university of tennessee at chattanooga engineering 329...

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


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.


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


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.


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


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


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


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.


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)


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%



0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

1

Gai

n (c

m-H

2O/%

)





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/%

)




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


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.


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)



Figure 13: Average Dead Time 30%-45%



0

0.1

0.2

0.3

0.4

0.5

0.6

1

Dea

d T

ime

(s)





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)




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


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.


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)



Figure 16: Average Time Constant 30%-45%



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)





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)




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


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


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.



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



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.


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).


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


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.


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


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:


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


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.


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


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°),


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.


Results: During the sine response experiment, described previously, data was collected at

several different frequencies.


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



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.



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



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


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.


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


-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




-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


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°.



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


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



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


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.


-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



-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



-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



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°.


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



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



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


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 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


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


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



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




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,


1/10th decay, 1/4th decay, and the ultimate K value. These values represent the effective

range of the controller gain.


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.



50-60% were determined to be .08cm-H2O/%, 2nd order, 1.2Hz, 77 %/cm-H2O, 0.62s,


time constant for the range from 50-60% were determined to be .08cm-H2O/%, 0.2s, and


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.



75-95% were determined to be .085cm-H2O/%, 2nd order, 1Hz, 250 %/cm-H2O, 0.26s,


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.


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.


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


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


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



0.041

0.042

0.043

0.044

0.045

1

Gai

n (c

m-H

2O)




0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

1

Gai

n (c

m-H

2O)





0.0910.0920.0930.0940.0950.0960.0970.0980.099

0.10.1010.1020.103

1

Gai

n (c

m-H

2O)




0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1

Dea

d T

ime

(s)





0

0.1

0.2

0.3

0.4

0.5

0.6

1

Dea

d T

ime

(s)




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)




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)



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)





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)





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



Δc

Δm


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



T

t


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


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



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)


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


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



-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


-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



-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



-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


-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



-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



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


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



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


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


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



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