Page 429 - Linear Control System D'Azzo (5th Edition)

Root-Locus Compensation: Design 411

Equation (10.51) shows that the transducer plus the unity feedback have intro-
duced the ideal proportional plus derivative control ð1 þ KtsÞ that can also be
achieved by a cascade compensator. The results should therefore be qualita-
tively the same.

Partitioning the characteristic equation (10.51) to obtain the general
format F(s) ¼ À1, which is required for obtaining a root locus, yields

Gx ðsÞH1ðsÞ ¼ KKtðs þ 1=KtÞ ¼ À1 ð10:52Þ
sðs þ 1Þðs þ 5Þ

The root locus is drawn as a function of KKt. Introduction of minor loop rate
feedback results in the introduction of a zero [see Eq. 10.52]. It therefore has
the effect of moving the locus to the left and improving the time response. This
zero is not the same as introducing a cascade zero because no zero appears in the
rationalized equation of the control ratio

CðsÞ K ð10:53Þ
¼

RðsÞ sðs þ 1Þðs þ 5Þ þ KKtðs þ 1=Kt Þ

The rate constant Kt must be chosen before the root locus can be drawn for
Eq. (10.52). Care must be taken in selecting the value of Kt. For example, if Kt ¼1
is used, cancellation of (s þ 1) in the numerator and denominator of Eq. (10.52)

yields

sðs K 5Þ ¼ À1 ð10:54Þ
þ

and it is possible to get the impression that C(s)/R(s) has only two poles. This is

not the case, as can be seen from Eq. (10.53). Letting Kt ¼ 1 provides the
common factor s þ1 in the characteristic equation so that it becomes

ðs þ 1Þ½sðs þ 5Þ þ KŠ ¼ 0 ð10:55Þ

One closed-loop pole has therefore been made equal to s ¼ À1, and the other
two poles are obtained from the root locus of Eq. (10.54). If complex roots are
selected from the root locus of Eq. (10.54),then the pole s ¼ À1is dominant and
the response of the closed-loop system is overdamped (see Fig. 10.4d).

The proper design procedure is to select a number of trial locations for
the zero s ¼ À1=Kt, to tabulate or plot the system characteristics that result,
and then to select the best combination. As an example, with z ¼ 0:45 as the
criterion for the complex closed-loop poles and Kt ¼ 0:4, the root locus is
similar to that of Fig. 10.31 with A ¼ 1. Table 10.3 gives a comparison of the
original system with the rate-compensated system. From the root locus of
Eq. (10.52) the roots are s1ó2 ¼ À1 Æ j2, and s3 ¼ À4, with K ¼ 20. The gain

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