wind turbine pitch adjustment scheme design exercise

7/29/2019 Wind Turbine Pitch Adjustment Scheme Design Exercise

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EE 3001

Control Engineering

Wind Turbine

Pitch Adjustment Scheme Design Excercise

Mark Roche

109479961

16/02/12


2/21

Pitch Adjustment Scheme Mark Roche

Design Exercise 109479961

Page | 1

Modelling the System

Modelling the wing turbine pitch control scheme yields:

Figure 1- Simulink Model for Open Loop System


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Page | 2

1)Show how the open-loop system responds to step changes in themotor voltage v(t) and disturbance torque

1] Response to step changes in motor voltage v(t)

The following time responses were obtained by setting the load torque to zero, andstepping the input voltage from 0V to 5V at 0 seconds.

Figure 2- Open Loop Time Responses (Step Input Voltage)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

14

16

Time(s)

Current(A)

Open Loop Current Response (Step Voltage Input)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

Time(s)

Voltage

(V)

Open Loop Potentiometer Voltage Response (Voltage Step Input)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

Time(s)

Voltage(V)

Open Loop Tachometer Voltage Response (Voltage Step Input)


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Page | 3

2] Response to step changes in disturbance torque

The following time responses were obtained by setting the input voltage to zero, and

stepping the disturbance torque from 0V to 5V at 0 seconds.

Figure 3- Open Loop Time Responses (Disturbance Torque Step Input)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Time(s)

Current(A)

Open Loop Current Response (Disturbance Torque Step Input)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Time(s)

Voltage(V)

Open Loop Potentiometer Voltage Response (Disturbance Torque Step Input)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Time(s)

Voltage(V)

Open Loop Tachometer Voltage Response (Disturbance Torque Step Input)


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Page | 4

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

Time(s)

Open Loop Responses for Step Input Voltage and subsequent Step Input Disturbance Torque

Current

Potentiometer Voltage

Tachometer Voltage

3] Response of the system to step changes in disturbance torque with

constant input voltage

The following graph shows the system reaction to a load torque of 100V for a constant input

voltage of 5V. The disturbance torque is set to input into the system at 5 seconds. It is clear

that each of the plots follows the same trend as shown above:

Figure 4- Open Loop Time Responses for constant input voltage and stepped

disturbance torque

- There is an initial spike in current in the motor due to the step input voltage, which

gradually tends to zero. The disturbance torque acts to lift the current level in the system

following a waveform which could be approximated as

.

-The potentiometer voltage follows a waveform which can be approximated as a

ramp function. There is an initial transient period from 0-1.5 seconds however. The load

torque step input acts to reduce the slope of said ramp.

-The tachometer voltage follows a second order sigmoidal trend due to the input

voltage step and levels off at a steady state value. However when the disturbance hits the

system, the tachometer voltage drops and stays at this lower value for an infinite time

period.

The Disturbance Rejection Capability of the open loop system is poor.N.B.

The tachometer voltage and potentiometer voltage are directly linked to the blade speed

and blade pitch. By dividing each response by K and K respectively, the responses for

blade speed and pitch can be easily obtained.


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Page | 5

2) Determine by experiment the frequency response . Plot this asa Bode Plot, a Nyquist Plot and a Nichols Chart

Data Gathering and Experimental Bode Plot of

However, clearly as these transfer functions are merely first order approximations, it is

necessary for to achieve a truer approximation to the actual transfer functions and

. To do this, the input voltage was set to a sinusoidal wave of 1V amplitude. The

amplitude and phase change of the output wave was then determined. This processwas carried out for numerous frequencies of input voltage. Hence in this way, data points

were gathered for the experimental bode plot of , which are shown in detail in Table 1

below.

Experimental Data

Frequency

(rad/s) Gain (V)

Time Input

Wave (s)

Time Output

Wave (s)

Phase

(degrees) Gain (dB)

V/V0.01 0.4948 785.4 786.289 -0.50935948 -6.11141

0.05 0.4943 157.0795 157.97 -2.551094583 -6.12019

0.1 0.4929 78.5405 79.4281 -5.08557339 -6.14482

0.3 0.4786 47.1225 47.996 -15.01435902 -6.40055

0.6 0.4382 44.5058 45.328 -28.26515395 -7.166550.8 0.4055 49.087 49.867 -35.75256642 -7.84018

1 0.3725 45.553 46.288 -42.11239794 -8.57747

2 0.2456 47.9093 48.4518 -62.16592077 -12.1954

3 0.1761 48.6947 49.1138 -72.03798358 -15.0848

4 0.1358 49.0874 49.4274 -77.92226014 -17.342

5 0.11 49.323 49.6091 -81.96161259 -19.1721

6 0.09223 49.4801 49.7274 -85.01547764 -20.7026

7 0.07927 49.5923 49.8104 -87.47346658 -22.0178

8 0.069343 48.891 49.0865 -89.61059916 -23.1799

9 0.0617 49.0437 49.2211 -91.47844157 -24.194310 0.05546 49.166 49.3295 -93.6785995 -25.1204

20 0.02675 49.7163 49.809 -106.2263752 -31.4535

30 0.01679 49.68952 49.756333 -114.8430875 -35.499

60 0.006608 49.7681 49.8072 -134.4158987 -43.5986

80 0.004195 49.8924 49.9236 -143.0102657 -47.5454

100 0.002872 49.5272 49.5532 -148.9690267 -50.8363

200 0.0007994 49.48795 49.5022 -163.2929716 -61.9447

300 0.0003634 49.6843 49.6941 -168.4495918 -68.7923

400 0.0002061 49.7039 49.7114 -171.8873385 -73.7184

500 0.0001324 19.9334 19.9395 -174.7521275 -77.5622600 0.00009215 19.9727 19.9778 -175.3250853 -80.7101


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700 0.00006779 19.9379 19.9423 -176.4710009 -83.3767

800 0.00005194 19.9118 19.9156 -174.1791697 -85.69

900 0.00004106 19.9474 19.9508 -175.3250853 -87.7316

1000 0.00003328 19.9884 19.9915 -177.6169165 -89.5563

Table 1- Experimentally Measured Gains and Phase Angles for

The Time Input Wave and Time Output Wave columns refer to the time at which a

certain peak occurs in the input/output sinusoid. This phase difference was then easily

determined by:

At first it was quite challenging to decide which peaks of the input/output waves should be

measured for each frequency. However, a number of things soon became clear:

At low frequencies, it was necessary to run the simulation for a lengthy time (in the caseof 0.01rads-1, more than 700 seconds). This is due to the fact that the wave itself is

travelling so slowly at these low frequencies. For these measurements minimizing the

step size to optimize measurement accuracy was not a major goal as if the minimum

step size was too small, it would result in a slowing down of the simulation (hence,

maximum step size was set to 0.001). For these frequencies, measurements were taken

from the second peak of the input/output waves, as the first peak of the output wave

still incorporated transient characteristics.

At medium to high frequencies, a time scale of 50 seconds was chosen for the durationof the simulation. This was thought to be a reasonable time scale as, for the final peaks

of the output wave, transient behaviour was seen to have died away and also, a

timescale of 50 seconds allowed for a small maximum step size to be set within the

simulation configuration parameters (maximum step size 0.0001, relative tolerance 1e-

7).

However, at very high frequencies, ( 500rads-1), the need to increase measurementaccuracy heightens due to the fact that the wave is moving so quickly. Hence, it was

necessary to reduce the maximum step size to 0.00001. However, it was observed that a

step size of this size resulted in a major retardation of the simulation when used with a

50 second time scale. It is for this reason that all data above 400rads-1

was measured

over a timescale of 20 seconds, which was chosen as it was seen to give optimum levels

of accuracy coupled with a relatively quick simulation time.

Subsequently, the data points in Table 1 above were plotted on a Bode Gain/Phase Plot as

shown below:


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Page | 7

Figure 5-Bode Gain and Phase Plots for

Experimental Bode Plot of

From the Simulink diagram shown earlier it is clear that the only thing separating the

potentiometer voltage from the tachometer voltage is a factor of:

Hence, in order to convert from gain and phase values for to those for , all that isneeded is to scale the gain (not in dB) by

and subtract 90

oof phase from the phase

reading of each datapoint. The resulting data points found are expressed on the table

below:

Experimental Data

Frequency

(rad/s) Gain (V)

Time Input

Wave

Time Output

Wave

Phase

(degrees) Gain (dB)

V/V0.01 98.96 1413.75 1414.6 -90.48701413 39.90919373

0.05 19.772 911.052 911.92 -92.48663683 25.92101204

0.1 9.858 1963.497 1964.3812 -95.06609282 19.87577628

0.3 3.190666667 486.9474 487.8152 -104.9163832 10.0776287

0.6 1.460666667 243.474 244.289 -118.0176362 3.29102237

0.8 1.01375 198.3142 199.089 -125.514216 0.118617344

10-2

10-1

100

101

102

10-90

-80

-70

-60

-50

-40

-30

-20

-10

0Experimentally Determined Bode Gain Plot (Vw(jw)/V(jw))

Gain(dB)

Frequency(rad/s)

10-2

10-1

100

101

102

103

-180

-160

-140

-120

-100

-80

-60

-40

-20

0Experimentally Determined Bode Phase Plot (Vw(jw)/V(jw))

Frequency(rad/s)

Degrees


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Page | 8

10

-2

10

-1

10

0

10

1

10

2

10

3-160

-140

-120

-100

-80

-60

-40

-20

0

20

40Experimentally Determined Bode Gain Plot (VB (jw)/V(jw))

Frequency(rad/s)

Gain(dB)

1 0.745 95.8185 96.5535 -132.1123979 -2.556874545

2 0.2456 3004.148 3004.687 -151.7648503 -12.19543275

3 0.1174 3003.8864 3004.3058 -162.0895498 -18.60663806

4 0.0679 9004.1975 9004.536 -167.5784855 -23.36260451

5 0.044 9004.119 9004.404 -171.6464858 -27.13094647

6 0.030743333 9004.0665 9004.3132 -174.8092128 -30.24498092

7 0.022648571 9001.336 9001.551 -176.2301482 -32.89918372

8 0.01733575 9001.645 9001.84 -179.381416 -35.22114729

9 0.013711111 9001.885 9002.0635 -182.0456698 -37.25854699

10 0.011092 9009.6165 9009.779 -183.1056417 -39.09980278

20 0.002675 9004.51175 9004.6035 -195.1377554 -51.45352427

30 0.001119333 9001.7625 9001.829 -204.3050801 -59.02081126

60 0.000220267 9001.841 9001.8805 -225.7909974 -73.14102441

80 0.000104875 9001.93925 9001.9705 -233.2394488 -79.58656052

100 0.00005744 9001.9353 9001.96125 -238.6825478 -84.81571138

200 0.000007994 900000.3611 900000.3754 -253.9003076 -101.9447171

300 2.42267E-06 9.3462 9.356 -258.4495918 -112.3141267

400 1.0305E-06 9.9 9.9074 -259.5955074 -119.7390401

500 5.296E-07 19.6444 19.6504 -261.8873385 -125.5210405

600 3.07167E-07 19.6585 19.6636 -265.3250853 -130.2525183

700 1.93686E-07 29.5601 29.5645 -266.4710009 -134.2580482

800 1.2985E-07 49.7569 49.7608 -268.7628321 -137.7311609

900 9.12444E-08 49.7576 49.761 -265.3250853 -140.7958714

1000 6.656E-08 49.5885 49.5916 -267.6169165 -143.5357337

Table 2- Experimentally Measured Gains and Phase Angles for

This yields bode plots for :


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Page | 9

10-2

10-1

100

101

102

103

-280

-260

-240

-220

-200

-180

-160

-140

-120

-100

-80Experimentally Determined Bode Phase Plot (VB (jw)/V(jw))

Frequency(rad/s)

Phase(Degrees)

Figure 6-Bode Gain and Phase Plots for

Nyquist Plot for

Figure 7-Nyquist Plot for (Second Plot shows close-up view of Im=0)

20

40

60

80

100

30

210

60

240

90

270

120

300

150

330

180 0

X: -0.01551

Y: 1.172e-010


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Page | 10

Nichols Chart for

Figure 8- Nichols Chart for

Hence, the Nyquist Plot shows that the extra gain needed to make the system marginally

stable is:

This is an approximation for the ultimate gain of the system.Hence the gain margin has beendetermined also form the Nyquist Plot to be 64.4745. The gain and phase margins can also

be determined from the Nichols Chart as shown above. As the Open Loop dB Gain at -180o

was found from the Nichols Chart to be -35.8dB, this implies that:

From the Nichols Chart the phase margin can also be determined as the phase difference

between the critical point and the point on the plot which corresponds to 0dB Open LoopGain. Hence, from the Nichols Chart shown above: o

-360 -315 -270 -225 -180 -135 -90 -45 0-200

-150

-100

-50

0

50

6 dB3 dB

1 dB0.5 dB

0.25 dB0 dB

-1 dB

-3 dB-6 dB

-12 dB

-20 dB

-40 dB

-60 dB

-80 dB

-100 dB

-120 dB

-140 dB

-160 dB

-180 dB

-200 dB

Nichols Chart

Open-Loop Phase (deg)

Open-LoopG

ain(dB) Gain Margin

Phase Margin


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Page | 11

3)Use the frequency response to determine an estimate of the open-loop transfer function

The transfer function can be obtained both directly from the

gain bode plot, or

indirectly from the gain bode plot. In homework tutorials, the direct method was

discussed, however the indirect method will be explained here, as this was the method that

the student used to approach the problem initially, and it was found to give an

approximation of equivalent accuracy to that of the direct method. Effectively, how the

student approached the problem was to obtain the transfer function from the

corresponding bode plot, as a question very similar was covered in the course notes. The

method of asymptotes was used as shown in the plot below:

Figure 9- Bode Gain Plot for including asymptotes

By inspection, the bode gain plot of is noticeably broken down into 3 key areas. That

which corresponds to an asymptote of 0dB/decade (low frequencies), that which

corresponds to an asymptote of -20dB/decade and that which converges to an asymptote of

-40dB/decade (high frequencies). Where these asymptotes cross corresponds to the corner

frequencies of the transfer function. By analysing the above plot:

and

10-2

10-1

100

101

102

103

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0Experimental Bode Gain Plot Vw/V

Radial Frequency (rad/s)

Gain

(dB)

Experimental Bode Plot

0dB/decade Asymptote




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Page | 12

It is known that the transfer function is to be second order as by analysing the Simulink

diagram it can be seen that the circuit consists of 2 integrators up until the tachometer.

Hence the transfer function will be of the form:

The system also has a constant gain for low frequencies of -6.1115dB

This implies that: => Therefore:

( )

Which simplifies down to:

From this point, it is not difficult to obtain - merely scale

by

. Hence:

By declaring this transfer function in Matlab, the actual gain and phase margins, and

crossover frequencies of this system can be obtained (which for the gain and phase margins

will obviously be slightly different to those obtained via the Nichols Chart). The code needed

is relatively simple:

[] [ ][ ] This Yields:

Gainmargin =61.8161, Phasemargin = 53.6001, Crossover1 = 8.2083,

Crossover2 =0.8042, Crossover3 = 1

Hence, the gain and phase margins obtained from the Nichols Chart were clearly accurate,

presuming that the frequency response data accurately describes the system.


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Page | 13

4)A proportional controller with gain Kp is proposed to control pitchangle. Show through simulation in Simulink, how the closed-loop

performance depends on the choice of gain Kp. Show how your

Nichols Chart could have predicted these results.In this part of the design, it was attempted to improve the performance characteristics of the system

by closing the loop and introducing a proportional controller. Hence, the equivalent circuit

becomes:

Figure 10- Diagram of Closed Loop Model

Where the subsystem contains:

Figure 11- Internal View of Subsystem

The input voltage is set to a step input of 5V, and by varying the value of Kp- the proportional gain-

changes in the closed loop performance are evident:


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Page | 14

Figure 12- Time Responses for Kp=10



0 10 20 30-150

-100

-50

0

50

100

150

200

Time(s)

Current(A)

Current

0 10 20 30-6

-4

-2

0

2

4

6

8

Time(s)

Voltage(V)

Tachometer

0 10 20 300

1

2

3

4

5

6

7

8

9

Time(s)

Voltage(V)

Potentiometer

0 2 4 6 8 10-3

-2

-1

0

1

2

3

4x 10

8

Time(s)

Current(A)

Current

0 5 10-4

-3

-2

-1

0

1

2

3

4x 10

6

Time(s)

Voltage(V)

Tachometer

0 2 4 6 8 10-6

-4

-2

0

2

4

6x 10

5

Time(s)

Voltage(V)

Potentiometer

0 10 20 30 40 50-500

-400

-300

-200

-100

0

100

200

300

400

500

Time(s)

Current(A)

Current

0 20 40 60-15

-10

-5

0

5

10

15

Time(s)

Tachometer

Voltage(V)

0 10 20 30 40 500

1

2

3

4

5

6

7

8

9

10

Time(s)

Potentiometer

Voltage(V)


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Page | 15

Hence, it is clear that for low values of proportional gain (Kp=10), the time responses of the current,

speed and pitch angle are stable waveforms. However, the peak overshoot and settle time are

undesirable. These qualities generally disimprove with increasing Kp

However, once Kpreaches the level of ultimate gain, the system becomes marginally stable and

subsequent to this, when Kp is increased past the level of ultimate gain, the system will become

unstable, which is demonstrated above for a proportional gain of 200. For stable choices of Kp there

is zero steady-state error for step inputs, due to the free integrator in the system which makes the

system itself Type 1, having a position error constant of infinity.

Figure 15-Time Response for Kp=61.6595

The above plot shows the time responses for a 5V step input, when the value of proportional gain is

equal to the value obtained for marginal stability for the Nichols Chart. Clearly, the gathered data

gives an accurate description of the system, as the gain margin ( which in this case equals the

ultimate gain of the system) , almost gives perfect marginal stability (which would be a continuoussinusoidal output of constant amplitude for a step input voltage). However in this case, the

amplitude of the sinusoid is increasing slightly with time, hence the system is slightly unstable.

In this manor, the Nichols Chart has already predicted the results shown above in Figures 12-15.

When the proportional gain is set to a value greater than 61.6595 instability occurs. But what about

the disturbance rejection capability of the new proportionally controlled closed-loop system? In the

0 5 10-1500

-1000

-500

0

500

1000

1500

Time(s)

Current(A)

Current

0 5 10-25

-20

-15

-10

-5

0

5

10

15

20

25

Time(s)

Voltage(V)

Tachometer

0 5 10-2

0

2

4

6

8

10

12

Time(s)

Voltage(V)

Potentiometer


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Page | 16

5)Design a first-order phase-lead compensator to provide what youbelieve is the best trade-off between steady-state accuracy and

dynamic performance. Demonstrate the performance of you pitch

control scheme, through simulation in Simulink.

The aim of this section is to design a phase-lead compensator for the system, so that the

system will track a ramp of the form

-with the best trade-off between steady-state error and dynamic performance.

From the material studied in lectures, the design specifications for a well configured phase-lead

compensator will follow:

Where PO% refers to the peak overshoot for a step change in the desired pitch angle, and ess is the

steady- state error. For a ramp input waveform:

The transfer function of a phase-lead compensator is:

Therefore the velocity error constant is:

As we want PO%=10%- need to find the corresponding damping of the system, which can be foundvia the equation:

This equation is difficult to solve- therefore the damping required was obtained from the graph

given in the course notes. The resulting value of is 0.6, which when subbed back into the aboveequation gives a peak overshoot of 10%. Hence, if the system is assumed to be second order

dominant,

Now the gain compensated bode plot is found (K=40.42078032):


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Page | 17

Figure 16- Gain Compensated Bode Plot for Phase-Lead Compensator Design

The value of can be found from this plot as the phase difference between the compensated gaincrossover frequency and -180

o. From the above plot:

Therefore:

The phase-lead compensator will inject a gain of magnitude:

has been determined on the above bode plot and is marked with the data cursor.

-150

-100

-50

0

50

100From: Subsystem1/1 To: Subsystem1/1

Magnitude(dB)

System: ModelI/O: Subsystem1/1 to Subsystem1/1Frequency (rad/sec): 6.69Magnitude (dB): -7.68e-008

System: ModelI/O: Subsystem1/1 to Subsystem1/1Frequency (rad/sec): 12.3Magnitude (dB): -10.6

10-2 10-1 100 101 102 103

-270

-225

-180

-135

-90

System: ModelI/O: Subsystem1/1 to Subsystem1/1Frequency (rad/sec): 6.69Phase (deg): -177

Phase(deg)

Bode Diagram

Frequency (rad/sec)


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Page | 18

Now take And as

can be found to equal 0.0240824All the necessary parameters for the transfer function of the phase lead compensator have been

determined and hence:

Now the phase lead compensator is added to the system (the subsystem shown is the same as that

shown in Figure 11) :

Figure 17- Diagram of Model with Phase-Lead Compensator Controller

With the input voltage

set to a ramp of slope

2V/s, Figure 18 shows

the response that was

achieved for the

potentiometer voltage:

Figure 18- Time

Response of

System with

Phase-Lead

Compensation for

Ramp Input

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

16

18

20

Time(s)

Voltage(V)

Time Response for Potentiometer Voltage (Ramp Input Voltage)

Input Voltage



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Page | 19

0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time(s)

Voltage(V)

Input Voltage


The potentiometer voltage does undergo an initial lock-in phase however during the first second of

simulation (approximately), however this can be expected in any system, and should not be

considered a reason to reconfigure

the controller. This initial lockin is

shown in Figure19. But how does the

system react to a stepped input

voltage?

Figure 20 shows the response of the

potentiometer voltage to a input

voltage of 5V stepped up at time= 1

second. The plot clearly shows that

the output has zero tracking error

relative to the input voltage, and has

a steady-state value of 5V after the

initial overshoot. The overshoot

however- designed to be 10% of the

setpoint- actually rises to 6.119V- an

overshoot of 22.38%. This may or

may not cause problems during use,

however, the controller is easily re-

calibrated. Figure 19- Potentiometer Voltage Lock-In Phase

Figure 20-Time

Response of Phase-

Lead Compensated

System for Step

Voltage Input

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

X: 1.227

Y: 6.119

Time(s)

Voltage(V)

Time Response for Potentiometer Voltage (Step Input Voltage)

Input Voltage



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Page | 20

Discussion and ConclusionIn conclusion, the design exercise has a whole has been a success. Both open-loop and closed-loop

models were analysed and it is now clear the array of methods which can be used in systems such as

pitch control schemes. The open-loop model obtained directly from the mathematical modelling

process was the starting point of the design. The dynamics of the model could not be adjusted andthe system had no disturbance rejection capability, as it was not under feedback control at this time.

It is for this reason that the student set out to determine the open loop frequency response of the

model- in an effort to design a controller for the pitch adjustment scheme. Once the Nyquist Plot

and Nichols chart for the open-loop system were obtained, these gave a better understanding of

how to go about closing the loop effectively. The gain margin of the system was determined and

from this point, the student attempted to design a proportional controller for the system. It was

found that for values of gain above the gain margin of the open-loop system, the pitch angle would

become unstable. However, for values of gain lower than the ultimate gain of the system, a stable

response was achieved. It was found that the lower the proportional gain, the shorter the settling

time for step response. Any stable value of Kp leads to perfect step tracking, due to the fact that theclosed-loop system under proportional gain has one free integrator, and hence, is Type 1. However,

a better controller design was possible, in the form of a phase-lead compensator. During the design

process, the peak overshoot for step response and steady-state error for ramp tracking are picked by

the designer, and hence, this method makes the system a lot more controllable. The phase-lead

compensator not only allows set points to be tracked with zero error, but also allows ramps to be

tacked with relatively small percentage error (designed by the user).

wind turbine pitch adjustment scheme design exercise

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