Page 253 - Electric Drives and Electromechanical Systems

250 Electric Drives and Electromechanical Systems

If we apply a unit step input to the system, we can write, from Table 10.1, that:
z
UðzÞ¼
1 z
Giving,
ð0:5z þ 0:5Þ z
XðzÞ ¼ 2
ðz 0:8z þ 0:4Þ ð1 zÞ
Applying the Final Value Theorem yields:
1
x ss¼ lim 1 z XðzÞ¼ 1:66
z/1
This value gives a steady state value of 1.66, which makes it unsuitable for many applications
in robotics and machine tools due to the overshoot.
If the impulse response of magnitude 10 is applied to the system, from Table 10.1:
UðzÞ¼ 10
Giving,

ð0:5z þ 0:5Þ
XðzÞ¼ 2 10
ðz 0:8z þ 0:4Þ
Applying the Final Value Theorem yields:

x ss¼ lim 1 z 1 XðzÞ¼ 0
z/1
These values can be confirmed by using a suitable control analysis package, that gives the
following responses to either an impulse or unit step:

Impulse Response Step Response
20 2
1.8
15 1.6
1.4
10 1.2
Amplitude 5 Amplitude 0.8 1

0.6
0 0.4
0.2
-5 0
0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2
Time (seconds) Time (seconds)
Response to an impulse of magnitude ten. Response to a unit step.

nnn