University of Tennessee at Chattanooga Step Response Modeling Control Systems Laboratory By Stephen Rue Tan Team (Stephanie Raulston, Stefan Hanley) Course: ENGR 3280L Section: 000 Date: 03/06/2013 Instructor: Dr. Jim M. Henry
1 Introduction: This report will cover the seven different lab experiments conducted so far this semester. The first two experiments were conducted to obtain steady state operating curves and step response curves with respect to certain motor input percentages. Beginning the experiments for the steady state operating curve (SSOC), trial and error was required to approximate the input percentage, in order to have the proper output for each particular range (low, middle, and high). Once the desired input and output were determined, graphs were generated to show the SSOC; this curve will help to approximate an output value given at any specific input value within the desired range. For the next two experiments, we learned how to model dynamic systems with Excel and how to obtain modeling results in graphical form by modeling first-order-plus-deadtime (FOPDT) with Excel. The object of these experiments was to observe the dynamic response of the approximate linear FOPDT model for a system, the impact of parameter values on the dynamic response, and to adjust the linear FOPDT model parameters to get the model results to agree with the experimental results. The next experiment that was preformed was the frequency and relay feedback experiment. In the frequency experiment we observed experimentally the time response of the output function of the system to a sine function input at a variety of different frequencies, the system's amplitude ratio, and the system's phase lag. This was performed over several experimental trials. One member of the team performed the alternate relay feedback experiment. We observed experimentally the time response of the output function of the system to a relay feedback control at different parameters of the relay feedback. The last experiment was the root locus plotting experiment. In this experiment we used the three parameters from the experimental data found in the FOPDT experiment: Gain, K, dead time, t o and first-order response time, t. Using these parameters we were able to find the K cu, the K c that gives about
2 1/4, 1/10 and 1/500 decay response and the K c that gives critical damped response for our system. Following this introduction, the background and theory is introduced in order to provide a better understanding of the lab setup for the experiments that have already been conducted. After the background and theory, the procedure will tell what was done in the lab; including the variables and constraints for the experiments. Succeeding the procedure is the results, which will show the data collected in the lab after which the discussion will explain the results and their significance. A summarization of the entire paper will be in the conclusions and recommendations section. The Appendix will be the last section and will include all remaining data and references. Background and Theory: The duct cooling station temperature system is operated via the Internet. The system is accessed by a link to weblab.utc.edu that can be found on the ENGR3280L homepage. The system allows for different input percentages to be applied. The user can vary the system inputs and outputs during the experiment. Figure 1. A block diagram representing the system variables
3 The block diagram in figure one above begins by an input placed into the paint spray booth system, the input is the percentage of motor power, and this percentage is manipulated in order to find the specific output. Once the experiment is running there is a heater in the duct and a blower that blows ambient temperature air over the heater cools the duct. In the step response experiment, the input percentage is set at a lower or higher value than used for the particular range in the SSOC experiment; this allows for a step percentage to be inputted for the system. Figure 2. Schematic diagram of the Dunlap Plant Spray-Paint Booth Figure 2 above is the schematic for the Dunlap Plant Spray-Paint Booth, the system begins when the weblab sight sends information via the internet to the temperature transmitter (TT 201), from there a signal is sent to the temperature rate recording controller (TRC 201) and then on to the temperature control actuator (TCZ 201). The temperature rate actuator sends a signal to the blower (B-205), the valves D1 and D2 are controlled by the weblab site for specific time intervals specified.
4 Procedure: All experiments begin with a screen that will control the input percentage, total time for the experiment, and the time for the valves to be closed or open. Our system requires outputs in the range of 44-56 with both dampers closed for the duration of the experiment. To initiate the steady state operating experiment, a basic value needs to be guessed to receive an output for the system. Trial and error is needed to pin point the input for our particular output this takes several iterations. Once the output is reached for several values throughout the desired range, the input and output can be used to predict a steady state operation curve. The curve predicts output given an input multiplied by an operation constant. The step response experiment has a place to input the beginning input percentage and an added step value on the screen. For the step response experiment, the step values can be either positive or negative, in order to get a step up or step down output vs. time. The step response uses the final input that falls between the individual ranges; the input used in the SSOC experiment will become the final value after the step is performed. A lower or higher value is placed in the initial input on the left side of the screen, with both valves closed and a step value must be inputted into the right side of the screen to allow for calculations to be produced after the experiments. The step value must be positive for a step down function and negative for a step up. Also inputted on the main screen of the step response is the time for the step to occur and total time for the experiment; the experiment worked well when the total experiment time was around 15 minutes with a step at 7 minutes, this time frame allowed the system to reach steady state before and after the step. At the bottom of the screen, both valves need to be closed for the entire experiment. Once the experiment has run the entire time set, the values from the sensor can be exported to Excel.
5 The next experiment that was preformed was the Modeling Experiment. In this experiment the team learned to model first order plus dead time (FOPDT) with excel. An excel file was generated per the instructions for modeling that used the data collected from the step up experiments. We adjusted the parameters so that the model curves fit the data curves as nearly as possible. The suggested order for doing this, using the experimental directions was: Set input baseline and output baseline so the baseline parts of the model curves agree with the experimental data. Set td and A so that the model input curve agrees entirely with the experimental input curve Set K, to and t so that the model output curve agrees with the experimental output curve as close as possible. Suggested values of these come from your previous analysis. Now, adjust K, to and t so that the model output curve agrees with the experimental output curve as best as possible. Once this was completed for the 3 up and 3 down steps the results were posted to blackboard. The next two experiments performed by the team were the frequency and relay feedback experiments. The frequency experiments were conducted to observe, experimentally, the time response of the output function of the system to a sine function input at a variety of different frequencies. First, we chose a value for the sine amplitude height of the "input" variable. Next, we chose a value for the "input" variable, M(t), that would be the base line value. Then, we chose the frequency of the sine wave. Finally, we chose how long the experiment would run. Then click run. One member of the team did the alternate experiment for the relay modeling experiment. In this experiment the team observed the time response of the output function of the system to a relay feedback control at different parameters of the relay feedback and
6 approximated the Kcu factor. The results for both of these experiments are listed below in results and the graphs for each experiment are listed in the appendix section. For the relay feedback experiment the entire linear operating range was used to conduct experiments, since very little output response can be seen in a smaller operating window. Results: report. The results are listed in the same sequence as they are presented in the procedure of the Output (Degrees C) 55 50 45 40 35 50% Input 30 0 1 2 3 4 5 6 7 Time (min) Figure 3: This figure is a test ran with a power input of 50% The figure above is an example of the data gathered from one of the SSOC experiments. The experiment was ran for 7 minutes and begins with an input of 50%, the output flow rate of the machine is shown by the red boxes on the graph. The temperature rate begins to increase before receding to the steady state. The average output is taken from this region, along with the standard deviation used in calculating the uncertainty.
7 Input (%) Output (lb/min) Uncertainty 44 58.6 0.4 45 57.7 0.8 46 56.5 1.2 47 56.3 0.4 48 55.8 0.4 49 50.2 0.8 50 50.1 0.6 52 49.8 0.2 55 48.3 0.3 Table 1: This table organizes the input, output, and uncertainties for the SSOC The table above has the inputs for the system from 9 experiments ran within our range. The uncertainty is found by multiplying the standard deviation by two, the uncertainty is shown in the SSOC in figure 4 below. Output Temperature (Degrees C) Input % vs. Output Temperature 60.0 58.0 56.0 54.0 52.0 50.0 48.0 Output = -0.7*Input + 89.18 46.0 44.0 42.0 40.0 42 44 46 48 50 52 54 56 INPUT (%) Figure 4: The steady state operating curve for the paint spray booth station
8 The figure above shows the SSOC for the system within our range of 44-56. Figure 4 and the formula help to estimate the specific output, given a certain input power percentage. The SSOC for our system is shown on the graph above. Figure 5: This figure shows the change in input power percentage m, the output change c, dead time (t 0 ), and the time constant (τ). The figure above is a graph of the step up response, the c and m shown above are used to calculate the gain for the system by using the equation Gain= c/ m. The graph also shows the area considered dead time (t 0 ), the dead time for this particular system was about 0.7 seconds. The time constant (τ) calculation for this particular system is shown at the top of the graph beside the y-axis. The example values above that were gathered from the step up and step down for our particular range are shown in the table 2 below. Regions Gain up Gain down τ up τ down t 0 up t 0 down High 0.26-0.07 0.53 1.03 0 0.17 Medium 0.61-0.25 0.33 0.7 0 0.4 Low 1.07-0.89 0.3 0.7 0 0.7
9 Uncertainty Gain up Gain down τ up τ down t 0 up t 0 down High 0.08 0.06 0.12 0.99 0 0.58 Medium 0.35 0.09 0.12 0.4 0 0 Low 0.56 1.12 0 0.72 0 0 Table 2: Values gathered for gain, time constant, and dead time. This table is a list of averages for each team member for the gain, time constant and dead time. The uncertainties for each of the characteristics of the step response are located in the lower set of tables. These uncertainties are used in the following figures as (I) shaped bars centered on the top of each colored bar. 1.2 1.0 0.8 0.6 K Variation K (deg/%) 0.4 0.2 0.0-0.2-0.4-0.6 Lower Up Lower Down Mid Up Mid Down Higher Up Higher Down Figure 6: Gain for the system from one of the team member s results including the uncertainty.
10 t0 Variation t0 (min) 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0-1.0-2.0 Lower Up Lower Down Mid Up Mid Down Hgher Up Higher Down Figure 7: Dead time for the system using the values from table 2 including the uncertainty 1.4 1.2 1.0 τvariation τ(min) 0.8 0.6 0.4 0.2 0.0 Lower Up Lower Down Mid Up Mid Down Higher Up Higher Down Figure 8: Time constant for the system using the values from table 2 including the uncertainty.
11 The results of the frequency experiment are listed in tables 3 and 4 below. For each range, a table was constructed containing values of frequency, amplitude ratio, and phase angle found from the sinusoidal response experiments. These tables include uncertainty that was found using a Student s T distribution with a 95% confidence level. This is outlined more in the appendix. Frequency Amplitude Ratio AR Uncertainty Phase Angle PA Uncertainty 0.4 0.34 0.04-267.2727273 49.99 0.5 0.35 0.06-222 20.78 0.6 0.31 0.05-268.2352941 88.16 0.7 0.27 0.03-257.1428571 51.43 0.8 0.31 0.05-304.6153846 55.38 0.9 0.31 0.05-221.5384615 55.38 1 0.16 0.05-180 72.00 Table 3: Results from week 6 lab K 0.35 tau 0.1 t0 0.5 fu 0.9 Table 4: Tabulated Results from week 6 lab Values for the ultimate gain, K cu, were determined experimentally by using relay feedback. In relay feedback, the input pump % oscillates between two set values, the ceiling and floor, which causes the output to oscillate with the changing input. The results are tabulated in table 6 below.
12 Discussion: In the step response experiment, the results throughout the output range of 44-56, the gains, time constant, and dead time for the system are pretty consistent from low range to the high range with the uncertainty bars for each set of data almost falling within the range of the other values for the system characteristic. The average gain for the system was 0.60 which is very similar to the constant value of 0.58 obtained in the steady state operating curve. The dead time and time constant for the system averaged around 0.4 seconds and 0.52 seconds respectively. The results for the modeling lab are depicted in the appendix. For the modeling experiment the models followed the step up and down results closely. For the frequency experiments the results are listed in the tables 3-5 above and throughout this experiment, the values for K, τ, and t o, ultimate controller gain, ultimate frequency, and order of the system were found. Conclusion: The objective of the experiment was to create a step response control system with outputs of 44-56 lbm/min with both dampers closed for the entire experiment, and to study three system characteristics for a step response. The inputs for the system were manipulated in order to control what the output for the system. The time constant and dead times throughout the system were similar within the high, medium, and low sections of our output range. The experiments ran well, the system took a few experiments to warm up and give consistent data; overall the system provided usable data. Through this experiment, the values for K, τ, and t o, ultimate controller gain, ultimate frequency, and order of the system were found. This experiment demonstrated that through sinusoidal response analysis, previously mentioned parameters for a system can be
13 determined through Bode plot analysis and formulation. The experimental results give us the definite values of the system, while the modeling only gives an estimate of what one might expect to see. I would recommend for future experimentation, the results from the Bode plot should be used as the base parameters when talking about the flow system with both valves closed.
14 Appendices: Figure 9: 44% input SSOC Figure 10: 45% SSOC
15 Figure 11: 46% INPUT SSOC Figure 12: 47% input ssoc
Figure 13: 48% input ssoc 16
17 Figure 14: Results of 44% - 48% inputs ssoc Figure 15: 48% ssoc
18 Figure 16: 49% input ssoc Figure 17: 50% input ssoc Figure 18: 52% input ssoc
Figure 19: 55% input ssoc 19
20 Figure 20: results 48% - 55% input Figure 21: step responce 55 53% to 73% Input 75 70 Output (Degrees C) 50 45 65 60 55 50 Input % 45 40 4 5 6 7 8 9 10 11 12 Time (min) 40
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23 0.30 t 0 Low Range (sec) 0.20 0.10 0.00-0.10 t0 up t0 down
Modeling FOPDT 24
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