Proportional Controller Performance for Aerator Mixer System

Transcription

1 1 Proportional Controller Performance for Aerator Mixer System By Nicholas University of Tennessee at Chattanooga ENGR Green Team (Monty Veal, TJ Hurless) April 2th, 21

2 2 Introduction- The experiment of the aerator mixer system involves a variable speed motor that is controlled by the operator through a computer interface called Lab View. The speed of the motor is altered by the operator s percent choice of input power. The experiment was done by selecting a desired bias for the system to operate at, with an appropriate set point, and changing the Kc values. This was done to observe the operation and behavior in the time domain of the aerator mixer system's approximate linear FOPDT model with proportional control. Also to observe the response of a closed loop controlled system to a set point change, and to observe the effect of the value of the proportional feedback gain, K c. While observing the limits of stable operation of the closed loop system, and to predict K c for critically damped response, quarter decay response and K cu. By analyzing the aerator mixers performance though proportion controlling and proportional integration controlling, the operator can receive the wants from the costumer, can produce an efficient output fulfilling the customer s needs. Following this is the Background and Theory section which describes all previous work done, beginning with the Steady-state operating curve and step response analysis. Then the theory behind the proportional controller performance will be discussed. Once the proportional controller performance has been discussed the proportional integration controller performance will be covered. The procedure comes after the background and theory, here will discuss the steps taken to perform the actual experiment and analysis. This will be followed by the results which gives the data acquired by both the experimentation and modeling of the proportional integration controller. A discussion

3 3 will follow the results where the data acquired will be analyzed, and compared to the model. A conclusion will be next summing up the system. The appendix will be last and is where all of the raw graphs will be for referencing all experiments.

4 4 Background and Theory The Aerator Mixer System consists of a motor that turn a mixing blade. One of the most common applications for an aerator mixer is at water treatment facilities. According to the EPA, waste water must be aerated to insure that the aerobic bugs that eat the waste have sufficient oxygen to survive. The motor and the mixing blade stir the waste to insure that the incoming oxygen is properly mixed. Figure 1 below represents the setup of the aerator mixer station. Figure 1. Aerator Mixer Schematic Diagram The computer sends a signal to the above motor drive (SCZ 247), and the motor speed is then determined by the speed transmitter (ST 247). The SRC 247 initiates the (SCZ 247) speed control drive. The SCZ 247 speed control drive is what allows the operator to vary the motor power input percentage.

5 5 Figure 2. Aerator Mixer Block Diagram Figure 2 above is a block diagram that represents the basic functions of the aerator mixer station. Figure 2 illustrates the basic function of the aerator mixer drive system. The input function m(t), is located on the left, and the output function c(t) is located on the right. For these functions, the m stands for manipulated, and the c stands for controlled. The m (manipulated variable) is what the controller has direct control over, for instance the motor power in a percentage. The c (controlled variable) is what the customer might want in a result of the system running. This input variable is constant and again is controlled by the operator. The output function c(t) in RPM s, is the drive speed of the system. The experiment of the aerator mixer was designed to measure the steady-state output speed of the motor in terms of a constant input percentage that the motor was given. The experiment was run using Lab View software and the data was compiled into Excel. Graphs of output motor speed vs. time were constructed, and as mentioned above the steady-state portion starts approximately one or two second after startup. The average output motor speed and standard deviation taken from the steady-state region of each graph was then used to construct the SSOC graph. The results indicated that the SSOC graph yielded a linear relationship between the motor output speed and the power percentage delivered to the motor. The significance of the SSOC graph is explained the in following paragraphs.

6 6 Steady-State Operating Curve (SSOC) The Steady-State Operating Curve (SSOC) and step response graphs were constructed from data that was collected through a data acquisition system which is connected to the aerator mixer station in the laboratory. The experiment was able to be run physically and through the internet. It was chosen to run the experiment through the internet. The Lab View interface allows the operator to choose the desired input power and experiment run time to be entered. The data from the experiment was then exported into Microsoft Excel and analysis was conducted. Within Excel, graphs were constructed using time, percent input power, output motor speed for each percentage power range. Data from these graphs were taken and used to construct the SSOC (Steady-State Operating Curve). The SSOC is constructed from the constant input% power to the motor, and the steady state portion of the output (RPM) of the motor. The average output (RPM) for each range is then graphed against its corresponding constant input% power, and the result is the SSOC graph. The operating range for both the input and output does exclude the initial start up because it is not steady state. Since graphing ΔC/ΔM produces the SSOC graph, then the slope of this graph yields the gain of the system. Graphs were constructed from each input percentage by graphing output (RPM s) vs. Time (s).graph 1 below represents one of the input percentage graphs and the rest can be found in the week 3 SSOC report.

7 7 MV Graph 1 Graph 1 above represents the speed (RPM) of the motor as a function of time. The steady state region at an 8% input power yields an average motor speed of 1376 RPM with a standard deviation of± 3.28 RPM. The average speed and standard deviation were taken from the portion of the graph that has reached steady state since this is what is of concern within this experiment. It should be noted that during the first 1-2 seconds of the experiment, the system is not at a steady state and therefore this portion of the graph and data are ignored. The average motor speed and standard deviation was collected from all input percentage experiments run, and were calculated based upon the steady state portion. In order to construct the SSOC graph, the above mentioned data was needed from all the experiments run. The input percentage, mean output motor speed, and twice the standard

8 8 deviation were used to construct the SSOC graph. Twice the standard deviation is used to insure a 95% confidence level. Table 1 below represents the data used to construct the SSOC graph. Average Motor Power (%) Output Speed Std. Dev (RPM) 2x S.D. (RPM) (RPM)

9 Table 1 From Table 1 above, the SSOC was constructed. The output RPM was graphed against input percent. Twice the standard deviation represented uncertainty within the experiment, therefore the uncertainty bars were also plotted on the SSOC graph. Graph 2 below represents the SSOC. Slope=17.6 RPM/% MV

10 1 Graph 2 The SSOC curve generated in graph 2 above indicates a strong linear relationship between the motor output speed and power input percentage to the motor. It should be noted that the first one or two seconds were ignored from each experiment due to the non steady-state start up. Table 1 indicates that the highest standard deviation is ±6.9 RPM from the mean. This standard deviation indicates a close relationship between the data and the mean. The slope or gain (K) of the SSOC curve was found to be 17.6 (RPM/%). FOPDT from Step Response The step response portion of the experiment is intended to approximate the FODPT parameters. The parameters being approximated are the gain K (RPM/%), dead time t o (s), and time constant τ (s). These parameters are approximated from the step response graph. The gain K is determined from change in output (RPM) denoted ΔC, divided by the change in input % power denoted ΔM. The dead time (t o ), is the time elapsed from when the step in input occurs to where the system begins to respond. The last parameter, time constant τ, is the time that the system takes to respond 63.2% of the total output change ΔC. The FOPDT Model was also used in approximating the parameters. The model uses the transfer function: (1)

11 11 Putting the transfer function into Microsoft Excel, the input (%) and output (RPM) can be graphed against time. The Transfer function is graphed over the top of the experimental input and output, and the FOPDT parameters were manipulated until the modeling data matches that of the experimental data. Graph 3

12 12 Graph 4 From Graph 3 above, the dead time t o and time constant τ was calculated to be.9 and.2 seconds. Graph 4 shows how the gain K is calculated, which was approximately 17.5 RPM/%. Each experiment was run twice for both the step up and step down, which enables the Student s T-Statistic to be used. The other step response graphs can be found in Appendix 1. The values for all of the FODPT parameters were approximated and the averages were tabulated in the following charts. Experimental and Model FOPDT Parameter Averages Gain K (RPM/%) Dead Time to (s) Time Constant τ (s) Experimental

13 13 Model Students T 17.4 ± Table 2 The comparison of the experimental and modeling values that Table 1 displays also shows the uncertainty between the two methods. The uncertainty between the two methods for the approximation of the FOPDT parameters is so small that it can be considered negligible. Close Loop Feedback The closed loop feedback for this equation is below. Figure 3. CLTF For Proportional Controller The CLTF for the proportional only controller is m( t) = m + K e( t). M (bar) is the bias, Kc is the controller gain, and m (t) is the controller output. The transfer function of the FOPDT is posted below. c (2) To get the characteristic equation Pade s approximation for e^(-to*s) was used.

14 14 e tos 1 1+ t o s 2 o s 2 t (3) So by multiplying the proportional controller CLTF by the FOPDT CLTF using Pade s approximation, the characteristic equation for the proportional controller is as follows. CE = 4Km 2 to sm + 4KK 2τt s o 2 c + 4τs + 2t c( t) 2t s + 4 o sk e( t) c (4) With these formulas, a mathematical model will be able to be used to predict what the experiment should act like. With this knowledge an error will be calculated to show how accurate the experiment actually was. Relay Feedback and Root Locus When operating a Relay Feedback the customer may want the motor of their aerator mixer to be rotating at 7 rpm. In order to achieve this need, the operator of the motor must operate at a specific motor power percentage. This percentage can be anywhere from 35% to 45% as the customer has specified. To satisfy the customer the operator must run the motor power at 45% to achieve the 7 rpm need. Once the aerator is fully operating at 7 rpm the speed will continually increase well above 7 rpm s if the motor percentage is not lowered. To keep the speed around 7 rpm the controller will drop the motor power percentage to 35% as the customer has specified. This will slow the speed of the aerator until it slows down to 7 rpm where the motor will then

15 15 pump 45% motor power back into the system. This is continued as shown in graph 5 below. Graph 5: Relay Feedback Once the Relay Feed back had been constructed the ultmate frequency (fu), the Amplitude Ratio (AR), and the Ultimate Gain (Kcu) could becalculated. The derivations of these are as followed: Ultimate Frequency fu= 1/ T (5) Amplitude Ratio

16 16 AR= ΔC/ΔM (6) Ultimate Gain Kcu= (4 ΔM) /(π ΔC) (7) These values will then be compaired to those constructed by the Sine Bode Diagrams. Bode Diagrams Bode Diagrams can be used to efficiently find values for Ultimate Frequencies, Amplitude Ratio, and Ultimate Gain for systems. As shown below, the Bode Diagram has been constructed and on the diagram the the fu, the AR, and Kcu have been determined.

17 17 Graph 6: Bode Diagram Analysis Proportion Controller A model was provided in the lab website. Using that model and putting in the FOPDT parameters for the aerator mixer system and parameters for a case where you have a change in set point (within the operating range of the output variable for your system) at a certain time. The "system" parameters are the ones that have been found in previous experiments for the system. For the underdamped case, determine the value of K c that gives sustained oscillation. This is Kcu. For the underdamped case, put in a K c that is the K c for quarter-decay that was gotten from Root Locus or the formula in Smith & Corripio. Observe what the decay ratio is for this value of K c. Make this observation by measuring the peak heights on the graph. For the underdamped case, put in a K c that is the K c for 1/1-decay that you got from Root Locus. Observe what the decay ratio is for this value of K c. Make this observation by measuring the peak heights on the graph. For the underdamped case, put in a K c that is the K c for 1/5-decay that you got from Root Locus. Observe what the decay ratio is for this value of K c. Make this observation by measuring the peak heights on the graph. This is the model that will be used to compare to the physical experiment that will be run in the lab. For the Physical lab, the experiment is using the values of K c that your consultant used in approximate modeling results for various tests. Observe the experimental system responses for the equivalent experiments. This means reproduce in your experiments the exact conditions that your consultant showed you graphs of the

18 18 model. Make a note of the offset, decay ratio, monotonic or oscillatory and settling time for each value of K c. After each of the experiments were ran and the proportional controller was finished, the experiments only prepared the proportional integral controller (PI) to be ran. Proportional Integral Controller A proportional integral controller is just a basic feedback controller that operates a specific system, in this case an aerator mixer. This operation runs between two specific points, these points are the difference between the desired output and the desired set-point clarified by the customer and the integral of the difference. A simple schematic of this operation is shown below in the feedback control loop where the controller is a proportional- integral controller: Figure 4: The FCL for a PI-Controller By running a PI- controller, the order of the system goes from a second order controlled system, to a third order control system. An example of a characteristic equation (CE) for a third order PI- Controller can be seen below:

19 19 (Tau*Tau I *t o )s 3 + (Tau I *t o + 2*Tau*Tau I - K C K*Tau I *t o )s (2*Tau I + 2*K C *K*Tau I - K C *K*t o )s + 2*(K C *K) = (8) Where Tau, to, and K are all parameters found for the specified systems being ran. The perimeters for the aerator mixer can be located in table 2a in the appendix. The value for Tau I can be found by using Page 231 of Principles and Practices of Automatic Process Control. But selecting Kc and Tau I values from past experiments, a modeling prediction for the system can be made. Using the Routh Method can also be used to analysis the system solving for then the value of Kc where system will become a stable system which the customer wants. If the customer wants to use a PI- Controller in his/her system, a good use of the controller would be have speed control for a D.C. Motor. The PI- Controller can cause the efficientcy of the system to actually increase 15 to 27% (according to the International Journal of Electronic Engineering Research [link found below diagram for reference.] A simple schematic of a D.C. motor using a PI- Controller is below: Figure 5: Schematic of an motor-load coupled driven system. This specific system produces an output speed that can be represented in the figure below:

20 2 Figure 6: Plot of the output speed for the system shown by figure 5 above. By using a PI- Controller, most customers will become more satisfied than using another type of controller. But a disadvantage of a PI- Controller may be that it could cause slower response times by it being a closed looped system not giving the customer the satisfaction immediately.

21 21 Procedure Part 1. (design and modeling) In order for the operator to fully satisfy the customer by fulfilling the customers wants and needs, the operator must design a PI- controller by constructing a root locus plot to validate values for Kc and t I. By choosing the Kc increments and ti values the root locus can be set up. This can be seen below in Results. Once the root locus has been displayed, Kc values can be determined by using the root locus plot in excel. The Kc values will fit in specific categories; Critical Damping, one- five hundredth decay, onetenth decay, quarter decay, and Ultimate damping. These values and the root locus plot can be found below in results. Part 2. (experimental data) Concluding part one, take the Kc and ti found and run the values in the PI- Controller experimental section of Chem.engr.utc.edu/329. Once each of the Critical Damping, onefive hundredth decay, one- tenth decay, quarter decay, and Ultimate damping experiments have been ran, analysis your graphed results in excel. Be sure to record the decay ratio, settling time, and offsets. After running each of those experiments, a windup and a wind-down experiment will be ran, by choosing a set-point change for the system, run the experiment and analysis the characteristics of the system. The change in set point can be determined by finding the value 5% higher than the minimum operating point, and 1% below the maximum operating point. To construct the wind-up make the change in set point a positive (making it jump up), and to construct the wind-down

22 22 make the change in set points a negative to create a step down. All of these results can be found below in Results.

23 23 Results Part 1. The FOPDT Parameters for Tau I used for part one of this procedure is located below: Table 3: FOPDT Parameters Using the FOPDT Parameters in Table 3, the following Root Locus Plot can be created:

24 24 Figure 7: Root Locus Plot Analyzing the plot, the Kc values can be determined. The Kc values are shown above in the plot but can also be seen below in the table for a better understanding of the Critical Damping, one- five hundredth decay, one- tenth decay, quarter decay, and Ultimate damping Kc s.

25 25 Table 4: Kc Results from Root Locus. Part 2. The data results for each of the experiments ran for each of the kc values are located below in Table 5. Trial Kc (%/rpm) Tau i Decay Ratio Osc./Mono Set. Time (s) Ultimate Damping Osc. 4.2 Quarter Decay Osc /1th Decay Osc /5th Decay Mono 6.3 Critical Damping.22 1 Mono 1.8 Table 5: Data Results for the PI- Controller Experiments Constructed The graphs corresponding to the data above in table 5 are located below:

26 26 Engineering Ultimate Damping Input (%) 8 6 N.M.T April Time (sec) Graph 7: Trial #1, Ultimate Damping Engineering Quarter Decay Input (%) N.M.T. 2 April Time (sec) 4 2 Graph 8: Trial#2, Quarter Decay

27 27 Engineering 1/1th Decay Input (%) N.M.T. April Time (sec) Graph 9: Trial #3, 1/1 th Decay Engineering 1/5th Input (%) N.M.T. April Time (sec) Graph 1: Trial #4, 1/5 Decay

28 28 Trial #5: Critical Damping Engineering Critical Damping Input (%) N.M.T. April Time (sec) Graph 11: Trial #5, Critical Damping Below for Graphs 12 and 13, the Wind up and Wind down experiments will be shown as trial #6 and #7.

29 29 Engineering Wind Up Input (%) 1 8 N.M.T April Time (sec) Graph 12: Trial #6, Wind UP In the Wind Up the System must over shoot the Clients need by about 2rpm to achieve their wanted results, with a settling time of just over 5 seconds. Engineering Input (%) N.M.T. April Time (sec) Graph 13: Trial #7, Wind Down

30 3 Discussion Looking at Part one in results, the FOPDT Parameters used for K, Tau, To, Tau I, and Kc increment are 17.6 RPM/%,.18s,.9s, 1s, and.2 RPM/% respectively. As a results of these parameters, the Root locus is plots, shown in figure 7. On the root locus plots are all of the Kc values in the boxes for each of the types of dampers and decays. These values are listed in a table below the root locus in table 4. Proceeding part 1, part 2 was done knowing each of the values found in part 1. In table 5 the results for each of the trials rain for each of the Kc values is shown. This table consists of the Kc values, the Tau I that I used for this experimental analysis, the decay ratio for each of the experiments, whether the data was oscillatory or if the data was monotonic, and also the settling time was also noted in table 5. The data in the table comes from the graphs In graph 7, the Kc Value was.28 %/RPM; it has a decay ratio of 1, settling time of 4.2 seconds, and it is oscillatory. In graph 8, the the Kc Value was.18 %/RPM; it has a decay ratio of.27, settling time of 5.1 seconds, and it is oscillatory. In graph 9, the Kc Value was.14 %/RPM; it has a decay ratio of.4, settling time of 5.6 seconds, and it is oscillatory.

31 31 In graph 1, the Kc Value was.58 %/RPM; it has a decay ratio of.1, settling time of 6.3 seconds, and it is monotonic. In graph 11, the Kc Value was.22 %/RPM; it has a decay ratio of, settling time of 1.8 seconds, and it is monotonic. After all of the experiments were ran using the different Kc values, the wind up and wind down experiments were ran. As shown in graph 12 for wind up from 5% to 9% and from the graph there is an over shoot of about 2 rpm to achieve the customers wants, and this process occurs in just over 5 seconds. As for graph 13, the wind down the graph goes from 9% to 5% and there is only about 1 rpm undershoot for about 5 seconds as well.

32 32 Conclusion As a conclusion to this Proportional Integral Controller, a lot is now known about the Green Team s Aerator Mixer. Coming into the Control Systems class, I had no clue about what a Control System even was. But Now proceeding this experiment it all has come clear to me. I know now, that if something is needed by a customer, say it is a speed of a mixer, the customer can not receive a perfect want. They can only receive a constant decrease of oscillation around their need. As a result of this particular experiment I have concluded that the aerator mixer is as close to a perfect system as you (%/rpm) of.28,.18,.14,.58,.22 for Ultimate Damping, Quarter Decay, 1/1 TH can get in this semesters class. It has a Gain of 17.6 rpm/% with Kc values decay, 1/5 TH decay, and Critical Damping, respectfully. As each of the Kc values got smaller the Decay Ratio did as well, but the time it took the system to settle was longer. For the aerator mixer this was not a big issue due to the small settling times but if the settlings times became a lot larger this may be a huge concern for the customer. Every customer is going to want an instant result of their want and need but in order to do that the Kc value will have to be high. So in order to satisfy the customer ultimately the operator should run a system at its quarter decay to maximize the Kc while having a relative low settling time. IN RESULTS MAKING THE CUSTOMER HAPPY!!!

33 33 Appendix Experiment 1 Kcu for Model Experiment 1 ¼ Decay for Model

34 34 Experiment 1 1/1 Decay for Model

35 35 Experiment 1 1/5 Decay for Model

36 36 Experiment 1 Over Damped for Model

37 37 Experiment 1 Critical Damped for Model

38 38 Experiment 2 Kcu for Model

39 39 Experiment 2 ¼ Decay for Model

40 4 Experiment 2 1/1 Decay for Model

41 41 Experiment 2 1/5 Decay for Model

42 42 Experiment 2 Over Damped for Model

43 43 Experiment 2 Critical Damped for Model

44 44 Experiment 3 Kcu for Model

45 45 Kcu Response OUTPUT/SET POINT TIME Experiment 3 1/1 Decay for Model

46 46 1/1th Decay 12 1 OUTPUT/SET POINT TIME Experiment 3 ¼ Decay for Model

47 47 1/4 Decay OUTPUT/SET POINT TIME Experiment 3. 1/5 Decay Model

48 48 1/5th Decay 12 1 OUTPUT/SET POINT TIME Experiment 3. Over Damping for Model

49 49 Over Damped 12 1 OUTPUT/SET POINT TIME Experiment 3. Critical Damping for Model

50 5 Critical Damped 12 1 OUTPUT/SET POINT TIME Experiment 1 Kcu for Physical Lab Proportional Control Experiment Ultimate Input (%) Time (s) Output(RPM) Experiment 1 1/1 Decay for Physical Lab

51 51 Proportional Control Experiment Tenth Decay Input (%) Time (s) Output(RPM) Experiment 1 1/25 Decay for Physical Lab Proportional Control Experiment Quarter Decay Input (%) Time (s) Output(RPM) Experiment 1 1/5 Decay for Physical Lab

52 52 Proportional Control Experiment 1/5 Decay Input (%) Time (s) Output(RPM) Experiment 1 Over Damping for Physical Lab Proportional Control Experiment Overdamped Input (%) Time (s) Output(RPM) Experiment 1 Critical Damping for Physical Lab

53 53 Proportional Control Experiment Critical Damped Input (%) Time (s) Output(RPM) Experiment 2 Kcu for Physical Lab Engineering Kcu Input (%) Output (rpm) Time (sec) Experiment 2 1/1 decay for Physical Lab

54 54 Engineering Input (%) Output (rpm) Time (sec) Experiment 2 1/4 decay for Physical Lab

55 55 Engineering Input (%) Output (rpm) Time (sec) Experiment 2 1/5 decay for Physical Lab

56 56 Engineering Input (%) Output (rpm) Time (sec) Experiment 2 Over Damped for Physical Lab

57 57 Engineering Input (%) Output (rpm) Time (sec) Experiment 2 Critical Damping for Physical Lab

58 58 Engineering Input (%) Output (rpm) Time (sec) Experiment 3 Kcu for Physical Lab

59 59 Experiment 3 1/1 Decay for Physical Lab

60 6 Experiment 3 ¼ Decay for Physical Lab Experiment 3 1/5 Decay for Physical Lab

61 61 Experiment 3 Over Damped for Physical Lab

62 62 Experiment 3 Critical Damping for Physical Lab

63 63

64 64 1/5th Decay 12 1 OUTPUT/SET POINT TIME Over Damped 12 1 OUTPUT/SET POINT TIME

65 65 Critical Damped 12 1 OUTPUT/SET POINT TIME The Results of the Relay feedback are in the table constructed below. Averages Uncertainties C (RPM) M (%) 1 AR (RPM/%) Period (s/cycle).17.1 Ultimate Frequency (cycles/s) Ultimate Gain Kcu (%/RPM) Time (s).17.24

66 66 Data from Proportion Controller This experiment was run using three different set of parameters. These parameters are included in Table 2. Table 8. Experiment 3 Physical Results Trials Kc (%/RPM) Decay Ratio Osc. Or Mono Settling Time. (s) Offset (rpm) Critical Damp.28 1/5th Decay.3 1/1th Decay.9 1/4th Decay.16 Over Damped.2 Decay ratio=. Monotonic.2 7 Decay ratio=. Monotonic.2 55 Decay ratio=.1 Oscillatory.3 2 Decay ratio=.21 Oscillatory.4 2 Decay ratio=. Monotonic.3 5 Kcu.18 Decay ratio=.9 Oscillatory Table 2. Parameters used for Proportion Controller Experiment Parameters Experiment Experiment Experiment

67 K (RPM/%) Tau (s) To (s) Delta R (RPM) R Base (RPM) Input Baseline (%)

68 68

69 69

70 7

71 71

72 72 Table 9. 2x Standard Deviation Between Model and Experimental Error Analysis Decay Ratio Settling Time (s) Offset (RPM) exp1 exp2 exp3 exp1 exp2 exp3 exp1 exp2 exp3 Critical Damp /5th Decay /1th Decay /4th Decay Over Damped Kcu

73 73 Recommendations The mechanics of the report look good, and all the wording is good. I may recommend on zooming into the interesting parts of the graphs in your results section Also each graph is supposed to be followed by a 2 to 3 sentence description of what the plot shows This may help you from losing some cheap points.