UNIVERSITY OF GLASGOW

Degrees of MEng, BEng, MSc and BSc in Engineering

CONTROL M (ENG5022)
Monday 10 December 2018
09:30 – 11:30

Answer ALL questions in Section A and ONE question from Section B and ONE
question from Section C.

The numbers in square brackets in the right-hand margin indicate the marks allotted to the
part of the question against which the mark is shown. These marks are for guidance only.

An electronic calculator may be used provided that it does not have a facility for either
textual storage or display, or for graphical display.

Data sheet included within paper.

Continued overleaf
Page 1 of 7
 SECTION A

Q1 (a) Compare and contrast the properties of open- and closed-loop control
structures. In your analysis, focus on the characteristics of each structure with
respect to plant disturbances, changes in the plant gain, and stabilisation. [5]

(b) Refer to the closed-loop system shown in Figure Q1. Define the sensitivity
function 𝑆𝑜 and the complementary sensitivity function 𝑇𝑜 and show explicitly
how these transfer functions determine the properties of the system with
respect to reference, disturbance and measurement noise signals. [5]

Figure Q1

(c) Consider the linear state space system

𝑥̇ (𝑡) = 𝑨𝑥(𝑡) + 𝑩𝑢(𝑡)
𝑦(𝑡) = 𝑪𝑥(𝑡) + 𝑫𝑢(𝑡)

with 𝑥(0) = 𝑥0. Show how a linear transfer function G(s) can be derived. [5]

(d) Given is the equation of motion of a pendulum as

𝑚𝑔𝑙
𝐼𝜃̈(𝑡) + 𝑐𝜃̇(𝑡) + 2 sin 𝜃(𝑡) = 𝜏(𝑡)

Linearise this equation and present it in the standard state-space form of a

differential equation,

𝑥̇ (𝑡) = 𝑨𝑥 (𝑡) + 𝑩𝑢(𝑡)

𝑦(𝑡) = 𝑪𝑥(𝑡) + 𝑫𝑢(𝑡)

with the angle θ as the output, and the external torque τ as the input. [5]

Continued overleaf
Page 2 of 7
Q2 (a) With the help of sketches as necessary, highlight the main differences of a
digital feedback controller, with respect to an analogue one. [6]

(b) Under which condition(s) the difference equation

uk = a1uk −1 + a2uk −2 + ... + b0ek + b−1ek −1 + b−2ek −2 + ... + b1ek +1 + b2ek + 2 + ... is
causal? What is the physical meaning of causality? [4]

U (s) a
(c) Consider a first order continuous transfer function H ( s ) = = .
E (s) s + a
Demonstrate that the backward rectangular numerical integration rule can be
z −1
implemented through the substitution s  [6]
Tz

(d) Derive how the stability region of a continuous transfer function maps into the
z-plane, using that rule, and use sketches to explain your results. What are the
consequences of your result, when applying this rule to a real system? [4]

Continued overleaf
Page 3 of 7
 SECTION B

Q3 (a) Describe design goals which one aims to achieve when using feedback
control. Relate these to requirements for the complementary sensitivity
function 𝑇𝑜 and the sensitivity function 𝑆𝑜 , and explain how these can be
translated into requirements of the magnitude and phase of the loop gain 𝐿. [6]

(b) Describe what is meant by controller design using loop-shaping. [4]

(c) Consider a PID controller in the context of frequency response design using
loop-shaping.

(i) What are the three components of the PID controller? Give their
responses in the Laplace domain. [3]

(ii) Derive the transfer function of a PID controller 𝐶(𝑠), and express the
result in terms of a gain 𝐾, a time constant relating to the integral term,
𝑇𝑖 , and a time constant relating to the derivative term, 𝑇𝑑 . What are the
poles and zeros of 𝐶(𝑠). [4]

(iii) Based on the transfer function derived in (ii), sketch the Bode
frequency response plot of an ideal PID controller. [3]

Continued overleaf
Page 4 of 7
Q4 (a) What is meant by state estimator feedback control. Use a block diagram to
illustrate your explanations and mark the elements which form the
compensator. What are the advantages of using a state estimator? [5]

(b) For the structure described in (a), describe in detail the structure of the state
estimator (observer) and discuss the behaviour of the state estimation error
𝑥̃(𝑡) = 𝑥(𝑡) − 𝑥̂(𝑡). [5]

(c) Consider the plant

 −10 1  0 
x(t ) =   x(t ) +   u (t )
 −20 0  2
y (t ) = 1 0 x(t )

Calculate the observer gain vector 𝐿 such that the closed loop observer poles
are located at -80 and -90. [5]

(d) Describe a test for observability of a state space system. Show whether the
following system is observable:

 x1   −3 1   x1  10
 x  =  −1 0  x  +  1  u
 2   2  
x 
y = 1 0  1 
 x2 

[5]

Continued overleaf
Page 5 of 7
 Section C

Q5 The following transfer function is a lead network:

𝑠+1
𝐻(𝑠) = 0.2𝑠+1

(a) Find the discrete equivalent of it, when preceded by a zero-order hold (ZOH),
for sample time T = 0.5 s. Use 4 significant digits for all numbers in the
solution. [8]

(b) Using the inverse z-transform, find the corresponding difference equation. [4]

(c) State a necessary and sufficient condition for BIBO stability and determine
whether the difference equation is BIBO stable. [8]

Q6 Consider a continuous transfer function of a first-order low pass filter with

steady-state gain of 20 dB and cutoff frequency (gain decays by -3 dB) at 10
rad/s.

(a) Design the discrete equivalent of it, using the Tustin rule, considering a
sampling time of 0.2 s. Compute the gain (in dB) of the digital filter at the
cutoff frequency, and compare it with the analogue version. [8]

(b) Re-design the same, but this time apply a pre-warping such that the gain is
preserved at the original cutoff frequency. Once designed, verify the gain
numerically. [8]

(c) Finally, re-design the discrete equivalent using the forward rectangular rule,
and compare the gain at the cutoff frequency. [4]

Continued overleaf
Page 6 of 7
TABLE 1

End of question paper

Page 7 of 7