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Cork Institute of Technology
Bachelor of Science (Honours) in Electrical Power Systems – Award
(NFQ Level 8)
Summer 2006
Control Systems Engineering
(Time: 3 Hours)
Instructions Answer FIVE questions. All questions carry equal marks. Laplace Tables available on request.
Examiners: Mr. M. Hennessy Prof. E. McQuade Mr.P. F.O’Murchu
Q.
Given that; V b = Kbddt^ θ and that armature torque T = KT Ia. Take I f to be constant.
(a) Prove that the transfer function which relates output shaft angular position to the input voltage is given by the following;
[ ( )( ) T b ]
T i s sJ B R sL K K
K
V s
s ()=^ + ( + +
θ () 12marks
(b) Prove that the transfer function which relates output shaft angular velocity to the input voltage is given by the following;
JL
BR KK
s sRL BJ
JL
K
V s
s b T
T i () ( ) ( )
ω 8marks
Vi Vb
R L^ J
B
ω
Ia I f
Fig.
Q2. Given that the closed loop transfer function of a plant is given (^1) + HC (( ss )) CG (( ss )) G ( s ) = T ( s ),
where C(s), G(s), and H(s) are the transfer functions for the controller, plant, and sensor respectively. (a) Show that the controller transfer function C(s) can be evaluated from the following
relationship. (^)
( )^1
T s
T s C s Gs 6marks
(b) Given a first order system with a transfer function (^2) s^4 + 1 , and using the above relationship in (a) determine the controller settings for a PI (proportional + integral) controller such that the closed loop system will be critically damped, have a loop gain of 4, and have a settling time t (^) s of 10 seconds. s n [ Take " t "= (^) ω^8 ] 14marks
Q3. An open loop plant which consists of two series connected elements G 1 (^) ( s )and G 2 ( s )
having individual transfer functions given by the following;
- 5 0. 5
( )^0.^01
G 1 (^) s = (^) s (^2) + s + and G s s ()^100 2 =^. The root-locus for the complete open loop system is shown in Fig. 3. (a) Calculate the gain K and the corresponding natural radiancy for marginal stability in closed 6marks (b) Show clearly on the diagram the values of the asymptote breakpoints and the locus departure value from the real axis. 6marks (c) Evaluate the value of gain K which will give a damping factor of 0.707 in closed loop mode. { N.B. Return Fig. 3 with the exam script} 8marks
s +^151
s +
s +
Cont T^ L
Valve Boiler Mixer
Sensor
V in
0 C
Fig. 4a
G ( s )
H ( s )
C ( s )
ss
s Es
U s
(a) Determine the equivalent discrete controller UE^ (( zz )), given that a Zero Order Hold
sampler is used with a sampling time of 1 second. 16marks (b) Determine the difference equation for the above discrete controller as would be implemented in a PLC system or a C-Controller. 4marks
Q7. (a) Verify that the state space representation, in vector form, for the circuit shown in Fig. 7 can be given by the following matrix equations.
1 (^ ())
1 1
2 .
. V t V t RC
i t C RC
L L
R
V t
i t lC i C
L
, as X = AX + BU
.
V ( t ) [ 0 1 ] (^) Vi (( tt )) [ ]( 0 V (^) i ( t )) c o L +
= as Y = CX + DU 10marks
(b) Given that VV ss CsI A B D i
C = ( − )−^1 +
( ) evaluate the system output response to a unit step
input. 10marks
Name______________________________ Exam No.______________________________
Fig. 7
Vi ( t ) Vc ( t )
R 1
L
R 2
C
Ii
I (^) L IC
0.1μF
2H
Frequency (rad/sec)
Phas
e(de
Mg); agnit
ude
B(d)
Bode Diagrams
0
20 Gm=20.695 dB (at 0.13333 rad/sec), Pm=94.123 deg. (at 0.025914 rad/sec)
-300 10 -3 10 -2 10 -1 100
0
Time (sec.)
Ampl
udeit
Step Response
(^00 50 100 150 200 250 300 )
1
2 From: U(1)
TYo:
(^1 )
N.B. Return this sheet with your exam script
Fig. 4b
Fig. 4c
