Page 665 - Mechanical Engineers' Handbook (Volume 2)

656 Controller Design

Solving these equations after the appropriate substitutions, the feedforward parameters are
K 0.90, 1.11 , 0.1 . Using these parameters, U /U frequency and step
1
8
8
7
8
c
1
responses are computed and presented in Figs. 33 and 34, along with the corresponding
U /U responses. In this example, the use of feedforward techniques offers a substantial
c
6
improvement in system bandwidth, by comparison with a prefilter alone. Of course, the
feedforward scheme suffers from sensitivity to forward-loop gain variations similar to those
of the prefilter scheme, as illustrated in Figs. 29 and 30.
As previously mentioned, feedforward techniques can also reduce following errors. In
general, the following error U can be calculated as
e
U
U e 1 (30)
1
˙
U s U
c c
For the example of Fig. 32, in which the closed-loop transfer function is determined by the
feed-forward pole, the following error is determined from

U 8e 1 1 1 7
˙
U c s s 1 s 1 (31)
7
7
˙
For steady command rates, the error is U . Without feedforward, the following error can
7
c
be determined from Eq. (8):

U 1e 1 1 1
˙
2
U c s ( /K )s (1/K )s 1
u1
u1
1
( /K )s (1/K )
1 u1 u1 (32)
2
/K )s (1/K )s 1
u1
1
u1
˙
For steady command rates, the error is U c /K . For this example, 0.1/K . In this case,
7
u1
u1
the proper use of feedforward has reduced the following errors by a factor of 10.
6 PID CONTROLLERS
A very popular form of controller is called PID (proportional-integral-differential). It is very
simple in concept and is relatively easy to mechanize. Many essays have been written that
describe rules of thumb for ‘‘tuning’’ the controller. Unfortunately these tuning procedures
can be rather tedious and are usually applicable for only very simple actuator–load dynamics.
The purpose of this section is to offer a unified rationale for applying PID controllers that
is useful in synthesizing a control system. This rationale also provides insight for system-
atically adjusting PID parameters on actual hardware.
6.1 Equivalence to Frequency Compensation
The basis for the ensuing discussion is that a PID controller is simply a particular form of
forward-loop frequency compensation. This can be seen from the generalized controller
shown in Fig. 35. Note that the differential path is filtered to limit the high-frequency am-
plitude ratio. Combining the parallel paths of Fig. 35, a single transfer function for the
controller can be written: