Image Processing Reference
In-Depth Information
1-
z
-μ
G
p
(
z
)
-
t
d
(
z
)
t
c
(
z
)
+
+
G
p
(
z
)
z
-μ
G
pi
(
z
)
e
(
z
)
u
(
z
)
e
s
(
z
)
-
FIGURE 9.34
Block diagram of the feedback system with Smith predictor.
m
of the open-loop TC system with a dispenser lag,
cycles, then the closed-loop
transfer function is given by
G
pi
(
z
)
G
p
(
z
)
z
m
1
G
c
(
z
) ¼
(
9
:
89
)
þ
G
pi
(
z
)
G
p
(
z
)
Equation 9.89 implies that the closed-loop system response of a PI controller designed
by ignoring time delay, as in Section 9.11.2, is governed by the transfer function of the
controller-plus-undelayed plant while the actual response of the controller is subjected
to a time delay of
m
cycles. With the Smith predictor, input to the PI controller is
z
m
) G
p
(z) as shown in Figure 9.34.
To implement the Smith predictor, the block diagram of Figure 9.34 is converted
to difference equations as follows.
First, let us write the important equations of the system with Smith predictor and
PI controller in the z-domain, which is easy to interpret. From Figures 9.31 and 9.34,
we can write
modi
ed by a feedback term (1
z
m
e
s
(
z
) ¼
e
(
z
)
ð
1
Þ
G
p
(
z
)
u
(
z
)
(
9
:
90
)
z
z
u
(
z
) ¼
K
p
e
s
(
z
) þ
1
K
i
e
s
(
z
)
(
9
:
91
)
u
(
z
) ¼ t
u
(
z
)
(
9
:
92
)
A useful form to implement in the computer is the difference equation form which
contains a sequence of numbers iterated from each measurement-process-update
cycle. To do this, the transfer function, G
p
(z), is replaced by g
=
1) in Equation
9.90 and inverse z-transform is applied on the resulting equation to yield the
difference equation form:
(z
e
s
(
k
) ¼
e
s
(
k
1
) þ
e
(
k
)
e
(
k
1
)
gu
(
k
1
) þ
gu
(
k
m
1
)(
9
:
93
)
u
(
k
) ¼
u
(
k
1
) þ (
K
p
þ
K
i
)
e
s
(
k
)
K
p
e
s
(
k
1
)
(
9
:
94
)
u
(
k
) ¼ t
u
(
k
)
(
9
:
95
)
e
(
k
) ¼
t
d
(
k
)
t
c
(
k
)
(
9
:
96
)
Equations 9.93 through 9.96 are recursive. For computation, we
first start with
Equations 9.96 and 9.95 with known desired TC and measured TC values for a
known dispense process. After this, we compute e
s
(k) and
u(k) from Equations 9.93
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