university of tennessee at chattanooga engineering 329...
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Matthew Chatham-Tombs - 1 - November 2, 2007
University of Tennessee at Chattanooga
Engineering 329
Paint Spray Booth Pressure System: Proportional Controller
Design
Matthew Chatham-Tombs
Eric L. Young
Jonathan Blanco
Nov. 2, 2007
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Matthew Chatham-Tombs - 2 - November 2, 2007
Introduction:
Three spray paint booths at an assembly plant require a feedback control system
to maintain a desired pressure. A blower that is powered by a variable speed motor
provides the pressure. The purpose of this lab is to observe the time response of the
output function of the system to a sine function input at different frequencies, to
determine the first order parameters for the mathematical model of the system, to
compare the experimental Bode plots with the approximate FOPDT model's Bode plots
for the system, and to determine the effective range of the controller gain. The response
of the system to the sine input will change according to the frequency of input to the
system. The main objectives of this lab is to observe the response of the output function
of the system to a sine input, to observe the system gain, K, the dead time, t0, and the time
constant, τ, create bode plots and root locus plots for the experiment, to observe effective
range of the controller gain, and to observe these parameters in several regions of the
steady-state curve.
The following report includes a background of the lab discussing the system, the
schematics, the steady-state curve, the effect of a step input on the system, the effect of a
sine function input on the system, and the methods for determining the system parameters
and controller range. Also included is the procedure for the lab and the method by which
the results were calculated. These results are then presented using tables and charts. The
results are then discussed and conclusions are made.
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Matthew Chatham-Tombs - 3 - November 2, 2007
Background:
A diagram of the blower, booth, and control system is shown below.
Figure 1: Schematic Diagram of the Dunlap Plant Spray-Paint Booths The input function for the blower-booth system is the power sent to the blower,
which varies from 0-100% of the rated power of the motor. The output function is the
pressure of the booth measured in cm-H2O. The input function is designated m(t) as it
represents the manipulated variable while the output function is designated c(t) as it
represents the controlled variable. The following diagram shows the input-output
relationship.
Figure 2: Block diagram of the paint Booth System
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Matthew Chatham-Tombs - 4 - November 2, 2007
The aim of the previous lab was to obtain step response data. Figure 3 shows a
typical input step function, m(t).
Figure 3: Step Input
The input function is initially at a base line input and abruptly “steps up” the
value of the step height. Notice that the input does not take time to reach the upper
operating value. The step of the input happens instantaneously.
Figure 4 shows a typical response of a system to a step input.
Figure 4: Step Response
Unlike the instantaneous change of the input, the output takes a certain amount of
time to respond to the input step. From the graph in Figure 4 one is able to determine the
parameters of the system. These parameters are the steady-state gain, K, the dead time,
t0, and the time constant, τ. These are also referred to as the First-Order-Plus-Dead-Time
(FOPDT) parameters. These parameters are part of the transfer function of the system.
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Matthew Chatham-Tombs - 5 - November 2, 2007
The transfer function of a first order system in the Laplace domain can be approximated
by the equation,
It is important to observe these parameters for different regions of the steady-state
curve. The steady-state curve was developed in a previous lab using average values of
the output for given values of the input. The steady-state curve for the pressure system is
shown in the graph below.
Steady State Operating Curve, Pressure
0
1
2
3
4
5
6
7
0 20 40 60 80 100
c, Input (%)
m, P
ress
ure
Out
put (
cm-H
20)
Series1
Figure 5: Steady-state operating curve for the paint Booth System This curve was created using the results of experiments conducted online. The
pressure outputs presented on the graph are the averages of the steady-state operating
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Matthew Chatham-Tombs - 6 - November 2, 2007
values for each input percentage. The uncertainty bars at each data point show two times
the actual standard deviation. The operating range for the pressure system input has been
determined to be 25% to 100%. The corresponding range for the output is 0.1 cm-H2O
to 5.61 cm-H2O. The slope of the steady-state curve is also a way to calculate the gain,
K, of the system. The slope of the steady-state curve appears rises continuously
throughout the operating range. The average slope from 30% to 45% is 0.04 cm-H2O/%.
The average slope from 50% to 60% is 0.074 cm-H2O/%. The average slope from 75%
to 95% is 0.096 cm-H2O/%.
In order to determine parameter values for several regions of the operating range
several experiments must be conducted. A sample of the resulting response to a step
input is shown below.
Figure 6: Step Up Response
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Matthew Chatham-Tombs - 7 - November 2, 2007
This graph shows the response of the pressure system to a step input of 15%. The
base line input value is 30% and at a time of 25 seconds the input instantaneously steps
up to 45%. It can be seen that ample time was given for the system to reach a steady
state before and after the step takes place. The parameters can also be determined by
means of a step down response. The pressure systems response to a step down is shown
in the graph below.
Figure 7: Step Down Response This graph shows the pressure systems response to a step down input. The base line
value is 45% with a step down of 15% at a time of 25 seconds. Again it can be seen that
enough time has been permitted for the system to reach a steady state before and after the
step. The calculations of the first order parameters using a step up or a step down should
be equal in the same region of the operating range. Once several experiments have been
conducted for each region of the steady-state curve Excel can be used to model the
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Matthew Chatham-Tombs - 8 - November 2, 2007
experimental results. A sample model created using Excel is shown
below.
FOPDT Model
0
5
10
15
20
25
30
35
40
45
50
0 10 20 30 40 50 60 70
Time (s)
Inpu
t (%
)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Out
put (
cm-H
2O)
Input Value(%)InputOutput(cm-H20)Output
Figure 8: Excel First-Order-Plus-Dead-Time Model In the graph shown above the purple output line is the result of the experimental
data. The blue output line is the resulting model created in Excel. If the input function to
an FOPDT is a step function, having a step equal to A and occurring at time equal to td,
the input function m(t)=A*u(t-td). The time response of this system is then,
c(t)=A*u(t-td-t0)*K*(1-e^-[(t-td-t0)/τ]). The derivation of these equations can be found on
page 237 of Principles and Practice of Automatic Process Control by Smith and
Corripio. Using Excel and the time response of the system a model of the experimental
data can be created. By manipulating the parameters K, the gain, t0, the dead time, and τ,
the time constant an accurate representation of the experimental data can be obtained.
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Matthew Chatham-Tombs - 9 - November 2, 2007
The blue output line in the figure above was created with the time response function and
the manipulation of the first order parameters.
The figure below shows the way in which the experimental values of the first-
order parameters were obtained in a previous lab.
Figure 9: Fit 2 Method for determining first order parameters. Figure 9 is a representation of the fit 2 method for determining the first order
parameters for the system. The gain, K, is calculated using the equation Δc/Δm. The
dead time, t0, is determined by drawing a line tangent to the steepest part of the rising
input and determining how long after the step occurred this tangent line crosses the input
baseline. The time constant, τ, is determined by the amount of time after the dead time it
takes for the output to reach 63.2% of the value of Δc.
Δc Δm .632(Δc)
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Matthew Chatham-Tombs - 10 - November 2, 2007
Figure 8 shows the results of the modeling done in Excel. By manipulating the
first order parameters a very accurate representation of the experimental data was created.
This was done for several experiments for a range of input operating values. The
parameters determined from this modeling were analyzed using the Student’s T method.
When such a small number of data points are collected the standard deviation is not a
desirable way to determine the accuracy of the results. Using the Student’s T method the
uncertainty= (c(t)max – c(t)min)*t/n. A table of the values of t/n, depending on the number
of experimental results, is presented on the website
http://chem.engr.utc.edu/engr329/Lab-manual/Students-T.htm.
The figures below show the results for experimental and modeling values
determined for the system parameter gain, K. These results are in the operating ranges
from 30%-45%, 50%-60%, and 75%-95%.
Average Gain, K 30%-45%
0
0.01
0.02
0.03
0.04
0.05
1
Gai
n (c
m-H
2O/%
)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
Figure 10: Average Gain, K 30%-45%
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Matthew Chatham-Tombs - 11 - November 2, 2007
Average Gain, K 50%-60%
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
1
Gai
n (c
m-H
2O/%
)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
Figure 11: Average Gain, K 50%-60%
Average Gain, K 75%-95%
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
1
Gai
n (c
m-H
2O/%
)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
Figure 12: Average Gain, K 75%-95%
These three figures show the average gain calculated in different regions of the
steady-state operating curve. The error bars are a representation of the error analysis
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Matthew Chatham-Tombs - 12 - November 2, 2007
conducting using the Student’s T method. The experimental averages differ from the
modeling values due to the methods in which they were obtained. The data falls within
the same range when the uncertainty is taken into account for the calculations.
The figures below show the results for experimental and modeling values
determined for the system parameter dead time, t0. These results are in the operating
ranges from 30%-45%, 50%-60%, and 75%-95%.
Average Dead Time, t0 30%-45%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1
Dea
d T
ime
(s)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
Figure 13: Average Dead Time 30%-45%
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Matthew Chatham-Tombs - 13 - November 2, 2007
Average Dead Time, t0 50%-60%
0
0.1
0.2
0.3
0.4
0.5
0.6
1
Dea
d T
ime
(s)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
Figure 14: Average Dead Time 50%-60%
Average Dead Time, t0 75%-95%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
Dea
d T
ime
(s)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
Figure 15: Average Dead Time 75%-95%
These three figures show the average dead time calculated for different input
values. The experimental and modeling values differ due to the method by which each
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Matthew Chatham-Tombs - 14 - November 2, 2007
were obtained. The experimental method, the fit 2 method, is very subjective. The
tangent line needed in order to obtain dead time is determined by the researcher. The
tangent line is placed tangent to the steepest part of the output increase, which may be
difficult to determine.
The figures below show the results for experimental and modeling values
determined for the system parameter time constant, τ. These results are in the operating
ranges from 30%-45%, 50%-60%, and 75%-95%.
Average Time Constant, τ 30%-45%
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1
Tim
e C
onst
ant (
s)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
Figure 16: Average Time Constant 30%-45%
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Matthew Chatham-Tombs - 15 - November 2, 2007
Average Time Constant, τ 50%-60%
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1
Tim
e C
onst
ant (
s)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
Figure 17: Average Time Constant 50%-60%
Average Time Constant, τ 75%-95%
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1
Tim
e C
onst
ant (
s)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
Figure 18: Average Time Constant 75%-95%
These three figures show the average time constants evaluated for different
regions of the steady state curve. The experimental results were determined using the fit
2 method while the modeling results were obtained by creating a model of the step
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Matthew Chatham-Tombs - 16 - November 2, 2007
response graph utilizing Excel. These values are fairly consistent throughout the
different ranges. The accuracy of the results were analyzed using the Student’s T method
and are shown using the error bars.
For the sine experiment, the input to the system is represented as a sinusoidal
wave with a specified frequency. These experiments were performed using the website
http://chem.engr.utc.edu/green-engineering/Booth-Pressure/Booth-Pressure-System-
Sine.htm. The experiment requires a specified baseline input value, the amplitude of the
sine wave A, the frequency of the sine wave f, and the length of the experiment. The
following graph is an example from one of the experiments:
Figure 19: Sine Input Response 75%-95%, 0.3Hz The graph above is a sine response for the pressure system. The baseline input value is
85%, the amplitude of the sine wave is 10, the frequency of the sine wave is 0.3Hz, and
the length of the experiment is 20 seconds. The input is shown in the blue and the output
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Matthew Chatham-Tombs - 17 - November 2, 2007
is shown in the red. Notice that it takes about seven seconds for the transients to die out.
It is important to run the experiment long enough to get enough cycles to get correct
information from the output sine wave. The transients take about 6*τ to die out and one
cycle takes about 1/f. For the experiments being performed the time used was
(6*τ)+(3∗1/f) so that three cycles could be obtained, three measurements could be taken,
and the error could be found using the student’s T described previously.
Sine Response: 0.3Hz
0
20
40
60
80
100
120
10 12 14 16 18 20
Time (sec)
Pow
er In
put (
%)
0
0.5
1
1.5
2
2.5
3
Pressure Output (cm-H2O/%)
Input Value(%)Output(cm-H20)
Δc
Δm
Figure 20: Amplitude Ratio Calculation, 75-95% Range
The figure above shows a graph made in Excel from the data collected from one
of the experiments performed online. The x-axis scale ranges from 10 to 20 seconds so
that the area of interest is easier to see. The green arrows show twice the amplitude of
m(t) and the red arrows show twice the amplitude of c(t). Similar to the way the steady-
state gain was found in previous labs, the amplitude ratio (AR) is the ratio of Δc to Δm or
(Δc/Δm). As the frequency of the input sine wave increases, the AR decreases.
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Matthew Chatham-Tombs - 18 - November 2, 2007
Sine Response: 0.3Hz
0
20
40
60
80
100
120
10 12 14 16 18 20
Time (sec)
Pow
er In
put (
%)
0
0.5
1
1.5
2
2.5
3
Pressure Output (cm-H2O/%)
Input Value(%)Output(cm-H20)
T
t
Figure 21: Phase Angle Calculation
Figure 22 shows the previous graph, but here the phase angle is being calculated
here. The time that it takes the input to complete one cycle is represented here by “T”.
The time from the peak of the input to the peak of the output is represented here by “t”.
The phase shift is the fraction of a cycle that the output lags behind the input, (t/T). This
can be represented in degrees as the phase angle (PA) by multiplying the phase shift by
360, or (360*t/T). As the frequency of the input sine wave increases, the PA decreases.
Repeating this experiment at varying frequencies and recording the AR and the
PA at these frequencies makes it possible to determine the gain (K) of the system, the
time constant (τ )of the system, the dead time (to) of the system, apparent order (m), the
ultimate frequency (f u), and the ultimate gain (Kcu) with a Bode Plot.
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Matthew Chatham-Tombs - 19 - November 2, 2007
Phase Angle vs. Frequency
-250.00
-200.00
-150.00
-100.00
-50.00
0.00
0.01 0.1 1 10
Frequency (Hz)
Phas
e A
ngle
-180°
fu
Figure 22: Calculation of Ultimate Frequency
The previous graph is an example graph of the PA versus frequency Bode plot. Notice
that the scale of the x-axis is logarithmic, but y-axis is not. The frequency where the
phase angle is -180° is known as the ultimate frequency (fu).
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Matthew Chatham-Tombs - 20 - November 2, 2007
Bode Plot, AR vs Frequency
0.00
0.00
0.00
0.01
0.10
1.00
0.01 0.1 1 10
Frequency (Hz)
Am
plitu
de R
atio
(cm
-H2O
/%)
Slope=-2
K
1/Kcu
Figure 23: AR vs Frequency, Bode Plot In the Bode plot above, the AR was graphed versus the frequency on a log-log plot. The
AR at the ultimate frequency is equal to 1/Kcu. As the frequencies become smaller, the
values for the AR approach an asymptote, which is the gain for the system. The order is
also found using this Bode plot, which is the negative slope of the plot at the high
frequencies. Using the values found using the Bode plots, the dead time and the time
constant can be found using the following equations:
ω=2πf
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Matthew Chatham-Tombs - 21 - November 2, 2007
Approximate FOPDT model Bode plots for the system can be made in Excel
using the parameters found using the experimental Bode plots as a starting point. The
input sine function to a FOPDT can be expressed m(t) = A*sin(2πft), where A is the
amplitude and t is the time. The output response of the system is represented as
. Because the variables m(t) and c(t) in these
equations are deviation variables, it is necessary to add input baseline and output baseline
to the values to get them to agree with the experimental data. The equations for modeling
the sine response experiment will then be AR= K/SQRT(1+(2*PI()*A10)^2*τ ^2) and
PA= (ATAN(-2*PI()*A10*τ)-2*PI()*A10*to)*180/PI(). Creating Bode plots with both
the experimental data and the modeling data will show the accuracy of the modeling data.
Now the parameters can be changed to find the values of the parameters that make the
model agree with the experimental results.
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Matthew Chatham-Tombs - 22 - November 2, 2007
Bode Plot, Pressure System
0.000
0.000
0.001
0.010
0.100
1.0000.001 0.1 10 1000
Frequency, m, (Hz)
Am
plitu
de R
atio
, c, (
cm-H
2O/%
)
Figure 24: AR vs Frequency, Bode Plot
-300
-250
-200
-150
-100
-50
00.00 0.01 0.10 1.00 10.00 100.00
Frequency, m, (Hz)
Phas
e A
ngle
, (de
gree
s)
Figure 25: PA vs Frequency, Bode Plot
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Matthew Chatham-Tombs - 23 - November 2, 2007
The two graphs above are examples of modeling the Bode plots in Excel. The blue line is
the experimental data and the pink line is the modeling data. The modeling parameters
will agree with experimental data when both sets of data agree at lower frequencies on
the AR vs. frequency Bode plot and when both curves agree at -180° on the PA vs.
frequency Bode plot.
The transfer function for a FOPDT system, mentioned previously, is the
following:
Pade’s approximation, found on page 219 of Principles and Practice of Automatic
Process Control by Smith and Corripio, can be used to simplify the exponential function
in the transfer function. Substituting in Pade’s approximation and simplifying
algebraically yields the following:
Notice that the denominator in the equation above is a second order polynomial with
descending powers of “s”. For a proportional feedback controller, the controller transfer
function is Gc(s)=Kc, so the open loop transfer function (OLTF) for a FOPDT with
proportional control becomes the following:
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Matthew Chatham-Tombs - 24 - November 2, 2007
The closed loop transfer function (CLTF)= OLTF/(1+OLTF). The denominator set equal
to zero is known as the characteristic equation.
To find the values for Kc that give critical damped response, under-damped
responses, and the ultimate value of Kc, Kcu, we can use the characteristic equation to
create a root locus plot. This is done by solving for the roots of the characteristic equation
using the quadratic equation, then plotting the real roots along the x-axis and the
imaginary roots along the y-axis.
ROOT LOCUS PLOT
-10
-8
-6
-4
-2
0
2
4
6
8
10
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2
REAL AXIS
IMA
GIN
AR
Y A
XIS
Kcu
Kcd
Under damped Kc's
Figure 26: Root Locus Plot
The graph above is an example of a root locus plot. The place where the smallest values
of imaginary roots are found is the Kc value for critical decay. When the imaginary roots
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Matthew Chatham-Tombs - 25 - November 2, 2007
cross where the x-axis is equal zero is known as the ultimate Kc value. Between the
critical decay point and the ultimate decay point is a region where the Kc values give
under-damped responses. In the under-damped region, certain Kc values for different
decay ratios correspond to certain angles. The Kc value for 1/500 decay is at 45°, so
where (imaginary roots/real roots) =tan-1(45°), the Kc value here will give 1/500 decay.
The Kc value is important for a proportional controller, which is the simplest type
of controller. The equation that describes the operation of this controller is m(t)=
m*+Kc[r(t)-c(t)], where m(t) is the controller output, m* is the bias, Kc is the controller
gain, r(t) is the set point, and c(t) is the controlled variable. In the previous equation, [r(t)-
c(t)] represents the error, and so the output of the controller is proportional to the error.
The fact that the controller will only have one tuning parameter is a strong point, but the
controller will have to operate with offset and will not return to the set point.
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Matthew Chatham-Tombs - 26 - November 2, 2007
Procedure:
The main objectives of this lab is to observe the response of the output function of
the system to a sine input, to observe the system gain, K, the dead time, t0, and the time
constant, τ, create bode plots and root locus plots for the experiment, and to observe these
parameters in several regions of the steady-state curve. In order to accomplish these
objectives it is necessary begin by performing sine response experiments at varying
frequencies for different regions of the SSOC using the web site
http://chem.engr.utc.edu/green-engineering/Booth-Pressure/Booth-Pressure-System-
Sine.htm. Once the phase angles and amplitude ratios have been found using the method
described previously, the Bode plots can then be made.
The Bode plots can now be used to mathematically model the sine response of
the system using Excel. Columns “A” and “D” of the Excel file contain the frequencies
that were used for the sine response experiment. Column “B” contains the AR’s and
column “E” contains the PA’s calculated during the sine response experiment. Below the
frequencies used in the experiment will be the model frequencies, beginning at 0.001Hz
and increasing by 25%. In column “H”, the values for K, t0 and τ found in the sine
response experiment are placed respectively. In column “C”, beginning in the same row
that the modeling frequencies begin, is where the model AR begins. The equation for the
model AR is =K/SQRT(1+(2*PI()*A10)^2*τ^2). In column “F”, beginning in the same
row that the model frequencies begins, is where the model PA begins. The equation for
the PA is =(ATAN(-2*PI()*A10* τ)-2*PI()*A10*t0)*180/PI(). Now the Bode plots can
be created, graphing the experimental values and the model values together. The
parameters K, t0, and t can then be manipulated until the model data agrees with the
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Matthew Chatham-Tombs - 27 - November 2, 2007
experimental data. The correct value for K has been found when the modeling data and
the experimental data agree at low frequencies on the AR vs. frequency Bode plot. The
correct value of τ has been bound when the modeling data and the experimental data
agree near the "corner" of the AR vs. frequency curve. The correct value of t0 has been
bound when the modeling data and the experimental data agree at -180° on the PA vs.
frequency Bode plot. The values found for K, τ, and t0 will be used from now on.
In order to create a Root Locus plot, the roots of the characteristic equation, described in
the background section, must be found. This was done using Excel and the quadratic
equation. The values for K, τ, and t0 are placed in column “H” of the Excel file, as well as
the change in Kc used to find and plot the roots. Column “I” contains the terms from the
characteristic equation with the term “s2”, or the “a” for the quadratic equation. The “J”
column contains the terms from the characteristic equation with the term “s”, or the “b”
for the quadratic equation. The “K” column contains the terms from the characteristic
equation with no “s” term, or the “c” for the quadratic equation. The “L” column
contains the portion of the quadratic equation “sqrt(b2-4ac)”. Column “A” will be the Kc
values, increasing by ΔKc. Column “B” and “D” contain the arguments =(-
J2+IF(L2<0,0,SQRT(L2)))/2/I2 and =(-J2-IF(L2<0,0,SQRT(L2)))/2/I2 respectively, to
determine the real roots. Columns “C” and “E” contain the arguments =IF(L2<0,SQRT(-
L2)/2/I2,0) and =IF(L2<0,-SQRT(-L2)/2/I2,0) respectively, to determine the imaginary
roots. Column “F” divides the absolute value of the second imaginary root row by the
absolute value of the second reel root row to determine different decay ratios. When
column “F”=tan(45°), the value of Kc gives a decay ratio of 1/500. When column
“F”=tan(70°), the value of Kc gives a decay ratio of 1/10. When column “F”=tan(77°),
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Matthew Chatham-Tombs - 28 - November 2, 2007
the value of Kc gives a decay ratio of 1/4. Because the roots of the characteristic equation
are real and imaginary, they are plotted on a graph where the x-axis is real and the y-axis
is imaginary. The place where the smallest values of imaginary roots are found is the Kc
value for critical decay. When the imaginary roots cross where the x-axis is equal zero is
known as the ultimate Kc value.
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Matthew Chatham-Tombs - 29 - November 2, 2007
Results: During the sine response experiment, described previously, data was collected at
several different frequencies.
Sine Response: 0.3Hz
0
20
40
60
80
100
120
10 12 14 16 18 20
Time (sec)
Pow
er In
put (
%)
0
0.5
1
1.5
2
2.5
3
Pressure Output (cm-H2O/%)
Input Value(%)Output(cm-H20)
Δc
Δm
Figure 27: Sine Response, 75-95%Range
The graph shown above is from a sine experiment performed over the 75-95% range of
the SSOC. The frequency used during the experiment was 0.3Hz. The first measurement
of the amplitudes is shown. The AR for this experiment is Δc/Δm=0.9/20=0.045. This
was repeated for the other two peaks so that error could be calculated.
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Matthew Chatham-Tombs - 30 - November 2, 2007
Sine Response: 0.3Hz
0
20
40
60
80
100
120
10 12 14 16 18 20
Time (sec)
Pow
er In
put (
%)
0
0.5
1
1.5
2
2.5
3
Pressure Output (cm-H2O/%)
Input Value(%)Output(cm-H20)
T
t
Figure 28: Sine Response 75-95% Range
The graph above shows a sine experiment performed over the 75-95% range of the
SSOC. The frequency used during the experiment was 0.3Hz. The first measurement of
the lag time is shown. Here, T=1/f=3.33s, t=1s, and PA=(t/T)*360=108°. This was
repeated for the other two peaks so that error could be calculated.
Frequency, Hz Amplitude Ratio (cm-H2O/%) AR Error (+/-) PA (°) PA Error (+/-)0.02 0.08 0.007 15 20.05 0.07 0.007 29 30.1 0.06 0.006 58 60.2 0.03 0.003 89 90.4 0.02 0.002 112 110.8 0.006 0.0006 167 171.6 0.002 0.0002 207 213.2 0.0001 0.00001 230 23
Figure 29: AR, PA, and Error for 75-95 %Range
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Matthew Chatham-Tombs - 31 - November 2, 2007
Frequency, Hz Amplitude Ratio (cm-H2O/%) AR Error (+/-) PA (°) PA Error (+/-)0.01 0.08 0.005 7 0.10.03 0.08 0.001 23 5.70.05 0.06 0.007 39 8.40.1 0.05 0.001 54 1.00.2 0.04 0.004 82 13.20.4 0.02 0.001 95 16.90.8 0.01 0.003 162 36.7
2 0.009 0.001 212 52.3
Figure 30: AR, PA, and Error for 50-60 %Range Frequency, Hz Amplitude Ratio (cm-H2O/%) AR Error (+/-) PA (°) PA Error (+/-)
0.01 0.04 0.004 2 0.20.02 0.04 0.004 5 0.50.03 0.04 0.004 13 10.05 0.04 0.004 32 30.1 0.03 0.003 51 50.2 0.02 0.002 60 60.4 0.01 0.001 86 90.8 0.005 0.0005 144 14
2 0.0009 0.0001 266 27 Figure 31: AR, PA, and Error for 36-40 %Range
The figures above show the data collected in the 75-95% range, the 50-60% range, and
the 36-40% range of the SSOC. The frequencies are arranged in ascending order. The
AR, PA, and the error are shown for each frequency.
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Matthew Chatham-Tombs - 32 - November 2, 2007
Bode Plot, Phase Angle vs. Frequency
-300
-250
-200
-150
-100
-50
0
0.01 0.1 1 10
Frequency (Hz)
Phas
e A
ngle
(deg
rees
)
-180°
fu
MCT10/4/07
Figure 32: PA vs. Frequency Bode Plot, 75-95% Range
Bode Plot, Phase Angle vs. Frequency
-300
-250
-200
-150
-100
-50
0
0.01 0.1 1 10
Frequency (Hz)
Phas
e A
ngle
(deg
rees
)
-180°
fu
JB 10/4/07
Figure 33: PA vs. Frequency Bode Plot, 50-60% Range
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Matthew Chatham-Tombs - 33 - November 2, 2007
Bode Plot, Phase Angle vs. Frequency
-300
-250
-200
-150
-100
-50
0
0.01 0.1 1 10
Frequency (Hz)
Pha
se A
ngle
(deg
rees
)
-180°
fu
EY 10/4/07
Figure 34: PA vs. Frequency Bode Plot, 34-40% Range
The graphs above were created using the sine response data collected for the 75-95%
range, the 50-60% range, and the 36-40% range of the SSOC. These graphs are semi-log
and show the PA vs. the frequency. The error bars are also shown in blue. The ultimate
frequency is shown on each graph at -180°.
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Matthew Chatham-Tombs - 34 - November 2, 2007
Bode Plot, AR vs Frequency
0.00
0.00
0.00
0.01
0.10
1.00
0.01 0.1 1 10
Frequency (Hz)
Am
plitu
de R
atio
(cm
-H2O
/%)
Slope=-2
K
1/Kcu
MCT 10/4/07
Figure 35: AR vs. Frequency Bode Plot, 75-95% Range
Bode Plot, AR vs Frequency
0.00
0.01
0.10
1.00
0.01 0.1 1 10
Frequency (Hz)
Am
plitu
de R
atio
(cm
-H2O
/%)
Slope=-2
K
1/Kcu
JB 10/4/07
Figure 36: AR vs. Frequency Bode Plot, 50-60% Range
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Matthew Chatham-Tombs - 35 - November 2, 2007
Bode Plot, AR vs Frequency
0.00
0.00
0.01
0.10
1.00
0.01 0.1 1 10
Frequency (Hz)
Am
plitu
de R
atio
(cm
-H2O
/%)
Slope=-2
K
1/Kcu
EY 10/4/07
Figure 37:AR vs. Frequency Bode Plot, 36-40% Range
The graphs above were created using the sine response data collected for the 75-95%
range, the 50-60% range, and the 36-40% range of the SSOC. These graphs ate log-log
and show the AR vs. the frequency. The error bars are so small that they are not
noticeable.
75-95% Range 50-60% Range 36-40% RangeK (cm-H20/%) 0.085 0.08 0.04System Order 2nd 2rd 2thfu (Hz) 1 1.2 1Kcu (%/cm-H2O) 250 77 330t0 (s) 0.26 0.62 0.27τ (s) 1.8 2.1 1.7
Figure 38: Sine Response Results
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Matthew Chatham-Tombs - 36 - November 2, 2007
The table above is the data collected from each range for the sine response experiment.
The upper and middle regions of the SSOC have similar K values, but the lower region is
almost half of the other two. The upper and lower regions have similar Kcu and t0 values,
but the middle region differs by more than fifty percent. It is possible that there was an
error during the experiments with one or more of the regions.
Once the experimental Bode plots are made, the modeling Bode plots can use the
experimental plots to find the correct parameters as described previously.
Bode Plot, Pressure System
-300
-250
-200
-150
-100
-50
00.00 0.01 0.10 1.00 10.00 100.00
Frequency, m, (Hz)
Phas
e A
ngle
, (de
gree
s)
Exp.Model
MCT10/4/07
Figure 39: Model PA vs Frequency Bode Plot, 75-95% Range
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Matthew Chatham-Tombs - 37 - November 2, 2007
Bode Plot, Pressure System
-300
-250
-200
-150
-100
-50
00.001 0.01 0.1 1 10 100
Frequency, m, (Hz)
Phas
e A
ngle
, (de
gree
s)
Exp.Model
JB10/4/07
Figure 40: Model PA vs Frequency Bode Plot, 50-60% Range
Bode Plot, Pressure System
-300
-250
-200
-150
-100
-50
00.001 0.01 0.1 1 10 100
Frequency, m, (Hz)
Phas
e A
ngle
, (de
gree
s)
Exp.Model
EY10/4/07
Figure 41: Model PA vs Frequency Bode Plot, 36-40% Range
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Matthew Chatham-Tombs - 38 - November 2, 2007
The graphs above were created using the sine response data collected and the modeling
data for the 75-95% range, the 50-60% range, and the 36-40% range of the SSOC. These
graphs are semi-log and show the PA vs. the frequency. The model parameters are correct
because the experimental data and the model data agree at a PA of -180°.
Bode Plot, Pressure System
0.000
0.000
0.001
0.010
0.100
1.0000.001 0.1 10 1000
Frequency, m, (Hz)
Am
plitu
de R
atio
, cm
-H2O
/%)
Exp.Model
MCT10/4/07
Figure 42: Model AR vs Frequency Bode Plot, 75-95% Range
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Matthew Chatham-Tombs - 39 - November 2, 2007
Bode Plot, Pressure System
0.00001
0.0001
0.001
0.01
0.1
10.001 0.1 10 1000
Frequency, m, (Hz)
Am
plitu
de R
atio
, cm
-H2O
/%)
Exp.Model
JB10/4/07
Figure 43: Model AR vs Frequency Bode Plot, 50-60% Range
Bode Plot, Pressure System
0.0001
0.001
0.01
0.1
10.001 0.01 0.1 1 10
Frequency, m, (Hz)
Am
plitu
de R
atio
, cm
-H2O
/%)
Exp.Model
EY10/4/07
Figure 44: Model AR vs Frequency Bode Plot, 36-40% Range
The graphs above were created using the sine response data collected and the modeling
data for the 75-95% range, the 50-60% range, and the 36-40% range of the SSOC. These
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Matthew Chatham-Tombs - 40 - November 2, 2007
graphs are log-log and show the AR vs. the frequency. The model parameters are correct
because the model data and the experimental data agree at lower frequencies.
75-95% Range 50-60% Range 36-40% RangeK= 0.08 0.08 0.04t0= 0.3 0.2 0.23τ= 1.8 1.95 1.4
Figure 45: Modeling Parameters for SSOC
The table above is of the modeling parameters for the upper, middle, and lower portion of
the SSOC. The gain is the first row, the second is the dead time, and the last row is the
time constant. These are the parameters that are used to create root locus plots of each
region.
To create root locus plots, the roots of the system characteristic equation must be
found using the method described previously.
ROOT LOCUS PLOT
-10
-8
-6
-4
-2
0
2
4
6
8
10
-9 -7 -5 -3 -1 1
REAL AXIS
IMA
GIN
AR
Y A
XIS
MCT10/24/07
Figure 46: Root Locus Plot, 75-95% Range
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Matthew Chatham-Tombs - 41 - November 2, 2007
ROOT LOCUS PLOT
-15
-10
-5
0
5
10
15
-12 -10 -8 -6 -4 -2 0 2
REAL AXIS
IMA
GIN
AR
Y A
XIS
JB10/24/07
Figure 47: Root Locus Plot, 50-60% Range
ROOT LOCUS PLOT
-15
-10
-5
0
5
10
15
-9.5 -7.5 -5.5 -3.5 -1.5 0.5
REAL AXIS
IMA
GIN
AR
Y A
XIS
EY10/24/07
Figure 48: Root Locus Plot, 36-40% Range
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Matthew Chatham-Tombs - 42 - November 2, 2007
The three plots above are the root locus plots for the 75-95% range, the 50-60% range,
and the 36-40% range of the SSOC. The x-axis is for the real roots and the y-axis is for
the imaginary roots. The place where the smallest values of imaginary roots are found is
the Kc value for critical decay. When the imaginary roots cross where the x-axis is equal
zero is known as the ultimate Kc value. Between the critical decay point and the ultimate
decay point is a region where the Kc values give under-damped responses.
Response to step change in set-point Symbol Value (cm-H2O/%)Critical Damping KCD 21.61/500th decay KC500 391/10th decay KC10 82Quarter-decay (under damping) KQD 104"Ultimate" (Marginal stability) Kcu 163
Figure 49: K Values for 75-95% Range
Response to step change in set-point Symbol Value (cm-H2O/%)Critical Damping KCD 361/500th decay KC500 62.41/10th decay KC10 126Quarter-decay (under damping) KQD 160"Ultimate" (Marginal stability) Kcu 253
Figure 50: K Values for 50-60% Range
Response to step change in set-point Symbol Value (cm-H2O/%)Critical Damping KCD 421/500th decay KC500 78.61/10th decay KC10 165Quarter-decay (under damping) KQD 209"Ultimate" (Marginal stability) Kcu 329
Figure 51: K Values for 36-40% Range
The tables presented above are K values for the 75-95% range, the 50-60% range, and the
36-40% range of the SSOC. The values calculated are for critical damping, 1/500th decay,
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Matthew Chatham-Tombs - 43 - November 2, 2007
1/10th decay, 1/4th decay, and the ultimate K value. These values represent the effective
range of the controller gain.
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Matthew Chatham-Tombs - 44 - November 2, 2007
Discussion: The results of the experimental data and the modeling data were similar for the
sine response experiment. The variances in these results were due to the methods by
which each were obtained. The sine response experimental values were determined using
the frequency sine response graphs. This method depends on the perspective of the
person collecting the data and can therefore vary slightly depending on the person. One
researcher may have different values from another on the same graph due to what they
see as the peak of the response. The following description of the average values of the
laboratory experiments come directly from the charts presented in the results section of
this report.
The sine response experimental average values for the gain, system order,
ultimate frequency, ultimate gain, the dead time, and the time constant for the range from
36-40% were determined to be .04cm-H2O/%, 2nd order, 1Hz, 330 %/cm-H2O, 0.27s,
and 1.7s respectively. The sine response model values for the gain, the dead time, and the
time constant for the range from 36-40% were determined to be .04cm-H2O/%, 0.23s,
and 1.4s respectively. The results for the gain obtained from each of the methods above
are similar and corresponded well to the slope of the steady-state curve in the same range.
The dead times and the time constants are also very similar.
The sine response experimental average values for the gain, system order,
ultimate frequency, ultimate gain, the dead time, and the time constant for the range from
50-60% were determined to be .08cm-H2O/%, 2nd order, 1.2Hz, 77 %/cm-H2O, 0.62s,
and 2.1s respectively. The sine response model values for the gain, the dead time, and the
time constant for the range from 50-60% were determined to be .08cm-H2O/%, 0.2s, and
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Matthew Chatham-Tombs - 45 - November 2, 2007
1.95s respectively. The results for the gain obtained from each of the methods above are
similar to each other and the slope calculated from the steady-state curve in the same
range. The time constants are also similar. The dead times differ by more than fifty
percent. This means that some error occurred when finding the dead time during the
experimental portion of the sine response experiment of the modeling portion.
The sine response experimental average values for the gain, system order,
ultimate frequency, ultimate gain, the dead time, and the time constant for the range from
75-95% were determined to be .085cm-H2O/%, 2nd order, 1Hz, 250 %/cm-H2O, 0.26s,
and 1.8s respectively. The sine response model values for the gain, the dead time, and the
time constant for the range from 75-95% were determined to be .08cm-H2O/%, 0.3s, and
1.8s respectively. The results for the gain obtained from each of the methods above are
similar to each other and the slope calculated from the steady-state curve in the same
range. The dead times and the time constants are also similar.
From the root locus plots, the effective range of the controller gain for each region
was determined. The effective range of the controller gain for the 36-40% region, the 50-
60% region, and the 75-95% region are 42-329 (cm-H2O/%), 36-253 (cm-H2O/%), and
22-163 (cm-H2O/%) respectively. Notice that the effective range of the controller gain
becomes smaller when moving from the lower range to the higher ranges of the SSOC.
The controller for the system will need to operate within a different effective controller
gain range depending on which region of the SSOC the system will be operating within.
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Matthew Chatham-Tombs - 46 - November 2, 2007
Conclusion:
The purpose of this lab was to observe the time response of the output function of
the system to a sine function input at different frequencies, to determine the first order
parameters for the mathematical model of the system, to compare the experimental Bode
plots with the approximate FOPDT model's Bode plots for the system, and to determine
the effective range of the controller gain. The method for evaluating the response of the
system was to provide an input pf varying frequencies. The response of the system to this
input is called the “sine input response” of the system. The main objectives of this lab
were to observe the response of the output function of the system to a sine input, to
observe the system gain, K, the dead time, t0, and the time constant, τ, create bode plots
and root locus plots for the experiment, to observe effective range of the controller gain,
and to observe these parameters in several regions of the steady-state curve. These
parameters were determined using Excel to create models of the sine input response and
create root locus plots. The manipulation of parameters in Excel was crucial in
developing an accurate model for the sine response experimental and in creating root
locus plots to determine the effective controller gain ranges. There did appear to be an
error during one or more of the sine input response experiments and repeating the sine
input response experiment should be taken into consideration.
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Matthew Chatham-Tombs - 47 - November 2, 2007
Appendix: Principles and Practice of Automatic Process Control by Smith and Corripio http://chem.engr.utc.edu/engr329/Lab-manual/Students-T.htm.
http://chem.engr.utc.edu/green-engineering/Booth-Pressure/Booth-Pressure-System-Sine.htm
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Matthew Chatham-Tombs - 48 - November 2, 2007
Steady State Operating Curve, Pressure
0
1
2
3
4
5
6
7
0 20 40 60 80 100
c, Input (%)
m, P
ress
ure
Out
put (
cm-H
20)
Series1
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Matthew Chatham-Tombs - 49 - November 2, 2007
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Matthew Chatham-Tombs - 50 - November 2, 2007
FOPDT Model
0
5
10
15
20
25
30
35
40
45
50
0 10 20 30 40 50 60 70
Time (s)
Inpu
t (%
)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Out
put (
cm-H
2O)
Input Value(%)InputOutput(cm-H20)Output
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Matthew Chatham-Tombs - 51 - November 2, 2007
Average Gain, K 30%-45%
0.041
0.042
0.043
0.044
0.045
1
Gai
n (c
m-H
2O)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
Average Gain, K 50%-60%
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
1
Gai
n (c
m-H
2O)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
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Matthew Chatham-Tombs - 52 - November 2, 2007
Average Gain, K 75%-95%
0.0910.0920.0930.0940.0950.0960.0970.0980.099
0.10.1010.1020.103
1
Gai
n (c
m-H
2O)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
Average Dead Time, t0 30%-45%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1
Dea
d T
ime
(s)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
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Matthew Chatham-Tombs - 53 - November 2, 2007
Average Dead Time, t0 50%-60%
0
0.1
0.2
0.3
0.4
0.5
0.6
1
Dea
d T
ime
(s)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
Average Dead Time, t0 75%-95%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
Dea
d T
ime
(s)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
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Matthew Chatham-Tombs - 54 - November 2, 2007
Average Time Constant, 30%-45%
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1
Tim
e C
onst
ant (
s)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
Average Time Constant, 50%-60%
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1
Tim
e C
onst
ant (
s)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
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Matthew Chatham-Tombs - 55 - November 2, 2007
Average Time Constant, τ 75%-95%
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1
Tim
e C
onst
ant (
s)
Experimental Step Up AverageModeling Step Up Average
Experimental Step Down AverageModeling Step Down Average
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Matthew Chatham-Tombs - 56 - November 2, 2007
Sine Response: 0.3Hz
0
20
40
60
80
100
120
10 12 14 16 18 20
Time (sec)
Pow
er In
put (
%)
0
0.5
1
1.5
2
2.5
3
Pressure Output (cm-H2O/%)
Input Value(%)Output(cm-H20)
Δc
Δm
Sine Response: 0.3Hz
0
20
40
60
80
100
120
10 12 14 16 18 20
Time (sec)
Pow
er In
put (
%)
0
0.5
1
1.5
2
2.5
3
Pressure Output (cm-H2O/%)
Input Value(%)Output(cm-H20)
T
t
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Matthew Chatham-Tombs - 57 - November 2, 2007
Phase Angle vs. Frequency
-250.00
-200.00
-150.00
-100.00
-50.00
0.00
0.01 0.1 1 10
Frequency (Hz)
Phas
e A
ngle
-180°
fu
Bode Plot, AR vs Frequency
0.00
0.00
0.00
0.01
0.10
1.00
0.01 0.1 1 10
Frequency (Hz)
Am
plitu
de R
atio
(cm
-H2O
/%)
Slope=-2
K
1/Kcu
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Matthew Chatham-Tombs - 58 - November 2, 2007
Bode Plot, Pressure System
0.000
0.000
0.001
0.010
0.100
1.0000.001 0.1 10 1000
Frequency, m, (Hz)
Am
plitu
de R
atio
, c, (
cm-H
2O/%
)
-300
-250
-200
-150
-100
-50
00.00 0.01 0.10 1.00 10.00 100.00
Frequency, m, (Hz)
Phas
e A
ngle
, (de
gree
s)
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Matthew Chatham-Tombs - 59 - November 2, 2007
ROOT LOCUS PLOT
-10
-8
-6
-4
-2
0
2
4
6
8
10
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2
REAL AXIS
IMA
GIN
AR
Y A
XIS
Kcu
Kcd
Under damped Kc's
Frequency, Hz Amplitude Ratio (cm-H2O/%) AR Error (+/-) PA (°) PA Error (+/-)
0.02 0.08 0.007 15 20.05 0.07 0.007 29 30.1 0.06 0.006 58 60.2 0.03 0.003 89 90.4 0.02 0.002 112 110.8 0.006 0.0006 167 171.6 0.002 0.0002 207 213.2 0.0001 0.00001 230 23
Frequency, Hz Amplitude Ratio (cm-H2O/%) AR Error (+/-) PA (°) PA Error (+/-)
0.01 0.08 0.005 7 0.10.03 0.08 0.001 23 5.70.05 0.06 0.007 39 8.40.1 0.05 0.001 54 1.00.2 0.04 0.004 82 13.20.4 0.02 0.001 95 16.90.8 0.01 0.003 162 36.7
2 0.009 0.001 212 52.3
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Matthew Chatham-Tombs - 60 - November 2, 2007
Frequency, Hz Amplitude Ratio (cm-H2O/%) AR Error (+/-) PA (°) PA Error (+/-)0.01 0.04 0.004 2 0.20.02 0.04 0.004 5 0.50.03 0.04 0.004 13 10.05 0.04 0.004 32 30.1 0.03 0.003 51 50.2 0.02 0.002 60 60.4 0.01 0.001 86 90.8 0.005 0.0005 144 14
2 0.0009 0.0001 266 27
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Matthew Chatham-Tombs - 61 - November 2, 2007
Bode Plot, Phase Angle vs. Frequency
-300
-250
-200
-150
-100
-50
0
0.01 0.1 1 10
Frequency (Hz)
Phas
e A
ngle
(deg
rees
)
-180°
fu
MCT10/4/07
Bode Plot, Phase Angle vs. Frequency
-300
-250
-200
-150
-100
-50
0
0.01 0.1 1 10
Frequency (Hz)
Phas
e A
ngle
(deg
rees
)
-180°
fu
JB 10/4/07
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Matthew Chatham-Tombs - 62 - November 2, 2007
Bode Plot, Phase Angle vs. Frequency
-300
-250
-200
-150
-100
-50
0
0.01 0.1 1 10
Frequency (Hz)
Phas
e A
ngle
(deg
rees
)
-180°
fu
EY 10/4/07
75-95% Range 50-60% Range 36-40% RangeK (cm-H20/%) 0.085 0.08 0.04System Order 2nd 2rd 2thfu (Hz) 1 1.2 1Kcu (%/cm-H2O) 250 77 330t0 (s) 0.26 0.62 0.27τ (s) 1.8 2.1 1.7
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Matthew Chatham-Tombs - 63 - November 2, 2007
Bode Plot, Pressure System
-300
-250
-200
-150
-100
-50
00.00 0.01 0.10 1.00 10.00 100.00
Frequency, m, (Hz)
Phas
e A
ngle
, (de
gree
s)
Exp.Model
MCT10/4/07
Bode Plot, Pressure System
-300
-250
-200
-150
-100
-50
00.001 0.01 0.1 1 10 100
Frequency, m, (Hz)
Phas
e A
ngle
, (de
gree
s)
Exp.Model
JB10/4/07
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Matthew Chatham-Tombs - 64 - November 2, 2007
Bode Plot, Pressure System
-300
-250
-200
-150
-100
-50
00.001 0.01 0.1 1 10 100
Frequency, m, (Hz)
Phas
e A
ngle
, (de
gree
s)
Exp.Model
EY10/4/07
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Matthew Chatham-Tombs - 65 - November 2, 2007
Bode Plot, Pressure System
0.000
0.000
0.001
0.010
0.100
1.0000.001 0.1 10 1000
Frequency, m, (Hz)
Am
plitu
de R
atio
, cm
-H2O
/%)
Exp.Model
MCT10/4/07
Bode Plot, Pressure System
0.00001
0.0001
0.001
0.01
0.1
10.001 0.1 10 1000
Frequency, m, (Hz)
Am
plitu
de R
atio
, cm
-H2O
/%)
Exp.Model
JB10/4/07
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Matthew Chatham-Tombs - 66 - November 2, 2007
Bode Plot, Pressure System
0.0001
0.001
0.01
0.1
10.001 0.01 0.1 1 10
Frequency, m, (Hz)
Am
plitu
de R
atio
, cm
-H2O
/%)
Exp.Model
EY10/4/07
75-95% Range 50-60% Range 36-40% RangeK= 0.08 0.08 0.04t0= 0.3 0.2 0.23τ= 1.8 1.95 1.4
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Matthew Chatham-Tombs - 67 - November 2, 2007
ROOT LOCUS PLOT
-10
-8
-6
-4
-2
0
2
4
6
8
10
-9 -7 -5 -3 -1 1
REAL AXIS
IMA
GIN
AR
Y A
XIS
MCT10/24/07
ROOT LOCUS PLOT
-15
-10
-5
0
5
10
15
-12 -10 -8 -6 -4 -2 0 2
REAL AXIS
IMA
GIN
AR
Y A
XIS
JB10/24/07
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Matthew Chatham-Tombs - 68 - November 2, 2007
ROOT LOCUS PLOT
-15
-10
-5
0
5
10
15
-9.5 -7.5 -5.5 -3.5 -1.5 0.5
REAL AXIS
IMA
GIN
AR
Y A
XIS
EY10/24/07
Response to step change in set-point Symbol Value (cm-H2O/%)Critical Damping KCD 21.61/500th decay KC500 391/10th decay KC10 82Quarter-decay (under damping) KQD 104"Ultimate" (Marginal stability) Kcu 163
Response to step change in set-point Symbol Value (cm-H2O/%)Critical Damping KCD 361/500th decay KC500 62.41/10th decay KC10 126Quarter-decay (under damping) KQD 160"Ultimate" (Marginal stability) Kcu 253
Response to step change in set-point Symbol Value (cm-H2O/%)Critical Damping KCD 421/500th decay KC500 78.61/10th decay KC10 165Quarter-decay (under damping) KQD 209"Ultimate" (Marginal stability) Kcu 329