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Transfer functions, their definition, development through an example, and their properties in the context of process control systems. Transfer functions are used to relate an input and output in a linear, dynamic model, enabling determination of the output response to any change in the input. The document also covers steady-state gain, order of a TF model, additive and multiplicative properties, and linearization of nonlinear models.
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linear
, dynamic model.
one
input and
one
output:
system
x t
y t
s^
s
The following terminology is used:
y output response “effect”
x input forcing function “cause”
(s
) denote the transfer function between an input,
x
, and an
output,
y
. Then, by definition
( )
(^
) ( ) Y
s
G s
s
where:
(^
)^
(^
)
( )
( )
s
y t
s^
x t ^
Development of Transfer Functions Example: Stirred Tank Heating System
Recall the previous dynamic model, assuming constant liquidholdup and flow rates:
(^
)^
i
dT
wC T
dt
Suppose the process is initially at steady state:
(^
)^
(^
)^
(^
)^
(^
)
i^
i
where
steady-state value of
etc. For steady-state
conditions:
(^
)
i
wC T
Subtract (3) from (1):
(^
)^
(^
)^
(^
)^
i^
i
dT
wC
dt
But,
(^
)^
because
is a constant
d T
dT
dt
dt
Thus we can substitute into (4-2) to get,
(^
)^
i
dT
wC T
dt
where we have introduced the following “
deviation variables
also called “perturbation variables”:
i^
i^
i
Take
of (6):
( )
(^
)^
( )
(^
)^
(^
)
i
sT
s^
t^
wC T
s^
s^
s
where two new symbols are defined:
(^
)
and
wC
Suppose
is constant at the steady-state value. Then, i T
Then we can substitute into
(10) and rearrange to get the desired TF:
( )
(^
)^
(^
)
i^
i^
i^
i
t^
t^
s
(^
) ( )
s
s
s
8
i T
Suppose that
is constant at its steady-state value:
(^
)^
(^
)^
(^
)
Q t
t^
s
Thus, rearranging
(^
) ( )
i T
s
T
s^
s
Comments:1. The TFs in (12) and (13) show the
individual
effects of
and
on
. What about
simultaneous
changes in both
and
?i T
i T
Properties of Transfer Function Models 1. Steady-State Gain
The steady-state of a TF can be used to calculate the steady-state change in an output due to a steady-state change in theinput. For example, suppose we know two steady states for aninput,
u
, and an output,
y
. Then we can calculate the steady-
state gain,
, from:
2
1
2
1
y
y
u
u −
For a linear system,
is a constant. But for a nonlinear
system,
will depend on the operating condition
(
) ,^
u y
Calculation of
from the TF Model:
If a TF model has a steady-state gain, then:
(^
)
0 lim
s
G s
This important result is a consequence of the Final ValueTheorem
-^
Note
: Some TF models do
not
have a steady-state gain (e.g.,
integrating process in Ch. 5)
in (4-38). Otherwise, the system
response to a step input will be an impulse. This can’t happen. Example:
n
m ≥
0
1
0
and step change in
du
a y
b
b u
u
dt
3. Additive Property
Suppose that an output is influenced by two inputs and thatthe transfer functions are known:
(^
) ( )
( )
( ) ( )
( )
1
2
1
2
and
s^
s
s^
s
s^
s
Then the response to changes in both
and
can be written
as:
1 U
2 U
(^
)^
(^
)^
(^
)^
(^
)^
(^
)
1
1
2
2
s^
s U
s
s U
s
The graphical representation (or
block diagram
) is:
G
( 1 s) G
( 2 s)
Y
(s
)
U
( 1 s) U
( 2 s)
Linearization of Nonlinear Models • So far, we have emphasized linear models which can be
transformed into TF models.
may be approximately linear.
-^
Conclude
: Linear approximations can be useful, especially
for purpose of analysis.
method called “linearization”. It is based on a Taylor SeriesExpansion of a nonlinear function about a specified operatingpoint.
variables,
u
and
y
(^
) ,^
dy
f^
y u
dt
Perform a Taylor Series Expansion about
and
and
truncate after the first order terms,
u
y
(^
)^
(^
)
y^
y
f^
f
f^
u y
f^
u y
u
y
u
y
where
and
. Note that the partial derivative
terms are actually constants because they have been evaluated atthe nominal operating point,Substitute (4-61) into (4-60) gives:
u
u
u
y
y
y
(^
) ,^
u y
(^
) ,
y^
y
dy
f^
f
f^
u y
u
y
dt
u
y
Combine (1) and (2),
i^
v
dh A
q
h
dt
Linearize
term,
(^
)
h
h
h
h
h
Or
h
h
h R
where:
h
h
h
h
Substitute linearized expression (5) into (3):
i^
v
dh A
q
h
h
dt
The steady-state version of (3) is:
i^
v
q
h
Subtract (7) from (6) and let
, noting that
gives the linearized model:
i^
i^
i
q
q
q
dh
dh
dt
dt
i
dh A
q
h
dt
