Proportional-Integral Controller Performance Silver Team Jonathan Briere ENGR 329 Dr. Henry 4/1/21 Silver Team Members: Jordan Buecker Jonathan Briere John Colvin
1. Introduction Modeling for the response of a Proportional-Only Controller was completed, to provide a basis for an experiment. The modeling was based on parameters previously acquired by Root Locus modeling, as well as former experiments throughout the semester. Then, experiments were completed using a Proportional-Only and Proportional-Integral controller. Each of these experiments was done to support the theory behind the model. In the case of the level-control station, the experiments could not be related back to the model, since the hardware was changed after modeling was completed. This changed the system parameters which modeling was based upon, resulting in an inaccurate model for the experiments. The report is organized into a background and theory section, followed by the procedure used for the experiments. After the procedure section, results and a discussion of the results will follow. The report will end with conclusions of the experiment and recommendations. The background and theory sections will present the technical background of the experiment as well as schematics of the system. First, the SSOC operation of the system will be discussed. Then, frequency modeling and experiments will be introduced. The Root-Locus theory and procedures for modeling will then be presented as a basis for the proportional controller design and experiments. The procedure section will provide a description of the steps used to complete the modeling and the physical experiments. The results section will provide tables and graphs obtained by models and experiments. The discussion section will look at the relevance, accuracy, and discrepancies of the results. The conclusions and recommendations section will provide a brief summary of the experiment and state anything that might be done to improve the 2 JB 4/14/1
experiment. Finally, the appendices will contain any intermediate data or results that are not pertinent to the presentation of results. 3 JB 4/14/1
2. Background and Theory 2.1 System Description The system used for all modeling and experiment is the Level Control Station.. Figure 1 shows physical representation of the system. Water is pumped from the basin, into the upper tank. Water is then routed from the upper tank to the lower tank, and back to the basin. Both tanks are equipped with overflow tubes, allowing water to be routed back to the basin without overflowing. Figure 1: System Diagram 4 JB 4/14/1
From the basin, water is pumped into an upper tank where it drains only under the force of gravity into the lower tank. The lower tank is equipped with a pressure sensor. The pressure sensor converts the water pressure into an electrical signal and relays that signal back to the LabVIEW software, through a data acquisition board. The LabVIEW program computes the height of the water in the tank based on the signal. The user may define a percentage of the maximum power input to the pump motor. The percentage of input to the pump corresponds to the steady state level that the water reaches in the tank. Figure 2: Level Control System Diagram Figure 2 depicts the process for the system. The input is the manipulated variable and is the percentage of power delivered to the motor. The system responds to manipulation of M(t) and the output, C(t), changes according to that response. C(t) is known as the control variable. 2.2 Steady State Operation Results The Steady State Operating Curve (SSOC) was determined by running different trials at specified increments. The user controls the duration of the experiment as well as the input percentage of the pump. From this, data returned to the operator by the system may be exported to Excel. 5 JB 4/14/1
The data was first graphed to plot the input (%) versus the time (sec) with the output (cm) on the secondary y-axis. This provided visual inspection of the start-up of the system and an approximation of where the system reaches steady-state. Graphs were constructed for all of the trials, which begin at 6% input and continue to 1% in 1% increments. Figure 3 provides the graph of the data for a 8% pump input. 8% 9 8 7 6 5 4 3 2 1 14 12 1 8 6 4 2 Output (cm) 5 1 15 2 Time (sec) Figure 3: Input versus time with output on secondary y-axis for 8% run. In the region of steady state, the mean and standard deviation were computed using the appropriate functions in Excel. The graphs for the other runs may be found in the appendix. The mean, standard deviation, and 2 times the standard deviation for the steady-state regions were computed for each run and tabulated to construct the steady state operating curve. Table 1 shows this data. Table xxxxx: Data for steady state operating curve. 6 JB 4/14/1
Input (%) Output (cm) StDev (cm) 2 X StDev (cm) 6.75.22 +.43 7.75.17 +.34 8 12.49.2 +.41 9 14.97.12 +.25 1 22.49.27 +.55 The average output was plotted against the input to construct the steady state operating curve below in figure 4. The slope of the line is the gain of the system. Uncertainty was taken to be 2 times the standard deviation and is represented on the graph by vertical error bars. There appears to be some discrepancy in the range between 7% and 9% as the slope varies significantly. The maximum deviation from the mean for this experiment is about 1.5% indicating that the height of water in the tank can be reasonably ascertained based on the input percentage to the pump in the operating range. 7 JB 4/14/1
Figure 4: Steady state operating curve 2.3 Step Response and Modeling Results The step response occurs when a system operating at steady state receives a sudden, definite, input change. The state between the two steady state portions is what is referred to as the step response curve. The first order plus dead time (FOPDT) parameters of the system are calculated from this section of the graph. The FOPDT parameters are: dead time (t), first order time constant (tau), and gain (K). The step response portion of the lab was performed in the same manner as the steady state lab. The only addition was to step the motor input up or down by 1% once the water level reached a steady state. This was performed from 7-8, 8-9, 9-1, 1-9, 9-8, and 8-7 percent motor inputs. After motor input was stepped, the water was allowed to reach steady state at the new motor input. Figure 5 shows a typical step response 8 JB 4/14/1
graph and associated parameters for the system. Graphs of the averages the parameters for the different regions may be found in the appendix. Figure 5: Typical step response for level control system Figure 6 shows the results for the step response modeling for the 9-1 motor input range. 9 JB 4/14/1
Figure 6: 9-1 Step Response Model The input function for both the model (light blue line) and the experimental (dark blue line) match up perfectly as shown in figure 6 above. This is to be expected since the input is the manipulated variable. In the case of the step response the only parameter that had to be changed was the time that the step occurred; in the case shown above this happens at approximately 2s. The model output shown above (green line) was fit to the experimental output s shape of the step response portion of the experimental output (red line) by changing the following parameters: dead time, gain, time constant, input baseline, output baseline. The dead time, gain, and time constant that were used in the modeling are shown below in Table 2.The models for the rest of the operating range can be found in the appendix. 2.4 Frequency Response Results The data were tabulated and analyzed using Excel. Figure 7 shows the frequency response at.3 hz for the 7-8% range. The other graphs are located in the appendix. 1 JB 4/14/1
Figure 7: Graph of input versus time for 7-8% sine response Table 3 shows the frequencies and corresponding amplitude ratios and phase shifts for the 7-8% range. These were obtained directly from the input percentage versus time graphs at the different frequencies. Table 1: Table of frequencies and corresponding AR and PS for 7-8% range Frequency AR PS.2.7-5.3.6-54.6.3-13.1.2-18 The system was allowed to run long enough to complete one full cycle (time > period). Because the systems had to run so long (more than 1 minutes for most trials) it was not reasonable to 11 JB 4/14/1
allow the system to run for multiple cycles. Because of these time restraints, it was not possible to have replications and data for statistical analysis. Bode diagrams were constructed using the data in table 4. Figure 8 shows the bode diagram for the 7-8% region. The diagram is the graph of the AR versus the frequency on a log-log scale imposed on top of the PS versus frequency graph (semi-log with frequency being logarithmic). The frequency axes are scaled identically to ensure the graph line up properly. From this diagram the FOPDT parameters were obtained. K was found to be.75cm / %, the ultimate gain (1/ARu) was found to be 5 % / cm the order of the system was found by the negative slope of the graph of AR versus frequency at lower frequencies and was found to be about.25. Finally, the ultimate frequency was found to be.1 cycles/second. 12 JB 4/14/1
Figure 8: Bode diagram for the 7-8% region Dr Henry provided instructions to model the frequency response using Excel. This modeling yielded the most accurate values for the FOPTD parameters. Figures 9 and 1 show the graphical results of this modeling. 13 JB 4/14/1
Frequency.1.1.1 1.1 Figure 9: Graph of AR versus frequency for frequency modeling.1 Frequency.1.1-5 -1-15 -2-25 Experimental Model Figure 1: Graph of PA versus frequency for frequency modeling. Table 4 gives the FOPDT parameters found from the frequency response modeling. The gain was found to be.88 cm / %. The dead time was 3 seconds and the time constant was 58 14 JB 4/14/1
seconds. These values will be the ones used for the FOPDT parameters for the remainder of the semester. Table 2: Table of FOPDT parameters found from frequency response modeling. Parameters K=.88 to= 3 tau= 58 2.5 Root Locus The open loop transfer function (OLTF) is the product of all of the transfer functions in the control loop. The system diagram is shown in figure 11. Figure 11: Control loop system diagram Ke t os For this loop, the OLTF is, K c τs +1 Ke t o s K c τs +1 The closed loop transfer function(cltf) is modeled by. Using Pade s Ke t os 1+ K c τs +1 K 1 t s 2 K c (τs +1) 1 + t s 2 approximation, the CLTF becomes K 1 t s 2 1+ K c (τs +1) 1 + t s 2 15 JB 4/14/1
The characteristic equation is the denominator of this function set equal to zero and through t algebraic manipulation becomes, τs 2 2 + τ + t 2 K K t c s + K 2 c K +1=, a quadratic which may be solved for various values of Kc and plotted on an imaginary coordinate system to produce the root locus plot. 2.5 Proportional-Only Controllers The error between the setpoint and the controlled variable output is known by e( t) r( t) c( t) =, Where r (t) is the setpoint of the system, and c (t) is the controlled variable. The controller output for a Proportional-Only Controller is described by m( t) = m + K e( t) C Where m (t) is the controller output and m is the bias; the output from the controller when the error is zero. 2.6 Proportional-Integral Controllers In Proportional-Integral Controllers, an extra amount of intelligence is added to the controller to solve for the correct controller output which will remove the offset. The controller output is described by whereτ I is the integral time. KC m( t) = m + KC e( t) + e( t) dt, τ I 16 JB 4/14/1
3. Procedure 3.1 Root-Locus Modeling Using a spreadsheet provided by Dr. Henry, five K values were determined: K CU, the ultimate frequency of the system, K CD, the quarter decay value, C1 K, the 1/1 th decay, K C5, the 1/5 th decay, and finally, K CD, the critical damping value, were plotted and behavior was observed. Parameters obtained from previous experiments were used as input to generate the model. Each controller gain value was found by using given ratios of imaginary over real, then matching these ratios for the equivalent K C in the root locus modeling. 3.2 Proportional-only Controller Design Using a spreadsheet provided by Dr. Henry, were plotted. Then, several of these K C values obtained from the Root-Locus modeling K C values were plotted. From these graphs, the Decay Ratio, Monotonic/Oscillatory response, Settling Time, and Offset were noted. 3.3 Proportional-only Controller Experiment As instructed by Dr. Henry, the team ran experiments for values of K C ranging from 1. to 7.. Since the pump in the system was replaced, the system parameters changed, and the system would not respond as it did in the model. To choose the set point, the team ran the system at an input level of 7%, and let the system reach steady-state. This steady state output value was then chosen as a basis for the set point of the system. The team initially ran the experiment for a set point 5 cm above this set point, let the system reach steady state, then lowered the set point back 17 JB 4/14/1
to the value determined by the initial steady-state run. Results were outputted to a spreadsheet, and the results were graphed, and checked for the offset, decay ratio, monotonic or oscillatory and settling time. 3.4 Proportional-Integral Controller Experiment K C values ranging from 1. to 7. were once again run for the proportional integral controller. Α τ I value of 3 seconds was used for all experiments. Since the system calibration parameters were lost, the set point value was run from 3cm to 6cm, as instructed by Dr. Henry. This range of output gave data around the mid span of the input operating range. 18 JB 4/14/1
4. Results 4.1 Root-Locus Modeling The results obtained for Root-Locus Modeling were introduced in the background of this report. 4.2 Proportional-only Controller Design Kc 1.1 24 22 2 18 16 14 12 1 1 2 3 4 5 6 7 TIME Figure 12: Modeled Proportional-Only controller response for range K C =1, 7-8% input Figure 12 shows a typical modeled Proportional-Only controller response for a K C value of 1.1. Table xxxxxxxx and xxxxxxxx shows the tabulated results of the experiments from two different runs. All models are contained in the Appendix under figures XXXXX through XXXXX. 19 JB 4/14/1
Table XXXXX: Summary of P-controller design modeling: 7-8% range Decay Settling K Description C Ratio Monotonic/Oscilliatory Time Offset Ultimate 5.5 1 Oscilliatory Never N/A Root Locus 1/5 Decay 3.6.3 Oscilliatory 3 sec 1 Root Locus 1/1 Decay 2.7.2 Oscilliatory 2 sec 1.5 1/K*(tau/t) 2.2.9 Oscilliatory 1 sec 2 Root Locus 1/4 Decay 1.1 Oscilliatory 15 sec 2.5 Underdamped.4 Monotonic 75 sec 3.5 Table XXXXX: Summary of P-controller design modeling: 8-9% range Decay Settling K Description C Ratio Monotonic/Oscilliatory Time Offset Ultimate 29 1 Oscilliatory Never N/A Root Locus 1/4 Decay 19.7 Oscilliatory Never.5 Root Locus 1/1 Decay 15.3 Oscilliatory 175 1.5 Root Locus 1/5 Decay 6.1 Oscilliatory 15 1.75 Critical 3.1 Oscilliatory 13 2.5 Underdamped 1 Monotonic 13 3.5 Table XXXXX: Summary of P-controller design modeling: 9-1% range Decay Settling K Description C Ratio Monotonic/Oscilliatory Time Offset Ultimate 5.5 1 Oscilliatory Never N/A Root Locus 1/5 Decay 3.6.3 Oscilliatory 5 sec.5 Root Locus 1/1 Decay 2.7.25 Oscilliatory 3 sec 1.5 1/K*(tau/t) 2.2.15 Oscilliatory 22 sec 1.75 Root Locus 1/4 Decay 1.1 Oscilliatory 2 sec 2.5 Underdamped.4 Monotonic 2 sec 3.5 2 JB 4/14/1
4.3 Proportional-only Controller Experiment Figure 13: P only controller experiment. Figure 13 depicts an experiment run at a K C value of 3. The setpoint was first set to 21cm, then lowered to 16 cm, output value corresponding to an output bias of 7%, as predicted by an earlier experiment. Experiments were run by two of the team members. Additional graphs for values run can be found under figures xxxx through xxx of the appendix. 4.4 Proportional-only Controller Experiment 21 JB 4/14/1
Figure 14: Proportional-Integral Controller Experiment with K C = 1 Figure 14 shows the results of an experiment run for a set point varying from 6 cm to 3 cm, with a controller gain of 1, and an integral time constant of 3 seconds. Additional experiments were run following the same parameters to reinforce the accuracy of behavior of the system, and can be seen in the appendix under figures xxxxxxxxx through xxxxxxxxxxx. 5. Discussion 5.1 Root-Locus Modeling 5.2 Proportional-only Controller Design 22 JB 4/14/1
5.3 Proportional-only Controller Experiment 5.4 Proportional-Integral Controller Experiment 6. Conclusions and Recommendations 5.1 Root-Locus Modeling 5.2 Proportional-only Controller Design 5.3 Proportional-only Controller Experiment 5.4 Proportional-Integral Controller Experiment 23 JB 4/14/1
Appendix 8% 9 8 7 6 5 4 3 2 1 5 1 15 2 Time (sec) 14 12 1 8 6 4 2 Output (cm) A3: Graph of input and output versus time for 8% run. 9% 1 9 8 7 6 5 4 3 2 1 18 16 14 12 1 8 6 4 2 Output (cm) 13 18 23 28 33 Time (sec) A4: Graph of input and output versus time for 9% run. 24 JB 4/14/1
1% 12 25 1 2 8 6 4 2 15 1 5 Output (cm) 5 1 15 2 25 3 Time (sec) A5: Graph of input and output versus time for 1% run A6: Step Response for 6-7 Motor Input 25 JB 4/14/1
A7: Step Response for 7-8 Motor Input A1: Step Response for 1-9 Motor Input 26 JB 4/14/1
A11: Step Response for 9-8 Motor Input A14: Step Response Model for 7-8 Motor Input 27 JB 4/14/1
A15: Step Response Model for 8-9 Motor Input 28 JB 4/14/1
A17: Step Response Model for 1-9 Motor Input A18: Step Response Model for 9-8 Motor Input A19: Step Response Model for 8-7 Motor Input 29 JB 4/14/1
Average Gain 1.9.8 UP Dow n UP Dow n.7 Gain (cm/%).6.5.4 Dow n.3 UP.2.1 7-8 8-9 9-1 Input Operating Range (%) A2: Average Gain Average Tau 15 13 Dow n 11 9 Tau (s) 7 5 UP UP Dow n UP 3 Dow n 1-1 7-8 8-9 9-1 Input Operating Range (%) A21: Average Tau 3 JB 4/14/1
A22: Average dead time A23: Bode diagram 31 JB 4/14/1
A24: Root locus plot Engineering 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 Time (sec) A25: 7-8%.2 frequency 32 JB 4/14/1
A26: 7-8%.3 frequency Engineering 9 8 7 6 5 4 3 2 1 5 1 15 2 25 3 Time (sec) A27: 7-8%.6 frequency 33 JB 4/14/1
Engineering 9 8 7 6 5 4 3 2 1 5 1 15 2 25 3 Time (sec) A28: 7-8%.1 frequency Frequency.1.1.1 1.1 A29: AR versus frequency model 34 JB 4/14/1
.1 Frequency.1.1-5 -1-15 -2-25 Experimental Model A3: AR versus frequency model 35 JB 4/14/1