FINAL EXAMINATION
JULY 2010 SEMESTER
COURSE :
DATE
:
TIME
:
EBB3053
CONTROL SYSTEMS
(3 hours)
ANSWER
SCHEME
Universiti Teknologi PETRONAS
EAB2053
1. System modelling is an important aspect of control system theory.
a.
Why do transfer functions for mechanical networks look identical to
transfer functions for electrical networks?
[3 marks]
b.
Given the network in FIGURE Q1b, find the transfer function,
VL (s) / V (s) .
[7 marks]
FIGURE Q1b
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c.
The DC motor system and torque-speed curve are shown in
FIGURE Q1c(i) and Q1c(ii). Find the transfer function,  L ( s) / Ea ( s) , if
the system parameters are:
J1 = 2 kg-m2 , J 2 = 18 kg-m2 , D1 = 2 N-m s/rad
D2 = 36 N-m s/rad , N1 = 50 , N 2 = 150
[10 marks]
FIGURE Q1c (i)
FIGURE Q1c (ii)
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2. Systems are commonly represented in various forms of system model.
a.
Give two reasons for modeling systems in state space.
[3 marks]
b.
Represent the following transfer function in state space. Give your answer
in vector-matrix form.
[10 marks]
C ( s)
5s  10
 4
3
R( s) ( s  2s  s 2  5s  10)
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c.
For the following state-space representation of a system, find the transfer
function, G ( s )  Y ( s ) / U ( s ) .

x 
 3  5 2
5
 1  8 7  x   3


 u
 3  6 2
 2 
y 
1
 4 3 x
[7 marks]
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3.
A space station, shown in FIGURE Q3 (i), will keep its solar arrays facing the
Sun. Assume that the simplified block diagram of FIGURE Q3 (ii) is representing
the solar tracking control system that will be used to rotate the array via rotary
joints called solar alpha rotary joints.
FIGURE Q3 (i)
FIGURE Q3 (ii)
a.
Obtain the closed-loop transfer function of the system.
[3 marks]
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b.
State the system type, and find K p , K v and
Ka .
[8 marks]
Kp
=∞
Kv =
∞
K a = 0.01
c.
Use your answers in part (b) to find the steady-state errors for the
standard step, ramp, and parabolic inputs.
[3 marks]
d.
Find the range of K c / J to make the system stable using the RouthHurwitz criterion.
[6 marks]
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4.
A simplified block diagram of a meter used to measure oxygen concentration is
given as an unity feedback system in FIGURE Q4
function of the plant, G ( s ) 
Gc ( s )  K
with a forward transfer
1
. It is desired to design a PID controller,
s 1
2
( s  a )( s  b)
.
s
R(s)
C(s)
+
Gc(s)
G(s)
FIGURE Q4
a.
Design a PID controller Gc (s) such that the dominant closed-loop poles
are located at s  1  j 3 and a  1 .
[8 marks]
b.
Use your answers in part (a) to determine the values of b and K .
[4 marks]
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c.
In the s-plane, sketch the root locus for the designed system as K
varies from 0 to infinity.
[6 marks]
d.
Is the compensated system stable?
[2 marks]
The compensated system is stable.
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5.
FIGURE Q5 shows the Bode plot for an electric ventricular assist device (EVAD).
The open-loop transfer function is given as, G ( s) 
K ( s  10)( s  11)
, where
s( s  3)( s  6)( s  9)
K  1.
FIGURE Q5
a.
Calculate the value of damping ratio if the system has 15% overshoot.
[4 marks]
b.
Find the phase margin.
[4 marks]
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c.
Find the value of crossover frequency for the phase margin value
obtained in part (b).
[6marks]
d.
Determine the value of gain, K, to yield a closed-loop step response with
overshoot value given in part (a). Justify the second-order approximation.
[6marks]
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