Page 350 - Modern Control Systems

324 Chapter 5 The Performance of Feedback Control Systems

The steady-state tracking error for a step input of magnitude A is thus given by

(5.26)

Hence, the steady-state error for a unit step input with one integration or more,
N a 1, is zero because

v
'" isSi + *rW(.v' nft)

Av' v
= 0. (5.27)
=
- 0 5 * + KUz f/UPk
Ramp Input. The steady-state error for a ramp (velocity) input with a slope A is

2
s(A/s )
lim lim = lim- (5.28)
.v-ol + G c(s)G(s) s-os + sG c{s)G(s) .S-M)SG C(S)G(S)

Again, the steady-state error depends upon the number of integrations, N. For a
type-zero system, AT = 0, the steady-state error is infinite. For a type-one system,
N = 1, the error is

A
e« = lim
*-i>sKjl( S + z,)/[sll(s + Pk)Y

or
A
Cc c (5.29)
KU^/UPk K»

where K v is designated the velocity error constant. The velocity error constant is
computed as

K v= \im sG c(s)G(s).
s—*\)
When the transfer function possesses two or more integrations, N a 2, we obtain a
steady-state error of zero. When JV = 1, a steady-state error exists. However, the
steady-state velocity of the output is equal to the velocity of input, as we shall see
shortly.

Acceleration Input. When the system input is r(t) = Ar/2, the steady-state error is

s(A/f) A
e« = lim lim 2 (5.30)
v-ol + G e(s)G(s) s->vs G c(s)G(s)