7/29/2019 Chap 04 Marlin 2002
1/44
CHAPTER 4 : MODELLING &
ANALYSIS FOR PROCESS CONTROL
When I complete this chapter, I want to be
able to do the following.
Analytically solve linear dynamic modelsof first and second order
Express dynamic models as transfer
functions
Predict important features of dynamic
behavior from model without solving
7/29/2019 Chap 04 Marlin 2002
2/44
Outline of the lesson.
Laplace transform
Solve linear dynamic models
Transfer function model structure
Qualitative features directly from model
Frequency response
Workshop
CHAPTER 4 : MODELLING &
ANALYSIS FOR PROCESS CONTROL
7/29/2019 Chap 04 Marlin 2002
3/44
WHY WE NEED MORE DYNAMIC MODELLING
T
A
I can model this;
what more do
I need?
T
A
I would like to
model elements
individually
combine as needed
determine key
dynamic features
w/o solving
7/29/2019 Chap 04 Marlin 2002
4/44
WHY WE NEED MORE DYNAMIC MODELLING
T
A
I would like to
model elements
individually
This will be a
transfer function
Now, I can combine
elements to model
many process
structures
7/29/2019 Chap 04 Marlin 2002
5/44
WHY WE NEED MORE DYNAMIC MODELLING
Now, I can combine
elements to model
many processstructures
Even more amazing,
I can combine toderive a simplified
model!
7/29/2019 Chap 04 Marlin 2002
6/44
THE FIRST STEP: LAPLACE TRANSFORM
==
0
dtetfsftfL st)()())((
s
Ces
CdtCeCLt
t
stst ====
=
00
)(:Constant
Step Change at t=0: Same as constant for t=0 to t=
t=0
f(t)=0
7/29/2019 Chap 04 Marlin 2002
7/44
THE FIRST STEP: LAPLACE TRANSFORM
==
0
dtetfsftfL st)()())((
+==
000
11 dteedtedte)e())e((L st/tstst/t/t
s/1=
/s
e
/s
dtet)s/(t)s/(
10
1
10
1 11
+
=
+
==
+
+
)s(sss/ss 11111
11 +=+=+=
We have seen
this term
often! Its the
step response toa first order
dynamic system.
7/29/2019 Chap 04 Marlin 2002
8/44
THE FIRST STEP: LAPLACE TRANSFORM
Lets consider plug flow through a pipe. Plug flow has no
backmixing; we can think of this a a hockey puck
traveling in a pipe.
What is the dynamic response of the outlet fluid property
(e.g., concentration) to a step change in the inlet fluid
property?
Lets learn a new
dynamic response
& its Laplace
Transform
7/29/2019 Chap 04 Marlin 2002
9/44
THE FIRST STEP: LAPLACE TRANSFORM
Lets learn a new
dynamic response
& its Laplace
Transform
time
Xin
Xout
= dead time
What is the value of
dead time for
plug flow?
7/29/2019 Chap 04 Marlin 2002
10/44
0 1 2 3 4 5 6 7 8 9 10-0.5
0
0.5
1
time
Y,outletfromd
eadtime
0 1 2 3 4 5 6 7 8 9 10-0.5
0
0.5
1
time
X,
inlettodeadtime
THE FIRST STEP: LAPLACE TRANSFORM
Lets learn a new
dynamic response
& its Laplace
Transform Is this a
dead time?
What is thevalue?
7/29/2019 Chap 04 Marlin 2002
11/44
THE FIRST STEP: LAPLACE TRANSFORM
Lets learn a new
dynamic response
& its Laplace
Transform
The dynamic model for dead time is
)t(X)t(X inout =
The Laplace transform for a variable after dead time is
)())(())(( sXetXLtXL ins
inout ==
Our plants have
pipes. We will
use this a lot!
7/29/2019 Chap 04 Marlin 2002
12/44
THE FIRST STEP: LAPLACE TRANSFORM
We need the Laplace transform of
derivatives for solving dynamic models.
First
derivative:
General:
0==
t
)t(f)s(sf
dt
)t(dfL
constant
+++=
=
=
=
0
1
1
0
1
0
1
t
n
n
t
n
t
nn
n
n
dt
)t(fd....
dt
)t(dfs)t(fs)s(fs
dt
)t(fdL
constant
I am in desperate
need of examples!
7/29/2019 Chap 04 Marlin 2002
13/44
SOLVING MODELS USING THE LAPLACE
TRANSFORM
Textbook Example 3.1: The CSTR (or mixing tank)
experiences a step in feed composition with all other
variables are constant. Determine the dynamic response.
(Well solve this in class.)
F
CA0VCA
AAAA VkC')C'F(C'
dt
dC'V = 0
AA kCr
BA
=
kVF
FKand
kVF
Vwith''
'
+=
+==+ 0AA
A KCCdt
dC
I hope we get the sameanswer as with the
integrating factor!
7/29/2019 Chap 04 Marlin 2002
14/44
SOLVING MODELS USING THE LAPLACE
TRANSFORM
AA kCr
BA
=
F
CA0V1CA1
V2CA2
Two isothermal CSTRs are initially at steady state and
experience a step change to the feed composition to the
first tank. Formulate the model for CA2.
(Well solve this in class.)
222212
2
111101
1
AAAA
AAAA
C'kV)C'F(C'dt
dC'V
C'kV)C'F(C'dt
dC'V
=
=
'''
'''
1222
2
0111
1
AAA
AAA
CKCdt
dC
CKCdt
dC
=+
=+
Much easier than
integrating factor!
7/29/2019 Chap 04 Marlin 2002
15/44
SOLVING MODELS USING THE LAPLACE
TRANSFORM
Textbook Example 3.5: The feed composition experiences a
step. All other variables are constant. Determine the
dynamic response of CA.
(Well solve this in class.)
2AA kCr
BA
=
F
CA0VCA
Non-linear!
7/29/2019 Chap 04 Marlin 2002
16/44
TRANSFER FUNCTIONS: MODELS VALID
FOR ANY INPUT FUNCTION
Lets rearrange the Laplace transform of a dynamic model
Y(s)X(s) G(s)Y(s) = G(s) X(s)
A TRANSFER FUNCTION is the output variable, Y(s),
divided by the input variable, X(s), with all initial
conditions zero.
G(s) = Y(s)/X(s)
7/29/2019 Chap 04 Marlin 2002
17/44
TRANSFER FUNCTIONS: MODELS VALID
FOR ANY INPUT FUNCTION
Y(s)X(s) G(s)G(s) = Y(s)/ X(s)
How do we achieve zero initial
conditions for every model?
We dont have primes on the
variables; why?
Is this restricted to a step input?
What about non-linear models?
How many inputs and outputs?
7/29/2019 Chap 04 Marlin 2002
18/44
TRANSFER FUNCTIONS: MODELS VALID
FOR ANY INPUT FUNCTION
Y(s)X(s) G(s)G(s) = Y(s)/ X(s)
Some examples:
?)()(
)(:CSTRsTwo
?)()(
)(:tankMixing
==
==
sGsC
sC
sGsC
sC
A
A
A
A
0
2
0
7/29/2019 Chap 04 Marlin 2002
19/44
TRANSFER FUNCTIONS: MODELS VALID
FOR ANY INPUT FUNCTION
Y(s)X(s) G(s)G(s) = Y(s)/ X(s)
Why are we doing this?
To torture students.
We have individual models that we can
combine easily - algebraically.
We can determine lots of information
about the system without solving the
dynamic model.
I chose the
first answer!
7/29/2019 Chap 04 Marlin 2002
20/44
TRANSFER FUNCTIONS: MODELS VALID
FOR ANY INPUT FUNCTION
T
open
sm
sv
sFsG
valve %
/10.)(
)()(
30 ==
1250
2.1
)(
)()(
0
1
+
==
ssF
sTsG /sm
K 3tank1
1300
01
1
2
+==
s
KK
sT
sTsG
/.
)(
)()(tank2 110
01
2
+=
=
s
KK
sT
sT
sGmeasured
sensor
/.
)(
)(
)(
(Time in seconds)
Lets see how to
combine models
7/29/2019 Chap 04 Marlin 2002
21/44
TRANSFER FUNCTIONS: MODELS VALID
FOR ANY INPUT FUNCTION
The BLOCK DIAGRAM
Gvalve(s) Gtank2(s)Gtank1(s) Gsensor(s)
v(s) F0(s) T1(s) T2(s) Tmeas(s)
Its a picture of the model equations!
Individual models can be replaced easily
Helpful visualization
Cause-effect by arrows
7/29/2019 Chap 04 Marlin 2002
22/44
TRANSFER FUNCTIONS: MODELS VALID
FOR ANY INPUT FUNCTIONCombine using BLOCK DIAGRAM ALGEBRA
Gvalve(s) Gtank2(s)Gtank1(s) Gsensor(s)
v(s) F0(s) T1(s) T2(s) Tmeas(s)
)()()()(
)(
)(
)(
)(
)(
)(
)(
)()(
)(
)(
sGsGsGsG
sv
sF
sF
sT
sT
sT
sT
sTsG
sv
sT
vTTs
measmeas
12
0
0
1
1
2
2
=
==
G(s)v(s) Tmeas(s)
7/29/2019 Chap 04 Marlin 2002
23/44
TRANSFER FUNCTIONS: MODELS VALID
FOR ANY INPUT FUNCTIONKey rules for BLOCK DIAGRAM ALGEBRA
7/29/2019 Chap 04 Marlin 2002
24/44
QUALITATIVE FEATURES W/O SOLVING
FINAL VALUE THEOREM: Evaluate the final valve of
the output of a dynamic model without solving for the
entire transient response.
sY(s)lim)(
=st
tY
Example for first order system
pA
pA
stA KC
ss
KCstC 0
0
0
)1(
lim|)( =+
=
7/29/2019 Chap 04 Marlin 2002
25/44
...)]sin()cos([...
..)(...)(
+++
++++++++=t
ttt
q
p
etCtC
etBtBBeAeAAtY
21
2210210
21
What about dynamics
can we determine
without solving?
QUALITATIVE FEATURES W/O SOLVING
We can use partial fraction
expansion to prove the following
key result.
Y(s) = G(s)X(s) = [N(s)/D(s)]X(s) = C1/(s- 1) + C2/(s- 2) + ...
With i the solution to the denominator of the transfer
function being zero, D(s) = 0.
...)]sin()cos([...
..)(...)(
+++
++++++++=t
ttt
q
p
etCtC
etBtBBeAeAAtY
21
2210210
21
...)]sin()cos([...
..)(...)(
+++
++++++++=t
ttt
q
p
etCtC
etBtBBeAeAAtY
21
2210210
21
Real, distinct iReal, repeated i
Complex i
q is Re( i)
7/29/2019 Chap 04 Marlin 2002
26/44
QUALITATIVE FEATURES W/O SOLVING
With i the solutions to D(s) = 0, which is a polynomial.
...)]sin()cos([.....)(...)(
+++
++++++++=t
ttt
q
p
etCtCetBtBBeAeAAtY
21
2
210210
21
1. If all i are ???, Y(t) is stable
If any one i is ???, Y(t) is unstable
2. If all i are ???, Y(t) is overdamped(does not oscillate)
If one pair ofi
are ???, Y(t) is
underdamped
Complete statements
based on equation.
7/29/2019 Chap 04 Marlin 2002
27/44
QUALITATIVE FEATURES W/O SOLVING
With i the solutions to D(s) = 0, which is a polynomial.
...)]sin()cos([.....)(...)(
+++
++++++++=t
ttt
q
p
etCtCetBtBBeAeAAtY
21
2
210210
21
1. If all real [ i] are < 0, Y(t) is stable
If any one real [ i] is 0, Y(t) is unstable
2. If all i are real, Y(t) is overdamped (does notoscillate)
If one pair ofi
are complex, Y(t) is underdamped
7/29/2019 Chap 04 Marlin 2002
28/44
AA kCr
BA
=
F
CA0
V1C
A1V2CA2
(Well solve this in class.)
'''
'''
1222
2
0111
1
AAA
AAA
CKCdt
dC
CKCdt
dC
=+
=+
QUALITATIVE FEATURES W/O SOLVING
1. Is this system stable?
2. Is this system over- or underdamped?
3. What is the order of the system?
(Order = the number of derivatives
between the input and output variables)
4. What is the steady-state gain?
Without
solving!
7/29/2019 Chap 04 Marlin 2002
29/44
QUALITATIVE FEATURES W/O SOLVING
FREQUENCY RESPONSE:The response to a sine input of
the output variable is of great practical importance. Why?
Sine inputs almost never occur. However, many
periodic disturbances occur and other inputs can be
represented by a combination of sines.
For a process without control, we want a sine input tohave a small effect on the output.
7/29/2019 Chap 04 Marlin 2002
30/44
QUALITATIVE FEATURES W/O SOLVING
0 1 2 3 4 5 6-0.4
-0.2
0
0.2
0.4
time
Y,outletfroms
ystem
0 1 2 3 4 5 6-1
-0.5
0
0.5
1
time
X,inlettosystem
input
output B
A
P
P
Amplitude ratio = |Y(t)| max / |X(t)| max
Phase angle = phase difference between
input and output
7/29/2019 Chap 04 Marlin 2002
31/44
QUALITATIVE FEATURES W/O SOLVING
Amplitude ratio = |Y(t)| max / |X(t)| max
Phase angle = phase difference between
input and output
For linear systems, we can evaluate directly using transfer function!
Set s = j , with = frequency and j = complex variable.
===
+===
))(Re(
))(Im(tan)(anglePhase
))(Im())(Re()(RatioAmp.
jG
jGjG
jGjGjGAR
1
22
These calculations are tedious by hand but easily performed in
standard programming languages.
7/29/2019 Chap 04 Marlin 2002
32/44
QUALITATIVE FEATURES W/O SOLVING
Example 4.15 Frequency response of mixing tank.
Time-domain
behavior.
Bode Plot - Shows
frequency response for
a range of frequencies
Log (AR) vs log( )
Phase angle vs log( )
7/29/2019 Chap 04 Marlin 2002
33/44
QUALITATIVE FEATURES W/O SOLVING
F
CA0V1CA1
V2CA2Sine disturbance with
amplitude = 1 mol/m3
frequency = 0.20 rad/min
Must have
fluctuations
< 0.050 mol/m3
CA2
Using equations for the frequency response amplitude ratio
0500120120011
1
2
2202
220
2
..).)(.(||)(
||||
)(|)(|
||||
>== +
=
+==
A
p
AA
p
A
A
C
KCC
KjGCC
Not acceptable. We need
to reduce the variability.
How about feedback
control?
Data from 2 CSTRs
7/29/2019 Chap 04 Marlin 2002
34/44
OVERVIEW OF ANALYSIS METHODS
We can determine
individual modelsand combine
1. System order
2. Final Value
3. Stability
4. Damping
5. Frequency response
We can determine
these features without
solving for theentire transient!!
Transfer function and block diagram
7/29/2019 Chap 04 Marlin 2002
35/44
Flowchart of Modeling Method
Goal: Assumptions: Data:
Variable(s): related to goals
System: volume within which variables are independent of position
Fundamental Balance: e.g. material, energy
Check
D.O.F.
Is model linear? Expand in Taylor Series
DOF = 0 Anothe r equa tion:-Fundamental balance
-Constitutive equations
DOF 0
No
Express in deviation variables
Group parameters to evalua te [gains (K), time-constants (), dead-times()]
Take Laplace transform
Substitute specific input, e.g.,
step, and solve for output
Ana lytical solution
(step)
Numerical solution
Analyze the model for:
- causality- order- stability
- damping
Yes
Combine several models into
integrated sys tem
We can use astandard modelling
procedure to
focus our
creativity!
Combining Chapters 3
and 4
7/29/2019 Chap 04 Marlin 2002
36/44
Too small to read here - check it out in the textbook!
7/29/2019 Chap 04 Marlin 2002
37/44
CHAPTER 4: MODELLING & ANALYSIS WORKSHOP 1
Example 3.6 The tank with a drain has a continuous flow in
and out. It has achieved initial steady state when a step
decrease occurs to the flow in. Determine the level as afunction of time.
Solve the linearized model using Laplace transforms
7/29/2019 Chap 04 Marlin 2002
38/44
CHAPTER 4: MODELLING & ANALYSIS WORKSHOP 2
1. System order
2. Final Value
3. Stability
4. Damping
5. Frequency response
The dynamic model for a non-
isothermal CSTR is derived in
Appendix C. A specific example
has the following transferfunction.
Determine the features
in the table for this
system.
)..()..(
)()(
80357918345076
2 ++=ss
ssFsT
c
T
A
7/29/2019 Chap 04 Marlin 2002
39/44
CHAPTER 4: MODELLING & ANALYSIS WORKSHOP 3
F
CA0V1CA1
V2CA2
Answer the following using the MATLAB program
S_LOOP.
Using the transfer function derived in Example 4.9,determine the frequency response for CA0 CA2. Check
one point on the plot by hand calculation.
7/29/2019 Chap 04 Marlin 2002
40/44
CHAPTER 4: MODELLING & ANALYSIS WORKSHOP 4
Feed
Vaporproduct
Liquid
productProcessfluid
Steam
F1
F2 F3
T1 T2
T3
T5
T4
T6 P1
L1
A1
L. Key
We often measure pressure for process monitoring and
control. Explain three principles for pressure sensors, select
one for P1 and explain your choice.
7/29/2019 Chap 04 Marlin 2002
41/44
CHAPTER 4 : MODELLING &
ANALYSIS FOR PROCESS CONTROL
When I complete this chapter, I want to be
able to do the following.
Analytically solve linear dynamic models of first
and second order
Express dynamic models as transfer functions
Predict important features of dynamic behavior
from model without solving
Lots of improvement, but we need some more study!
Read the textbook
Review the notes, especially learning goals and workshop
Try out the self-study suggestions Naturally, well have an assignment!
7/29/2019 Chap 04 Marlin 2002
42/44
LEARNING RESOURCES
SITE PC-EDUCATION WEB
- Instrumentation Notes
- Interactive Learning Module (Chapter 4)
- Tutorials (Chapter 14) Software Laboratory
- S_LOOP program
Other textbooks on Process Control (see course
outline)
7/29/2019 Chap 04 Marlin 2002
43/44
SUGGESTIONS FOR SELF-STUDY
1. Why are variables expressed as deviation variables
when we develop transfer functions?
2. Discuss the difference between a second order reaction
and a second order dynamic model.
3. For a sine input to a process, is the output a sine for aa. Linear plant?
b. Non-linear plant?
4. Is the amplitude ratio of a plant always equal to or
greater than the steady-state gain?
7/29/2019 Chap 04 Marlin 2002
44/44
SUGGESTIONS FOR SELF-STUDY
5. Calculate the frequency response for the model in
Workshop 2 using S_LOOP. Discuss the results.
6. Decide whether a linearized model should be used for
the fired heater for
FT
1
FT
2
PT
1
PIC
1
AT
1
TI
1
TI
2
TI
3
TI
4
PI
2
PI
3
PI
4
TI
5
TI
6
TI7
TI
8
TI
9
FI
3
TI
10
TI
11
PI
5
PI
6
a. A 3% increase in thefuel flow rate.
b. A 2% change in the
feed flow rate.
c. Start up from ambienttemperature.
d. Emergency stoppage
of fuel flow to 0.0.
fuel
feed
air