DC2.pdf

MECH2006Dynamics and Control

DC2Analogue Servo System

Abdul Latif Ismail

March 17, 2013

Abdul Latif Ismail MECH 2006- DC2

Contents

1 Summary 1

2 Results 12.1 Experiment 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Experiment2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Experiment 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Experiment 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Calculations and Discussion 53.1 Derivation of the transfer function of the motor . . . . . . . . . . . . . . . . . . 53.2 Time Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Calculation of T , km and kg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Potentiometer Gain k3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5 Overall transfer function relating Vi to Vo and expressions for n, , Tp and

percentage overshoot in terms of k3, km, P1 and T for the proportional controller 93.6 Relationship of n, , Tp and percentage overshoot on varying P1 . . . . . . . . . 113.7 Critically damped system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.7.1 Theoretical value for P1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.7.2 Experimentally value for P1 . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.8 Can the design criteria be met? . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.9 Exploration in the sensitivity of Tp and percentage overshoot to change in P1

and P2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.10 Optimum P1 and P2 combination . . . . . . . . . . . . . . . . . . . . . . . . . . 153.11 Transfer function of Vi to Vo and derivation for Tp and percentage overshoot in

terms of k3, km, P1, P2, and T for velocity feedback controller . . . . . . . . . . 163.12 Range of P1 and P2 that satisfy design objectives . . . . . . . . . . . . . . . . . 183.13 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Conclusion 19


1 Summary

The main objective of this laboratory session was to design a controller for the Puma robot,which rotates about the zaxis. The main design objective was to reduce the percentageovershoot to a maximum of 10% and a maximum peak time of 0.8s. The main tasks of thislaboratory session were:

Find the values of T and km Finding the potentiometer gain k3 Proportional controller Velocity feedback controller

By following these tasks the design criterion was achieved.

2 Results

2.1 Experiment 1

Table 1: Results from first experiment.

Vi Tacho Speed m Vo T[V] [rpm] [rad/s] [V] [s]

- + - + - + - + - +0.780 0.976 8.4 7.9 28.15 26.47 0.829 0.558 0.302 0.3571.902 2.049 19.8 18.8 66.35 63.00 1.829 1.537 0.424 0.3942.866 2.976 29.3 28.3 98.19 94.83 2.683 2.366 0.441 0.4244.829 5.073 50.2 48.8 168.22 163.53 4.610 4.050 0.436 0.4407.829 8.049 80.5 78.6 269.76 263.39 7.268 6.683 0.457 0.475

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2.2 Experiment2

Table 2: Results obtained from second experiment.

p m V[deg] [rad] [deg] [rad] [V]

20 0.35 19 0.33 0.95140 0.70 38 0.66 2.20760 1.05 58 1.01 3.20880 1.40 78 1.36 4.415100 1.75 100 1.75 5.817120 2.09 112 1.95 6.56

2.3 Experiment 3

Table 3: Results obtained from third experiment.

P1 Tp A B Overshoot[%] [s] [V] [V] [%]

20 1.678 0.695 4.878 14.2525 1.434 1.085 4.561 23.7930 1.242 1.244 4.866 25.5740 1.015 1.695 4.781 35.4550 0.881 1.952 4.817 40.5260 0.805 2.171 4.878 44.5170 0.747 2.451 4.829 50.7680 0.705 2.537 4.817 52.6790 0.679 2.561 4.854 52.76100 0.663 2.731 4.805 56.84

2


(a) P1=0% (b) P1=20% (c) P1=25%

(d) P1=30% (e) P1=40% (f) P1=50%

(g) P1=60% (h) P1=70% (i) P1=80%

(j) P1=90% (k) P1=100%

Figure 1: Graphs of sensitivity of Tp and percentage overshoot with respect to P1.

2.4 Experiment 4

3


(a) P1=0%, P2=0% (b) P1=50%, P2=0% (c) P1=100%, P2=0%

(d) P1=0%, P2=50% (e) P1=50%, P2=50% (f) P1=100%, P2=50%

(g) P1=0%, P2=100% (h) P1=50%, P2=100% (i) P1=100%, P2=100%

Figure 2: Graphs of sensitivity of Tp and percentage overshoot with respect to P1 and P2.

Figure 3: Optimum combination of P1 and P2.

4


3 Calculations and Discussion

3.1 Derivation of the transfer function of the motor

For this derivation, the inertia of the motor will be I, the damping b and the torque Q. Byassuming the torque produced by the motor is proportional to the error signal . [1]

Q = k (1)

Q = Id2mdt2

+ bdmdt

=(ID2 + bD

)m (2)

Where m is the motor output. By combing both equations 1 and 2 gives:

k =(ID2 + bD

)m (3)

Since the input in this case is Vi, equation 3 can be simplified to:

kVi =(ID2 + bD

)m (4)

mVi

=k

ID2 + bD(5)

As the motor speed is required, equation 5 has to be differentiated:

mVi

=DmVi

=kD

ID2 + bD(6)

By diving by the D term:mVi

=k

ID + b(7)

By dividing by b:mVi

=kb

IDb

+ 1(8)

Therefore:mVi

=km

1 + TD(9)

3.2 Time Constant

The time constant is a measure of how long it takes a systems output to reach 63.2% of itspeaktopeak value. Therefore by considering the maximum and minimum voltages of theTacho output, it is possible to calculate the time constant.

The graphs in figure 4 shows the way in which the output (blue curve) behaves to a squarewave voltage input (red curve). To calculate the time constant, the maximum and minimumvoltages had to be identified. So by looking at both (a) and (b) in figure 4, the maximum andminimum values are 0.558V and -0.829V respectively. Therefore by considering the differencebetween the maximum and minimum, it gives a value of 1.387V. Thus, the time constant has

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(a) Positive

(b) Negative

Figure 4: Print screen of voltage against time at 1V.

to be 63.2% of the peakto peakvalue. Which in this case is 0.877V. Therefore, this differencehas been drawn in both (a) and (b) of figure 4 as shown by the blue line. Now by seeing thetime difference between the step input to reach this voltage difference, two brown vertical lineshave been drawn, which will give the time constant. In this case the time constant was 0.357sand 0.302s for positive and negative respectively.

3.3 Calculation of T , km and kg

By looking at table 1, the time constants for the input voltage are shown in the last column.By taking an average of the time constants, T can be obtained. This has been shown in table4.

By plotting a graph of output velocity m against the input voltage Vi the gain km can beidentified by the gradient of the line.

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Table 4: Table of time constants for Vi

T[s]

- +0.302 0.3570.424 0.3940.441 0.4240.436 0.4400.457 0.475

T=0.415s

Figure 5: Graph of motor speed against input voltage.

Therefore the gradient shows the gain km is approximately 33.461 rad/Vs.

By plotting a graph of the tachogenerator output voltage against motor speed m, the tachogain kg can be obtained by looking at the gradient of the line.

In this case the kg value obtained is 0.0261 Vs/rad.

7


Figure 6: Graph of tacho output voltage against motor speed.

Figure 7: Initial block diagram.

By replacing these variables for the calculated values in figure 7 the following block diagramwill be achieved:

Figure 8: Block diagram with corresponding numerical values.

3.4 Potentiometer Gain k3

By plotting a graph of change in output voltage against thirtytwo times the output motorangle, the slope will give the gain k3.

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Figure 9: Graph of change in output voltage against thirtytwo time the output angle.

Therefore, by looking at figure 9 the gain k3 is shown to be 0.1069V/rad.

3.5 Overall transfer function relating Vi to Vo and expressions forn, , Tp and percentage overshoot in terms of k3, km, P1 and Tfor the proportional controller

Figure 10: Block diagram of proportional controller

By looking at figure 10, it can be seen that:

VoVi Vo =

3.3P1kmk3D (1 + TD)

(10)

Vo

[1 +

3.3P1kmk3D (1 + TD)

]=

3.3P1kmk3D (1 + TD)

Vi

Vo [D (1 + TD) + 3.3P1kmk3] = 3.3P1kmk3Vi

VoVi

=3.3P1kmk3

TD2 +D + 3.3P1kmk3(11)

9


Since the form required is:2n

D2 + 2nD + 2n(12)

So by dividing by T in equation 11:

VoVi

=3.3P1kmk3

1T

D2 + 1TD + 3.3P1kmk3

1T

(13)

Therefore by comparing coefficients of equations 12 and 13, it can be seen that:

2n =3.3P1kmk3

T

n =

3.3P1kmk3

T(14)

Also:

2n =1

T

=1

2Tn(15)

By using the expression from equation 14, equation 15 can be written as:

=1

2T

T

3.3P1kmk3

=1

2

3.3TP1kmk3(16)

Since the expression for Tp is known, the derived terms in equations 14 and 16 can be substitutedto replace n and respectively:

Tp =pi

d(17)

Tp =pi

n

1 2

Tp =

T

3.3P1kmk3 pi

1 113.2TP1kmk3

Tp =

T

3.3P1kmk3 2pi

3.3TP1kmk3

13.2TP1kmk3 1Which gives:

Tp =2piT

13.2TP1kmk3 1(18)

The equation for percentage overshoot, is given as:

Overshoot = 100exp

[pi1 2

]% (19)

10


Where A is the peak value, and B is the value at steady state.Substituting equation 16:

Overshoot = 100exp

pi 123.3TP1kmk31 1

13.2TP1kmk3

%Using the same simplification as used between equations 17 and 18 gives:

Overshoot = 100exp

[ 2pi3.3TP1kmk323.3TP1kmk3

13.2TP1kmk3 1

]%

Which gives:

Overshoot = 100exp

[ pi13.2TP1kmk3 1

]% (20)

3.6 Relationship of n, , Tp and percentage overshoot on varyingP1

The data gathered in the experiment is shown in table 3.To identify the sensitivity of n, , Tp and percentage overshoot to P1 a series of graphs havebeen drawn to see how each parameter varies with respect to the change in P1.

Figure 11: Graph of natural frequency n against P1.

Looking at figure 11, it can be seen that the natural frequency n is proportional to P1. Byreferring to equation 14, it can be seen that n is a function of

P1. Therefore, increases with

an increasing P1 value. Since the natural frequency n ranges from: 2.38 < n < 5.34, it canbe said that the natural frequency is sensitive to the change in P1.

11


Figure 12: Graph of damping ratio against P1.

By looking at figure 12 it can be seen that the damping ratio decreases with respect to P1. Thisis because, referring back to equation 16, it can be seen that is a function of 1/

P1. Therefore

causing it to decay. Also, it can be seen in figure 12 that after a P1 value of approximately 0.55the decay becomes linear, this could be because of the relationship between n and P1. Therange of the damping ratio is: 0.51 < < 0.22, which clearly shows that the damping ratio isvery sensitive to the change in P1.

(a) Theoretical (b) Experimental

Figure 13: Graph of peak time Tp against P1.

By looking at figure 13, it can be seen that both the theoretical and experimental graphs lookvery similar. Thus resulting that Tp follows the theoretical relationship predicted. This can bebacked up because, at a P1 value of 100%, the theoretical Tp value is approximately 0.6s andthe experimental value is approximately 0.66s, which shows a similarity between the two. Infigure 13 (b) it can also be seen that up to a P1 value of approximately 60%, the Tp reducesvery quickly, whereas after, the Tp reduction is very slow. By looking at the range of theexperimental data, which is: 1.6 < Tp < 0.67s, it can be said that the sensitivity of peak timeTp to the change in P1% is relatively high.

12


(a) Theoretcial (b) Experimental

Figure 14: Graph of percentage overshoot against P1.

Now looking at figure 14, it can be seen that the trend between the theoretical and experimentalgraphs is similar, however not as identical to that in figure 13, as there are many anomalies.These anomalies can be seen very clearly at approximately 25% and 70% of P1, and can becaused due to the errors in the apparatus used and human errors. Lastly, looking the exper-imental percentage overshoot data shown in figure 14, it can be seen that the values range:14.2 < overshoot < 56.9. Which shows a large sensitivity.

3.7 Critically damped system

3.7.1 Theoretical value for P1

Referring to equation 16, the theoretical P1 for the critical damping ratio can be calcu-lated:

=1

2

3.3TP1kmk3Since the numerical values for , T , km and k3 are known, it gives:

1 =1

2

3.3 0.415 P1 33.461 0.1069

P1 =1

4 3.3 0.415 33.461 0.1069 = 0.051 5.1%

3.7.2 Experimentally value for P1

Looking at figure 15, it can be seen that this graph has been extrapolated to the xaxis, thisis because the equation that was generated by the trend line was:

y = 4 1012x6 2 109x5 + 3 107x4 3 105x3 + 0.0014x2 0.0442x+ 1.0044 (21)Which implies that the yintercept is approximately 1.0044. This indicates that the criticaldamping occurs at approximately 0% P1 value, which is not a true indication of when thesystem is critically damped.

13


Figure 15: Graph of damping ratio against P1, extrapolated.

Figure 16: Graph of voltage against time, with P1 at 15%.

However, using the data obtained in the laboratory session a graph of 15% P1 was drawn andis shown in figure 16. Looking at figure 16 it can be seen that this graph is a very goodapproximation to a critically damped system. Thus, it can be said that experimentally thatthe system is critically damped at a P1 value of 15%. The error produced by this value isapproximately 193.92%.

3.8 Can the design criteria be met?

The main design criteria of this proportional controller:

Percentage overshoot 10% Tp 0.80s

By looking at figure 17 it can be seen that post 20% P1, the percentage overshoot will exceedthe criteria. Thus, for the percentage overshoot criteria to be met, P1 has to be less than 20%.However, the peak time Tp only reaches a value less than 0.8s after a P1 value of approximately

14


Figure 17: Graph of overshoot and peak time against P1, superimposed.

60%, which implies that the second criteria where Tp 0.80s can only be achieved after a P1value of 60%. Therefore, it can be concluded that this proportional controller will not be ableto reach its desired requirements.

3.9 Exploration in the sensitivity of Tp and percentage overshoot tochange in P1 and P2

In this part of the laboratory session, it was required to see how the peak time Tp and percentageovershoot varied with respect to both P1 and P2. Graphs of the different combinations havebeen shown in figure 2.

In reference to figure 2, the first observation that can be made is that by having P1 at 0%,and changing P2 it has no effect on the behaviour on the system. However, with an increasedP1, it can be seen that P2 helps reduce the percentage overshoot of the system. Whereas, P1increases the percentage overshoot, but helps in reducing Tp of the system.

3.10 Optimum P1 and P2 combination

The graph shown in figure 3, shows the optimum combination that was supposed to meet thedesign criteria. The P1 and P2 values selected were 100% and 70%, respectively. However, thepercentage overshoot calculated is approximately 10.5%, which shows that the design criteriawas missed by 5%. This was due to the lack of iterations conducted in the laboratory session.Whereas, the time Tp produced is approximately 0.6s, which is under the required time.

15


3.11 Transfer function of Vi to Vo and derivation for Tp and percent-age overshoot in terms of k3, km, P1, P2, and T for velocityfeedback controller

Figure 18: Block diagram for the velocity feedback in a position control system.

Using the block diagram in figure 18, the overall transfer function can be obtained. Theequations that can initially be generated are:

A = Vi Vo (22)

B = A E (23)

C =3.3P1km

D (1 + TD)B (24)

E = P2kgDC (25)

Vo = Ck3 (26)

Combining equations 22, 23 and 25:

B = Vi Vo P2kgDC (27)

Simplifying equation 26 gives:

C =Vok3

Which can simplify equations 24 and 27 to:

B =D (1 + TD)

3.3P1km Vok3

(28)

B = Vi Vo P2kgDVok3

(29)

Equating equations 28 and 29 gives:

D (1 + TD)

3.3P1km Vok3

= Vi Vo P2kgDVok3

16


[D (1 + TD)

3.3P1kmk3+P2kgD

k3+ 1

]Vo = Vi[

D (1 + TD) + 3.3P1P2kmkgD + 3.3P1kmk33.3P1kmk3

]Vo = Vi

Therefore:VoVi

=

[3.3P1kmk3

D (1 + TD) + 3.3P1P2kmkgD + 3.3P1kmk3

]VoVi

=

[3.3P1kmk3

TD2 + (3.3P1P2kmkg + 1)D + 3.3P1kmk3

](30)

Since the form required is the same as that in equation 12:

2nD2 + 2nD + 2n

Equation 30 can be written as:

VoVi

=

3.3P1kmk3TD2 +

(3.3P1P2kmkg+1

T

)D + 3.3P1kmk3

T

(31)Which shows:

n =

3.3P1kmk3

T(32)

Also:

2n =3.3P1P2kmkg + 1

T

=3.3P1P2kmkg + 1

2Tn(33)

Substituting equation 32:

=3.3P1P2kmkg + 1

2T

3.3P1kmk3T

=3.3P1P2kmkg + 1

2

3.3TP1kmk3(34)

As seen in equation 17, it can be seen that Tp can be calculated by:

Tp =pi

d

Tp =pi

n

1 2

Tp =pi

3.3P1kmk3T

1

[3.3P1P2kmkg+1

23.3TP1kmk3

]2Tp =

pi3.3P1kmk3

T 1

4T 2(3.3P1P2kmkg + 1)

2

17


Tp =2piT

13.2TP1kmk3 (3.3P1P2kmkg + 1)2(35)

Now obtaining an expression for the percentage overshoot, shown in equation 19:

Overshoot = 100exp

[pi1 2

]%

Replacing :

Overshoot = 100exp

3.3P1P2kmkg+123.3TP1kmk3 pi1

(3.3P1P2kmkg+1

23.3TP1kmk3

)2%

Overshoot = 100exp

23.3TP1kmk3 3.3P1P2kmkg+123.3TP1kmk3 pi13.2TP1kmk3 (3.3P1P2kmkg + 1)2

%Overshoot = 100exp

(3.3P1P2kmkg + 1) pi13.2TP1kmk3 (3.3P1P2kmkg + 1)2

% (36)3.12 Range of P1 and P2 that satisfy design objectives

In subsection 3.8, the design objectives have been mentioned. In order to see how, the peaktime and percentage overshoot reacted to a change in P1 and P2 values inequalities had beenformed.

2piT13.2Txkmk3 (3.3xykmkg + 1)2

0.8 (37)

exp

(3.3xykmkg + 1) pi13.2Txkmk3 (3.3xykmkg + 1)2

0.1 (38)Where x is equivalent to P1 and y being equivalent to P2.Both equations can then be shown graphically using Grapher.

Looking at figure 19 two regions can be identified, the overshoot region that is shown in red andthe peak time region, which is shown in blue. However, the region in which the design criteriacan be picked is when both red and blue regions overlap. This only occurs when: 0.83 < P1 < 1and 0.56 < P2 < 0.79.

18


Figure 19: Graphical representation of P1 and P2.

3.13 Errors

The initial sources of errors caused in this experiment can be clearly seen in figures 5, 6 and9. This is because a line of best fit had been taken of these values, which does not mean thatall the points lie on the line of best fit. Also, looking at figures 13 and 14, it can be seen thatboth experimental curves are similar to the theoretical curves, but not still give a slight error.Additionally the points in figure 14 (b) are not on the line resulting in a slight error to becaused.

Now looking at figure 15, the extrapolated line for the damping ratio against P1 does not givea suitable value for a critically damped system, thus taking more values in future experimentswould be advised. Additionally, no readings were repeated, so to ensure reliability it would bebest if readings were repeated.

Lastly, during the experiment, it was not clear whether the readings taken for the time constant,peak time, initial peak and initial steady state value were accurate due to the noise in thesystem. Therefore accumulating to the error caused in this experiment.

4 Conclusion

Overall, this experiment was a good insight to the operation of a Puma robot, and how thevelocity feedback controller ensures the design criteria to be achieved. As mentioned in sub-section 3.8, the design specification could not be reached due to the inverse relation betweenthe peak time and overshoot produced. Thus, by adding a velocity feedback, it allowed for thedesign requirements to be achieved.

19


References

[1] Dr A.R. Greig. Mech 2006 Dynamics & Control Linear Control Theory. University CollegeLondon.

[2] Dr A.R. Greig. Analogue Servo System. University College London.

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SummaryResultsExperiment 1Experiment2Experiment 3Experiment 4

Calculations and DiscussionDerivation of the transfer function of the motorTime ConstantCalculation of T, km and kgPotentiometer Gain k3Overall transfer function relating Vi to Vo and expressions for n, , Tp and percentage overshoot in terms of k3, km, P1 and T for the proportional controllerRelationship of n, , Tp and percentage overshoot on varying P1Critically damped systemTheoretical value for P1Experimentally value for P1

Can the design criteria be met?Exploration in the sensitivity of Tp and percentage overshoot to change in P1 and P2Optimum P1 and P2 combinationTransfer function of Vi to Vo and derivation for Tp and percentage overshoot in terms of k3, km, P1, P2, and T for velocity feedback controllerRange of P1 and P2 that satisfy design objectivesErrors

Conclusion

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