ME2142E Feedback and Control Lab -Frequency Response

7/21/2019 ME2142E Feedback and Control Lab -Frequency Response

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by

LIN SHAODUN A0066078X

Lab Group 6B

Date 1st

Apr 2011


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1

TABLE OF CONTENTS

OBJECTIVES 2

EXPERIMENT PROCEDURE 2

TABLE 1 3

NICHOLSCHART 4

TABLE2 5

BODE PLOT 6

SAMPLE CALCULATION 8

DISCUSSION 8


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OBJECTIVES

To perform a frequency response test on an aircraft electro-hydraulic servo-actuator and to determine the phase and gain margins of the servo.

Understand how to obtain open-loop gain and phase from Nichols Chart. Practice deriving transfer function from Bode plot. To verify that increasing the gain causes system instability.

EXPERIMENT PROCEDURE

1. Verify that the equipment is complete and familiarize the connections and thelayout.

2. Set the function generator output as follow: Sine wave frequency = 0.5 Hz, Amplitude of output = 5 VPP

Leave the amplitude setting unchanged throughout the experiment.

3. Open the hydraulic supply valve and note the supply pressure. The actuatorshould now be moving in and out.

4. Measure the voltage output Vo and change in timetime by oscilloscope. Derivethe gain and phase angle.

5.

Repeat step 4 at increasing frequencies of 1Hz until reach 12Hz. Tabulate theresults in Table 1 and make a plot on the Nichols chart at each frequency.

6. Having completed the plot on the Nichols chart read off the gain and phasemargins.

7. Reset the function generator for 5V peak-to-peak square wave at about 0.5 Hz.Gradually increase the loop gain by increasing the summing amplifier feedback

resistor. At each increase observe the systems response to a step input.

8. Increase the gain K until the system has reached instability. Convert the gain K to(dB) using Gain = 20 log10 K. Compare this value of gain margin with that

obtained from the Nichols chart.

9. From the Nichols chart, write down the values of the open loop transfer functionin Table 2.

10.Draw the Bode gain plot and Bode phase plot on the same graph. Estimate theorder of the system and its time constants.


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TABLE 1

(Hz) (rad/s)=2f

Vi (V) Vo (V) Gain (dB)

=20 logi

o

V

V

time (s) Phase (deg)=time x x360o

0.5 3.142 5.0 4.641 -0.647 0.160 28.8

1. 6.283 5.0 4.20 -1.514 0.132 47.52

2. 12.566 5.0 3.081 -4.206 0.106 76.32

3. 18.850 5.0 2.241 -6.971 0.088 95.04

4. 25.133 5.0 1.701 -9.365 0.074 106.6

5. 31.416 5.0 1.321 -11.561 0.070 126.0

6. 37.699 5.0 1.121 -12.987 0.062 133.9

7. 43.982 5.0 0.904 -14.856 0.058 146.2

8. 50.265 5.0 0.792 -16.005 0.054 155.5

9. 56.549 5.0 0.720 -16.833 0.0472 152.9

10. 62.832 5.0 0.512 -19.794 0.0448 161.3

11. 69.115 5.0 0.448 -20.954 0.0432 171.1

12. 75.398 5.0 0.380 -22.384 0.0408 176.3


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NICHOLS CHART

Step 8:

System become unstable when K=28, Gain = 20log10K = 28.9.

From the Nichols Chart, the best fit line of Gain vs. Phase plot

meets 180 at Gain=25.0, the predicted result is close to the

actual K during experiment.

Phase margin

Gain margin


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TABLE 2

Take the readings of the Open Loop Gain and Open Loop Phase from the Nichols

Chart and then plot on the Bode plot. Bode Plot consists of 2 graphs; Gain against and

Phase against

.

(rad/s) Open Loop Gain (dB) Open Loop Phase (Deg)

3.142 5.7 -95.0

6.283 1.0 -101.6

12.566

-4.4 -110.0

18.850 -7.7 -116.5

25.133 -10.2 -121.0

31.416 -12.7 -136.0

37.699 -14.0 -142.0

43.982 -15.8 -150.0

50.265 -16.4 -154.0

56.549 -17.2 -157.0

62.832 -19.5 -162.0

69.115

-21.0 -171.0

75.398 -23.1 -176.0


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BODE PLOT

-30

-25

-20

-15

-10

-5

0

5

10

15

20

1.0E+00 1.0E+01 1.0E+02

OpenLoopGain(dB)

Frequency ( Rad / s )

Open Loop Gain vs.

-180

-170

-160

-150

-140

-130

-120

-110

-100

-90

-80

1.0E+00 1.0E+01 1.0E+02

Op

enLoopPhase(Deg)

Frequency ( Rad / s )

Open Loop Phase vs.

()

()

()

()


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Matlab Bode Plot of Transfer Function

Mathematica Bode Plot of Transfer Function

-60

-40

-20

0

20

40

Magnitude(dB

)

Bode Diagram

Frequency (rad/sec)

10-1

100

101

102

103

-180

-135

-90

Phase(deg)


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SAMPLE CALCULATION

Note: The sample calculation is based on table 1, f=0.5Hz.

1. Frequency Rad/s

2. Gain()

3. Phase()

DISCUSSION

1. Discuss the results including the response obtained with the increased gain.a) Result of Nichols chart show the gain margin of the system is about 25dB, which

is very close to experiment result (5VPP square wave @ 0.5Hz). When increase the

loop gain, the shape of output waveform become closer and closer to the input

square waveform, until the gain reach 28 system become oscillating which can be

observed from oscilloscope display.

b) The Bode plot using data obtained from Nichols chart shows this is a second ordersystem.

The Gain plot consist two lines with -20dB/decade and -40dB/decadeslope.

The intersection point of two lines is the time constant, The intersection point of first line and Y axis is the K value.

With this information, its easy to obtain the transfer function of this system.

Using Matlab to do the Bode plot of this transfer function, the shape fits with

experiment data, this shows the transfer function derived from the Bode plot is

correct.

ME2142E Feedback and Control Lab -Frequency Response

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