comparission of results for proportional controller...

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.


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


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


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.


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


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


Table 14. Summary of Results for Dead Time

t0 (sec)

Step Up


Modeling Step Down


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


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


Table 15. Summary of Results for Time Constant

τ (sec)

Step Up


Modeling Step Down


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.


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


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%.


00.5

11.5

22.5

33.5

44.5


%/v




Table 17. Summary of Results for Controller Gain at an Input Range of 87%-90%


Root Locus



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


11


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%.


0

1

2

3

4

5

6


%/v





Root Locus



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


12


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


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


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


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


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APPENDIX:


17

OUTPUT VS. TIME (83% INPUT)

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TIME (s)

INPU

T, M

(t) (%

)

0

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80

OU

TPU

T, C

(t) (V

)

Figure 13. Voltage vs. Time for 83% Input Power


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0 5 10 15 20 25 30 35

TIME (s)

INP

UT,

M(t)

(%)

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OU

TPU

T, C

(t) (V

)


JRA 1/20/10

JRA 1/20/10


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OUPUT VS. TIME (87% INPUT)

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TIME (s)

INPU

T, M

(t) (%

)

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OU

TPU

T, C

(t) (V

)



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TIME (s)

INPU

T, M

(t) (%

)

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TPU

T, C

(t) (V

)


JRA 1/20/10

JRA 1/20/10


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TIME (s)

INPU

T, M

(t) (%

)

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OU

TPU

T, C

(t) (V

)



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TIME (s)

INPU

T, M

(t) (%

)

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OU

TPU

T, C

(t) (V

)


JRA 1/20/10

JRA 1/20/10


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TIME (s)

INPU

T, M

(t) (%

)

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TPU

T, C

(t) (V

)


JRA 1/20/10


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.

comparission of results for proportional controller...

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