2 process characteristics n response
TRANSCRIPT
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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: If M = 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 PulseIt represents a brief, sudden change in a process variable:
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4. Sinusoidal InputProcesses are also subject to periodic, or cyclic, disturbances. They can be approximated by a sinusoidal disturbance:
sin0 for 0
(5-14)sin for 0
tU t
A t t
Examples:
1. 24 hour variations in cooling water temperature.
where: A = amplitude, ω = angular frequency
sin 2 2( ) AU ss
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Response of first order system• First order differential equation
• General first order transfer function
)()()(01 tbXtYa
dttdYa
ctbxtyadt
tdya )()()(01
)()()( tKXtYdt
tdY
)(1
)( sXsKsY
0
01
//abKaa
inputoutput/response
Smith & Corripio pg 41
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sx
sKsY
1)(
1.Step response
)(1
)( sXsKsY
5.2.1 page 108 Seborg,Inverse laplace
Response in time domain,y(t)
Coughanowr page 79
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So, do you understand
how process constant,
corresponds to 0.632?1 2 3𝜏 4 5
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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
xyK
Gain,
Time constant, – value of t which the response is 63.2% complete
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2. Ramp response
)(1
)( sXsKsY
21)(
sa
sKsY
y(t)=??
Inverse Laplace, SeborgPage 110
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Ramp response for first order system
The normalized output lags the input by exactly one time constant
Rampinput Ramp
output
Seborg page 111
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22)(
s
sU
222
2210
22p
sss
1ss1sK
)s(Y
1K
1K
1K
22p
2
22p
1
22
2p
0
3. Sine input
By partial fraction decomposition,
)tsin(1
Ke
1K
)t(y22
pt22
p
Where )(tan 1
)(1
)( sUsKsY
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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
Θ=Time delay/dead time
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1. Step response
)(1
)(0
sXs
KesYst
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
tdyadt
tyda )()()()(012
2
2
)()()()(012
2
2 tbXtYadt
tdYadt
tYda
)()()(2)(2
22 tKXtY
dttdY
dttYd
)(12
)(22
sXss
KsY
0
20
1
0
1
0
2
22
abK
aaa
aaaa
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1+)s(+sK=G(s)
212
21
21
1s2sK=G(s) 22
21
21
2=
2nd order ODE model(overdamped)
Composed of two first order subsystems (G1 and G2)
roots: 12
damped critically 1dunderdampe 10
overdamped 1
11)(
21
21
ssKK
sY
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1. Step response
)(12
)(22
sXss
KsY
s
xss
KsY
12
)( 22
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Second Order Step Changea. Overshoot – fraction 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 sine wave is reduced during one complete cycle
21pt
2
22
2exp1
c aa b
os=2
exp1
ab
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d. period of oscillation, P – time between two successive peaks of the response.
e. Rise time, tr – time taken for the process output to first reach the new steady state value.
f. Settling time – time it takes for the output to come within a band of the final steady-state value and remain in this band
2
2
1p
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Ideal response: The desired process response is achieved at an instantaneous time.
SP1
PV1
PV2
SP2
Time
Idealresponse
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
Idealresponse
Terminology
© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)
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Unstable: The process response could not be stabilized at the set point.
SP1
PV1
PV2
SP2
Time
Idealresponse
Terminology
© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)
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SP1
SP2
PV
Time01. LCL = Lower control (quality) limit.
2. UCL = Upper control (quality)
limit.
Out of spec
Out of spec LCL
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
© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)
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QAD
Underdamped
Overdamped
Oscillatory
Offset
Various shapes of process responses under automatic control.
© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)
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Settling criteria: A response curve that meet any of the following criteria (criterion) is considered settle.
1. Response time2. Settling time3. Rise time4. Quarter Amplitude Damping (QAD)5. Quality limits (BEST for product quality control)6. No overshoot or no undershoot (BEST for
temperature and pH control)7. Minimum IAE, ITSE, etc.
Unit 1: Process settling criteria
Terminologies
© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)