ENGR406 (506)
Week 21 Revision
Professor James Taylor
Engineering Department
ENGR406 (506)
Revision Contents/Plan
I have prepared some slides…
• Brief recap of the syllabus
• Reminder of what you need to know for sections 5 and 6
• Exam Rubric for 2023 & Practice Question
(largely borrowed from the 2019 exam paper)
In addition (at any time today)…
• Do you have any questions for me?
• Would you like me to recap any specific problem areas?
• I can hang around at the end for individual conversations
Part 1 – Terminology and
Objectives of Control
Contents
• Section 1.1 – Introduction
• Section 1.2 – Syllabus
• Section 1.3 – Feedback
• Section 1.4 – Control Objectives
• Section 1.5 – Module Details
3
Part 2 – Discrete–Time
Transfer Function Models
Contents
• Section 2.1 – Why Model?
• Section 2.2 – Types of Model
• Section 2.3 – Difference Equations
• Section 2.4 – Backward Shift Operator
• Section 2.5 – Steady state Gain
• Section 2.6 – Poles and Stability
• Section 2.7 – Block Diagrams
4
1 2 3
1 2 3
1 2 3
1 2 31
b z b z b z
a z a z a z
− − −
− − −
+ +
+ + +
Part 3 – System Identification
Contents
• Section 3.1 – Least Squares
• Section 3.2 – Response Error
• Section 3.3 – Equation Error
• Section 3.4 – Estimation Accuracy
• Section 3.5 – Summary & Recap
• Section 3.6 – Identification
5
Comparison of
EE (left) and RE (right)
1 1ˆ ˆ( ) ( )k k kE A z y B z u
− −= −
1
1
( )
( )
B z
A z
−
−
system
kxku
kξ
+
ky
− +
1ˆ( )B z− 1ˆ( )A z−
1
1
( )
( )
B z
A z
−
−
system
kxku
ˆky
kξ
+
ky
1
1
ˆ( )
ˆ( )k k k
B ze y u
A z
−
−= −
− +
1
1
ˆ( )
ˆ( )
B z
A z
−
− 1.0
ky
6
Part 4 – Discrete–Time
Control & Pole Assignment
Contents
• Section 4.1 – Proportional Action
• Section 4.2 – Integral Action
• Section 4.3 – Motorway Traffic
• Section 4.4 – Pole Assignment
7
Pole Assignment
is a key, key, key topic!
Brings together PI control,
block diagram analysis,
closed loop Transfer Function,
characteristic equation, poles,
steady state gain. Also, the
maths here links back to the
objectives of control and looks
ahead to state space methods.
Part 5 – State Variable
Feedback
Contents
• Section 5.1 – Minimal State Space
• Section 5.2 – State Variable Feedback
• Section 5.3 – NMSS: Integral-Of-Error
• Section 5.4 – NMSS: Output States
• Section 5.5 – NMSS: Input States
• Section 5.6 – NMSS: General Form
(NMSS: Non-Minimal State Space)
8
Part 5 Exam Revision Tip 1
Sections 5.1 and 5.2 are about minimal state
feedback. Concepts are important, but the
mathematical equations do not have to be
memorised to answer numerical questions in the
exam (any questions would give the info required).
Anyway, the main focus of the module is
on non-minimal state feedback in section 5.3
onwards.
9
Part 5 Exam Revision Tip 2
Exam questions often ask students to write down
a non-minimal state space representation of a
Transfer Function.
In the closed book exam, the state vector will be
stated in the question – this is to avoid ambiguity
over the type of state space model.
10
Syllabus
11
1Concept of Feedback, Terminology and
Objectives of Control
2Transfer Functions represented using the
Backward Shift operator
3System Identification
4Block diagram analysis, pole assignment and
discrete-time PI control
5
Introduction to
“Modern Control”
Theory
State variable
feedback using
non-minimal state
space models
CORE TOPIC
6
Optimal control
Multivariable & Non-linear
Robust control
Feed-forward control
Stochastic control
Syllabus(2)
12
1Concept of Feedback, Terminology and
Objectives of Control
2Transfer Functions represented using the
Backward Shift operator
3System Identification
4Block diagram analysis, pole assignment and
discrete-time PI control
5
Introduction to
“Modern Control”
Theory
State variable
feedback using
non-minimal state
space models
CORE TOPIC
6
Optimal control
(e.g. LQ, LQG, LQR)
Multivariable & Non-linear
Robust control
Feed-forward control
Stochastic control,
model predictive
control (MPC), etc.
Part 6 – Additional Topics &
Case Studies
Contents
• Section 6.1 – Optimal & Predictive Control (29 mins)
• Section 6.2 – Multivariable Control (24 mins)
• Section 6.3 – Implementation Forms (3 mins)
• Section 6.4 – Adaptive & Nonlinear Control (9 mins)
13
An example of nonlinear control was
considered in the vibro-lance and
nuclear robotics research seminars in
week 13 (slides are on Moodle)
Part 6 Exam Revision Tip
• Part 6 was not in the coursework but is examinable
• Mathematical details not essential (do not have to be
memorised to answer numerical questions in the exam)
• Concepts however are important, see videos on
Moodle, especially for the following topics –
• Optimal control
• Multivariable control
• Some open ended MEng / MSc exam questions allow
for students to optionally mention additional topics
that add relevant content to their answers
14
Summer Exam 2023
• Be aware that the exam rubric (format/instructions)
has changed several times over the past few years
• The syllabus/content has not changed
• Exam 2023 for Engr406/506 is closed book i.e. you
cannot bring books, notes, course materials etc. into
the exam
• Examination PRACTICE Paper on Moodle is a practice
paper using the correct 2023 rubric
• Selected worked solutions are provided below
15
See PDF on Moodle
16
Question 1
Consider the following first order difference equation,
where y(k) represents the blood clotting speed in a human
patient, u(k) is the dose of warfarin (mg) and s(k) represents
an unknown stochastic component. Warfarin is a drug used to
regulate the blood clotting speed of patients with chronic
medical conditions and is called the International Normalised
Ratio (INR). The INR of a healthy patient is usually within 2 to
3 units. For this question, the sampling rate dt is daily, i.e.
once per day, the blood clotting speed is measured and the
drug dosage adjusted.
17
1 2( ) ( 1) ( 2) ( )y k a y k b u k s k= − − + − +
Question 1 (a)
• Assuming zero initial conditions, a1=-0.4, b2=0.25 and
s(k>0)=0, find the unit step response for k=1…8.
Develop the Transfer Function form of the model i.e.
using the backward shift operator. What is the time-
delay, time constant, steady state gain, characteristic
equation, pole and stability condition of the model?
• Sketch (graph paper is not required) the response to a
unit step input. With the help of your sketch, explain
the physical meaning of the terms time-delay, time
constant and steady state gain
18
Solution to Q1 (a) (1)
• Use difference equation, similar to section 2.3 slide 12.
• Output k=1…8 for u(k>0)=1:
0, 0, 0.25, 0.35, 0.39, 0.406, 0.412, 0.415
• Or possibly output k=1…8 for u(k>=0)=1:
0, 0.25, 0.35, 0.39, 0.406, 0.412, 0.415, 0.416
• Find the Transfer Function, pole and stability similar to
the examples in sections 2.4 and 2.5
19
Solution to Q1 (a) 2
20
2 2
2
1 1
1
0.25( ) ( ) ( )
1 1 0.4
b z zy k u k u k
a z z
− −
− −= =+ −
2 2
2
1 1 1 1
1 1
1 0.25 1( ) ( ) ( ) ( ) ( )
1 1 1 0.4 1 0.4
b z zy k u k k u k k
a z a z z z
ξ ξ
− −
− − − −= + = ++ + − −
11 0.4 0z−− =
0.4 0 0.4z z− = → = (Stable)
Solution to Q1 (a) 3
• Time constant days
• Steady state gain
• Time delay = 2 samples i.e. 2 days
21
1 1.091
log (0.4)e
− =
0.25 0.4167
1 0.4
=
−
Solution to Q1 (a) 4
22
Time k
y(k)
( ) 0.42y k →∞ =
Time-delay
Time-constant
0.42 0.63 0.26× =
Question 1 (b)
• Assume that a1=-0.4 and b2=0.25 have been estimated
using the data from 100 patients. Estimate the dosage
required to obtain an INR of 2.5 at steady state.
• Discuss the limitations of this approach i.e. on giving a
new patient the dose you have just obtained.
• Do you consider this to be an example of open-loop or
closed-loop control?
23
Solution to Q1 (b)
• Use the steady state gain determined in part (a)
y(steady state)=0.4167 x u0
where u0 is a time-invariant dose
• Hence, the required u0 = 2.5/0.4167 = 6 mg
• Open-loop controller
• Limitations – patient variability not accounted for
by this fixed dose. Also be numerous cofactors
including diet, alcohol, exercise and co-medication
that might change over time for a given patient.
24
Question 1 (c)
• Assume that a data-set for one patient consists of y(k)
and u(k) measurements for 20 consecutive days. In this
case, define the Response Error associated with the
model.
• With the aid of a block diagram, explain how Least
Squares-based Response Error methods can be used to
estimate the parameters of the model.
• Explain the potential disadvantages of the Response
Error approach.
25
Solution to Q1 (c)
• Response error
• Least squares cost function
• Rest of the question is ‘bookwork’ – you need to be
able to describe the concept and be aware of the
pros/cons of response error vs. equation error
(section 3.1 through 3.4 videos).
26
2
2
1
1
ˆ
( ) ( ) ( )
ˆ1
b ze k y k u k
a z
−
−= − +
20
2
1

k
k
k
J e
=
=
= ∑
Question 1 (d)
• Using the above model as an example, explain the
difference between the data-based and mechanistic
modelling approaches.
27
Solution to Q1 (d)
• You need to be aware of the model types briefly
introduced in section 2.2.
• Here you should briefly describe the data-based and
mechanistic modelling approaches.
28
Question 2
29
1370mm
15
00
m
m
72
0m
m
Potentiometer4
Potentiometer1 Potentiometer3
Potentiometer2
Dipper
Bucket
Boom
3
5
4
1
2
Question 2 (1)
The Lancaster University Computerised Intelligent
Excavator (LUCIE), illustrated in Figure X, was developed
some years ago to dig foundation trenches on a building
site. LUCIE was based on a commercial excavator, but an
on-board computer system was used to control the
hydraulics. Figure X indicates several actuators: hydraulic
cylinders to control the Boom and Dipper arms [1];
rotation of the bucket [2] and cab [3]; movement of the
dozer blade at the front of the cab [4]; and movement of
caterpillar tracks [5]
30
Question 2 (3)
Analysis of experimental data collected at dt=0.1 s
suggests that the dynamics of the boom manipulator
joint is well represented by the following discrete-time
Transfer Function:
where y(k) represents a scaled angle relative to an initial
operating condition, while u(k) is the applied voltage
scaled between -1 and +1, with negative values implying
that the manipulator is to be lowered and positive values
for raising it.
31
1
1
1
( ) ( )
1
Tb zy k u k
a z
−
−= +
Question 2 (4)
• A proportional-integral (PI) controller for the Boom
takes the following Transfer Function form
32
( )1( ) ( ) ( ) ( )1
I
P
Ku k r k y k K y k
z−
= − −
−
Question 2 (a)
• Draw the block diagram of the PI controller, as applied
to Transfer Function form of the Boom model.
33
Solution to Q2 (a) (1)
34
Question 2 (b) (2)
• Assuming b1=0.7 and T=1, use the rules of block
diagram analysis (or other methods) to show that the
closed-loop characteristic equation is:
• Design (i.e. determine the control gains) a PI controller
that yields closed–loop poles of 0.75 and 0.8 on the
real axis of the complex z-plane
35
( )2 0.7 2 0.7 1 0.7 0P I Pz K K z K+ − + + − =
Solution to Q2 (b) (3)
36
1
1
1 1
1 1
1 11
1 11
1
1
1
1
1
1
P
P
b z
a z b z
a z K b zb z K
a z
−
− −
− −−
−
 
 +  =
+ + +
+  + 
1
1
1 1 1
1 1
1
1
1 1 1
1 1
1 1
( ) ( )
1
1 1
I
P
I
P
K b z
z a z K b z
y k r k
K b z
z a z K b z
−
− − −
−
− − −
  
  − + +  =
  +   − + +  
Solution to Q2 (b) (4)
37
( )( )
1
1
1 1 1 1
1 1 1
( ) ( )
1 1
I
P I
K b zy k r k
z a z K b z K b z
−
− − − −
+
=
− + + +
( ) ( )
1
1
1 2
1 1 1 1 1
( ) ( )
1 1
I
P I P
K b zy k r k
a K b K b z a K b z
−
− −= + + − + + − −
Solution to Q2 (b) (5)
• Characteristic equation
• Equivalently
38
( ) ( )1 21 0.7 2 0.7 1 0.7 0P I PK K z K z− −+ − + + − =
( )2 0.7 2 0.7 1 0.7 0P I Pz K K z K+ − + + − =
Solution to Q2 (b) (6)
• We desire the following characteristic equation
• Equating the coefficients
39
( )( ) 2( ) 0.75 0.8 1.55 0.6 0D z z z z z= − − = − + =
0.7 2 0.7 1.55 ; 1 0.7 0.6P I PK K K− + = − − =
0.5714PK = 0.0714IK =
Question 2 (c)
• For the case that T=2, develop the non-minimal state
space representation of the boom model. Here:
• Briefly comment on why such a state space model
might be used for the case that T=2
40
[ ]( ) ( ), ( 1), ( ) Tk y k u k z k= −x
( ) ( 1) ( 1) ( ) ; ( ) ( )dk k u k y k y k k= − + − + =x Fx g d hx
( ) ( 1) ( ) ( )dz k z k y k y k= − + −
Solution to Q2 (c)
41
1 2( ) ( 1) ( 2)y k a y k b u k= − − + −
1 2( ) ( 1) ( ) ( 1) ( 2)dz k z k y k a y k b u k= − + + − − −
1 2
1 2
0 0 0( ) ( 1)
( 1) 1 0 0 ( 2) 0 ( 1) 0 ( )
( ) ( 1)1 0 1
d
a by k y k
u k u k u k y k
z k z ka b
− −        
− = − + − +        
  −               
[ ]( ) 1 0 0 ( )y k k= x
Question 2 (d)
Suggest likely control objectives associated with the
problem of fully autonomous digging of trenches on
building sites and categorise these using a hierarchical
framework of your own devising. Hint: your answer
should mention both technical control objectives from
the lectures (e.g. closed-loop stability) in addition to
speculating on wider issues such as safety, among other
considerations.
42
Solution to Q2 (d)
• Something like the discussion we had in class for the
control objectives of a vibro-lance system
• Section 1.4 (video, slides and the associated exercise)
43
Conclusions
• Revision and example paper… best of luck