Page 496 - Mechanical Engineers' Handbook (Volume 2)

7 Steady-State Performance and System Type 487

Figure 32 Closed-loop configuration.

K GG (0) (87)
p
c
p
as the position error constant. With this definition the steady-state error due to a step input
of magnitude c can be written as
c
e (88)
ss
1 K p
Thus a high value of K , corresponds to a low steady-state error. If the steady-state error is
p
to be zero, then K . The only way that K is if the open-loop transfer function has
p
p
at least one pole at the origin, that is, G G (s) must be of the form
p
c
m
1 i 1 (s z )
i
GG (s) n (89a)
p
c
N
j
s j 1 (s p )
where N 1. When N 1,
m
1 i 1 z i
GG (0) n → (89b)
p
c
0 j 1 p j
Hence, it can be concluded that for the steady-state error due to a step input to be zero, the
open-loop transfer function must have at least one free integrator. The value of N specifies
the type of system. If N 1 it is called a type I, when N 2 it is called a type II system,
and so on. So to get zero steady-state error for a step input, the system loop transfer function
must be at least type I.
7.2 Ramp Input
If the reference input is a ramp ctu (t), where u (t) is the unit step, then from Eq. (85)
s
s
1 c c
e lim s lim (90)
ss 2
s→0 1 GG (s) s s→0 sG G (s)
c
p
p
c
From Eq. (90) for small steady-state errors lim s→0 sG G (s) K must be large, where K v
p
c
v
is the velocity error constant and
c
e (91)
ss
K v
As in the case with the step input, for e to be small, K must be very large. For zero
v
ss
steady-state error with a ramp input, K . From Eq. (90) it is clear that for K ,
v
v
G G (s) must be at least type II. Thus
c
p
m
1 i 1 (s z )
i
K lim sG G (s) lim s 2 n
p
v
c
s→0 s→0 s j 1 (s p )
j