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┃    p3  p2                ┃              ┃
┗━━━━━━━━━━━━━┻━━━━━━━┛
Fig. 5.25    System operating at the desired pointA.
LINEAR SYSTEMS, THEORY, AND DESIGN:A BRIEF REVIEW       489
a) Integral compensator
┏━━━━━━━━━━━┳━━━━━┓
┃                jco A ┃ I        ┃
┃          \           ┃/7        ┃
┃    \\A               ┃          ┃
┃-~:-. -.-\/<          ┃          ┃
┣━━━━━━━━━━━╋━━━━━┫
┃";; p:~               ┃       a  ┃
┃                      ┃~         ┃
┗━━━━━━━━━━━┻━━━━━┛
compensator
┏━━━━━━━━━━━━━━━━┳━━━━━┓
┃                         J(D J  ┃ l        ┃
┃                                ┃          ┃
┃                                ┃7         ┃
┣━━━━━━━━━━━━━━━━╋━━━━━┫
┃'13 /2'~                        ┃       a  ┃
┃                                ┃~         ┃
┗━━━━━━━━━━━━━━━━┻━━━━━┛
c) Root-Iocus of PI
compensator
Fig.5.26 Proportional-integralcompensator.
""t
 jt
 t
 'J
   !
j
 .
'!
490            PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Fig. 5.27    Implementation of a PI compensator.
The schematic implementation of a PI compensator is shown in Fig. 5.27 with
      Gc(s)=k,+(k)    . (5.164)
                                        = k,[l+~(1)]                       (5.165)
In this implementaion,ki - 1 and k2 - Zc. Note that the first term on the right-hand
side is the "proportional" part and the second term is the "integral" part.
    Generally, the derivative compensation is used when a simple gain adjustment
alone cannot give the desired transient response of the closed-loop system. This
concept is illustrated in Fig. 5.28. In Fig. 5.28a, a simple gain adjustment is suffi-
cient because the root-locus passes through the desired operating point A. However,
in Fig. 5.28b, the root-locus cannot pass through A fo~ any value of the gain k.
The addition of'a compensator zero close to the origin modifies the root-locus so
thatit is made to pass through point A as shown in Fig. 5.28c. Such a compensator
that produces a zero in the forward path is called a proportional-derivative (PD)
compensator.
     The transfer function of a PD compensatoris of the form
Gc(s) : s +Zc
(5.166)
which is essentially the sum of a differentiator s and a gain zc.
     The implementation of a PD compensator is schematically shown in Fig. 5.29.
　
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