UTC. Engineering 329. Frequency Response for the Flow System. Gold Team. By: Blake Nida. Partners: Roger Lemond and Stuart Rymer

Transcription

1 UTC Engineering 329 Frequency Response for the Flow System Gold Team By: Blake Nida Partners: Roger Lemond and Stuart Rymer March 9, 2007

2 Introduction: The purpose of the frequency response experiments was to observe the time response for the output function of the control system to a sinusoidal input at various frequencies in the flow system. After running the experiments each group had to find the amplitude ratio and phase angle for their control system s operation range. Each group then modeled the output frequency response using the sinusoidal model equation. The data was then remodeled in Microsoft Excel to determine the true behavior of our system. This report will discuss the theory behind the experiment and the actual laboratory set up, including a discussion about the steady state operation and step response. Then conclusions and recommendations will be presented at the end of the report. Following the introduction, the report covers the Background and Theory, which includes a discussion about the steady state operation and step response. This section also contains the schematic diagram and block flow diagram that will be used to model the system. The report will then introduce the experiments and their objectives. The results section will demonstrate how the sinusoidal frequency response was found. Then, the frequency response will be modeled using the sinusoidal equation in Excel. The model can be used to find a desired output for the flow control system. The discussion will include a description of the data and an analysis of the results. Finally, the conclusions and recommendation will be presented.

3 Background and Theory: The flow system is located on UTC s campus in the Engineering, Math and Computer Science building. It is controllable through the internet. The internet control page allows users to specify constant input power (%) over a specified test time (s) and manipulate 2 valves to be opened or closed for a specified length of time. Water is fed to these by a pump that is in a feedback control loop to maintain the wash water flow rate. These filters can be turned off using valves. Filter 2 and 3 are closed for our experiments. A schematic diagram of the flow system is located in Figure 1. The Flow Transmitter (FT301) receives the information from the computer about the motor speed. The information is then passed to the Flow Recording Control (FRC301) which is used to operate the Flow Rate Actuator (FCZ301) which directs the pump motor (P301) to a certain input percentage. The water is then pumped out of the reservoir into the three filter stations where the flow rate is recorded with respect to time. A graph is displayed which shows the flow rates before the data reaches steady state and the averaged steady state flow rate at successive power inputs. The data of flow verses time will be used to find the steady state operating curve.

4 Figure 1: Schematic Diagram of the Flow System Figure 2, shown below, shows the basic process that occurs during a single flow rate test. The Motor Power percentage m(t) is input by the operator and sent to the pump. The pump sends water though the filter wash pump system after which the output or flow rate c(t) is recorded by the computer and Labview. Both input and output data recorded by Labview are a function of time and in the time domain. Figure 2: Block Diagram for Flow System From previous experiments and analysis, the steady state operating region of the flow system was determined. The table of the collected data is located in the Appendix. This steady state is when output is consistent at a certain input. The SSOC diagram

5 shown in Figure 3 shows the linear operation of the output verses input from 0 to 100% input. When looking at the SSOC graph, one can tell that the graph does not become linear until an input of 27%. Steady State Operating Curve Output Flow Rate (lb/min) Output Operating Range c = 0.48m Input Operating Range Input Pow er (%) Figure 3: Flow Steady State Operating Curve The SSOC graph shows a linear region after an input of 27%. However, at an input of 35% the linear region has a bend or knee in the region. This causes a change in the slope of the SSOC. To prevent any slope changes our group decide to choose an input operating range of %. This portion of the graph is where the slope of the steady state operating curve is found. By adding a trend line in Excel, the slope of the linear region was found to be c = 0.48m Using this equation with m being the input %, a steady state flow rate (lb/min) c can be found in our operating range.

6 From previous experiments, observations of the Step Response were also made. When changing the input percentage of an experiment, the transient or non steady state portion or the output could be observed. When running a step response experiment, the output reaches steady state before and after the step occurs. Three FOPDT parameters can describe the system after the step occurs. An experimental run of a step response is located in Figure 4. Transient Portion Figure 4: Step Response Graph from 50-75% Input From Figure 4, one can see that the output reaches steady state before and after the step occurs. When the step occurs, the output reaches a transient stage, where the output is changing and is not at steady state. From this graph, one can calculate the gain (K), dead time (t o ), and time constant (τ). The next paragraph will describe how to find these parameters.

7 To calculate the gain, we had to take the change of the control variable (Time) and divide it by the change in the manipulated variable (Output). For the time constant, we had to look at the transient portion of the graph highlighted in Figure 4. The change in the control variable was first multiplied by and graphed as a horizontal line. Where this line met the manipulated variable a vertical line was drawn to the axis. Then another horizontal line was drawn for the steady state of the initial value and again where this met the manipulated variable a horizontal line was drawn to the axis. The difference from the first line to the second is the time constant of the graph. For the dead time, the second line for the time constant is used as the starting point and time where the step occurred in the input value is used as the second. The difference is the dead time of the Step Response. Figure 5 can be used to help explain this. 48 Dead Time and Time Constant 46 Final Avrerage: O utput (Q ) (lb/min) t = (Q + 239)/ % Line: 42.8 Initial Avrerage: 35.3 Time (t) (sec) Dead Time = = 0.60 Time Constant = = 0.85 Figure 5: Diagram for Dead Time and Time Constant

8 Figure 5 describes how to calculate the dead time and time constant for a step response. Several experiments were run in our groups operating range to find the average FOPDT parameters. The transfer function can be defined after finding the FOPDT parameters for a t Ke o s control system. The equation that describes the transfer function is: C( s) = M ( s), τs + 1 in which C(s) is the controlled variable and output function in the Laplace domain, M(s) is the manipulated variable and input function in the Laplace domain, and that transfer function is the part of the equation that contains our FOPDT parameters. The step response can be modeled through Excel. The equation that models a t t d t o step response is = [ ] τ c( t) A u t t, where A is the height of the d to K 1 e step and t d is the time in which the step occurs. Figure 6 shows a graph with an experimental run with the model. 80 Gold Team - FOPDT (50-75%) Input (%) Output (lbs/min) FOPDT MODEL K = 0.47 (lb/min)/% t o = 0.75 seconds τ = 0.70 seconds EXPERIMENTAL K = 0.48 (lb/min)/% t o = 0.75 seconds τ = 0.70 seconds Time (sec) Figure 6: The experimental run and model of a step response from 50-75%

9 Figure 6, shows an experimental run and model for a step response. From the graph, one can calculate the gain, dead time, and time constant. When looking at the graph, one can see that the model and experimental run have about the same parameters and the model plot is directly on top of the experimental data.

10 Procedure: The purpose of the frequency response experiments was to determine a model allowing an operator of the flow system to attain a desired output by setting the input and frequency. This data was found in several steps. The first step was observation. When observing a frequency response, the system was controlled at several different frequencies within the input range. The next step was the analysis. The analysis included observing the experimental graph s output verses input. The graph s showed the peak to peak of the input and output, the amount of time for one oscillation or period, and the amount of time the output lags the input. Figure 7 shows an experimental run of the frequency response in the input range of 40-60%. Frequency Response (f = 0.2 Hertz, Input = 40-60%) Input (%) time (sec) Output (lb/min) Input Value(%) Output(lb/min) Figure 7: Frequency Response for a frequency of 0.2 Hz in the 40-60% input range. From Figure 7, one can find the peak to peak amplitude for both the input and output. These peak to peak values will be used to find the amplitude ratio. The amplitude ratio can be found by dividing the output peak to peak by the input peak to

11 peak. The phase angle can also be found from Figure 7. To find the phase angle, one would subtract the lag time and the oscillation time and then multiply by 360 degrees. Now the phase angle and amplitude ratio were plotted on separate logarithmic graphs against frequency in a Bode Chart, shown in Figure 8. From the Bode Chart, the ultimate frequency (f u ) can be found at a phase angle of -180 degrees. At this frequency, one can determine the ultimate controller gain (K cu ) by drawing a line vertically to the experimental plotted line and recording the data point that corresponds to the amplitude ratio (y-axis). This point equals 1/ K cu so K cu can be sound by taking the inverse of the determined point from the graph. A Bode Chart was formed for the input ranges 40-60% and 60-80%. The values for gain were then averaged and can be used to model the sinusoidal response for the flow system using this equation, where ω is the frequency in hertz.

12 Results: From the frequency response that ultimate gain and ultimate frequency can be obtained. To compare some parameters with the step response, Table 1, lists the average values obtained from an experiment such as Figure 6. Input % K (lb/min)/% t o (sec) τ (sec) Average Table 1: The parameters found from the step response Figure 7, is an experiment test of a frequency response. There was several other experiments performed so the Bode Chart could be developed. Our group developed a Bode Chart are the input ranges 40-60% and 60-80%. The Bode chart in Figure 8 is from the 60-80% input range and plots the amplitude ratio and phase angle for frequencies of 4 Hz to 0.05 Hz. The vertical line is drawn where the phase angle is -180 degrees to the amplitude ratio curve. The Gain and K cu were then found by the method outlined in the procedure section.

13 Bode Chart (60-80%) 1 AR (lbs/min)/% 0.1 1/K cu f u Frequency (Hz) 0 Phase Angle (degrees) Frequency (Hz) Figure 8: Bode Chart for Input Range of 60-80%. The vertical red line denotes the ultimate frequency on the upper and lower graph and the Fu Amplitude Ratio on the upper graph. Bode Charts for the input range of 40-60% was created and can be found in the Appendices. The average experimental values for gain, ultimate frequency, ultimate controller gain, time constant, and dead time, were then placed into the frequency model equation to give a model of the plotted output from each experiment. The variable values for K, τ, and t o were then modified so the curve would accurately fit the data. The model

14 curve is graphed with the experimental data for each input range to represent the fit. Figure 9 is an example of the model and experimental data plotted on the same graph for the 60-80% input range. These values can be compared in Figure 8. Bode Chart (60-80%) 1 AR (lbs/min)/% 0.1 1/K cu f u Frequency (Hz) 0 Phase Angle (degrees) Frequency (Hz) Figure 9: The bode plot with experimental values for Amplitude Ratio and Phase angle plotted next to the modeled line for each using the altered gain, time constant, and dead time values for 70-85% input. The model equations were modified to fit the overall slope of the data points and represent the system in a more consistent way.

15 The experimental and model parameters for Gain, τ, and t o for each input range were then averaged to be the parameter applying to the entire operating range. Experimental parameters and averages and model parameters and averages are shown in Table 2. Input Range (%) Experimental Average K (lb/min)/% N/A t o (sec) N/A τ (sec) N/A f u (Hz) K cu (%/(lb/min)) Model Average K (lb/min)/% t o (sec) τ (sec) Table 2: Experimental and Model Parameters found by frequency response analysis. All parameters are averaged and shown in right column. These average parameters can be used to describe the transient period of the entire operating range. Comparing the parameters in Table 2, one can tell that the time constants are identical, but the gain and dead time are off. From modeling our frequency data, our group now knows the FOPDT parameters for the frequency response. Table 2 also contains the ultimate frequency and ultimate controller gain.

16 Discussion: From our results, our group now knows what the ultimate frequency and ultimate controller gain. When running an experiment our group needs to stay well below the ultimate frequency and ultimate controller gain. If one runs experiments close to these ultimate values then the control system will oscillate and the experiment will give data that is very tough to analyze and possibly could be inaccurate. From Table 2, one can see that the values of the ultimate parameters for the different input ranges are different. The big error is possibly from the amount of experiments that we run in the input range of 60-80% compared to 40-60%. Three experiments were run for each frequency in the range of 60-80%, but only one experiment was run for the input range of 40-60%. Another instance where error was found was in the Bode Charts. When looking in the Appendix at the phase angle verses frequency one can tell that the experimental data at -180 degrees is 0.4 Hz which is half of the modeled value at 0.8 Hz. This is another reason why the ultimate frequency and ultimate controller gain were different for the two input ranges. The purpose of the experiment was to obtain parameters which can be used to describe output when operating the flow system. Based on the data provided, the system output may be modeled using the transfer function and the frequency parameters.

17 Conclusions and Recommendations: From the frequency response parameter, our group concludes that there several different ways we can find the parameters to match our system. The first way was using the step response. After running experiments for the frequency response our group has found another was to find FOPDT parameters. The gain (K), or output per unit input, should operate at 0.57 (lb/min)/%. The time constant (τ), or time for output to completely change per input change to be 0.70 seconds. Dead time (t o ), the time it takes to have any response after an input change will be 0.40seconds. These values can then be used to describe the transient or changing period of output during a changing input. This experiment was designed to introduce a different way to analyze a change in output during a change input. The goal was accomplished by attaining the frequency model parameters of K, τ, and t o. A recommendation to reduce the error in the Bode Charts would be to run more experiments. I feel this would bring the ultimate frequency and ultimate controller gain for the two input ranges to similar values.

18 Appendix: Input Pump Speed (%) Output Flow Rate (lb/min) Standard Deviation (lb/min)

19

20 Table 3. SSOC experimental data Gold Team - FOPDT (30-50%) Input (%) K = (lb/min)/% t o = 0.8 seconds τ = 1.05 seconds FOPDT Model from Equation Experimental Data time (sec) Output (lb/min) Figure 10: Step Response Experimental and Model data for 30-50% input

21 Bode Chart (40-60%) AR (lbs/min)/% /K cu f u Frequency (Hz) Phase Angle ( o ) Frequency (Hz) Figure 11: Bode Chart for input range of 40-60%