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The transfer function of such a compensator can also be written in the following
form:
Gc(s) : ki +k2S = k2 (;c + s) (5.167)
The first term on the right-hand side is the "proportional" part, and the second term
is the "derivative" part. The gains ki and k2 are design variables to be determined
so that the system attains the specified transient response.
LINEAR SYSTEMS, THEORY, AND DESIGN:A BRIEF REVIEW 491
┏━━━━━━━━━━━━━━━┳━━┓
┃ ┃ \ ┃
┣━━━━━━━━━━━━━━━╋━━┫
┃-- -- -J.--- :. -. ┃ ┃
┃ ┃(J ┃
┗━━━━━━━━━━━━━━━┻━━┛
a)
┏━━━━━━━━━━━━━━┳━━━━━━━┓
┃ A\\ Jco I ┃ L ┃
┃ ---X ┃V ┃
┃-p3 '/2'~ ┃'pl a - ┃
┣━━━━━━━━━━━━━━╋━━━━━━━┫
┃ ┃~ ┃
┗━━━━━━━━━━━━━━┻━━━━━━━┛
┏━┳━━━━━━┓
┃ ┃Compensatc ┃
┃ ┃/ ┃
┃ ┃'pl ┃
┣━╋━━━━━━┫
┃ ┃ ┃
┗━┻━━━━━━┛
c)
Zero
Fig.5.28 Concept of proporhonal-derivative compensation.
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492 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Fig.5.29 Implementation ofPD compensator.
5.9.3 LeadVLag Compensator
The implementation of a PI or a PD compensator requires active elements.
Instead, we can use passive networks to achieve nearly the same objective. A
passive network usually produces a pole-zero combination.
The transfer function of a lead/lag compensator is of the form
s+zc
Gc(s) = s+ p
(5.168)
A schematic diagram of a lead/lag compensator is shown in Fig. 5.30. By a suit-
able choice of the locations of pole and zero, we can have either a lag or lead
compensator. For example, if the zero is close to the origin (zc _ 0) and the pole
is farther away to the left of the zero, then it is a lead compensator. On the other
hand, if both the pole and zero are located close to the origin with the pole located
to the right of the zero, then it is a lag compensator.
Note that for both the lead and lag compensators, the pole and the zero are
supposed to be located in the left half of the s-plane.
5.9.4 Proportional-Integraland Derivative Controller
Suppose we want to improve the transient response as well as reduce the steady
state error, then we use the PID (proportional-integral and derivative) controller.
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