Upload
syed-nazrin
View
21
Download
7
Embed Size (px)
DESCRIPTION
good
Citation preview
Ch
ap
ter
4
Chapter 4
Transfer Functions
Ch
ap
ter
4
Chapter Objectives
End of this chapter, you should be able to:
• Define what is a transfer function
• Develop transfer functions from mathematical models
• Use properties of transfer functions in simplifying and
analyzing models
• Use linearization to derive transfer functions for
nonlinear processes
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 2
Transfer Functions
• An algebraic expression
• Convenient representation of a linear,
dynamic model
• A transfer function (TF) relates one
input and one output:
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 3
Ch
ap
ter
4
system
x t y t
X s Y s
Transfer Functions
• Independent of initial conditions
• Independent of particular choice of forcing
functions
The following terminology is used:
x
input
forcing function
“cause”
y
output
response
“effect”
Ch
ap
ter
4
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 4
where:
Y sG s
X s
Y s y t
X s x t
= L
= L
Ch
ap
ter
4
Definition of transfer function
• Let G(s) denote the transfer function
between an input, x, and an output, y. Then,
by definition
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 5
Figure 2.3 Stirred-tank heating process with constant holdup, V.
Development of Transfer Functions
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 6
Ch
ap
ter
4
Example: Stirred Tank Heating System
Recall the previous dynamic model, assuming constant liquid
holdup and flow rates:
(2-36)i
dTV C wC T T Q
dt
Suppose the process is at steady state:
0 (2)iwC T T Q
Subtract (2) from (2-36):
(3)i i
dTV C wC T T T T Q Q
dt
Ch
ap
ter
4
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 7
But,
(4)i
dTV C wC T T Q
dt
where the “deviation variables” are
, ,i i iT T T T T T Q Q Q
0 (5)iV C sT s T wC T s T s Q s
Take L of (4):
Ch
ap
ter
4
At the initial steady state, T′(0) = 0.
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 8
Rearrange (5) to solve for
1
(6)1 1
i
KT s Q s T s
s s
Ch
ap
ter
4 where
1and
VK
wC w
)()()()()( 21 sTsGsQs=GsT i
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 9
G1 and G2 are transfer functions and independent of the
inputs, Q′ and Ti′.
Note G1 (process) has gain K and time constant τ.
G2 (disturbance) has gain=1 and time constant τ.
Gain = G(s=0). Both are first order processes.
If there is no change in inlet temperature (Ti′= 0),
then Ti′(s) = 0.
System can be forced by a change in either Ti or Q
(see Example 4.1).
Ch
ap
ter
4
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 10
Conclusions about TFs
1. Note that (6) shows that the effects of changes in both Q
and are additive. This always occurs for linear, dynamic
models (like TFs) because the Principle of Superposition is
valid.
iT
2. The TF model enables us to determine the output response to
any change in an input.
3. Use deviation variables to eliminate initial conditions for TF
models.
Ch
ap
ter
4
6/18/2014 CCB3013 Chemical Process Dynamics,
Instrumentation and Control
11
Ch
ap
ter
4
Example: Stirred Tank Heater
0.05K 2.0
0.05
2 1T Q
s
No change in Ti′
Step change in Q(t): 1500 cal/sec to 2000 cal/sec
500Q
s
0.05 500 25
2 1 (2 1)T
s s s s
What is T′(t)?
/ 25( ) 25[1 ] ( )
( 1)
tT t e T ss s
/ 2( ) 25[1 ]tT t e
From line 13, Table 3.1
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 12
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 the input. For example, suppose we know two
steady states for an input, u, and an output, y. Then we can
calculate the steady-state gain, K, from:
2 1
2 1
(4-38)y y
Ku u
For a linear system, K is a constant. But for a nonlinear
system, K will depend on the operating condition , .u y
Ch
ap
ter
4
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 13
Calculation of K from the TF Model:
If a TF model has a steady-state gain, then:
0
lim (14)s
K G s
• This important result is a consequence of the Final Value
Theorem
• Note: Some TF models do not have a steady-state gain (e.g.,
integrating process in Ch. 5)
Ch
ap
ter
4
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 14
2. Order of a TF Model
Consider a general n-th order, linear ODE:
1
1 1 01
1
1 1 01(4-39)
n n m
n n mn n m
m
m m
d y dy dy d ua a a a y b
dtdt dt dt
d u dub b b u
dtdt
Take L, assuming the initial conditions are all zero. Rearranging
gives the TF:
0
0
(4-40)
mi
i
in
ii
i
b sY s
G sU s
a s
Ch
ap
ter
4
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 15
The order of the TF is defined to be the order of the denominator
polynomial.
Note: The order of the TF is equal to the order of the ODE.
Definition:
Physical Realizability:
For any physical system, 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 (4-41)du
a y b b u udt
Ch
ap
ter
4
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 16
General 2nd order ODE:
Laplace Transform:
2 roots
: real roots
: imaginary roots
y=Kudt
dy+b
dt
yda
2
2
KU(s)Y(s)bs+as 12
1)(
)()(
2
bsas
K
sU
sYsG
a
abbs ,
2
42
21
2b 1
4a
2b 1
4a
Ch
ap
ter
4
2nd order process
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 17
Ch
ap
ter
4
Examples
1. 2
2
3 4 1s s
2 161.333 1
4 12
b
a
2 13 4 1 (3 1)( 1) 3( )( 1)3
s s s s s s
3 , ( )t
ttransforms to e e real roots
(no oscillation)
2. 2
2
1s s
2 11
4 4
b
a
2 3 31 ( 0.5 )( 0.5 )
2 2s s s j s j
0.5 0.53 3cos , sin
2 2
t ttransforms to e t e t
(oscillation) 6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 18
Ch
ap
ter
4
From Table 3.1, line 17
2 2
2
( )
1 3
2
es b
s s
L- bt
2
2
sin t
2 2=
(s+ 0.5)
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 19
Two IMPORTANT properties (L.T.)
A. Multiplicative Rule
B. Additive Rule
Ch
ap
ter
4
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 20
Example 1:
Place sensor for temperature downstream from heated
tank (transport lag)
Distance L for plug flow,
Delay time:
V = fluid velocity
Tank:
Sensor:
V
Lθ
Ch
ap
ter
4
1)(
)(
1
11
s
K
sU
sTG
s+τ
eK=
T(s)
(s)T=G
s-
s
2
22
1
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 21
Tank:
Sensor:
Overall transfer function:
2 is very small – can be neglected
sτ
eKKGG
U
T
T
T
U
T θs
ss
1
2112
1
Overall Transfer Function C
ha
pte
r 4 1)(
)(
1
11
s
K
sU
sTG
1 1
2
2
22 K
s+τ
eK=
T(s)
(s)T=G
s-
s
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 22
Ch
ap
ter
4
Example 2: Consider the system shown below.
The system consists of two liquid surge tanks in
series so that the outflow from the first tank is the
inflow to the second tank. 6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 23
For tank 1,
Ch
ap
ter
4
48)-(4 11
1 qqdt
dhA i
1
1
1
1h
Rq (4 - 49)
1
1
1
1h
Rq
Putting (4-49) and (4-50) into deviation variable form gives
(4 -52)
51)-(4 1
1
1
11 h
Rq
dt
hdA i
Substituting (4-49) into (4-48) eliminates q1:
50)-(4 1
1
1
11 h
Rq
dt
dhA i
6/18/2014 CCB3013 Chemical Process Dynamics,
Instrumentation and Control
24
The transfer function relating )(1 sH to )(sQi is found
by transforming (4-51) and rearranging to obtain
(4-53)
11)(
)(
1
1
11
11
s
K
sRA
R
sQ
sH
i
Ch
ap
ter
4
Similarly, the transfer function relating )(1 sQ to
)(1 sH is obtained by transforming (4-52).
(4-54) 111
1 11
)(
)(
KRsH
sQ
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 25
Ch
ap
ter
4
• The desired transfer function relating the outflow from
Tank 2 to the inflow to Tank 1 can be derived by
forming the product of (4-53) through (4-56).
The same procedure leads to the corresponding
transfer functions for Tank 2,
11)(
)(
2
2
22
2
1
2
s
K
sRA
R
sQ
sH
(4-55)
222
2 11
)(
)(
KRsH
sQ
(4.56)
6/18/2014 CCB3013 Chemical Process Dynamics,
Instrumentation and Control
26
Ch
ap
ter
4
or
which can be simplified to give
11
1
)(
)(
21
2
sssQ
sQ
i (4-59)
)(
)(
)(
)(
)(
)(
)(
)(
)(
)( 1
1
1
1
2
2
22
sQ
sH
sH
sQ
sQ
sH
sH
sQ
sQ
sQ
ii
(4-57)
1
1
1
1
)(
)(
1
1
12
2
2
2
s
K
Ks
K
KsQ
sQ
i (4-58)
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 27
Linearization of Nonlinear Models
• Required to derive transfer function.
• Good approximation near a given operating point.
• Gain, time constants may change with operating point.
• Use 1st order Taylor series.
),( uyfdt
dy
)()(),(),(,,
uuu
fyy
y
fuyfuyf
uyuy
uu
fy
y
f
dt
yd
ss
Subtract steady-state equation from dynamic equation
(4-60)
(4-61)
(4-62)
Ch
ap
ter
4
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 28
Example 3:
q0: control variable,
qi: disturbance variable
...(1) 0qqdt
dhA i
Use L.T.
0( ) ( ) ( )iAsH s q s q s
(in deviation variables)
Suppose q0 is constant: 00 Δq
...(2) 1
As(s)q
H(s)
(s)qAsH(s)
i
i
• Pure integrator (ramp) for step
change in qi
Ch
ap
ter
4
• Example of non-self regulating
system 6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 29
Nonlinear element
More realistically, if q0 is manipulated by a flow
control valve, ...(3) 0 hCq v
nonlinear element
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 30
Figure 2.5
RV: line resistance
linear ODE : eq. (4-74)
hR
qV
1
...(4) 1
hR
qdt
dhA
V
i
...(5) hCq Vif
Ch
ap
ter
4
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 31
Substitute linearized expression of
(5) into (1):
The steady-state version of (3) is:
0 (7)i vq C h
1(8)i
dhA q h
dt R
Ch
ap
ter
4
Linearised version of (5) is
(6) 1
hR
hCqdt
dhA vi
hR
hC
hhh
ChC
hhdh
dqq
hCq
v
vv
h
v
1
1
2
1
)(
Subtract (7) from (6) and let , noting that
gives the linearized model:
dh dh
dt dt
iii qqq
6/18/2014 CCB3013 Chemical Process Dynamics,
Instrumentation and Control
32
Ch
ap
ter
4
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 33
Conclusions
• Definition of transfer functions
• Development of transfer functions
• Properties of transfer functions
• 1st order process
• 2nd order process
• Integrating process (Non-self regulating)
• Examples
6/18/2014 CCB3013 Chemical Process Dynamics, Instrumentation and Control 34