Page 655 - Mechanical Engineers' Handbook (Volume 2)

646 Controller Design

transducers, they offer more flexibility in modifying the servo dynamics than does frequency
compensation alone. The following sections discuss the merits of feeding back derivatives
of the controlled variable, feeding back variables dynamically different than the controlled
variable, and nonelectronic mechanizations of inner loops.

4.1 Derivatives of the Controlled Variable
Section 3 illustrates the benefits of lead compensation but also shows that its effectiveness
is limited by the high-frequency gain amplification of the compensator itself. This problem
can be alleviated by the use of transducers that directly measure derivatives of the controlled
variable. To illustrate the potential benefits, consider again the example of Fig. 6b with
additional feedback of the first and second derivatives of the controlled variable, as illustrated
in Fig. 22a. As shown in Fig. 22b, the three feedbacks can be mathematically combined into
a single 1oop having a pure second-order pair of zeros, without the added lag normally
associated with lead compensators. The result is that much higher forward-loop gains can
be achieved for a given level of closed-loop stability. Of course, each transducer will intro-
duce higher frequency dynamic effects that will eventually limit the maximum forward-loop
gain, but these effects are usually less restrictive than those imposed by electronic lead
compensators. Note that the natural frequency and damping ratio of the feedback zeros in
Fig. 22 are determined by the magnitude of the derivative feedback gains relative to the
primary feedback gain, which is 1.0 in this case.
There are a variety of uses for derivative feedback loops, including improved closed-
loop accuracy, bandwidth, and stability, as well as reduced sensitivity to changes in system
parameters. These uses can be illustrated with the aid of Figs. 23 and 24, which show the
effects of various sets of feedback zeros on the closed-loop dynamics of Fig. 22b. Figure 23
gives root loci for the different zero locations. Because electrical noise and excitation of
high-frequency dynamic modes usually limit the gain in the highest derivative loop, a
forward-loop gain is selected for each locus that holds K K ƒ2 constant. Closed-loop fre-
u2
quency responses are then given in Fig. 24. It can be seen that closed-loop damping is
improved in all cases. Placement of the feedback zeros at high frequency (low derivative
feedback gains) yields high closed-loop bandwidth, but the second-order poles can easily be
destabilized by a reduction in forward-loop gain or the presence of the inevitable higher
order modes described in Section 3.3. On the other hand, low-frequency feedback zeros
(high derivative feedback gains) result in lower closed-loop bandwidth but offer greatly
reduced gain sensitivity and are little affected by high-frequency modes. Also in the latter
case, the closed-loop response is dominated by the low-frequency second-order poles, which
are nearly the same as the feedback zeros. Therefore, the closed-loop characteristics do not
, , ). A good compro-
vary significantly with large changes in the plant parameters (K u2 2 2
mise between bandwidth and parameter sensitivity is to set ƒ2 approximately equal to .
2
Comparison of Figs. 23 and 24 with Figs. 18 and 19 show that derivative feedback
offers better stability and more flexibility than lead compensation, particularly when the
forward loop has an inherent integration. In addition, higher forward-loop gains can generally
be used with derivative feedback, which improves the static accuracy of the system. However,
forward-loop lead compensation can often produce higher closed-loop bandwidth if the
forward-loop integration is electronic in nature, as illustrated in Fig. 18.
A useful way to optimize the system of Fig. 22a is to first establish rough estimates of
the feedback and forward-loop gains using the combined feedback approach of Figs. 22b
and 23. It is often helpful to include a first-order approximation of the phase lags caused by
the higher order modes [Eq. (20)]. Once the approximate gains are established, the closed-