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The following bar chart shows how the results of the experimental proportional controller
compare for all three input ranges.
Figure 36. Result Summary for Controller Gain for Proportional Controller Experiment
Comparission of Results for Proportional Controller Experiment
0
0.2
0.4
0.6
0.8
1
1.2
KCU KQD K10 K500 KCD
%/v
Kc @83-86%Kc @87-90%Kc @91-94%
JRS 03/31/2010
Jered Swartz 03/29/2010
2
DISCUSSION:
Steady-State
The steady-state operating curve in Figure 5 details that the control system is in
its' operating range within the input power limits of 83% and 95% outlined in the
experiment, because of the linear relationship of the curve. This is important because this
region could be represented by a first-order linear differential equation. This means that
one could predict the output voltage in this range with this type of equation. The gain, or
slope of the steady-state operating curve, was estimated to be 1.5 V/%. The uncertainties
of the voltage output were between 0.20% and 0.31%, which would indicate that the
mean voltage output calculated for each input power percentage is a very close
representation of the data acquired. It was determined that within the specified power
input range of 83% and 95% that the system is in steady-state operation.
Step Response
The average gains of the system for several step up, and step down responses are
shown in Figure 10 and indicate that the gain is nearly constant for step up responses.
The gain in the step down responses increases as the input power percentage increases.
The gain is also higher for the step up responses than the step down responses.
Jered Swartz 03/29/2010
3
The average dead times of the system for the same step up and step down
responses are shown in Figure 11 and indicate that the dead time in the step up responses
is greater than its corresponding step down response. There were high levels of
uncertainty associated with the average dead time values. The average time constants of
the system for the same step up and step down responses are shown in Figure 12 and
indicate that the time constant is greater with a step up response than with a step down
response. There were high levels of uncertainty associated with the average time
constant. The three FOPDT parameter values were higher for the step up responses than
the step down responses.
The modeling methods outlined in the background for the step response correlate
closely with the values determined by pure analytical means. As with the graphical
method the step up parameters are higher than the step down counterparts. Value wise
they tend to line up closely. Tables 2 and 5 support this conclusion.
Sine Input
The sine input response outlines important characteristics for the system, by
allowing the values for the ultimate gain and ultimate frequency to be determined. This
method is somewhat subjective however, and lends itself to variations within the data.
As can be seen in Table 6 the average ultimate gain values tend to fluctuate, with an
average of 1.1 %/Volt for the entire set. Ultimate frequency for the system was
determined to around 8 Hertz.
Modeling methods employed on the Bode plots allow the FOPDT parameters to
be calculated from the sine input response as well. Table 7 summarizes these findings
Jered Swartz 03/29/2010
4
with Gain, dead time, and time constant values of 1.37 V/%, .045 seconds and .99
seconds, respectively.
Relay Feedback
The relay Feedback control method is an invaluable tool which allows the
ultimate gain and ultimate frequency to be determined with less calculation time than the
sine input response method. Values obtained by this method over the range chosen tend
to differ from those obtained by the sine method. The ultimate frequency as seen in
figure 21 is found to be 7.1 Hertz. Comparing this to the value of 7.5 Hertz found over
the same range in the sine experiment we see a strong correlation. The Ultimate gain
differs substantially, with the relay feedback estimating its value at .86 %/V and the sine
method determining it to be 1.11%/V.
Root Locus
The root locus plot is a very helpful graph. This graph and excel spread sheet
allow the user to find various gains based upon what the customer wants. The excel
spreadsheet also quickly calculates the real and complex roots of the system a lot more
quickly then a human could. The graph also shows the effect zeta has on the controller
gain. Overall the Root locus is a good tool to predict the controller gain that will be
needed for a specific request from a customer. The data from the root locus modeling
matched up well with the data from the proportional controller gain.
Proportional Controller
The modeling from the proportional controller gain matched up nicely with the
results from the root locus and proportional controller experiment. Since the OLTF for
the proportional controller is known along with the FOPDT parameter, it is easy to
Jered Swartz 03/29/2010
5
estimate the controller gains of the system. Even using the eyeball method instead of
calculating out the decay ratios for the modeling, the values were still close to the values
that were calculated in the experiment.
The proportional controller experiment took a little longer to conduct then the
modeling because the decay ratios were being found as closely as possible. The
proportional controller experiment data did not match up with the root locus data or the
proportional controller modeling. The reason the data did not match up is that a
proportional controller can not be used with the voltage system to obtain the right data.
Under most circumstances doing the modeling first would give the user a good idea of
what values to use for the controller gain to get the appropriate decay ratio. Without
doing the modeling first it would have taken a long time to find each of the decay ratios.
The settling time and offset are also easy to obtain from both the modeling and
experimental graphs.
For the SSOC, step, and sine experiments the team obtained the FOPDT
parameters of τ , 0t , and k. The following figure shows how the values of k related to
each other for each of the SSOC, step, and sine experiments.
Jered Swartz 03/29/2010
6
Comparission of Results for K
00.20.40.60.8
11.21.41.61.8
2
83%-86% 87%-90% 91%-94%
k (v
/%)
SSOCStep Up ExperimentalStep Up ModelingStep Down ExperimentalStep Down ModelingSine
Figure 37. Result Summary for Gain for All Experiments
The table below is a data table showing the values on the graph.
Table 13. Summary of Results for Gain
K (v/%)
SSOC Step Up
Experimental Step Up
Modeling Step Down
Experimental Step Down Modeling Sine
83%-86% 1.7 1.85 1.81 1.33 1.44 1.4 87%-90% 1.7 1.8 1.85 1.4 1.5 1.37 91%-94% 1.7 1.79 1.77 1.52 1.45 1.35 From the table above it can be seen that the gain varied from experiment to experiment.
The step down experiment and modeling matched up with the sine experiment. The gain
of 1.4 v/% was used for the root locus, proportional controller experiment, and
Jered Swartz 03/29/2010
7
proportional controller modeling based upon the results above. The following figure is a
bar chart depicting the results of dead time 0t for the system.
Results Comparission for t0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
83%-86% 87%-90% 91%-94%
Tim
e (s
ec) Step Up Experimental
Step Up ModelingStep Down ExperimentalStep Down ModelingSine
Figure 38. Result Summary for Dead Time for All Experiments
The table below is a data table showing the values on the graph.
Table 14. Summary of Results for Dead Time
t0 (sec)
Step Up
Experimental Step Up
Modeling Step Down
Experimental Step Down Modeling Sine
83%-86% 0.07 0.04 0.02 0.03 0.06687%-90% 0.06 0.04 0.04 0.01 0.03 91%-94% 0.03 0.04 0.03 0.01 0.06 From the table above it can be seen that the dead time varied a little bit from experiment
to experiment. Since the dead times during the sine experiment were close to 0.06 sec, a
dead time of 0.06 sec was used for the root locus, proportional controller modeling, and
Jered Swartz 03/29/2010
8
proportional controller experiment. The following figure is a bar chart depicting the
results of time constant τ for the system.
Comparission of Results for τ
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
83%-86% 87%-90% 91%-94%
Tim
e (s
ec) Step Up Experimental
Step Up ModelingStep Down ExperimentalStep Down ModelingSine
Figure 39. Result Summary for Time Constant for All Experiments
The table below is a data table showing the values on the graph.
Table 15. Summary of Results for Time Constant
τ (sec)
Step Up
Experimental Step Up
Modeling Step Down
Experimental Step Down Modeling Sine
83%-86% 0.33 0.13 0.07 0.07 0.1 87%-90% 0.24 0.05 0.07 0.1 0.1191%-94% 0.22 0.22 0.11 0.1 0.08
The table above shows that the time constant also varied through out the different
experiments. Since the dead times for the sine input were 0.1 sec, 0.08 sec, and 0.11 sec
a dead time of 0.09 sec was used for the remainder of the experiments.
Jered Swartz 03/29/2010
9
For the remaining experiments (root locus, proportional controller modeling, and
proportional controller experiment) a gain of 1.6 v/%, a dead time of 0.06 seconds, and a
time constant of 0.09 seconds were used. The values were determined from the tables
above. Using these FOPDT parameter the root locus, proportional controller experiment,
and proportional controller modeling were used to find the ultimate controller gain, the
critical dampening, 1/500th decay, 1/10th decay, and 1/4th decay. The following figure
compares the results of controller gain for an input range of 83%-86%.
Controller Gains for Input Range of 83%-86%
0
0.5
1
1.5
2
2.5
3
3.5
KCD KQD K10 K500 KCU
%/v
Root LocusProportional Controller ModelingProportional Controller Experiment
Figure 40. Result Summary for Controller Gain for an Input Range of 83%-86%
The following Table shows the results from the bar chart above.
Table 16. Summary of Results for Controller Gain at an Input Range of 83%-86%
Controller Gain for Input Range 83%-86%
Root Locus
Proportional Controller Modeling
Proportional Controller Experiment
KCD 0.16 0.34 0.25 KQD 2 2.1 0.6 K10 1.5 1.6 0.42 K500 0.58 0.6 0.31 KCU 3.1 3 0.9
Jered Swartz 03/29/2010
10
From the table above it can be seen that the modeling and experimental values for the
input range of 83%-86% due not match up. The following figure compares the results of
controller gain for an input range of 87%-90%.
Controller Gains for Input Range of 87%-90%
00.5
11.5
22.5
33.5
44.5
KCD KQD K10 K500 KCU
%/v
Root LocusProportional Controller ModelingProportional Controller Experiment
Figure 41. Result Summary for Controller Gain for an Input Range of 87%-90%
The following Table shows the results from the bar chart above.
Table 17. Summary of Results for Controller Gain at an Input Range of 87%-90%
Controller Gain for Input Range 87%-90%
Root Locus
Proportional Controller Modeling
Proportional Controller Experiment
KCD 0.25 0.42 0.3 KQD 2.3 2.3 0.67 K10 1.76 1.8 0.46 K500 0.72 0.8 0.34 KCU 4.1 3.6 0.9
Jered Swartz 03/29/2010
11
From the table above it can be seen that the modeling and experimental values for the
input range of 87%-90% due not match up. The following figure compares the results of
controller gain for an input range of 91%-94%.
Controller Gains for Input Range of 91%-94%
0
1
2
3
4
5
6
KCD KQD K10 K500 KCU
%/v
Root LocusProportional Controller ModelingProportional Controller Experiment
Figure 41. Result Summary for Controller Gain for an Input Range of 91%-94%
The following Table shows the results from the bar chart above.
Controller Gain for Input Range 91%-94%
Root Locus
Proportional Controller Modeling
Proportional Controller Experiment
KCD 0.38 0.38 0.38 KQD 2.8 2.8 0.71 K10 2.2 2.2 0.5 K500 0.93 0.93 0.41 KCU 5.3 4.4 1
Jered Swartz 03/29/2010
12
From the table above it can be seen that the modeling and experimental values for the
input range of 91%-94% due not match up.
From all of the experiments conducted it is obvious that the modeling data and the
experimental data do not match up. The experimental data is what actually happened,
and the modeling data is what is expected to happen based upon given equations. From
the information above it has been determined that the voltage system cannot be properly
model for the voltage range of 70-95 volts.
CONCLUSIONS AND RECOMMENDATIONS:
Steady-State
The objective of the experiment was to acquire data from a system and develop a
steady-state operating curve for that system. This curve was created by inputting
different power percentages to an electric motor which powers a generator and receiving
output voltages from the generator. LabVIEW was used for data acquisition and EXCEL
Jered Swartz 03/29/2010
13
was used in creating graphs for data analysis. The steady-state portion of each graph was
used to form a steady-state operating curve. The graphs in the Appendix indicate a slight
time delay before the system reaches steady-state operation. The results show that
between 83% and 95% input power, that the system is operating under steady-state
conditions.
Step Response
The objective of the experiment was to acquire data from the voltage system and
determine the first-order plus dead time (FOPDT) parameters for the system. The
FOPDT parameters are the gain, dead time, and time constant. These parameters were
found by graphing the experimental data of the step responses. LabVIEW was used for
data acquisition and EXCEL was used in creating graphs for data analysis. The results
for average gain, dead times, and time constants can be seen in Figures 10, 11, and 12,
respectively. The results show that the three FOPDT parameter values were higher for
the step up responses than the step down responses. There is a high amount of
uncertainty associated with the dead times, and time constants.
Modeling this data in Excel provides another method in which to compare the
results found by analytical means. The two methods are strongly correlated which lends
a high amount of validity to the results found.
Sine Input
While the primary concern of the step input and constant input techniques are to
find the FOPDT parameters, the sine method provides a means by which to determine
more useful characteristics of the voltage system, namely ultimate frequency and ultimate
controller gain. Other than the relay feedback method no other methods for determining
Jered Swartz 03/29/2010
14
these parameters has been explored. It is apparent from the results however that range of
interest plays a strong part in the results one can expect to find. These parameters seem
strongly tied to the set up of the experiment so consistency of method is key.
Sine modeling on the other hand does yield parameters that can be compared to
existing data. However as stated before in order to maintain an accurate assumption of
the systems response consistency it paramount.
Relay Feedback
The Relay Feedback is a quick way to delve into the intricacies of the control
system without tedious hand calculations and subjective graphical analysis. This method
does appear to not be without its own set of shortcomings. While it does give the user
another useful way to compare data regarding the ultimate frequency and ultimate
controller gain, the choice of input range must be thoroughly considered. Set point and
ceiling/floor values can greatly affect the output results. It could be concluded that this
type of analysis should be performed over the entire expected operating range as opposed
to a small portion thereof.
Root Locus
The objective of the modeling was to acquire the real and complex roots of s
when the FOPDT parameters were plugged into the OLTF. The roots were then used to
generate a curve on which various different controller gains exist for different decay
ratios. The graph was created by placing the negative and positive real roots on the x-
axis and the negative and positive complex roots on the y-axis. The data was generated
from the FOPDT parameters that were discovered in the sine input experiment. From the
Jered Swartz 03/29/2010
15
root locus plot it is easy to find the controller gain of the system for a specific outcome
the customer wants.
Proportional Controller gain
The objective of the experiment was to use the FOPDT parameters along with a
known bias, set point, and change in set point to reach an outcome selected by the
customer. The change in set point is based upon what the customer wants. Using
LabView a plot of the data was created in excel. From the graph the settling time, state,
controller gain, and offset could all be calculated. The outcome that should be selected is
the one that gives the customer the output closer to what they desire. In this system the
quarter decay gives the customer the closet and most accurate result for all three input
ranges. For the proportional controller gain experiment, the data did not come out
properly in any of the operating ranges. The reason the data was so different from the
modeling and the root locus, is that a proportional controller can not be used with the
voltage system.
The proportional controller was also modeled. The modeling was done by
inputing the FOPDT parameters, set point, change in set point, and time of the change in
set point. The modeling showed completely different results than the experiment did. If
a proportional controller worked with the voltage system the data would have matched
closely between the proportional controller experiment and the proportional controller
modeling
Jered Swartz 03/29/2010
16
APPENDIX:
Jered Swartz 03/29/2010
17
OUTPUT VS. TIME (83% INPUT)
0
10
20
30
40
50
60
70
80
90
0 5 10 15 20 25 30 35
TIME (s)
INPU
T, M
(t) (%
)
0
10
20
30
40
50
60
70
80
OU
TPU
T, C
(t) (V
)
Figure 13. Voltage vs. Time for 83% Input Power
OUTPUT VS. TIME (85% INPUT)
0
10
20
30
40
50
60
70
80
90
0 5 10 15 20 25 30 35
TIME (s)
INP
UT,
M(t)
(%)
0
10
20
30
40
50
60
70
80
OU
TPU
T, C
(t) (V
)
Figure 14. Voltage vs. Time for 85% Input Power
JRA 1/20/10
JRA 1/20/10
Jered Swartz 03/29/2010
18
OUPUT VS. TIME (87% INPUT)
0
10
20
30
40
50
60
70
80
90
100
0 5 10 15 20 25 30 35
TIME (s)
INPU
T, M
(t) (%
)
0
10
20
30
40
50
60
70
80
90
OU
TPU
T, C
(t) (V
)
Figure 15. Voltage vs. Time for 87% Input Power
OUTPUT VS. TIME (89% INPUT)
0
10
20
30
40
50
60
70
80
90
100
0 5 10 15 20 25 30 35
TIME (s)
INPU
T, M
(t) (%
)
0
10
20
30
40
50
60
70
80
90
OU
TPU
T, C
(t) (V
)
Figure 16. Voltage vs. Time for 89% Input Power
JRA 1/20/10
JRA 1/20/10
Jered Swartz 03/29/2010
19
OUTPUT VS. TIME (91% INPUT)
0
10
20
30
40
50
60
70
80
90
100
0 5 10 15 20 25 30 35
TIME (s)
INPU
T, M
(t) (%
)
0
10
20
30
40
50
60
70
80
90
100
OU
TPU
T, C
(t) (V
)
Figure 17. Voltage vs. Time for 91% Input Power
OUTPUT VS. TIME (93% INPUT)
0
10
20
30
40
50
60
70
80
90
100
0 5 10 15 20 25 30 35
TIME (s)
INPU
T, M
(t) (%
)
0
10
20
30
40
50
60
70
80
90
100
OU
TPU
T, C
(t) (V
)
Figure 18. Voltage vs. Time for 93% Input Power
JRA 1/20/10
JRA 1/20/10
Jered Swartz 03/29/2010
20
OUTPUT VS. TIME (95% INPUT)
0
10
20
30
40
50
60
70
80
90
100
0 5 10 15 20 25 30 35
TIME (s)
INPU
T, M
(t) (%
)
0
10
20
30
40
50
60
70
80
90
100
OU
TPU
T, C
(t) (V
)
Figure 19. Voltage vs. Time for 95% Input Power
JRA 1/20/10
Jered Swartz 03/29/2010
21
Table 3. Summary of Step Response Data
K, V/% ±, V/% To, s ±, s τ, s ±, s Tr, s ±, s 83% UP 1.85 0.54 0.07 0.02 0.33 0.32 1.65 1.58 85% DOWN 1.33 0.34 0.02 0.01 0.07 0.02 0.37 0.12 86% UP 1.71 0.21 0.04 0.04 0.37 0.52 1.83 2.58 88% DOWN 1.40 0.29 0.04 0.02 0.07 0.20 0.37 1.01 89% UP 1.80 0.45 0.06 0.02 0.24 0.17 1.22 0.87 91% DOWN 1.51 0.14 0.03 0.00 0.06 0.02 0.32 0.12 92% UP 1.79 0.24 0.03 0.00 0.22 0.03 1.12 0.17 94% DOWN 1.52 0.20 0.03 0.02 0.11 0.20 0.53 1.02
NOTE: 72 graphs were used to acquire the data in Table 3.