1314-1_Final SKEE3133

-2SKEE 3133
Question 1
a)
A closed-loop control system has an open-loop transfer function
G(s) =
(i)
K(s + 3)(s + 8)
s(s + 50)(s +100)
Normalise the transfer function G(s) .
(3 marks)
(ii)
Sketch the magnitude response of the Bode plot for the system.
(5 marks)
(iii)
Estimate the final phase (at the highest frequency) of the system.
(2 marks)
b)
A unity feedback control system is given in Figure Q1(b)(i) with
G(s) =
R(s)
(s + 5)
s(s +1)(s 2 + 7s +12)
C(s)
+
K
G(s)
-
Figure Q1(b)(i)
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(i)
For K = 1 , the Bode plot is given in Figure Q1(b)(ii). Evaluate the stability of
the system.
(5 marks)
Figure Q1(b)(ii)
(ii)
The gain margin is then increased by 20% from its original value. Determine the
new gain K .
(5 marks)
c) The phase and gain margin of a system can be altered by using the gain adjustment
method. Discuss the effects of the gain adjustment on the steady state error, time
response, frequency response and stability of the system.
(5 marks)
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Question 2
a)
Mathematical modeling is one of the necessary steps in a control system design
process. Give two reasons why this step is needed.
(3 marks)
b)
Find the transfer function, G(s) =
θ1 (s)
for the rotational system shown in Figure
T (s)
Q2(b).
T(t) θ1(t)
θ4(t)
N1=25!
N4=100!
θ3(t)
θ2(t)
N2=100!
50!N%m%s/rad!
1!N%m/rad!
N3=20!
Figure Q2(b)
(8 marks)
-5SKEE 3133
c)
A motor is an electromechanical component that yields an angular displacement
output θ L (t) for a voltage input ea (t) , and has a schematic diagram shown in
Figure Q2(c)(i).
When the current carrying armature, ia (t) , is rotating in a
magnetic field, the voltage vb (t) (back electromotive force, e.m.f) is proportional to
the speed:
vb (t) = K b
dθ m (t)
dt
The torque developed by the motor is proportional to the armature current ia (t) :
Tm (t) = K t ia (t)
where K b is a constant of the back e.m.f and K t is the motor torque constant.
Fixed
field
Ra
ia(t)
Tm(t)
vb(t)
θm(t)
N1
J1
N2
motor
θL(t)
J2
D
Figure Q2(c)(i)
(i)
Prove that the transfer function of the system is
Kt
θ m (s)
(Ra J m )
=
'
Ea (s)
1 !
K K $*
s )s + # Dm + t b &,
Ra %+
( Jm "
where J m and Dm are the equivalent inertia and viscous damping of the
system.
(7 marks)
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The motor Torque-Speed characteristics when driving a load is shown in
Figure Q2(c)(ii). Find the transfer function,
θ L (s)
. The mechanical system
Ea (s)
parameters are as follows:
D=
J1 =
J2 =
N1 =
N2 =
2 N-m-s/rad
2 kg-m2
10 kg-m2
10
30
(7 marks)
Tm
370!
Torque (N-m)
(ii)
Ea = 75 V
40!
Speed (rad/s)
Figure Q2(c)(ii)
ωm
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Question 3
a)
Block diagram is widely used in the engineering practice. Give two reasons to
support this statement.
(3 marks)
b)
For the block diagram in Figure Q3(b), find the single block equivalent of
T ( s) =
C ( s)
R( s)
(12 marks)
H1
R(s)
+
G1
G2
+
-
G4
+
+
+
G7
C(s)
G3
G6
+
G5
Figure Q3(b)
c)
From the signal flow graph in Figure Q3(c), obtain the transfer function of the
system,
C(s)
, using Mason’s Rule.
R(s)
(10 marks)
-8SKEE 3133
-H1
-H3
R(s)
1
G1
G4
1
G2
G5
G2
G3
-H2
Figure Q3(c)
G6
1
C(s)
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Question 4
a) Briefly discuss three main objectives in designing a control system.
(3 marks)
b) Consider the position servomechanism used in antenna tracking system as shown in
Figure Q4(b). In this system, an electric motor is used to rotate a radar antenna,
which automatically tracks an aircraft. A unit step input is used to investigate the
performance of the antenna tracking capability.
R(s)
+
_
+
K
C(s)
_
Figure Q4(b)
(i)
Prove that the closed-loop transfer function of the system is
C(s)
0.4K
=
2
R(s) 0.8s + 2s +1.2 + 0.4K
(3 marks)
(ii)
For a critically damped response, find the value of K and the natural
undamped frequency, ω n .
(5 marks)
(iii)
Find the steady-state error, ess , of the system by using the value of K
obtained from (b)(ii) above.
(4 marks)
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c) A plant with the following transfer function is controlled by a proportional
controller, as shown in Figure Q4(c).
G(s) =
R(s)
1
s(s + 2s + 2)(s + 3)
2
C(s)
+
K
G(s)
-
Figure Q4(c)
(i)
Discuss the stability of the closed-loop system based on the Routh-Hurwitz
criterion.
(3 marks)
(ii)
Determine the range for the proportional controller gain K for the system to
be stable. If the system is marginally stable, determine the oscillation
frequency of the system.
(7 marks)
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Question 5
a) (i)
Explain the importance of conducting the frequency domain analysis on a
linear control system.
(1 marks)
(ii)
Give the definitions of ‘gain margin’ and ‘phase margin’.
(2 marks)
b) A unity feedback control system has an open-loop transfer function as follows:
G(s) =
15K(s +10)
(s + 6)2 (s 2 +14s + 33)
The Bode plot for the system is shown in Figure Q5(b).
Figure Q5(b)
From the Bode plot given in Figure Q5(b) above, find the values of:
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(i)
Gain margin.
(1 marks)
(ii)
Phase margin.
(1 marks)
(iii)
Gain crossover frequency.
(1 marks)
(iv)
Phase crossover frequency.
(1 marks)
(v)
Comment on the stability of the system.
(3 marks)
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c) The Bode plot of an open-loop transfer function, G(s) , is given in Figure Q5(c).
Figure Q5(c)
(i)
From the Bode plot given in Figure Q5(c) above, determine G(s) .
(6 marks)
(ii)
What is the system type?
(1 marks)
(iii)
What is the steady-state error?
(2 marks)
(iv)
If you were to reduce the steady-state error obtained in (c)(i) above by 20%,
what should the gain, K , be?
(6 marks)