Chemical Process Dynamics, Instrumentation and Control

CCB3013

Chapter 3Laplace Function Models

Process Dynamics and ControlSeborg . Edgar . Mellichamp . Doyle

Third Edition

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

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Transfer Functions of a Constant, Step Function and DerivativesC

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ExamplesC

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Partial Fraction Expansion

Example 1: Find f(t).

Example 2: Solve

Laplace Transforms for Various Time-Domain FunctionsC

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3f(t) F(s)

Transfer Functions• Convenient representation of a linear, dynamic model.

• A transfer function (TF) relates one input and one output:

The following terminology is used:

u

input

forcing function

“cause”

y

output

response

“effect”

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)(system

)(

)(

sY

ty

sU

tu

Definition of the transfer function:

Let G(s) denote the transfer function between an input, x, and an output, y. Then, by definition

where:

Development of Transfer Functions

Example: Stirred Tank Heating System

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)()(

)(sUsY

sG

)(L)(

)(L)(

tusU

tysY

Stirred-tank heating process with constant holdup, V.

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Development of Transfer Functions

Example: Stirred Tank Heating System

Recall the previous dynamic model, assuming constant liquid holdup and flow rates:

(2-36)idT

V C wC T T Qdt

Suppose the process is at steady state:

0 (2)iwC T T Q

Subtract (2) from (2-36):

(3)i idT

V C wC T T T T Q Qdt

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But,

(4)idT

V C wC T T Qdt

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):

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At the initial steady state, T′(0) = 0.

Rearrange (5) to solve for

1(6)

1 1 iK

T s Q s T ss s

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1and

VK

wC w

G1 and G2 are transfer functions and independent of the inputs, Q′ and Ti′.Note G1 (process) has gain K and time constant G2 (disturbance) has gain=1 and time constant 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 4.3).

(s)T(s)G(s)Q(s)(s)=GT i 21

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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.

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Q’(s)

T’i(s)+

+

+T’(s)

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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 s

s s

/ 2( ) 25[1 ]tT t e

Properties of Transfer Function (TF) 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

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Calculation of K from the TF Model:

If a TF model has a steady-state gain, then:

0

lim (14)s

K G s

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)()(

)(sUsY

sG

Final Value Theorem:

If f(s) is the Laplace transform of f(t), then

Example: Find the final value of the function f(t) for which the Laplace transform is:

Initial Value Theorem:

If f(s) is the Laplace transform of f(t), then

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Example: Find the initial value of the function that has the following transform:

Transform of an Integral:

If f(s) is the Laplace transform of f(t), then

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Translation of Transform:

If f(s) is the Laplace transform of f(t), then

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Translation in Time (Time Delay)

If f(s) is the Laplace transform of f(t), then

Provided that: f(t)=0 for t<0

(a) A time function without time delay (b) A time function with time delay.

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

iin

ii

i

b sY s

G sU s

a s

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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. Otherwise, the system response to a step input will be an impulse. This can’t happen.

n m

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General 2nd order ODE:

Laplace Transform:

2 roots

: real roots

: imaginary roots

Ku=ydt

dyb+

dt

yda

2

2

)()(1+bsas2 sKUsY

1)(

)()(

2

bsas

K

sU

sYsG

a2

a4bbs

2

2,1

2b 1

4a

2b 1

4a

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2nd order process

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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)3s s s s s s

3 , ( )t ttransforms to e e real roots

(no oscillation)

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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 2t ttransforms to e t e t

(oscillation)

2 2

2

( )

1 32

es b

s s

L- bt

22

sin t

2 2=

(s+0.5)

Two IMPORTANT properties (L.T.)

A. Multiplicative Rule

B. Additive Rule

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Example 1:Example 1:

Place sensor for temperature downstream from heatedtank (transport lag)

Distance L for plug flow,

Dead time

V = fluid velocity

Tank:

Sensor:

Overall transfer function:

11

1

KT(s)G = =

U(s) 1+ s

V

L

222

s-2s

2 1,K s+1

eK=

T(s)

(s)T=G

is very small(neglect)

s1

eKKGG

U

T

T

T

U

T

1

s21

12ss

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h1

R1

qi

h2

R2

q1

q2

For tank 1, 48)-(4 11

1 qqdt

dhA i

11

1

1h

Rq (4 - 49)

11

1

1h

Rq

Putting (4-49) and (4-50) into deviation variable form gives (4 -52)

51)-(4 1

11

11 h

Rq

dt

hdA i

Substituting (4-49) into (4-48) eliminates q1:

50)-(4 1

11

11 h

Rq

dt

dhA i

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• 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)

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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)

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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

(3-61)

(3-62)

(3-63)

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h

qi (t)

qo(t)

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0if Vq C hC

hap

ter

3

i v

dhA q C h

dt

Perform Taylor series of right hand side

1i

dhA q h

dt R

0.52 / vR h C

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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

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