University of Tennessee at Chattanooga Steady State and Step Response By: Alex Bedley Engineering 328L Buff (Alexander Hudson, Ashley Poe) February 1, 13
Introduction In the past two experiments, we were conducting experiments to obtain steady state operating curves and step response curves with respect to certain motor input percentages. To begin the experiments for the steady state operating curve (SSOC), trial and error is needed 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 have been figured, a graph was generated to determine the SSOC; this curve will help to approximate an output value given at any specific input value within the desired range. The SSOC can help with the next lab, the objective of the second experiment is to determine the system steady state gain, time constant, and dead time for the flow system with a step response using a variety of different step sizes and input values. Following the introduction, the background and the theory will be discussed in order to provide a further detailed understanding of the lab experiment and setup process. After the background and the theory, the operating procedure is explained in order to run the experiment and understand the included operating variables. Results and discussion then follow and state the values gathered in the lab in tabular and graphical form, emphasize and the importance of the values gathered. Finally, the conclusion and recommendations will give an overview of the results and the possible ways to improve the experiment. Alex Bedley February 1, 13 2
Background and Theory The filter wash flow control system is operated via the internet. The flow system is accessed by a link to weblab.utc.edu which can be found on the ENGR328L homepage. The system allows for different input percentages to be applied. These different inputs proved specific flow rate outputs. The system inputs, the valve positions (open or closed), and experimental time duration can be varied during the experiment by the user. Figure 1: Block diagram for the filter wash station The block diagram in figure one above begins by an input placed into the filter wash pump 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 the pump system uses sensors in order to find the flow rate coming from the system in pound mass per minute. 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. Alex Bedley February 1, 13 3
Figure 2: The schematic diagram for the filter wash station. Figure 2 above is the schematic for the filter wash station, the system begins when the weblab sight sends information via the internet to the flow rate transmitter (FT 1), from there a signal is sent to the flow rate recording controller (FRC 1) and then on to the flow control actuator (FCZ 1). The flow rate actuator sends a signal to the pump (P ), the valves MV 2 and 3 are controlled by the weblab site for specific time intervals specified. Alex Bedley February 1, 13 4
Procedure Both 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 19 31 lbm/min with both valves closed for the entire experimental time duration. To begin 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 will take 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 negative for a step down function and positive for a step up. Also inputted on the main screen of the step response is the time for the step to Alex Bedley February 1, 13
occur and total time for the experiment; the experiment worked well when the total experiment time was around seconds with a step at seconds, 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. Alex Bedley February 1, 13 6
Results 6 Input (%) Input (%) Output (lbm/min) **Steady State Region Output (lbm/min) Time (s) Figure 3: This figure is a test ran with a power input of % The figure above is an example of the data gathered from one of the SSOC experiments. The experiment was ran for seconds and begins with an input of %, the output flow rate of the machine is shown by the red boxes on the graph. The flow rate begins to increase and overshoots before receding to the steady state inside of the oval on the graph. The average output is taken from this region, along with the standard deviation used in calculating the uncertainty. Alex Bedley February 1, 13 7
Input (%) Output (lbm/min) Uncertainty 19.9 1.8 46. 1.7 47 21 1.8 49 22.4 1.7 23 2.23 1 23.7 1.8 3 1.8 26. 1.8 7 27. 1.9 9 28.8 1.8 6 29. 1.8 61. 1.7 63 31. 1.8 66 33. 1.8 Table 1: This table organizes the input, output, and uncertainties for the SSOC The table above has the inputs for the system from 14 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. Alex Bedley February 1, 13 8
Flow Rate (lbm/min)=.64* Input (%) Output (lbm/min) 6 6 7 Input (%) Figure 4: The steady state operating cure for the filter wash flow station The figure above shows the SSOC for the system within our range of 19 31 lbm/min. Figure 4 and the formula in yellow helps to estimate the specific output, given a certain input power percentage. The SSOC for our system is shown in yellow on the graph above; [Flow rate (lbm/min)=.64*input(%)]. Alex Bedley February 1, 13 9
Figure : This figure shows the change in input power percentage Δ m, the output change Δc, dead time (t), and the time constant (τ). The figure above is a graph of the step up response, the Δc and Δm shown above are used to calculate the gain for the system by using the equation Gain=Δc/Δm. The graph also shows the area considered dead time (t), the dead time for this particular system was about. 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. Alex Bedley February 1, 13
Regions Gain up Gain down τ up τ down t up t down Low.62.71.83.87.48.7 Medium.66.63.8.87.7.3 High.63.67.77.3.6.67 Uncertainty Gain up Gain down τ up τ down t up t down Low.9.1.12.12.6.12 Medium.2.4..12.12.12 High.12.12.12.23..23 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 or textured bar. Alex Bedley February 1, 13 11
1..9.8.7.6...... Gain (lbm/min/%) Low Medium High Gain up Gain down Figure 6: Gain for the system using the values from table 2 including the uncertainty. In figures 6, the step up response is shown in the graphs as the blue textured block, and the step downs are the red boxes. The uncertainties for each system characteristic are shown in the graphs as the black (I) centered at the top of the graph as explained before. Alex Bedley February 1, 13 12
1. 1. τ (sec) τ up τ down.8.6... Low Medium High Figure 7: Time constant for the system using the values from table 2 including the uncertainty. In figures 7, the step up response is shown in the graphs as the blue textured block, and the step downs are the red boxes. The uncertainties for each system characteristic are shown in the graphs as the black (I) centered at the top of the graph as explained before. Alex Bedley February 1, 13 13
1..9.8.7.6..... t (sec) t up t down. Low Medium High Figure 8: Dead time for the system using the values from table 2 including the uncertainty. In figures 8, the step up response is shown in the graphs as the blue textured block, and the step downs are the red boxes. The uncertainties for each system characteristic are shown in the graphs as the black (I) centered at the top of the graph as explained before. Alex Bedley February 1, 13 14
Discussion In the first experiment, the steady state operating curve showed that the flow rate was.64*input (%). For the output range that we were trying to fall within, the required motor input was between 66%. For the flow station, the SSOC shows that the system is very linear and consistent; this mathematical model gives fairly accurate results when trying to predict the output for the system when specifying the input. In the step response experiment, the results throughout the output range of 19 31 lbm/min, 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.6 which is very similar to the constant value of.64 obtained in the steady state operating curve. The dead time and time constant for the system averaged around.8 seconds and. seconds respectively. The uncertainties were slightly larger in the higher end of our range except for the step up dead time. Alex Bedley February 1, 13
Conclusions and Recommendations The objective of the experiment was to create a steady state operating curve for the filter wash flow control system with outputs of 27 31 lbm/min with both valves 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 linear fit for the SSOC became Flow rate (lbm/min)=.64(lbm/min/%)*input(%). The linear fit slope of the steady state operating curve was within a hundredth of the average gain from the step response. 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. Alex Bedley February 1, 13 16
Appendices Input (%) 9 8 7 6 Time (s) Output (lbm) Figure 9: Low range steady state input % Input (%) 9 8 7 6 Time (s) Output (lbm) Figure : Low range steady state input 46% Alex Bedley February 1, 13 17
Input (%) Time (s) Output (lbm) Figure 11: Low range steady state input 46% Input (%) 6 9 8 7 6 Time (s) Output (lbm) Figure 12: Mid range steady state input 49% Alex Bedley February 1, 13 18
Input (%) 6 6 Time (s) Output (lb/min) Figure 13: High range steady state % input Input (%) 6 6 Time (s) Output (lb/min) Figure 14: High range steady state 8% input Alex Bedley February 1, 13 19
Input (%) 7 6 6 Time (s) Output (lb/min) Figure : High range steady state 6% input Input (%) 7 6 6 Time (s) Output (lb/min) Figure 16: High range steady state 61% input Alex Bedley February 1, 13
Input (%) 7 6 6 Time (s) 9 8 7 6 Output (lb/min) Figure 17: High range steady state input 61% 7 Input (%) 6 Output (lb/min) 6 Time (s) Figure 18: High range steady state 63% input Alex Bedley February 1, 13 21
7 Input (%) 6 Output (lb/min) 6 Time (s) Figure 19: High range steady state 66% input Input (%) 8 7 7 6 6 Output (lbm/min) Time (sec) Figure : Low range step down Alex Bedley February 1, 13 22
Input (%) 8 7 7 6 6 Output (lbm/min) Time (sec) Figure 21: Low range step down Input (%) 8 7 7 6 6 Output (lbm/min) Time (sec) Figure 22: Low range step down Alex Bedley February 1, 13 23
Input (%) 48 24 46 22 44 42 18 38 16 36 14 34 32 12 Time (sec) Figure 23: Low range step up Output (lbm/min) Input (%) 48 24 46 22 44 42 18 38 16 36 14 34 32 12 Time (sec) Figure 24: Low range step up Output (lbm/min) Alex Bedley February 1, 13 24
Input (%) 48 24 46 22 44 42 18 38 16 36 14 34 32 12 Time (sec) Figure : Low range step up Output (lbm/min) Output(lb/min) 6 Output(lb/min) Input Value(%) 8 8 7 7 6 6 input (%) Time (seconds) Figure 26: Mid range step down Alex Bedley February 1, 13
6 8 output(lb/min) Output(lb/min) Input Value(%) 8 7 7 6 6 input (%) Time (seconds) Figure 27: Mid range step down output (lb/min) 6 Output(lb/min) Input Value(%) 8 8 7 7 6 6 input (%) Time (seconds) Figure 28: Mid range step down Alex Bedley February 1, 13 26
output (lb/min) 28 26 24 22 18 16 14 12 Output(lb/min) Input Value(%) 6 input (%) Time (seconds) Figure 29: Mid range step up 6 output (lb/min) Output(lb/min) Input Value(%) input (%) Time (seconds) Figure : Mid range step up Alex Bedley February 1, 13 27
6 output (lb/min) Output(lb/min) Input Value(%) input (%) Time (seconds) Figure 31: Mid range step up Step Response Down Input Value(%) 72 7 68 66 64 62 6 28 32 34 36 38 Time(sec) Output(lb/min) Figure 32: High range step down Alex Bedley February 1, 13 28
Input Value(%) 72 7 68 66 64 62 Step Response Down Output(lb/min) 6 28 32 34 36 38 Figure 33: High range step down Time(sec) Input Value(%) 72 7 68 66 64 62 6 Step Response Down 28 32 34 36 38 Time(sec) Output(lb/min) Figure 34: High range step down Alex Bedley February 1, 13 29
Input Value(%) Figure : High range step up Input Value(%) Figure 36: High range step up Step Response Up 6 26 27 28 29 31 32 33 34 Time(sec) Step Response Up 6 26 27 28 29 31 32 33 34 Time(sec) 32 28 26 24 22 18 16 14 12 8 6 4 2 28 26 24 22 18 16 14 12 8 6 4 2 Output(lb/min) Output(lb/min) Alex Bedley February 1, 13
Input Value(%) Figure 37: High range step up Step Response Up 6 26 27 28 29 31 32 33 34 Time(sec) 28 26 24 22 18 16 14 12 8 6 4 2 Output(lb/min) Alex Bedley February 1, 13 31