Page 117 - Mechatronics with Experiments

CLOSED LOOP CONTROL 103

.
X = 0 f Wind Constant
d
K m disturbance force
D
+ f d (t)
-
.
X

Wind Constant
m disturbance force
c f d (t)

x(t)
f d (t) 1 K
1 1 D
1.0 K 2
D

FIGURE 2.45: Mass force system under velocity feedback control, and response to a constant
disturbance.

is to reject the disturbances and reduce the steady-state error, as will be shown in the next
section.

2.12.4 PI Control

Let us consider the mass-force system under a proportional plus integral (PI) control. The
control algorithm is

t
u(t) = K (x (t) − x(t)) + K (x ( ) − x( ))d
p d I ∫ 0 d
and its Laplace transform is

1
u(s) = K (x (s) − x(s)) + K (x (s) − x(s))
p d I d
s
( )
K I
= K + (x (s) − x(s))
p d
s
Let’s take the Laplace transform of the mass-force system and substitute the PI controller
for u(s)
̈ x = u(t) − w (t)
d
( )
K I
2
s x(s) = u(s) − w (s) = K + (x (s) − x(s)) − w (s)
p
d
d
d
s