THE UNIVERSITY OF AUCKLAND
SECOND SEMESTER, 2019 Campus: City
ELECTRICAL & ELECTRONIC ENGINEERING Systems and Control
(Time Allowed: THREE hours)
NOTE:
• Answer any FIVE out of SIX questions.
• All questions are of equal mark value.
• If you answer 6 questions, only the first five you answer will be marked.
• Cross out any work you do not want marked.
• A table of inverse Laplace transforms is provided as an Appendix.
• An answer sheet for Question 5 is provided as an attachment.
ELECTENG 303
Page 1 of 12
1. (a)
ELECTENG 303
The switch in the circuit of Fig. Q1(a) has been closed for a long time. At t = 0 the switch opens.
R1
+
t=0
C R2
Vdc + −
o(t) L v-
Fig. Q1(a)
(i) Derive expressions for the initial conditions, 𝑣𝑣𝑐𝑐(0−) and 𝑖𝑖𝐿𝐿(0−), on the capacitor and inductor, respectively, for 𝑡𝑡 < 0.
Note: ensure that the polarity of 𝑣𝑣𝑐𝑐(0−) and the direction of 𝑖𝑖𝐿𝐿(0−) are clearly indicated in your answer. (3 marks)
(ii) Draw the s-domain circuit for t > 0. (3 marks)
(iii) Hence derive an expression for Vo(s) for t > 0 in terms of 𝑉𝑉𝑑𝑑𝑐𝑐, 𝑅𝑅1, 𝑅𝑅2, 𝐶𝐶, and 𝐿𝐿. Ensure
that it is expressed in the most appropriate form for inversion to vo(t). (6 marks)
(b) The switch in the circuit of Fig. Q1(b) has been open for a long time. At t = 0 the switch closes. Assuming there is no energy stored in the capacitor prior to t = 0, derive, using the convolution integral, an expression for 𝑣𝑣(𝑡𝑡), for 𝑡𝑡 ≥ 0. Ensure that you set out the steps in your working clearly. (8 marks)
t=0
R
Vdc
+
C v(t)
–
Note: the following expression may be useful.
Fig. Q1(b)
∞∞
𝑦𝑦(𝑡𝑡) = � 𝑥𝑥(𝜏𝜏)h(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝜏𝜏 = � h(𝜏𝜏)𝑥𝑥(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝜏𝜏
−∞
−∞
Page 2 of 12
2. The transfer function for a linear time-invariant (LTI) system is given by
𝐻𝐻(𝑠𝑠) = 20 (𝑠𝑠+5)(𝑠𝑠2 +3𝑠𝑠+10)
(6 marks)
(d) Is this system stable? Give your reasons.
(e) Using whatever approach you wish, as well as the information gained from your previous
𝑥𝑥(𝑡𝑡) = 15cos(2.5𝑡𝑡 + 𝜋𝜋2) Volts.
(4 marks)
ELECTENG 303
(a) If a constant input voltage, 𝑥𝑥(𝑡𝑡) = 10 Volts, is applied to the system, determine the resulting output, 𝑦𝑦(𝑡𝑡), once all transients have decayed to zero. (4 marks)
(b) If 𝑥𝑥(𝑡𝑡) = 𝛿𝛿(𝑡𝑡) Volts, determine 𝑦𝑦(𝑡𝑡).
(c) Determine 𝑦𝑦(𝑡𝑡) (once all transients have decayed to zero) if
(2 marks) answers, sketch the function |𝐻𝐻(𝑗𝑗𝑗𝑗)| for 0 ≤ 𝑗𝑗 ≤ 10 r/s. Use linear scales for both axes.
(Note: You need to clearly explain how you have arrived at your plot.)
(4 marks)
Page 3 of 12
3. (a)
ELECTENG 303
For the motor’s feedback system shown in Fig. Q3(a), find the steady state errors for the following test inputs: 𝑢𝑢(𝑡𝑡), 𝑡𝑡𝑢𝑢(𝑡𝑡) and 2𝑡𝑡2𝑢𝑢(𝑡𝑡). The function 𝑢𝑢(𝑡𝑡) is the unit step.
1 20
2
Fig. Q3(a)
(6 marks) (b) A simplified closed-loop transfer function for an airplane’s pitch system is given by
𝑇𝑇(𝑠𝑠) = 10(𝑠𝑠 + 1)(𝑠𝑠 + 5) . (𝑠𝑠+4)(𝑠𝑠+1.01)(𝑠𝑠2 +𝑠𝑠+1)
What percentage of overshoot, peak time, settling time and steady state value for a step input do you expect?
(6 marks)
(8 marks)
(c) Using the Routh-Hurwitz criterion, find the range of gain, 𝐾𝐾, to ensure stability in the feedback system of Fig. Q3(c). If you consider that there is no range of 𝐾𝐾 that stabilises the system, clearly give your reasons.
Fig. Q3(c)
where 𝐺𝐺(𝑠𝑠) = (𝑠𝑠+1) and 𝐻𝐻(𝑠𝑠) = 𝑠𝑠+10 . (𝑠𝑠+5) 𝑠𝑠2+𝑠𝑠+9
Page 4 of 12
4. (a)
How can you tell from the root locus if a closed loop system is unstable? (2 marks)
(b) Sketch the general shape of the root locus for each of the open-loop pole-zero plots shown in Fig. Q4(b).
𝑗𝑗ω O𝑗𝑗ω XO𝜎𝜎XX𝜎𝜎
ELECTENG 303
X
(i) (ii)
XOOX
𝑗𝑗𝑗𝑗
O X
𝜎𝜎XX XXX
𝜎𝜎
(8 marks)
ω
ω
X OX
Fig. Q4(b)
(iii)
(iv)
(QUESTION 4 CONTINUED NEXT PAGE)
Page 5 of 12
(c) Consider the closed loop system shown in Fig. Q4(c)
Fig. Q4(c)
where 𝐺𝐺(𝑠𝑠) = (𝑠𝑠−2) and 𝐻𝐻(𝑠𝑠) = (s−1) . (𝑠𝑠+5) (𝑠𝑠+3)
(i) Sketch the root loci of the closed loop system showing all of the relevant information where applicable (i.e. asymptotes, break-in point, breakaway point, imaginary axis crossing points). (8 marks)
(ii) Using the root locus, find the range of 𝐾𝐾 > 0 for the closed-loop stability of the given system. If you consider that there is no range of 𝐾𝐾 that stabilises the system, clearly give your reasons. (2 marks)
ELECTENG 303
Page 6 of 12
5. (a)
A unity feedback control system has an open loop transfer function
𝐺𝐺(𝑠𝑠) = 100(𝑠𝑠 + 2)(𝑠𝑠 + 3) . 𝑠𝑠(𝑠𝑠 + 1)(𝑠𝑠 + 4)(𝑠𝑠 + 10)(𝑠𝑠 + 20)
(i) Determine the phase angle of the Bode plot at very high frequencies. (1 mark)
(ii) Sketch the Bode magnitude and phase plots of the open loop transfer function.
Note: You should use the Bode plot provided as an attachment to this exam to sketch your answers. You must attach this to your answer book. A spare Bode plot has been provided if
you require.
(b) Shown in Fig. Q5(b) are the Bode magnitude and phase plots of an open loop plant.
40 20 0 -20 -40 -60
–80 -100 -120 -140
10 -2
-90 -135 -180 -225 -270
10 -2
10 -1
10 0
Frequency (rad/s)
10 1
10 2
10 3
10 -1
10 0
Frequency (rad/s)
Fig. Q5(b)
10 1
10 2
10 3
Bode Diagram/Plot
Determine the velocity error constant, the gain margin, the phase margin, the phase margin frequency and the gain margin frequency.
(5 marks)
Page 7 of 12
ELECTENG 303
(14 marks)
Phase (degree) Magnitude (dB)
6. (a)
(b) When sketching a Nyquist diagram, what must be done with open loop poles on the imaginary
(2 marks)
Briefly state the Nyquist criterion. (2 marks)
axis?
(c) Sketch the Nyquist diagram for the system shown in Fig. Q6(c), if
(i) 𝐺𝐺(𝑠𝑠) =
(iii) 𝐺𝐺(𝑠𝑠) = (𝑠𝑠+2)(𝑠𝑠+3)
1
(ii) 𝐺𝐺(𝑠𝑠) = (𝑠𝑠+2)(𝑠𝑠+3)
𝑠𝑠
ELECTENG 303
1 𝑠𝑠(𝑠𝑠+2)(𝑠𝑠+3)
Fig. Q6(c)
Page 8 of 12
(QUESTION 6 CONTINUED NEXT PAGE)
(12 marks)
𝐾𝐾 = 1.
(d) Shown in Fig. Q6(d)(ii) is the Nyquist diagram of the system depicted in Fig. Q6(d)(i) with
𝑅𝑅(𝑠𝑠) +
−
𝐾𝐾(𝑠𝑠+5) 𝐶𝐶(𝑠𝑠) (𝑠𝑠 − 6)
Fig. Q6(d)(i)
ELECTENG 303
(𝑠𝑠 − 4)
Fig. Q6(d)(ii)
Using the Nyquist stability criterion, find the range of K > 0 for the closed-loop stability of the given system. If you consider that there is no range of K that stabilises the system,
clearly give your reasons.
Page 9 of 12
(4 marks)
APPENDIX FOLLOWS
(𝑠𝑠 + 1)(𝑠𝑠 − 2)
APPENDIX
ELECTENG 303
Table of Inverse Laplace Transforms f(t)
k δ(t) k u(t)
k t u(t) 𝑘𝑘 𝑡𝑡2 𝑢𝑢(𝑡𝑡)
F(s) k
k
s k
s2 k s3
k
(s+a) k
B + B* (s− p) (s− p*)
B + B* (s − p)2 (s − p* )2
B + B* (s− p)3 (s− p*)3
2!
k e−at u(t) k t e−at u(t)
k t2 e−at u(t) (s+a)3 2!
(s+a)2 k
where:
a is real
2 B e−σt cos(ωt+φ) u(t)
2 B t e−σt cos(ωt+φ) u(t) 2 t2 e−σt cos(ωt+φ) u(t)
B= B ejφ
p = −σ + jω
ANSWER SHEET FOLLOWS
B
2!
where:
Page 10 of 12
Answer Sheet
ELECTENG 303
60 40 20
0 -20 -40 -60 -80 -100 -120 -140
10-2 10-1
-90
-135
-180
-225
-270
Name:_____________________________________ ID Number: ______________________________
Bode Diagram/Plot
100 101 102 103
-300-2 -1 0 1 2 3 10 10 10 10 10 10
Frequency (rad/sec)
Page 11 of 12
ELECTENG 303
Phase (deg) Magnitude (dB)
ANSWER SHEET
ELECTENG 303
60 40 20
0 -20 -40 -60 -80 -100 -120 -140
10-2 10-1
-90
-135
-180
-225
-270
Name:_____________________________________ ID Number: ______________________________
Bode Diagram/Plot
100 101
102 103
-300-2 -1 0 1 2 3 10 10 10 10 10 10
Frequency (rad/sec)
Page 12 of 12
ELECTENG 303
Phase (deg) Magnitude (dB)