Automating Manufacturing Systems - Process Control and ...

plc pid - 25.9
Summation Block:
e = C desired
– C actual
(1)
Controller:
V c
=
K p e
(2)
Current Amplifier:
Servomotor:
Ball Screw:
Encoder:
V m
= V c
----
d
⎝
⎛ dt⎠
⎞ ⎛ K 2 ⎞ K ω + ⎜---------
m ⎟ ω = ⎛------
m
⎝ JR ⎠
⎝JR⎠
ω = ----θ
d
dt actual
x
=
C actual
θ
------------- actual
TPI
= PPR( θ actual )
⎞ Vm
(3)
(4)
(5)
(6)
(7)
Figure 25.9
A Servomotor Feedback Controller
The system equations can be combined algebraically to give a single equation for
the entire system as shown in Figure 25.10. The resulting equation (12) is a second order
non-homogeneous differential equation that can be solved to model the performance of the
system.
----
d
⎝
⎛ dt⎠
⎞ 2 θ ⎛ K 2 ⎞
--------- m actual ⎜ ⎟
⎛----
d ⎞ K
(21.4), (21.5) + θactual = ⎛------
m ⎞ Vm
(21.8)
⎝ JR ⎠
⎝dt⎠
⎝JR⎠
(21.2), (21.3) = K p e
(21.9)
V m
(21.1), (21.9) V m = K p ( C desired – C actual )
(21.10)
d
⎝
⎛ ----
dt⎠
⎞ 2 θ ⎛ K 2 ⎞
--------- m actual ⎜ ⎟
⎛----
d ⎞ K
(21.8), (21.10) +
m
⎝ JR ⎠
⎝dt⎠
θactual = ⎛
⎝
------⎞ JR⎠
Kp ( C desired
– C actual
) (21.11)
(21.7), (21.11)
d
⎝
⎛ ----
dt⎠
⎞ 2 θ ⎛ K 2 ⎞
--------- m d
+ actual ⎜ ⎟
⎛----
⎝ JR ⎠
⎝dt⎠
⎞ θactual
K m
= ⎛
⎝
------⎞ JR⎠
Kp ( C desired
– PPRθ actual
)
d
⎝
⎛ ----
dt⎠
⎞ 2 θ K 2
+ ⎛ ⎞
--------- m actual ⎜ ⎟
⎛----
d ⎞ K m ( PPR)K p
⎝ JR ⎠
⎝dt⎠
θactual + ⎝
⎛ ------------------------------
JR ⎠
⎞ θactual =
(21.12)
⎛K p K
------------- m ⎞
⎝ JR ⎠ Cdesired
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