Engineering Data book

EG4311
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January Examinations 2015
DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE
CHIEF INVIGILATOR

Department ENGINEERING
Module Code EG4311
Module Title Robust Control (Semester 1)
Exam Duration Two Hours

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Number of Pages 5
Number of Questions 4
Instructions to Candidates Answers are expected to three questions.
Answers to only three questions will be marked. Attempted solutions which
you do not wish to submit should be crossed out. If you do attempt more than
three questions, and do not identify which three you want to be marked, only
the first three in the answer book will be marked.
For each question, the distribution of marks out of 20 is indicated in brackets.
For this exam you are allowed to use the following
Calculators Casio FX83 or Casio FX85 (any variant)
Books/Statutes Engineering Data book
Additional Stationary Not required

Page 1 of 5

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EG4311
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1. i. Give two daily-life or engineering device/process examples to explain what is meant
by an open-loop control system and by a closed-loop system, respectively.
[6 marks]
ii. Further, using your example, compare the advantages and possible disadvantages

between the open- and closed-loop system structures. [6 marks]
iii. What is meant by a control system being robust? Why is the closed-loop structure
essential to robust control? What kind of uncertainties are usually considered in
robust control? Please use your example to illustrate your viewpoints. [8 marks]

Page 2 of 5

EG4311
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2. a. i. What is meant by Bounded-Input-Bounded-Ouput (BIBO) stability?
[3 marks]
ii. A control system is modelled by a transfer function matrix G(s). How do you find
out if G(s) is BIBO stable? [3 marks]
iii. Consider the feedback system depicted in Figure 1, where G(s) represents the
plant to be controlled and K(s) the controller to be designed. r is the reference
input signal, y the system output, u the control signal, d the disturbance at the
system output and n the sensor noise. Please define what is meant by internal

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stability, using this feedback system.
b. Consider the feedback system depicted in Figure 1.
[5 marks]

i. In the case that G(s) = (s+2s-)(1s+3), one at first trial chooses a controller K(s) =
s+2
s-1. Derive the transfer function Tyr(s) that describes the relationship between

r and y, and further determine the stability of Tyr(s).
ii. Can such a controller K(s) be used in practice? Give your reasons.
[4 marks]

[5 marks]
K(s) G(s)
r
u
y
+

d +
+
+
+
n
Figure 1:
Page 3 of 5

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EG4311
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3. a. Gain and phase margins are commonly used measures of robust stability in classical
control theory.

i. How are gain and phase margins defined? [3 marks]
ii. State the major limitation of these robustness measures. Draw a Nyquist plot to
show that they are not always reliable measures. [3 marks]

b. The Small Gain Theorem is a much more powerful tool in robust control than the gain
and phase margins.
i. With the aid of a block diagram, state the Small Gain Theorem. [4 marks]
ii. Why can the Small Gain Theorem sometimes give conservative results? Describe a circumstance in which the Small Gain Theorem would be a necessary
condition as well, i.e. it is not conservative at all in that circumstance.
[5 marks]
iii. Consider the closed-loop system depicted in Figure 2, where the plant dynamics
is represented by the nominal dynamics G(s) and an unstructured, multiplicative
perturbations D(s) at the (plant) input end. Use the Small Gain Theorem to
derive a condition for the design of K(s) to achieve robust stability. [5 marks]
r + e K(s) u G(s) y

+
+
v

Figure 2:
Page 4 of 5

EG4311
All candidates

4. Consider the H¥ Mixed Sensitivity control system design problem shown in Figure 3.
i. Define the various signals labelled in Figure 3, i.e. r; d; e; u; y; z1; z2. [3 marks]
ii. Derive the relationship between z1 and d and between z2 and r, respectively.
[6 marks]
iii. Derive the generalised interconnection plant P for the feedback system considered,

as in Figure 4. [4 marks]
iv. Use the (lower) Linear Fractional Transformation (LFT) to derive a cost function for
the H¥ optimal control problem for minimum disturbance rejection and limited control
effort. [4 marks]
v. Does the cost function formulated in (iv) also consider reference tracking? Give your
reasons. [3 marks]

d
P
u
G
K
+
+

+
W1
W2
r
z1
z2
e
y
Figure 3:
P(s)
K(s)
dr
u e
z z1 2
Figure 4:
END OF EXAMINATION
Page 5 of 5

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