Revision_Set_with_solution

8/7/2019 Revision_Set_with_solution

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Question 1

a) Explain the following control terminologies with the aid of respective plotting:i) Stability.

ii) Steady-state error.

iii) Time constant for a first order step response.

iv) Settling time for a second order step response.

(8 marks)

b) For the block diagram shown in Figure Q1(b), determine the relationship betweenthe output variable C(s) and the input variable R(s) by analyzing the systems

variables.

(7 marks)

Figure Q1(b)

+++

_

R(s)

G1(s)

C(s)

G2(s)

G3(s)

q(s)

p(s)

y(s)

x(s)


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c) For the electrical network shown in Figure Q1(c), determine the relationshipbetween the output voltage Vo(s) and the input voltage Vi(s).

(6 marks)

Figure Q1(c)

d) The following lines in a MATLAB program are written:

num = 5;

den = [1 3 12];

step(num,den)

Sketch the response obtained when the above program is run and state the type of

the response.

(4 marks)

L2

L1

R

C

Vi(t) Vo(t)


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Question 2

a) Give a brief explanation of a control system.(4 marks)

b) A car equipped with a cruise control system can maintain its speed according to adrivers set speed, despite the presence of any disturbances (for example, different

road condition, etc.). Explain whether this is an open loop or a closed loop control

system. Draw the corresponding block diagram describing its operation.

(7 marks)

c) In the analysis and design of a control system, three criteria, namely the transientresponse, the steady state response and the stability, should be considered.

Describe briefly those criteria in the analysis and design of a control system, as

mentioned above.

(6 marks)

d) Figure Q2(d) shows the unit steps responses for the systems A, B and C. Based onthe description in question 2(c) above, discuss the performance of each system.

Hence, which system has the best performance? Give reason to support your

choice.

(8 marks)


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Figure Q2(d)

Step Response

Time (sec)

Amplitude

0 1 2 3 4 5 6 0

0 .2

0 .4

0 .6

0 .8

1

1 .2

1 .4

C

B

A


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Question 3

a) i) Define what is meant by a transfer function.(3 marks)

ii) How can we change a differential equation into another form of equation

in the s-domain?

(1 mark)

iii) What assumption is made concerning the initial conditions when dealing

with a transfer function?

(1 mark)

b) Figure Q3(b) is a representative of a situation involving the vibration of machinescontaining unbalanced components. M1 and M2 are the masses, K1 and K2 are the

spring constants, B1 and B2 are the coefficients of viscous damping.

The parameters M2, K2, and B2 may be chosen so that the main mass M1 does not

vibrate in response to foundation vibration.

Assuming that the gravitational acceleration is neglected,

i) Obtain a set of differential equations describing the dynamics of the

system.

(5 marks)

ii) Transform the differential equations in s-domain and re-write them in

matrix form.

(5 marks)


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Figure Q3(b)

c) Find the transfer function,)(

)(1

sT

sU, for the rotational system shown in Figure

Q3(c).

(10 marks)

Figure Q3(c)

F(t)


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Question 4

a) i) What are the three familiar forms that subsystems are normally connectedin a block diagram?

(3 marks)

ii) What are the two basic components in a signal flow graph?

(2 marks)

b) Find the overall transfer function for the system shown in Figure Q4(b) using theblock diagram reduction method.

(10 marks)

Figure Q4(b)

c) Change the block diagram in Figure Q4(b) into an equivalent signal flow graph.Hence, obtain the overall transfer function using the Masons Rule.

(10 marks)

G1 G3

H2 H1

G4

R(s) C(s)+ +

+

+

-

-

G2


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Question 5

a) Explain the meaning of step response of a system.(2 marks)

b) An open loop control system is shown in Figure Q5(b), where the R(s) is a unitstep input and C(s) is the systems output.

Figure Q5(b)

i) Identify the system order and type, plot the systems poles on an s-plane,and state the type of the output response.

(4 marks)

ii) If a negative feedback with 65)( ! ssH is added into the system, does

the stability of the system change? What would be the type of the output

response now?

(6 marks)

c) Figure Q5(c) shows the unit step response of an unknown open loop controlsystem. From this response, obtain the transfer function of the system.

(4 marks)

2

502

ss

R(s) C(s)


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12 ssK

R(s) C(s)+

-

K

s 65

-

Figure Q5(c)

d) For the system shown in Figure Q5(d), find K so that for an input of 2u(t), therewill be a 0.01 error in the steady-state output response.

Note:

01

00)(

u!

!

tfor

tfortu

(9 marks)

Figure Q5(d)

t(sec)0.1

0.5

1.0

0.1 0.2 0.5 0.7 1.0 1.5

c(t)

PK-UTM-FKE-(O)-05


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Solutions


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Q2)

a)

A control system consists of subsystems and processes/plant for the purpose of

controllingthe outputofthe plant.

A control system provide an outputorresponse fora given input.

b)

c)Transient response:A control system responds to an inputby undergoing a transient

response before reaching a steady-state response. Several criteria are consideredto

determine the performance of systems transient response: Rise time, settling time,

overshoot, peaktime.

Input,

Desiredresponse

Control system

Output,

Actualresponse

4

2


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Steady-state response: The response determine the steady-state error of a control

system. It is desirable tohave a zero steady-state errorandthus, the actualresponse

= desiredresponse.

Stability: It is importantto design a control system which is stable. In this case, the

transientresponse approach zero andthus leavingonly the forcedresponse.

d)System A: fastestrise time, highovershoot, zero error, good settlingtime as B.

System B: noovershoot, settlingtime as A, zero error.

System C: noovershootbutslowest, error= 0.3.

Bestperformance: System B

2

2

2

2

2

2


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3

1

1


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2.5

1

2.5

1


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2


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2

3

2

1


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1

1

3

2


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2

2

1

1

1


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1

2


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3

2

1


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Q5)

a)

b)

i)

2

1

1

2

4


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ii)

c)

2

1

1

1

1

1

1

1

1


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d)

3

2

1

2

1

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