Page 119 - Mechatronics with Experiments

CLOSED LOOP CONTROL 105
After a few algebraic manipulations, the transfer function between position, desired posi-
tion, and disturbance force is found as

K s + K I s
p
x(s) = x (s) − w (s)
d
d
3
3
s + K s + K I s + K s + K I
p
p
Consider the case that the commanded position is zero, x (t) = 0 and there is a constant
d
1
step disturbance, w (s) = . Any non-zero response due to the disturbance would be an
d
s
error.
s 1
x(s) =−
3
s + K s + K s
p
I
1
x(s) =−
3
s + K s + K I
p
3
If Δ (s) = s + K s + K has stable roots, the response of the system will be zero despite
I
p
cls
a constant disturbance.
Using the final value theorem,
lim e(t) = e (∞) = lim se(s)
ss
t→∞ s→0
1
= lim s
3
s→0 s + K s + K
p I
e (∞) = 0
ss
The steady-state error due to a constant disturbance is zero under the PI type control. If
there is no integral control action, K = 0, the steady-state error would have been
I
s 1
e(s) =
s(s + K ) s
2
p
1 1
lim se(s) = = → ≠ 0
2
s→0 s + K p K p
Therefore it is clear that it is the integral of position error used in feedback control which
enables the control system to reject the constant disturbance and keep x(t) = x (t) in steady
d
state. The transient response to a step command change in desired position under no
disturbance condition,
w (t) = 0
d
x (t) = 1(t)
d
K s + K I
p
x(s) = x (s)
d
3
s + K s + K
p I
The closed loop system has a zero at
K I
−
K
p