AMME3500/ AMME8501/ AMME 9501 Systems Dynamics and Control
Design Project 1
Due: 23.59, Sunday Week 6
Weight: 15% of your total mark.
Approved Late Submissions: In case you receive an approval for a submission extension, then you
should add a comment along with your submission on Canvas stating your extended due date (When you
submit the work, on the top-right corner of the Canvas portal you will be able to see a button “Add
Comment”). You may also indicate your extension under the title of your report.
Late Submissions: According to our University policy, late submissions without approval will be sub-
jected to penalties: the penalty will be 5% of the total assignment mark per day. If the submission is late
by more than ten calendar days, then a mark of zero for the assignment will be awarded.
Project Summary
This project asks you to design some of the basic components of an autonomous car: the cruise control
system and a controller for automatically changing lanes. For the parameters of the vehicle model (masses,
lengths, etc), look up or estimate numbers for your car if you own one, or the car of a family member
or friend. This assignment draws most directly on knowledge of linearisation, second-order dynamical
systems and second-order control systems.
The approach you should take is that your tutor is your supervisor at your first job after graduation, and
they have asked you to prepare a design proposal. Therefore, the report should be of a high professional
standard. The following two sections contain the project description, split into “Cruise Control” and
“Lateral Control.”
1 Cruise Control
Let a vehicle be moving along a straight line with its velocity at time t described by v(t). We assume
that an engine controller has been designed, so that the control input u(t) is the force demanded from the
engine. The situation can be modelled via the following nonlinear differential equation:
Mv˙ +
1
2
AρcD(v(t))
2 = u(t). (1)
Here, M is the mass of the car, ρ is density of air in kg/m3, cD is a dimensionless drag coefficient, and A
is cross-sectional area of the vehicle in m2 (looking from the front). Reasonable values for cD for a car are
about 0.25 to 0.45. 1 For your car, look up, measure, or estimate A and cD.
As a first step towards the design of as cruise control system, you are asked to complete the following
design and testing tasks:
1Wikipedia has an interesting list.
1
Task 1 (Linearization): Select three pairs of equilibra (ve, ue). Linearize the system dynamics (1) about
these three pairs of equilibra. Select initial conditions for v(0), and simulate the three linearized dynamics
to obtain three trajectories of v(t). Plot the three trajectories and explain their similarities and differences.
Task 2 (Controller Design): Choose an equilibrium from any of the three ones found in Task 1 and
derive a linear dynamic model. For this linear model, design a controller that will precisely achieve any
desired speed (reference). Select references as r = 10 km/h, 50 km/h, and 100 km/h. Demonstrate the
effectiveness and limitations of your design by numerical experiments on the linear model for all three
reference values.
Task 3 (Validations): The controller designed in Task 2 needs to be tested before real-world validations.
There are three main challenges to consider:
1. The controller is designed from the linear model, but the “true” system dynamics in (1) is nonlinear.
2. There may be disturbances acting on the vehicle. In particular, suddenly the vehicle may suddenly
reach a hill and encounter a steep slope.
3. The mass of the vehicle is uncertain, e.g., due to the number of passengers being unknown.
Carry out the following analysis and design for Task 3:
(1) Establish an equation of motion of the vehicle by extending Equation (1) to the case with the slope
accounted for. Show why and how the new equation of motion is of the form
Mv˙ +
1
2
AρcD(v(t))
2 = u(t) + d(t), (2)
where d(t) is the slope disturbance.
(2) Combine your linear controller for reference tracking from Task 2 with the system model (2), and
investigate the resulting closed-loop dynamics.
For this purpose, suppose that the vehicle encounters a sudden transition from flat ground to a
very steep uphill slope of 6% grade2. Simulate the closed-loop dynamics for reference speeds of
r = 10 km/h, 50 km/h, and 100 km/h.
Plot your results for the three reference speeds and draw a conclusion on the performance of your
controller in this validation. Discuss how the feedback gains in the controller affect the system
response characteristics, such as disturbance compensation.
Examine and discuss the effect of uncertaintly in mass, and provide plots that you consider necessary.
Assumed knowledge and tools available to you: Lecture 2 covers linearisation, Lecture 3 treats
controller design for first-order systems, Prelab in Week 3 shows how to build Simulink blocks for a first-
oder system, Lab 1 in Week 4 will continue to help you on how to use Simulink for controller design.
You may also want to study parts of the textbook, available on https://fbswiki.org/wiki/index.php/
Feedback_Systems:_An_Introduction_for_Scientists_and_Engineers.
Support: In case you face issues that are not addressed in our Canvas Ed forum, you may ask questions
about the project in the timetabled Lab sessions during weeks 3–5. The Week 6 timetabled Lab session is
especially dedicated to helping you with this assignment.
2Note that the “grade” (in percent) of a slope is not the angle of its inclination, but rather the tangent of the
angle of inclination times 100.
2
2 Lateral Control
Here we examine the lateral (side-to-side) motion of the vehicle, in particular for automatic lane change
control.
A schematic of the vehicle with relevant quantities is shown below. Feel free to consult the textbook
Chapter 3, Example 3.11 and Chapter 6, Example 6.13, “Vehicle steering” for a more detailed analysis.
For this part, you should assume that the velocity of the vehicle is constant, and that the control input is
δf (t), the steering wheel angle, see figure.
The motion of the centre of mass (CoM) position (x, y) is described by the following differential equations:3
x˙ = v(t) cos
(
ψ(t) + β(t)
)
y˙ = v(t) sin
(
ψ(t) + β(t)
)
ψ˙ =
v(t)
lr
sin
(
β(t)
)
.
Note the coupling to longitudinal dynamics in (1) and (2) through the vehicle velocity. In addition, we
have the following algebraic relationship between the steering wheel angle and the CoM rotation angle β:
tan
(
β(t)
)
=
lr
lf + lr
tan
(
δf (t)
)
.
For your car, look up the wheelbase lr + lf . For simplicity you may assume that lr = lf .
We assume the vehicle is mostly moving in the x direction (meaning: the first differential equation can be
ignored). Thus, it is the lateral position y(t) that we want to control.
Task 1 (Linearization): Linearise the dynamics aaround an operating point with constant speed motion
v(t) ≈ v0 and small angles, i.e., ψ ≈ 0, β ≈ 0, δf ≈ 0. Document the entire linearisation procedure and
further show that one obtains
• a second-order differential equation describing how y(t) depends on δf (t); and thereby
• a transfer function from steering-wheel angle to lateral position of the form
G(s) =
as+ b
s2
.
Calculate the values of a and b for your car, noting that a and b will depend on v0.
3You might like to verify this, but it is not part of the assignment.
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Task 2 (Controller Design): For the second-order differential equation describing how y(t) depends on
δf (t) established in Task 1, design a controller for δf (t) so that y(t) can be changed from one position to
another, i.e., y(t) should be able to track a reference of ry = 2m, 3m and 4m. Explain why this suggests
that the controller will steer the vehicle for smooth and accurate transitions between lanes.
Task 3 (Validations): Simulate and plot the closed-loop system response of the linear model for lane-
change manoeuvres at speeds of 10 km/h, 50 km/h and 100 km/h. Plot the trajectories of y(t) when its
reference changes to ry = 4m. Explain the performance of the controller in terms of achieving its goal of
smooth and accurate lane changes.
Test the closed-loop system response when the vehicle is reversing at v0 = −8 km/h and v0 = −16 km/h.
In comparison with the responses obtained with v0 being positive, discuss the effect and physical meaning
of the (open-loop) system zero4 when the vehicle is reversing.
Assumed Knowledge: Lecture 4 and Lecture 5 cover how to design controllers for second-order linear
systems. Lab 2 in Week 5 will help you practice building Simulink blocks for second-order systems.
Support: You may ask questions about the project during the timetabled Lab sessions in weeks 3–5. You
may also ask questions about the project description in our Canvas Ed forum. The timetabled Lab session
in Week 6 is dedicated to helping you with this assignment.
3 Report Format
You must submit a professional-quality report as a machine-readable pdf –don’t use scanned images–
through Canvas. By professional-quality report, we mean that your report should be a self-contained,
consistent, and coherent article, instead of a collection of equations, numerical plots, and answers to
design questions.
The report must use the template for double-column IEEE Conference Articles. The template, in Word
or Latex, can be found at IEEE Templates. Your report has to obey the following structure:
1. Introduction
2. Cruise Control
2.1 Linearization
2.2 Controller Design
2.3 Validations
3. Lateral Control
3.1 Linearization
3.2 Controller Design
3.3 Validations
4. Discussion and Conclusions
Subsections 2.1, 2.2, 2.3, and 3.1, 3.2, 3.3 must fully address the required tasks in the project description.
4zero of the transfer function G(s)
4
The entire report must be no more than 8 pages including EVERYTHING, i.e., the cover page and
appendices. Your marks will depend not only on the technical correctness, but also the way that you
motivate your design choices, and the way that you analyse and present your results.
The report must be entirely your own work, except where clearly indicated otherwise. Any references to
external material (papers, books, websites or genAI) must follow the academic honesty guidelines.
Further information on academic honesty, academic dishonesty, and the resources available to all students
can be found on the academic integrity pages on the current student website. Further information about re-
search integrity and ethics for postgraduate research students and students undertaking research-focussed
coursework such as honours and capstone research projects can be found on the current student website.
4 Marking Criteria and Procedure
4.1 Mark Breakdown and Criteria
The mark breakdown is indicated below. The marks should serve as a rough guideline for how much space
to allocate to each section.
Section 1: Introduction (5%): Clear explanation of the motivation of study; Precise and comprehensive
introduction to project scope; Organization of report.
Section 2: Cruise Control (45%): Thorough investigation, clear explanation of the working, and complete
and correct presentation of the required results.
• Subsection 2.1: Linearization (10%)
• Subsection 2.2: Controller Design (15%)
• Subsection 2.3: Validations (20%)
Section 3: Lateral Control (45%): Thorough investigation, clear explanation of the working, and complete
and correct presentation of the required results.
• Subsection 3.1: Linearization (10%)
• Subsection 3.2: Controller Design (15%)
• Subsection 3.3: Validations (20%)
Section 4: Conclusions (5%): Summary of the project and results; Highlight the most significant discover-
ies/understandings; Discussion about possible improvements and future directions.
4.2 Marking Procedure
You report will be assigned to a marker from our teaching staff, and marking will follow strictly the above
criteria. To ensure consistency between markers we will carefully calibrate the marking.
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4.3 Feedback
You may receive two types of feedback on your report:
• A detailed mark breakdown of your total mark under Canvas rubrics. This will include the score for
each of the above items listed above. Thus, you will be able to see how well you have been doing in
each part of the report.
• Additional comments and/or suggestions from the marker.
We will also publish a document containing general feedback about the reports on our Canvas site.
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