DUE DATE: 10th January, 2006
COURSE: MSCSY 460
CONTROL SYSTEMS THEORY II
TAKEHOME EXAM
INSTRUCTIONS TO CANDIDATES
Please prepare a comprehensive report answering the following FIVE questions. All
necessary working must be shown. You are reminded of the necessity for proper
English and orderly presentation of your answers.
QUESTION 1
(a) The electrical circuit shown in Figure Q1 is used in a specific control system.
R1
L
Vi
R2
Vo
Fig. Q1a
(i)
(ii)
Express the mathematical relations of the input and output voltages, Vi and
Vo, respectively in the form of linear differential equations.
(5 marks)
Obtain the system’s transfer function G(s).
(5 marks)
(Given: R1=R2=1 , L=0.5 H)
(b) Figure Q1b shows a negative feedback control system, where G(s) is the transfer
function of the circuit shown in figure Q1a.
R(s)
+
G(s)
-
C(s)
2
Fig. Q1b
(i)
Obtain an expression for the closed loop transfer function, T(s). (5 marks)
(ii)
Obtain an expression for the unit step response, c(t).
(iii)
Calculate the closed loop system steady state response and time constant
(5 marks)
(5 marks)
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MSCSY460.TAKEHOME.EXAM.FALL2006
COURSE: MSCSY 460-CONTROL SYSTEMS THEORY II
TAKEHOME EXAM
QUESTION 2
(a) The block diagram of a second order control system is shown in figure Q2a,
R(s)
K
+
C(s)
G(s)
-
n2
where: G(s)=
s(s  2 n )
Fig.Q2a
Determine the closed loop transfer function of the above system.
(5 marks)
(b) The closed loop transfer function of the system shown in Fig.Q2a is given by:
G (s) 
6.4
s  0.8s  0.64
2
Determine:
(i) the constant gain, K,
(ii) the natural undamped frequency and
(ii) the damping ratio of the above system
(10 marks)
(c) The unit step response of the second order system in (b) is shown in figure Q2b.
12
10
8
6
Determine, approximately:
Fig. Q2b
4
(i)
The maximum percentage overshoot, Mp%.
(ii)
The peak time, tp
(iii)
The rise time, tr.
(iv)
The settling time,0
ts
(v)
The damped natural frequency, d.
2
5
10
(10 marks)
Time (second)
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MSCSY460.TAKEHOME.EXAM.FALL2006
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20
COURSE: MSCSY 460-CONTROL SYSTEMS THEORY II
TAKEHOME EXAM
QUESTION 3
(a) The characteristic equation of a closed loop system is given as:
q(s) = (s+1)(s4 + s3 + 2s2 + 2s +3)
By using the Routh-Hurwitch Stability Criterion determine the stability of the above
system.
(10 marks)
(b) Consider the block diagram of a control system shown in figure Q.3.
R(s)
+
10
(s 2  s  10)
K/s

C(s)
1
s2
Fig. Q3
Use the Routh-Hurwitz stability criterion to determine the range of K (where K>0),
for which the system is stable.
(15 marks)
QUESTION 4
(a) Reduce the block diagram of the system shown in figure Q4a, and obtain the
overall transfer function relating C (s ) and R(s) .
R(s)
C(s)
+
-
+
-
G1(s)
+
H1(s)
-
G2(s)
H2(s)
H3(s)
Fig.Q4a
(10 marks)
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MSCSY460.TAKEHOME.EXAM.FALL2006
COURSE: MSCSY 460-CONTROL SYSTEMS THEORY II
TAKEHOME EXAM
(b) For the speed-control system, shown in Fig. Q.4b:
r(t)
+
-
1
1  3s
6
c(t)
4
Fig. Q.4b
(i)
Determine and plot the response to a unit step input
(ii)
The value of the steady state error and closed loop time constant
QUESTION 5
Figures 5.a and 5.b: Rigid body PID control block diagram
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MSCSY460.TAKEHOME.EXAM.FALL2006
(10 marks)
(5 marks)
COURSE: MSCSY 460-CONTROL SYSTEMS THEORY II
TAKEHOME EXAM
The block diagrams of two control systems are shown in Figures 5.a and 5.b above.
Determine the following:
(i)
The closed loop transfer functions for the two systems.
(ii)
Determine and plot the response to a unit step input of both systems, under
the following conditions:
(iii)
(30 marks)
(50 marks)
Constants
Case # 1
Case # 2
Case # 3
kp
0
0
0
kd
0
0
0
ki
0
0
0
Discuss the effects of the choice of the PID controller on the response of
the two systems.
(20 marks)
NOTE: For simplicity use : k hw  1 and m  1 kgr in your analytical
calculations and plots. The choice of the constants k p ,
is up to you.
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MSCSY460.TAKEHOME.EXAM.FALL2006
k d , and k i