082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

666 Chapter 21 The Laplace transform
Q d (s)
+
–
K
0.5
s(s + 1)
Controller Servo-motor +
drive amplifier
Q a (s)
Figure21.16
Position control system.
anewdesiredpositionisrequestedthenthesystemwilltakesometimetoattainthis
new position. The engineer can choose a value of the controller gain to obtain the
besttypeofresponsefromthecontrolsystem.Wewillexaminetheeffectofvarying
K on the response of the servo-system.
We can use Rules 1 and 2 to obtain an overall transfer function for the system.
The forward transfer function is
G(s) = 0.5K by Rule 1
s(s +1)
The overall transferfunction a (s) isobtained by Rule 2 withH(s) = 1.So,
d
(s)
0.5K
a
(s)
d
(s) = s(s +1) 0.5K
1 + 0.5K(1) =
s(s +1) +0.5K = 0.5K
s 2 +s+0.5K
s(s +1)
Let us now examine the effect of varying K. We will consider three values,
K = 0.375,K = 0.5,K = 5, and examine the response of the system to a unit step
input ineach case.
ForK = 0.375
a
(s)
d
(s) = 0.1875
s 2 +s+0.1875
With d
(s) = 1 s , then
a
(s) =
usingpartialfractions. So,
0.1875
(s 2 +s+0.1875)s = 1 s + 0.5
s +0.75 − 1.5
s +0.25
θ a
(t) = 1 +0.5e −0.75t −1.5e −0.25t t 0
This is shown in Figure 21.17. Engineers usually refer to this as an overdamped
response. The response does notovershoot the final value.
ForK = 0.5
a
(s)
d
(s) = 0.25
s 2 +s+0.25
a
(s) =
0.25
s(s 2 +s +0.25) = 1 s − 1
s+0.5 − 0.5
(s +0.5) 2
θ a
(t) = 1 −e −0.5t −0.5te −0.5t t 0