2 - process characteristics n response.pdf

7/25/2019 2 - process characteristics n response.pdf

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Type of response


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Common input changes


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1. Step Input

A sudden change in a process variable can be approximated by

a step change of magnitude,M:

Special Case:IfM= 1, we have a unit step change. We

give it the symbol, S(t).

Example of a step change:A reactor feedstock is suddenly

switched from one supply to another, causing sudden

changes in feed concentration, flow, etc.

The step change occurs at an arbitrary time denoted as t= 0.


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We can approximate a drifting disturbance by a ramp input:

2. Ramp Input

Industrial processes often experience drifting

disturbances, that is, relatively slow changes up or down

for some period of time. The rate of change is approximately constant.


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Examples:

1. Reactor feed is shut off for one hour.

2. The fuel gas supply to a furnace is briefly interrupted.

0

h

URP tw Time, t

3. Rectangular Pulse

It represents a brief, sudden change in a process variable:


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4. Sinusoidal Input

Processes are also subject to periodic, or cyclic, disturbances.

They can be approximated by a sinusoidal disturbance:

sin

0 for 0(5-14)

sin for 0

tU t

A t t

Examples:

1. 24 hour variations in cooling water temperature.

where: A = amplitude, = angular frequency

sin 2 2( )

AU s

s


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Response of first order system

First order differential equation

General first order transfer function

)()(

)(01 tbXtYadt

tdY

a

ctbxtyadt

tdya )()(

)(01

)()()(

tKXtYdt

tdY

)(1

)( sXs

KsY

0

01

/

/

abK

aa


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s

x

s

KsY

1)(

1.Step response

)(1

)( sXs

KsY


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All first order systems forced by a step function will have

a response of this same shape.

Step response for first order system


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To calculate the gain and time constant

from the graph

x

yK

Gain,

Time constant, value of t which the response is

63.2% complete


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2. Ramp response

)(1

)( sXs

KsY

21)(

s

asKsY


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Ramp response for first order system

The normalized output

lags the input by exactly

one time constant


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22)(

ssU

22

2

22

10

22

p

ss

s

1ss1s

K)s(Y

1

K

1

K

1

K

22

p

2

22

p

1

22

2

p

0

3. Sine input

By partial fraction decomposition,

)tsin(1

Ke

1

K)t(y

22

pt

22

p

Where )(tan 1


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First order response to the sine wave


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Response with time delay

X(t)

Y(t)

t=0 t=t0

=Time delay/dead time


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1. Step response

)(1

)(0

sXs

KesY

st

First-order-plus-dead-time (FOPDT)


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Response of second order system

Second order differential equation

General second order transfer function

ctbxtyadt

tdya

dt

tyda )()(

)()(012

2

2

)()()()(

012

2

2 tbXtYadt

tdYa

dt

tYda

)()()(

2)(

2

22 tKXtY

dt

tdY

dt

tYd

)(12

)(22

sXss

KsY

0

20

1

0

1

0

2

22

a

bK

aaa

aa

a

a


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1+)s(+s

K=G(s)

21

2

21

21

1s2sK=G(s)

22

21

21

2=

2nd order ODE model

(overdamped)

Composed of two first order subsystems (G1and G2)

roots:

12

dampedcritically1

dunderdampe10overdamped1

11)(

21

21

ss

KKsY


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1. Step response

)(

12)(

22 sXss

KsY

sx

ss

KsY

12)(

22


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Second Order Step Changea. Overshootfraction of the final steady-state change

by which the first peak exceeds this change

b. time of first maximum-time required for the output

to reach its first maximum value

c. decay ratio-ratio which the amplitude of the sinewave is reduced during one complete cycle

21pt

2

22

2exp

1

c a

a b

os=2

exp1

a

b


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d. period of oscillation, Ptime between twosuccessive peaks of the response.

e. Rise time, trtime taken for the process output to

first reach the new steady state value.

f. Settling timetime it takes for the output to come

within a band of the final steady-state value andremain in this band

2

2

1p

process responses under automatic control


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I deal response:The desired process response is achieved at an instantaneous time.

SP1

PV1

PV2

SP2

Time

Ideal

response

process responses under automatic control.

Terminology

Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)


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Stable:The process response stabilized at (near) the set point .

SP1

PV1

PV2

SP2

Time

Ideal

response

Terminology



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Unstable:The process response could not be stabilized at the set point.

SP1

PV1

PV2

SP2

Time

Ideal

response

Terminology



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SP1

SP2

PV

Time01. LCL = Lower control (quality)

limit.

2. UCL = Upper control (quality)

limit.

Out of spec

Out of specLCL

UCL

Quality limits:A range, set values above and below the set point, whereby the process

is allowed to oscillate. Product quality is acceptable within these limits.

Terminology



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QAD

Underdamped

Overdamped

Oscillatory

Offset

Various shapes of process responses under automatic

control.


Unit 1: Process settling criteria


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Settli ng cri ter ia:A response curve that meet any of the following criteria (criterion)

is considered settle.

1. Res po nse t ime

2. Sett l ing t ime

3. Rise t im e

4. Quar ter Amp lit ude Damp ing (QAD)

5. Quali ty l im its ( BEST for product quality control)

6. No overshoo t o r no undershoo t ( BEST for

temperature and pH control)

7. M in imum IAE, ITSE, etc.

Unit 1: Process settling criteria

Terminologies

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