1 University of Tennessee at Chattanooga Engineering 3280L Using Root Locus Modeling for Proportional Controller Design for Spray Booth Pressure System By:
2 Introduction: The objectives for these experiments are to take measurements of the Spray Paint Booth Pressure Control System with damper two closed and find the system characteristics for the Spray Paint Booth Pressure Control System (detailed below). As shown in Figure, 1 in the Background and Theory section of this report, the air will be limited to booths one and two, with damper two closed. To achieve this objective, several different experiments were run under specific parameters for several trials using
3 the Spray Paint Booth Pressure Control System using varying motor input ranges, to obtain an output from 29 to 44 cm H 2 O. The Background and Theory section of this report will contain a schematic and block diagram of the Spray Paint Booth Pressure Control System that demonstrates the configuration of the system. There will be a discussion on the theory behind the experiments as well as general background information on the Spray Paint Booth Pressure Control System. The procedure portion of the report explains the different processes in which the experiments are conducted. The results section contains graphical data results of an example trial at a given input value and the final pertinent data, presented graphically that deliver information about the system characteristics. The discussion section will be a detailed discussion of the results. The conclusion and recommendation section will discuss the meaning of the results. The appendix contains all graphical results that are not represented in the body of the report and the references. Background and Theory: In Figure 1, the schematic diagram of the Dunlap plant spray-paint booth system is shown. The significance of a steady-state operating curve for control systems is represented by the relationship of the motor speed (input) and booth pressure (output). The data is obtained through electronic sensors that send the data to the computer for processing. The experiment can be remotely controlled by inputting the conditions of the experiment such as the time, the damper being open or closed, and the input power.
This image cannot currently be displayed. 4 Figure 1. Schematic diagram of the spray paint booth pressure system. The booths are not always being used at the same time. As shown in Figure 1, D-1 and D-2 are solenoid-operated valves that prevent air from entering booth 2 or booth 3 when closed. Figure 2. Block diagram of paint booth system. Figure 2: Block diagram of paint Booth System (Steady State Operating Curve) Figure 2 represents the open loop system for the spray booth system. M(t) in this diagram represents the percentage of power supplied to the blower motor and it is a function of time. C(t) in this
This image cannot currently be displayed. This image cannot currently be displayed. This image cannot currently be displayed. This image cannot currently be displayed. This image cannot currently be displayed. 5 diagram represents the pressure measured in cm-h 2 O and is also a function of time. The values determined from these experiments are the steady-state gain, K, which is expressed as a relationship between the input and the output of the system, the deadtime, t o, which is the time after the system s input is changed to twhen the output starts to change, and the first-order time constant, τ, which is the time of the system to reach 63.2% of its final value output. These characteristics are vitally important to FOPDT (first order plus dead-time) modeling. Another way to obtain the FOPDT system characteristics, is through step-response experiments. These experiments start at a certain baseline input value and then change the input by either stepping up or stepping down the input value and using the resulting information to obtain the same FOPDT characteristics through different means. This is achieved through the graphical interpretation of the results, and can be seen below in figure three. Figure 3. Graphical means of FOPDT characterstics. Two final methods of determining the FOPDT characteristics of the system involve using mathematical modeling techniques. For a system where the input function is a step function:, then the rime response of the FOPDT system is, where these variable are obtained from the excel sheets of the step experiment results. Another
This image cannot currently be displayed. The linked image cannot be displayed. The file may have been moved, renamed, or deleted. Verify that the link points to the correct file and location. 6 mathematical method to obtain these characteristics is to determine the characteristics with the areas above and below the step-response curve. This can be seen below in figure 4. Figure 4. Step response curve areas for FOPDT characteristics. The characteristics can be determine by using the equation where, with K being the gain of the system, L being the dead time, and T being the first order time constant. The areas are determined through the trapezoidal rule approximation, and all values used in the calculations are determined with the excel output of the step-response experiments. Another way to determine these characteristics is through frequency response experimentation. A frequency response is the response of the output of the function of the system to a sine function. This allows the determination of the FOPDT characteristics as well as the amplitude ratio and the phase lag of the system. The results of these experiments are similar to those in figure 5 below.
This image cannot currently be displayed. This image cannot currently be displayed. 7 Figure 5. Low frequency sine response graph and Lissajous figure. The results of this frequency response can be used to create a Bode plot, which relates frequency to the amplitude ratio, in a log-log plot, and the frequency to the phase angle in a semi-log plot. These can be used to help model the FOPDT characteristics and can be seen below in figure 6.
This image cannot currently be displayed. The linked image cannot be displayed. The file may have been moved, renamed, or deleted. Verify that the link points to the correct file and location. 8 Figure 6. Bode plot examples. The FOPDT characteristics can be found in the following equations as a results of the frequency response experiments:. This allows all of the characteristics to be determined as well as the apparent order, m, the ultimate frequency, f u, and the controller gain K cu. A similar experimental set-up focuses on the relay feedback of the system. This results in a output similar to that seen in figure 7 below.
This image cannot currently be displayed. The linked image cannot be displayed. The file may have been moved, renamed, or deleted. Verify that the link points to the correct file and location. 9 Figure 7. Relay feedback example output. The relay response allows you to determine the average values for the systems due to the results being given as different periods and allows for the calculation of independent uncertainties. The amplitude ratio and the controller gain can be found from this relay feedback as well, with the K cu being calculated as: The linked image cannot be displayed. The file may have been moved, renamed, or deleted. Verify that the link points to the correct file and location.. The frequency response can then be modeled by recognizing the input function as an FOPDT sine function with amplitude = A and frequency = f: The linked image cannot be displayed. The file may have been moved, renamed, or deleted. Verify that the link points to the correct file and location., then the steady oscillation of the output can be modeled as. The characteristics for the frequency modeling can be found using the same equations for the simple stepresponse modeling, producing the same type of Bode plots and being able to compare the resulting data.
The linked image cannot be displayed. The file may have been moved, renamed, or deleted. Verify that the link points to the correct file and location. This image cannot currently be displayed. The linked image cannot be displayed. The file may have been moved, renamed, or deleted. Verify that the link points to the correct file and location. 10 One use of these characteristics that can be found through the aforementioned ways, is through root locus plotting. For a FOPDT system, a transfer function of the form: where a proportional feedback controller with a transfer function of The linked image cannot be displayed. The file may have been moved, renamed, or deleted. Verify that the link points to the correct file and location. then it can be inserted into a simplified function of G(s) to obtain the OLTF, open loop transfer function: The characteristic equation becomes 1+OLTF = 0. The roots of this equation gives us the point to make the Root Locus plot. This plot can be seen below. Figure 8. Root Locus Plot This plot allows the determination of different K values to obtain different damping factors.
The linked image cannot be displayed. The file may have been moved, renamed, or deleted. Verify that the link points to the correct file and location. 11 Procedure: The steady-state operating curve experiment was conducted using LabVIEW for system controls. LabVIEW operates data acquisition to the pressure system in the laboratory to run and collect data for the experiment. Duration of the experiment was set to be 30 seconds. Damper 2 was closed for 30 seconds and opened for 0 seconds. Damper 1 was closed for 0 seconds and opened for 30 seconds. For this experiment input power was between 67 percent and 81 percent, in order to produce an output in the range of 29-44 cm H 2 O. Readings were taken in 2 percent intervals. The recorded data was then transferred to Excel from LabVIEW. Like the steady-state operating curve experiment, the step-response, and all subsequent experiments, used LabVIEW to operate system controls and obtain the pertinent information. For the step-response experiments, the same value range of the output was desired. The step-up experiments started at a baseline input value of 35% and stepped up to the appropriate values between 67 and 81% in order to obtain the desired output. The experiments were run for a total of 30 seconds, with the step up occurring at 10 seconds. The step down experiments were conducted in a similar manner, starting at an input value of 100% and stepping down to the appropriate input level for the desired output. The results are then output to excel for analysis. The frequency response experiment requires the experiment to be run for about ten different frequencies, starting at a frequency of, being sure to keep the frequencies at one significant figure. A good approximation for when to stop using lower frequencies is when the output is nearly in phase with the input, and a good approximation for when to stop using higher frequencies is when the output has no perceptible steady oscillation. Experiments need to be run after
The linked image cannot be displayed. The file may have been moved, renamed, or deleted. Verify that 12 the transients die out which is described as: The linked image cannot be displayed. The file may have been moved, renamed, or deleted. Verify that the link points to the correct file and location.and an additional the link points to the correct file and location.. So the experiment must be run for at least T 1 +T 2. Using a baseline input as a middle of the normal operating range of the input function and an amplitude ratio such that the peak and valley of the sine wave input are at the limits of the normal operation range. The results are then output to excel for analysis. For the relay feedback, the experiments should be run for the same T 1 +T 2 length as the frequency response experiments. The other parameters for the experiment set-up should be set so that a nominal point from the SSOC (steady-state operating curve) output and use the c(t) value as your setpoint, and the m(t) input as the input base-line. Then set the upper and lower limits for being 10% above and below the input base-line. The results are then output to excel for analysis of the results. Results: An example of the steady-state results of this experiment is shown in Figure 9. Figure 9 shows the output pressure at 76 percent input with damper two closed for 30 seconds generated by the Spray Paint Booth Pressure Control System. The blue line indicates the input power for the motor on the primary axis. The output is shown by the scatter plot in red on the secondary axis and is measured in cm-h 2 O. The duration of the experiment is shown on the horizontal axis and is measured in seconds.
This image cannot currently be displayed. 13 Figure 9. Steady state response @ 76% Input The first five seconds can be disregarded, because there is lag in the motor from bringing it from 0 percent to 76 percent. The same process was used for each trial and is represented in Appendix-B. Table 1 lists the experimental data which includes the input, output, and uncertainty. The uncertainty generally increases as the input increases. The uncertainty was obtained by determining the standard deviation of the steady-state output. Table 1. Steady State input to output values and uncertainties Input (%) Output Pressure (cm-h2o) Uncertainty (±) 66 31 2.5 67 32 2.3
14 68 32 2.3 69 33 2.4 70 35 2.4 71 36 2.5 72 36 2.4 73 38 2.6 75 40 2.8 76 41 2.7 77 42 2.7 78 43 2.7 79 44 2.8 80 45 2.9 81 46 3.0 Comment [1]: The steady-state operating curve is shown in Figure 10 below. The average output pressure is plotted against the percent of input power. The linear curve fit gives a slope of 1.1 input value, which indicates that the plot is considerably linear. The uncertainty of each trial can be seen as the bold black bars above and below the slope, which indicates the positive and negative values for the standard deviation.
15 Comment [2]: Figure 10. SSOC curve generated for all three regions. Below are an example of both the step-up and step-down results operating at a 37% input jumping up 37% and 100% jumping down 29%, respectively
This image cannot currently be displayed. This image cannot currently be displayed. 16 Figure 11. Step-up results. Figure 12. Step-down results.
17 The next three figures show the system characteristics for the three experimental regions, grouped by step-up and step-down results.
18 Figure 13. Steady State Gain, K (cm-h20/%) - Step Response Results Figure 14. Dead time, t (seconds) - Step Response Results
19 Figure 15. Time Constant, tau (seconds) - Step Response Results The step results can also be modeled. This can be done in two ways, with FOPDT modeling and with area curve fit. The results for the area under the curve modeling of step response are shown below in figures 16.
This image cannot currently be displayed. 20 Figure 16. Area Curve Fit values for gain, time constant, and dead time. The curve fit modeling can be seen below in figure 17. Figure 17. FOPDT Model
21 The results of the sine response testing can be seen below in table 2 for the upper region, with full graphs being available in the appendix.. Table 2. Sine Response results. Frequency Amplitude Ratio AR Uncertainty Phase Angle PA Uncertainty 0.3 1.2 0.1 23 0.1 0.75 1.1 0.5 56 0 1.5 0.8 0.2 137 0.1 2.5 0.5 0.1 180 0 3 0.3 0.3 221 0.2 These results can be plotted using a Bode plot as shown below for the phase angel and amplitude ratio.
22 Amplitude Ratio Figure 18. Bode Plot Phase Angle Figure 19. Bode Plot for The results of the relay feedback can be seen below.
23 Figure 20. Relay Feedback These results are tabulated in table 3. Characteristic Value Uncertainty Units tₒ 0.2 0 seconds τ 0.3 0.1 seconds Ku 1.7 0.2 %/cm-h2o K 0.8 0.1 cm-h2o/% F 0.8 0.3 Hz PA 2 0.4 radians The root locus model can be seen below.
24 Root locus plot Figure 21. Discussion: The linear curve fit of the steady state operating curve gives a slope of 1.1 input %, (or cm H2O Comment [3]: per percent). The objective for this laboratory was to find the steady-state operating curve, which requires the input level and output pressure to be linear. This shows that the output in the range of 29-44 cm H 2 O is in the operating range for the system. The maximum standard deviation in the steady-state operating curve for the range of 67 percent and 81 percent is 1.5 percent. This minute percentage shows the data is close to the average values and is not scattered. This gives the data a high confidence level for the data.
25 The results of the step-response show us that the system has statistically different K values for step-up and step-down responses, being statistically the same for all regions, statistically similar values of dead time and the first order time constant for both the step-up and step-down response for the middle and higher regions, with statistically different values for the lower region. The results for the area curve fit show and the FOPDT modeling show good agreement with the steady state and step response analyses. The Sine response and relay feedback show good agreement with all of the other models and experiments as well. The root locus plot models this response well. Conclusions and Recommendations: The objectives for this experiment were to take measurements of the Spray Paint Booth Pressure Control System with damper two closed and find the steady-state operating curve (SSOC) for the Spray Paint Booth Pressure Control System. The steady-state operating curves, sine responses, bode plots, and all supplemental experimental data for the Spray Paint Booth Pressure Control System can be found in the results section of the report. The students of the pink team would recommend that the option to kill a currently operating experiment be considered for addition to this program. Appendix-A: References Henry, Jim. Paint Spray Booth Pressure Control System-- Experiments and Analysis. University of Tennessee at Chattanooga, System Dynamic and Control in Green Engineering. 2004. Web.
26 01/21/2013. http://chem.eng.utc.edu/greenengineering/booth-pressure/booth-pressure- System-SSOC-student.htm Henry, Jim. Steady State Operating Curve, Green Engineering, University of Tennessee at Chattanooga, ENGR-3280L Webpage Week 1. Web. 01/21/2013. http://chem.eng.utc.edu/engr328l/labmanual/week-1-experiment.htm Henry, Jim. Week 3 Step Response Testing. Green Engineering, University of Tennessee at Chattanooga, ENGR-3280L Webpage Week 3. Web. 03/17/2013. http://chem.engr.utc.edu/engr328l/lab-manual/week-3-experiment.htm Henry, Jim. Week 4 Modeling. Green Engineering, University of Tennessee at Chattanooga, ENGR- 3280L Webpage Week 4. Web. 03/17/2013. http://chem.engr.utc.edu/engr328l/labmanual/week-4-assignment.htm Henry, Jim. Week 5 Modeling Approximate Linear FOPDT Model. Green Engineering, University of Tennessee at Chattanooga, ENGR-3280L Webpage Week 5. Web. 03/17/2013. http://chem.engr.utc.edu/engr328l/lab-manual/week-5-assignment.htm Henry, Jim. Week 6 Frequency Response. Green Engineering, University of Tennessee at Chattanooga, ENGR-3280L Webpage Week 6. Web. 03/17/2013. http://chem.engr.utc.edu/engr328l/labmanual/week-6-experiment.htm Henry, Jim. Week 6 Experiment: How to calculate Amplitude Ratio (AR) and Phase Angle (PA) Uncertainties. Green Engineering, University of Tennessee at Chattanooga, ENGR-3280L Webpage Week 6. Web. 03/17/2013. http://chem.engr.utc.edu/engr329/lab-manual/week-6- Experiments-Uncertainties.htm
27 Henry, Jim. Week 6 Relay Feedback. Green Engineering, University of Tennessee at Chattanooga, ENGR-3280L Webpage Week 6. Web. 03/17/2013. http://chem.engr.utc.edu/engr328l/labmanual/week-6a-experiment.htm Henry, Jim. Week 7 Modeling Frequency Response. Green Engineering, University of Tennessee at Chattanooga, ENGR-3280L Webpage Week 7. Web. 03/17/2013. http://chem.engr.utc.edu/engr328l/lab-manual/week-7-assignment.htm Henry, Jim. Week 7 Analyzing Relay Feedback Response. Green Engineering, University of Tennessee at Chattanooga, ENGR-3280L Webpage Week 7. Web. 03/17/2013. http://chem.engr.utc.edu/engr328l/lab-manual/week-7a-assignment.htm Henry, Jim. Week 9 Root Locus Plotting. Green Engineering, University of Tennessee at Chattanooga, ENGR-3280L Webpage Week 9. Web. 03/17/2013. http://chem.engr.utc.edu/engr328l/labmanual/week-9-assignment.htm Åström, Karl J, and Tore Hägglund. Automatic Tuning of Pid Controllers, pp. 32-33. Research Triangle Park, NC: Instrument Society of America, 1988. Print. Appendix-B: Graphs
28 Figure 22. Steady State operation
29 Figure 23. Steady State Operation
30 Figure 24. Steady State Operation
This image cannot currently be displayed. 31 Figure 25. Steady State Operation
32 Figure 26. Steady State Operation
33 Figure 27. Steady State Operation Figure 28. Steady State Operation
34 Figure 29. Steady State Operation Figure 30. Steady State Operation
This image cannot currently be displayed. This image cannot currently be displayed. 35 Figure 31. Steady State Operation Figure 32. Steady State Operation
36 Figure 33. Steady State Operation Figure 34. Steady State Operation
37 Pressure (cm-h2o) 40 35 30 25 20 15 10 5 Pressure = 1.1543 Input % 0 65.5 66 66.5 67 67.5 68 68.5 Input (%) 45 40 35 Figure 35. SSOC for lower region. Pressure (cm-h2o) Linear (Pressure (cm- H2O)) Pressure cm H2O) 30 25 20 15 10 Pressure (cm H2O)= 1 cm H2O per percent Pressure (cm H2O) Linear (Pressure (cm H2O)) 5 0 69 70 71 72 73 74 75 76 Input (%) Figure 36. SSOC for middle region.
38 50 Output (cm-h2o) 48 46 44 42 40 38 Pressur 76 77 78 79 80 81 82 Input (%) Figure 37. SSOC for upper region. 85 83 81 Low Frequency of 0.3 60 50 79 77 Input (%) 75 73 71 69 67 65 0.5 2.5 4.5 6.5 8.5 Time (sec) Figure 38. Sine Response Input Outp Output (cm-h2o) 40 30 20 10 0
39 85 83 81 79 77 Input (%) 75 73 71 69 67 Low Frequency of 0.75 Input 60 50 Output (cm-h2o) 40 30 20 10 65 0 0.5 1 1.5 2 2.5 3 3.5 4 85 83 81 79 Input (%) 77 75 73 71 Time (sec) Figure 39. Sine Response High Frequency of 1.5 69 67 65 0.5 1 1.5 2 2.5 3 Time (sec) Figure 40. Sine Response Input Output 50 45 40 Output (cm-h2o) 35 30 25 20 15 10 5 0
40 85 83 81 79 Input (%) 77 75 73 71 69 67 Input 65 0.5 1 1.5 2 85 83 81 79 Input (%) 77 75 73 High Frequency of 2.5 Time (sec) Figure 41. Sine Response High Frequency of 3 71 69 67 Inpu 65 0.5 1 1.5 2 Time (sec) Figure 42. Sine Response 50 45 40 Output (cm-h2o) 35 30 25 20 15 10 5 0 50 45 40 Output (cm-h2o) 35 30 25 20 15 10 5 0
This image cannot currently be displayed. This image cannot currently be displayed. 41 Figure 43. Bode Plot from MatLAB for amplitude ratio Figure 44. Bode plot from matlab for phase angle
This image cannot currently be displayed. 42 Figure 45. Root Locus sine response from MatLab relay feedback Figure 46. Root locus based on
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