BRUNE
DEPARTMENT OF ELECTRONIC AND COMPUTER ENGINEERING
SAMPLE EXAMINATION PAPER – JANUARY 2003
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EE2006A – CONTROL OF CONTINUOUS-TIME LINEAR SYSTEMS
________________________________________________________
Time allowed - 2 HOURS
Answer any THREE questions out of 5
Tables of Laplace Transforms are provided
Linear and Polar graph paper are available
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Examiner:
Dr. P.J. Turner
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1.
A unity feedback control system is shown in Figure 1.
Rs +
K
-
C s 
G s
plant
Figure 1.
The transfer function of the plant is:
G s  
1
s  2s  10
a)
Plot the open-loop poles on a pole/zero diagram in the s-plane and
hence sketch the root locus diagram for 0  K  .
[6 marks]
b)
From the root locus diagram, show that a value of K = 80 results in a
closed-loop damping ratio of 0.6.
[6 marks]
c)
Calculate the closed-loop transfer function using the value of K in part
b). Hence determine the steady state value of the output of the closedloop system when a unit step input is applied. What is the resulting
steady state error?
[4 marks]
d)
The proportional controller is now replaced with a controller of the
form:
K s  4
s
Sketch the new root locus diagram and explain the significance of the
pole at the origin of the s-plane.
[4 marks]
2.
A dynamical system is modelled by the differential equation:
dy
 5 y t   10u t 
dt
where u t  is the input and y t  is the output.
a)
Determine the transfer function for the system and calculate the
output response y t  when a unit step input is applied with zero
initial conditions.
Show that the output reaches a steady state value of 2.
[8 marks]
b)
Calculate the output response y t  to an initial condition of y 0  4
with no input applied.
[4 marks]
c)
The system is to be controlled by using unity negative feedback as
shown in Figure 2. with a compensator in the forward path having a
transfer function:
D s  
Rs +
-
0.25s  1.8
s  0.5
D s
Us
System
Transfer
Function
Y s 
Figure 2.
Find the closed-loop transfer function and hence calculate the closedloop output response when a unit step input is applied.
[8 marks]

3.
A system to be controlled has the following open-loop transfer function:
GH s 
a)
K
1 0.1s1 0.5s
Write down the open-loop frequency function, GH  j  and hence
determine expressions for the gain and phase of the system.
[5 marks]
b)
The following table shows the gain and phase of the system at selected
frequencies when K  5.
 (rad/s)
0
1
2
4
8
16
Gain
5.0
Phase (˚)
0˚
3.5
-56˚
0.9
0.3
-115˚
-141˚
Confirm some of the values in the table and complete it by calculating
the gain and phase at the frequencies shown.
Plot the polar (Nyquist) diagram on the polar graph paper provided.
[6 marks]
c)
From the polar diagram in part b), determine the phase margin of the
closed-loop system.
[2 marks]
d)
To what value may K be increased if the closed-loop damping ratio  is
to be equal to 0.5?
The identity:
tan 1 A  tan1 B  tan 1
 A  B 
1 AB 
may be used without proof.
[7 marks]
4.
An RLC circuit is shown in Figure Q4.
I s
2ž
4H
0.5 F
Vin  s
2ž
Vout  s
Fig. Q4
(a)
Use Kirchhoff’s Voltage Law to obtain an expression relating the
current I s  and the input voltage V in s .
[8 marks]
(b)
Determine the relationship between the output voltage V out s  and the
current I s . Hence show that the overall transfer function relating
output voltage to input voltage is:
V out s 
0.5
 G s   2
.
V in s 
s  1.5s  1
[8 marks]
(c)
Calculate the values of  and n and hence predict the percentage
overshoot and frequency of oscillations of the unit step response.
[4 marks]

5.
The open-loop transfer function of a DC servo system can be written as:
G s  
o s 
K

 i s  s 1  sT

where i denotes the input angle and o denotes the output angle.
(a)
Given that the gain K = 0.5 and the time constant T = 2 s, sketch the
open-loop response of angular velocity when a unit step input is
applied.
[6 marks]
b)
Predict the damping ratio  and natural frequency n of the closedloop system obtained by applying unity negative feedback to the openloop system. Sketch the output response of the closed-loop system when
a unit step input is applied.
[10 marks]
c)
Velocity feedback as shown in Figure 5 is used to improve the closedloop step response.
 i ( s) +
+
-
-
K
1 + sT
Ýo s 
1
s
 o (s)
Kv
Figure 5
Show that the overall closed-loop transfer function is now:
o s 

i s 
K
T

1

KK
v 
s 2  
s K

T
 T

Hence comment on the effect of varying K v on the closed-loop step

response.
[4 marks]
TABLE OF LAPLACE TRANSFORMS
Time function
f t
f t
Laplace Transform


Property
F s 

 f t e
 st
0
dt
Definition
f 1t  f 2 t 
F 1s   F2 s 
Linearity
k f t 
k F s 
Linearity
t 
H t

f t   H t   
e at H t 
sin t H t 
cos t H t 
e at f t H t 
e at sin t H t 
e at cos t H t 
t H t 
t n H t 
t n f t H t 
t e at H t 
df t 
dt
d n f t 
dt n
Unit Impulse
1
1
s
Unit Step
e s  F s 
1
s a

2
s  2
s
2
s  2
F s a 

2
s  a    2
s  a 
s  a 2   2
1
s2
n!
s n 1
dn
1n ds
n F s 
1
s  a 
2
s F s  f 0
s n F s   s n 1 f 0 
 f   d
1
F s 
s
 x   h t    d
Lim f t 
H s X s 
t
0
t
0
t 
Delayed time function

Lim s F s 
s 0
Differentiation
d
n 1
f 0
dt
n 1
Differentiation
Integration
Convolution
Final Value Theorem
(for stable systems)