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

Matthew Chatham-Tombs

-1-

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.

Matthew Chatham-Tombs

-2-

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

Matthew Chatham-Tombs

-3-

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.

Matthew Chatham-Tombs

-4-

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

m, Pressure Output (cm-H20)

7 6 5 4 3 Series1

2 1 0 0

20

40

60

80

100

c, Input (%)

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

Matthew Chatham-Tombs

-5-

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

Matthew Chatham-Tombs

-6-

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 Matthew Chatham-Tombs

-7-

November 2, 2007

experimental results. A sample model created using Excel is shown below.

50

1

45

0.9

40

0.8

35

0.7

30

0.6

25

0.5

20

0.4

15

0.3

10

0.2

5

0.1

0

Output (cm-H2O)

Input (%)

FOPDT Model

Input Value(%) Input Output(cm-H20) Output

0 0

10

20

30

40

50

60

70

Time (s)

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.

Matthew Chatham-Tombs

-8-

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 firstorder parameters were obtained in a previous lab.

Δm

.632(Δc)

Δc

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.

Matthew Chatham-Tombs

-9-

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%

Gain (cm-H2O/%)

0.05

0.04

0.03

Experimental Step Up Average Modeling Step Up Average

0.02

Experimental Step Down Average Modeling Step Down Average

0.01

0 1

Figure 10: Average Gain, K 30%-45%

Matthew Chatham-Tombs

- 10 -

November 2, 2007

Average Gain, K 50%-60% 0.08

Gain (cm-H2O/%)

0.07 0.06 Experimental Step Up Average

0.05

Modeling Step Up Average

0.04 Experimental Step Down Average

0.03

Modeling Step Down Average

0.02 0.01 0 1

Figure 11: Average Gain, K 50%-60%

Average Gain, K 75%-95% 0.11 0.1

Gain (cm-H2O/%)

0.09 0.08 Experimental Step Up Average Modeling Step Up Average

0.07 0.06 0.05

Experimental Step Down Average Modeling Step Down Average

0.04 0.03 0.02 0.01 0 1

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 Matthew Chatham-Tombs

- 11 -

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% 0.7

Dead Time (s)

0.6 0.5

Experimental Step Up Average Modeling Step Up Average

0.4 0.3

Experimental Step Down Average Modeling Step Down Average

0.2 0.1 0 1

Figure 13: Average Dead Time 30%-45%

Matthew Chatham-Tombs

- 12 -

November 2, 2007

Average Dead Time, t0 50%-60% 0.6

Dead Time (s)

0.5 Experimental Step Up Average Modeling Step Up Average

0.4 0.3

Experimental Step Down Average 0.2

Modeling Step Down Average

0.1 0 1

Figure 14: Average Dead Time 50%-60%

Average Dead Time, t0 75%-95% 1 0.9

Dead Time (s)

0.8 0.7

Experimental Step Up Average Modeling Step Up Average

0.6 0.5

Experimental Step Down Average Modeling Step Down Average

0.4 0.3 0.2 0.1 0 1

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

Matthew Chatham-Tombs

- 13 -

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% 2

Time Constant (s)

1.8 1.6 1.4

Experimental Step Up Average Modeling Step Up Average

1.2 1

Experimental Step Down Average Modeling Step Down Average

0.8 0.6 0.4 0.2 0 1

Figure 16: Average Time Constant 30%-45%

Matthew Chatham-Tombs

- 14 -

November 2, 2007

Average Time Constant, τ 50%-60% 1.8

Time Constant (s)

1.6 1.4 Experimental Step Up Average Modeling Step Up Average

1.2 1 0.8

Experimental Step Down Average Modeling Step Down Average

0.6 0.4 0.2 0 1

Figure 17: Average Time Constant 50%-60%

Average Time Constant, τ 75%-95% 2

Time Constant (s)

1.8 1.6 1.4

Experimental Step Up Average Modeling Step Up Average

1.2 1

Experimental Step Down Average Modeling Step Down Average

0.8 0.6 0.4 0.2 0 1

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

Matthew Chatham-Tombs

- 15 -

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-SystemSine.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

Matthew Chatham-Tombs

- 16 -

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

Power Input (%)

Pressure Output (cm-H2O/% ) 120

3

100

2.5 Δc

80 60

2 1.5

Δm

40

1

20

0.5

0

Input Value(%) Output(cm-H20)

0 10

12

14

16

18

20

Time (sec) 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 steadystate 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.

Matthew Chatham-Tombs

- 17 -

November 2, 2007

Sine Response: 0.3Hz Pressure Output (cm-H2O/% ) 120

3 T

Power Input (%)

100

2.5

80

2 t

60

1.5

40

1

20

0.5

0

Input Value(%) Output(cm-H20)

0 10

12

14

16

18

20

Time (sec) 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.

Matthew Chatham-Tombs

- 18 -

November 2, 2007

Phase Angle vs. Frequency 0.00

Phase Angle

-50.00

-100.00

-150.00

-180°

-200.00

-250.00 0.01

0.1

1

fu

10

Frequency (Hz)

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).

Matthew Chatham-Tombs

- 19 -

November 2, 2007

Bode Plot, AR vs Frequency 1.00 K

Amplitude Ratio (cm-H2O/%)

0.10

0.01

0.00

1/Kcu Slope=-2

0.00

0.00 0.01

0.1

1

10

Frequency (Hz)

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

Matthew Chatham-Tombs

- 20 -

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.

Matthew Chatham-Tombs

- 21 -

November 2, 2007

Bode Plot, Pressure System

0.001

0.1

10

1000

Amplitude Ratio, c, (cm-H2O/%)

1.000 0.100 0.010 0.001 0.000 0.000 Frequency, m, (Hz)

Figure 24: AR vs Frequency, Bode Plot

0.00

0.01

0.10

1.00

10.00

100.00 0

-100 -150 -200

Phase Angle, (degrees)

-50

-250 -300 Frequency, m, (Hz)

Figure 25: PA vs Frequency, Bode Plot

Matthew Chatham-Tombs

- 22 -

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:

Matthew Chatham-Tombs

- 23 -

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

Kcu 10

Under damped Kc's

8 6

IMAGINARY AXIS

4

Kcd 2 0 -8

-7

-6

-5

-4

-3

-2

-1

0

1

2

-2 -4 -6 -8 -10 REAL AXIS

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

Matthew Chatham-Tombs

- 24 -

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.

Matthew Chatham-Tombs

- 25 -

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-SystemSine.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

Matthew Chatham-Tombs

- 26 -

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