Page 481 - Linear Control System D'Azzo (5th Edition)

464 Chapter 12

FIGURE 12.2 Location of the desired dominant pole.

CASE 2. It may be possible to satisfy the specifications by use of the
following third-order model:

AA ð12:3Þ
MT2ðsÞ ¼ s þ d MT1ðsÞ ¼ ðs þ d Þðs2 þ 2zons þ o2nÞ

The dominant complex poles of Eq. (12.3) are determined in the same manner

as for case 1; i.e., the values of z and on are determined by using the specifica-
tions for Mp , tp , Ts , and Km. Because the model of Eq. (12.1) cannot satisfy
all four specifications, the known effects of the real pole s ¼ Àd in Eq. (12.3)

can be used to try to achieve these specifications. With a CAD program

(see appendixes C and D), various trial values of d are used to check whether

or not the desired specifications can be achieved. Note that for each value of d,
the corresponding value of A2 must be equal to A2 ¼ don2 in order to satisfy the
requirement that e(t)ss ¼ 0 for a step input. As an example, the specifications
Mp ¼1.125, tp ¼ 1.5s, and Ts ¼ 3.0s cannot be satisfied by the second-order
control ratio of the form of Eq. (12.1), but they can be satisfied by

Y ðsÞ ¼ 22 ð12:4Þ

RðsÞ ðs þ 1 Æ j3Þðs þ 2:2Þ

For a third-order all-pole open-loop plant, the model of Eq. (12.3) must be
used. Then, if the model of Eq. (12.1) is satisfactory, the pole Àd is made a
nondominant pole.

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