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XE220B_SEM2/2012/2013 of 14 Printing date: 19/09/2013

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SCHOOL OF COMPUTING, ENGINEERING AND MATHEMATICS

SEMESTER 2 EXAMINATIONS 2012/2013

XE220

Mathematics and Control

Time allowed: TWO hours

Answer: Answer any FOUR questions

Total number of questions is SIX

Each question carries 25 marks

Items permitted: Calculator

Aide memoire (candidates are allowed to

consult their pre-prepared, approved, aide

memoire sheet during the examination)

Items supplied: Graph papers

Appendix 1 - Table of Laplace transforms

Appendix 2 - Question 2 Answer Sheet

Marks for whole and part questions are indicated in brackets ( )


Question 1

The schematic diagram shown in Figure Q1 represents a tank system with three

valve-controlled feeds set to fluid flow rates , . The discharge flow

rate out of the tank is . The discharge flow rate is determined by the height of

fluid in the tank and the discharge coefficient of the valve located at the base of the

tank. The discharge flow rate , in which is the height of fluid in the

tank as a function of time.

(a) Write down an expression for the volume of fluid in the tank at a particular time

seconds, given that the tank has a cross-sectional area of value

and is uniform throughout the height of the tank.

(1 mark)

Question 1 is continued on the next page

Figure Q1 𝑓 𝑡

Discharge valves

Tank cross sectional area= 𝐴

ℎ 𝑡 Fluid height=

𝑓 𝑡

[Code]/2011/2012 of 14 Printing date:

Question 1 (continued)

(b) Write down an expression for the rate at which the volume of fluid in the tank is

changing with time , given the expression you have arrived at in part (a)

above.

(1 mark)

(c) Write down an expression for the rate at which the total volume of fluid is

entering the tank.

(1 mark)

(d) Given that the fluid discharge rate is write down a mathematical

expression for the rate at which the volume of fluid in the tank is changing with

time , using the expressions you have arrived at, in parts (b) and (c) above.

The expression should include the terms , .

(3 marks)

(e) Assuming that the Laplace transform of is F the Laplace transform of

is F and that the Laplace transform of is F write down the

expression for the Laplace transform of the expression derived above in part

(d). Assume that the Laplace transform of can be written as Note:

the Laplace transform of the function dt

dxty )( is given by

)()0()( ssXtxsY .

(4 marks)

(f) Assuming that feeds and are step inputs of amplitude

respectively, derive the expression for the Laplace transform of the height of

fluid in the tank . You may assume that the feed is set to zero

(3 marks)

(g) Assuming that the tank is empty initially, derive an expression for the height of

fluid in the tank as a function of time, given that the feeds and have

step input amplitudes of and respectively. The cross-

sectional area of the tank has a value of and the discharge coefficient

of the valve located at the base of the tank is .

(6 marks)

(h) Sketch a graph of as a function of time and indicate on the graph the time

constant of the tank system.

(6 marks)


Question 2

A crude oil delivery system to a petroleum oil processing plant maintains a desired

oil flow rate within an oil pipe by means of an electrically controlled electro-

mechanical valve. The oil flow delivery valve is operated by a signal which is a

measure of the delivered crude oil flow rate and forms a unity-feedback closed-loop

control system. The open-loop step-input response of the delivery system results in

the oil flow rate depicted in Figure Q2.

Figure Q2

NOTE: Figure Q2 is replicated at the end of this examination paper for you to

include with your answer book.


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 2 4 6 8 10 12 14 16 18 20

flo

w r

ate

(M

3/s

)

Time (seconds)

Open loop step response of oil delivery system



(a) For the open-loop operation, determine the following:

(i) The damped natural angular frequency

d .

(2 marks)

(ii) The transient response time.

(1 mark)

(iii) The s-plane pole plot and the position of the poles.

(6 marks)

(iv) The system’s forward transfer function.

(4 marks)

(b) For the closed-loop operation, determine the following:

(i) The system’s forward transfer function.

(2 marks)

(ii) The natural angular frequency

n .

(2 marks)

(iii) The damping ratio .

(2 marks)

(iv) The steady error for a step demand change in flow rate of 20 m3.s-1.

(6 marks)


Question 3

Two system block diagrams are shown in Figures Q3a and Q3b, each with feedback.

(a) Formulate the closed-loop system forward transfer function for the block

diagram in Figure Q3a and show that it is given by 10

16

)(

)(

ssYi

sY o.

(8 marks)

(b) Determine the transient response time of this system and show that the open-

loop steady-state error is 0.60 for a unit step demand change in input.

(9 marks)

(c) If the closed-loop system forward transfer function for the block diagram in

Figure Q3a is given by 10

16

)(

)(

ssYi

sY o, then redraw the functional block diagram

of the system shown in Figure Q3b so that it has a single feedback loop.

(3 marks)


0.25

Figure Q3 a

16 s + 6

0.25

Yi(s) Yo(s)

Figure Q3 b

16 s + 6



(d) Determine the forward transfer function of the system shown in Figure Q3b.

(5 marks)


Question 4

Figure Q4 shows the schematic diagram of a permanent-magnet DC motor driving a

load which has inertia and viscous friction. Assume that the motor winding

inductance is zero Henries.

Figure Q4

(a) Show that the transfer function for the DC-motor in Figure Q4 is:

)(

)(

)(

)(

JRkkJbs

JRK

sV

s

te

t

(8 marks)


v(t)

+

R Ω

e = ke Ө

Inertia ‘J’ Nm

Drag coefficient ‘b’ Nm/rad/s

.

T = kt Ia

Speed = Ө rad/s .

Ia



(b) If the motor is considered to be completely at rest with an input voltage of zero,

and then the input voltage is suddenly raised to 48 Volts, use the transfer

function derived in part (a) and the enclosed Laplace tables to show that the

value of the output speed after one time constant is approximately 327 radians

per second. Sketch the output-velocity versus time graph and label significant

features. The parameters for the system above are:

kt = 0.4 Nm/A R = 5 Ω

J = 0.01 kgm2 b = 0.005 Nms/rad

ke = 0.03 Vs/rad

(13 marks)

(c) For the equation in part (b) above, if viscous friction is now treated as an

independent variable rather than a constant, sketch how the time-constant

changes as viscous friction is increased.

(4 marks)


Question 5

(a) On the separate log graph-paper provided, sketch the system Bode-plot (using

piecewise-linear approximations) of an open-loop system with the following

transfer function. Do not forget to label the axes of your graphs fully:

)50(

5000)(

sssH

(12 marks)

(b) Determine the approximate gain and phase margins if

H(s) above is to be used

in a closed-loop system which possesses unity-gain feedback. Show clearly on

your Bode plot how you have arrived at these values.

(4 marks)

(c) Comment on the stability of a system with a phase margin of 90 degrees.

(2 marks)

(d) Comment on the stability of a system with a gain margin of 0 dB.

(3 marks)

(e) State advantages and disadvantages of using Bode plots for the purposes of

controller design.

(4 marks)


Question 6

(a) Sketch a schematic diagram of a position-control system which uses a DC

motor that drives a load (which possesses inertia and viscous friction) through a

gearbox. Include in your schematic a potentiometer that would allow the user

to control the amount of tachometer feedback. Explain the purpose of each

component of your diagram.

(12 marks)

(b) Assume that the subscript ‘M’ (e.g. as used in “TM”) indicates an electric motor

and ‘L’ indicates an inertial load that needs to be rotated against viscous

friction. Draw a diagram to illustrate a motor driving a load through a gearbox

and label the diagram with each of the terms in the gearbox equation below.

)()(

22

sN

Nbbs

N

NJJT

L

MLM

L

MLMM

(6 marks)

(c) For the equation in part (b) above, explain BRIEFLY what each term means.

(5 marks)

(d) Why are gearboxes used in practical systems?

(2 marks)


APPENDIX 1 – TABLE OF LAPLACE TRANSFORMS

x_

(s) estx(t)dt0

x(t)

1

s 1 or H(t)

1

s a

eat

1

s2

t

1

s a 2

teat

1

s a s b

eat ebt

b a

1

s s a

1

a1 eat

1

s2 a2

1

asin(at)

1

s2 a2

1

asinh(at)

1

s a 2 b2

1

beat sin(bt)

1

s a 2 b2

1

2be(ba )t e(ba )t

s

s a 2

1 at eat

s b

s a 2

(b a)t 1 eat


APPENDIX 1 (CONTINUED)

x_

(s) estx(t)dt0

x(t)

s

s a s b

aeat bebt

ab

s c

s a s b

(c a)eat (c b)ebt

b a

(s b)

s s a atebab

a

)(1

s

s2 a2

cos(at)

s b

s2 a2

baatab

a

122 tansin1

s

s a 2 b2

a

bbtebab

at

122 tansin

1

1

s3

t 2

2

1

s a 3

1

2t 2eat

1

s s a 2 ateat

a

)1(11

2

1

s2 s a

1

a2eat

t

a1

a2

1

s s2 a2 )cos(1

12

ata

Y(s) x(t 0) sX(s)

y(t) dx

dt


APPENDIX 2 - ANSWER SHEET FOR QUESTION TWO

Student Number: ……………………………………….

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 2 4 6 8 10 12 14 16 18 20

flo

w r

ate

(M

3/s

)

Time (seconds)

Open loop step response of oil delivery system

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