). At a sampling rate of only three times the damped natural frequency, the aliasing error is pronounced, as shown in Fig. 10-4(C).
To obtain the match between the difference equation solutions and the exact solution, shown in Fig. 10-4, the time after initiation of the step input is calculated by t = nT assuming n = 0 at the instant the step is initiated
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