# Engineering Data book

January Examinations 2015
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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]

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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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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

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

d +
+
+
+
n
Figure 1:
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3. a. Gain and phase margins are commonly used measures of robust stability in classical
control theory.

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:
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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,

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
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