
Negative feedback systems 125
with L
′
(0) = 1. Now consider a step reference input at time t = 0 with
amplitude ¯r. The steady-state error can be computed by a pplying the initial
value theorem
∗
, which yields
e
ss
= ¯r − lim
t→∞
y(t)
=
1 − lim
s→0
T (s)
¯r
=
1 − lim
s→0
C/s
n
1 + C/s
n
¯r = lim
s→0
s
n
s
n
+ C
¯r .
Hence, if L(s) has no pole at the origin (i.e. L(s) contains no integrators), the
steady-state error is
1
1 + C
¯r.
If, however, n ≥ 1, then e
ss
= 0 regardless of the values of C o r ¯r — an
extremely robust level of p erformance! This is the basis for the use of integral
control in many industrial feedba ck systems. By using the controller to in-
troduce a n integrator ...