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┣━━━━━━━━━━━━━╋━━━━━━━┫
┃ - "- ┃ (J- ┃
┃ 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)
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