Page 497 - Mechanical Engineers' Handbook (Volume 2)

488 Closed-Loop Control System Analysis

7.3 Parabolic Input
If the reference input is a parabolic input of the form
t 2
r(t) c u (t)
s
2
then the steady-state error becomes
c
e lim 2 (92)
ss
s→0 sG G (s)
c
p
From Eq. (92) for small steady-state errors
K lim sG G (s) (93)
2
a
c
p
s→0
must be very large, where K is the parabolic error constant or the acceleration error constant.
a
For zero steady-state error due to a parabolic input, K . Therefore the system open-loop
a
transfer function must be at least type III.
Table 4 shows the steady-state errors in terms of the error constants K , K , and K .
p v a
In steady-state performance considerations it is important to guarantee both closed-loop
stability and the system type. It should also be noticed that having a very high 1oop gain
as given by Eq. (89b) is very similar to having a free integrator in the open-loop transfer
function. This notion is useful in regulation-type problems. In tracking systems the problem
is compounded by the fact that the system type must be increased. Increasing the system
type is usually accompanied by instabilities. To illustrate this, consider the following ex-
ample.
Example 10 Synthesize a controller for the system shown in Fig. 33a so that the steady-
state error due to a ramp input is zero.
The open-loop system is unstable. For tracking purposes transform the system to a
closed-loop one as shown in Fig. 33b. To have zero steady-state error for a ramp, the system’s
open-loop transfer function must be at least type II and must be of a form that guarantees
closed-loop stability. Therefore let
G (s)
G (s) 1 (94)
c
s 2

Table 4 Steady-State Error in Terms of Error Constants
t 2
u (t)
Type cu s (t) ctu s (t) 2 s
c
0
1 K p
c
1 0
K v
c
2 0 0
K a
3 0 0 0