CHAPTER 4 : MODELLING & ANALYSIS FOR PROCESS CONTROL · 2017-10-16 · CHAPTER 4 : MODELLING &...

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

Outline of the lesson.

• Laplace transform

• Solve linear dynamic models

• Transfer function model structure

• Qualitative features directly from model

• Frequency response

• Workshop


WHY WE NEED MORE DYNAMIC MODELLING

TA

I can model this;what more do

I need?TA

I would like to• model elements

individually• combine as needed• determine key

dynamic featuresw/o solving


TA

I would like to• model elements

individually• This will be a

“transfer function”

Now, I can combine elements to modelmany process structures


Now, I can combine elements to modelmany process structures

Even more amazing,I can combine toderive a simplifiedmodel!

THE FIRST STEP: LAPLACE TRANSFORM

∫==∞

−

0dtetfsftfL st)()())((

sCe

sCdtCeCL

t

t

stst =−==∞=

=

−∞

−∫00

)(: Constant

Step Change at t=0: Same as constant for t=0 to t=∞

t=0f(t)=0


∫==∞

−

0dtetfsftfL st)()())((

∫∫∫∞

−−∞

−∞

−−− −+=−=−000

11 dteedtedte)e())e((L st/tstst/t/t τττ

s/1=

ττττ

/se

/sdte t)s /(t)s /(

10

1

10

1 11+

−=+

=−=∞

+−∞

+−∫

)s (ss s/ss 11111

11 +=

+−=

+−=

τττ

τ

We have seenthis term

often! It’s thestep response to

a first orderdynamic system.


Let’s 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?

Let’s learn a newdynamic response

& its LaplaceTransform




time

Xin

Xout

θ = dead timeθ

What is the value ofdead time for

plug flow?

0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

time

Y, o

utle

t fro

m d

ead

time

0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

time

X, in

let t

o de

ad ti

me




• Is this a dead time?

• What is thevalue?




The dynamic model for dead time is

)t(X)t(X inout θ−=

The Laplace transform for a variable after dead time is

)())(())(( sXetXLtXL ins

inoutθθ −=−=

Our plants havepipes. We willuse this a lot!


We need the Laplace transform of derivatives for solving dynamic models.

First derivative:

General:

0=−=

t)t(f)s(sf

dt)t(dfL

constant

+++−=

=−

−

=

−=

−

01

1

0

10

1

tn

n

t

nt

nnn

n

dt)t(fd....

dt)t(dfs)t(fs)s(fs

dt)t(fdL

constant

I am in desperateneed of examples!

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.

(We’ll solve this in class.)

FCA0 VCA

AAAA VkC')C'F(C'

dtdC'V −−= 0

AA kCrBA

=−→

kVFFK and

kVFV with ''

'

+=

+==+ ττ 0AA

A KCCdt

dC

I hope we get the same answer as with theintegrating factor!


AA kCrBA

=−→

FCA0 V1CA1

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.


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

=+

=+

τ

τ

Much easier thanintegrating factor!


Textbook Example 3.5: The feed composition experiences a step. All other variables are constant. Determine the dynamic response of CA.


2AA kCr

BA

=−

→

FCA0

VCA

Non-linear!

TRANSFER FUNCTIONS: MODELS VALID FOR ANY INPUT FUNCTION

Let’s 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)


Y(s)X(s) G(s)G(s) = Y(s)/ X(s)

• How do we achieve zero initial conditions for every model?

• We don’t have “primes” on the variables; why?

• Is this restricted to a step input?

• What about non-linear models?

• How many inputs and outputs?



Some examples:

?)()()( :CSTRs Two

?)()()( :tank Mixing

==

==

sGsCsC

sGsCsC

A

A

A

A

0

2

0



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 thefirst answer!


T

opensm

svsF

sGvalve %/10.

)()(

)(30 ==

1250

2.1

)()()(

0

1

+

−==

ssFsTsG /sm

K 3tank1

130001

1

2

+==

sKK

sTsTsG / .)()()(tank2 110

012

+=

=

sKK

sTsTsG measured

sensor

/ .

)()()(

(Time in seconds)

Let’s see how tocombine models


The BLOCK DIAGRAM

Gvalve(s) Gtank2(s)Gtank1(s) Gsensor(s)

v(s) F0(s) T1(s) T2(s) Tmeas(s)

It’s a picture of the model equations!

• Individual models can be replaced easily

• Helpful visualization

• Cause-effect by arrows


Combine using BLOCK DIAGRAM ALGEBRA

Gvalve(s) Gtank2(s)Gtank1(s) Gsensor(s)

v(s) F0(s) T1(s) T2(s) Tmeas(s)

)()()()( )()(

)()(

)()(

)()()(

)()(

sGsGsGsGsvsF

sFsT

sTsT

sTsTsG

svsT

vTTs

measmeas

12

0

0

1

1

2

2

=

==

G(s)v(s) Tmeas(s)


Key rules for BLOCK DIAGRAM ALGEBRA

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

pApA

stA KCss

KCstC 0

0

0 )1(lim|)( ∆=

+

∆=

→∞→

τ

...)]sin()cos([...

..)(...)(

+++

++++++++=t

ttt

q

p

etCtC

etBtBBeAeAAtYα

ααα

ωω 21

2210210

21

What about dynamics can we determinewithout 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-α1) + C2/(s-α2) + ...

With αi the solution to the denominator of the transfer function being zero, D(s) = 0.

...)]sin()cos([...

..)(...)(

+++

++++++++=t

ttt

q

p

etCtC

etBtBBeAeAAtYα

ααα

ωω 21

2210210

21

...)]sin()cos([...

..)(...)(

+++

++++++++=t

ttt

q

p

etCtC

etBtBBeAeAAtYα

ααα

ωω 21

2210210

21

Real, distinct αiReal, repeated αi

Complex αiαq is Re(αi)


With αi the solutions to D(s) = 0, which is a polynomial.

...)]sin()cos([...

..)(...)(

+++

++++++++=t

ttt

q

p

etCtC

etBtBBeAeAAtYα

ααα

ωω 21

2210210

21

1. If all αi are ???, Y(t) is stable

If any one αi is ???, Y(t) is unstable

2. If all αi are ???, Y(t) is overdamped (does not oscillate)

If one pair of αi are ???, Y(t) is underdamped

Complete statements based on equation.


With αi the solutions to D(s) = 0, which is a polynomial.

...)]sin()cos([...

..)(...)(

+++

++++++++=t

ttt

q

p

etCtC

etBtBBeAeAAtYα

ααα

ωω 21

2210210

21

1. If all real [αi] are < 0, Y(t) is stable

If any one real [αi] is ≥ 0, Y(t) is unstable

2. If all αi are real, Y(t) is overdamped (does not oscillate)

If one pair of αi are complex, Y(t) is underdamped

AA kCrBA

=−→

FCA0 V1CA1

V2CA2


'''

'''

1222

2

0111

1

AAA

AAA

CKCdt

dC

CKCdt

dC

=+

=+

τ

τ


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?

Withoutsolving!


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 to have a small effect on the output.


0 1 2 3 4 5 6-0.4

-0.2

0

0.2

0.4

time

Y, o

utle

t fro

m s

yste

m

0 1 2 3 4 5 6-1

-0.5

0

0.5

1

time

X, i

nlet

to s

yste

m

input

outputB

A

P

P’

Amplitude ratio = |Y’(t)| max / |X’(t)| max

Phase angle = phase difference between input and output


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 = jω, with ω= frequency and j = complex variable.

=∠==

+===

−

))(Re())(Im(tan)(angle Phase

))(Im())(Re()(Ratio Amp.

ωωωϕ

ωωω

jGjGjG

jGjGjGAR

1

22

These calculations are tedious by hand but easily performed in standard programming languages.


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(ω)• Phase angle vs log(ω)


FCA0 V1CA1

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

pAA

p

A

A

C

KCC

KjG

CC

τω

τωω Not acceptable. We need

to reduce the variability.How about feedback

control?

Data from 2 CSTRs

OVERVIEW OF ANALYSIS METHODS

We can determine individual models

and combine

1. System order

2. Final Value

3. Stability

4. Damping

5. Frequency response

We can determine these features without

solving for the entire transient!!

Transfer function and block diagram

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

CheckD.O.F.

Is model linear? Expand in Taylor Series

DOF = 0 Another equation:-Fundamental balance-Constitutive equat ions

DOF ≠ 0

No

Express in deviation var iables

Group parameters to evalua te [ga ins (K), time-constants (τ), dead-times(θ)]

Take Laplace transform

Substitute specific input, e.g.,step, and solve for output

Ana lytical solution(step)

Numerical solution

Ana lyze the model for:- causali ty- order- stability- damping

Yes

Combine severa l models intointegrated system

We can use astandard modelling

procedure tofocus ourcreativity!

Combining Chapters 3 and 4

Too small to read here - check it out in the textbook!

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 a function of time.

Solve the linearized model using Laplace transforms


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 transfer function.

Determine the features in the table for this system.

)..()..(

)()(

80357918345076

2 ++−−

=ss

ssFsT

c

T

A


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.


Feed

Vaporproduct

LiquidproductProcess

fluidSteam

F1

F2 F3

T1 T2

T3

T5

T4

T6 P1

L1

A1L. Key

We often measure pressure for process monitoring and control. Explain three principles for pressure sensors, select one for P1 and explain your choice.


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

Lot’s 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, we’ll have an assignment!

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)

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 a

a. Linear plant?b. Non-linear plant?

4. Is the amplitude ratio of a plant always equal to or greater than the steady-state gain?

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

FT1

FT2

PT1

PIC1

AT1

TI1

TI2

TI3

TI4

PI2

PI3

PI4

TI5

TI6

TI7

TI8

TI9

FI3

TI10

TI11

PI5

PI6

a. A 3% increase in the fuel flow rate.

b. A 2% change in the feed flow rate.

c. Start up from ambient temperature.

d. Emergency stoppage of fuel flow to 0.0.

fuel

feed

air

CHAPTER 4 : MODELLING & ANALYSIS FOR PROCESS CONTROL · 2017-10-16 · CHAPTER 4 : MODELLING &...

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