AMME3500 Systems Dynamics and Control
Design Project 1
Due: 23.59, Sunday Week 6
Weight: 15% of your total mark.
Approved Late Submissions: If you receive an approval for a submission extension, you should add a comment along with your submission over 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; and when it is more than ten calendar days late, 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. This assignment draws most directly on knowledge of linearisation, second-order systems and second-order control systems. The approach you should take is that your tutor is your boss at your first job after graduation, and they have asked you to prepare design proposal. Therefore the report should be of a professional standard.
1 Project Description: Cruise Control
Let a vehicle be moving in a straight line with its velocity described by v(t) at time t. We assume an engine controller has been designed, so that the control input u is the force demanded from the engine:
Here P 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 (Wikipedia has an interesting list). For your car, look up, measure, or estimate A and cD . You are asked to complete the following design and testing tasks.
Task 1 (Linearization): Select three pairs of equilibriums (ve, ue). Linearize the system dynamics (1) under the three pairs of equilibriums, respectively. 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 diferences.
Task 2 (Controller Design): Now take an equilibrium from any of the three choices in Task 1, and we get a linear model. For this linear model, design a controller that will precisely achieve any desired speed (reference). Select references as r = 10km/h, 50km/h, 100km/h. Demonstrate the efectiveness of your design by numerical experiments on the linear model for all three references.
Task 3 (Validations): The controller designed in Task 2 needs to be tested before real-world validations.
There are two challenges: the controller is designed from the linear model, but the true system dynamics in (1) is nonlinear; there may be disturbances. We suppose the vehicle encounters a sudden transition from flat ground to a very steep uphill slope of 6% grade. The carry out the following analysis and design for Task 3.
(1) Establish the corresponding equation of motion of the vehicle by extending the equation (1) to the case with the slope accounted for. Show why and how the new equation of motion is of the form.
where d is a disturbance.
(2) Substitute your linear controller for reference tracking from Task 2 into the system (2), and ob- tain the closed-loop dynamics. Simulate the closed-loop dynamics for reference speeds at r = 10km/h, 50km/h, 100km/h. Plot your results for the three reference speeds and draw a conclu- sion on the performance of your controller in this validation, and discuss how the feedback gains in the controller afect the system response characteristics such as steady-state error.
Assumed Knowledge and Tools: Lecture 2 covers linearisation, Lecture 3 covers controller design for first-order systems, Prelab in Week 3 covers how to build simulink blocks for a first-oder system, Lab 1 in Week 4 will continue to help you learn Simulink controller design for first-order systems.
Supports: You may ask questions about the project in Week 3 - 5 timetabled Lab sessions; you may also ask questions around the project description in our Ed forum. Week 6 timetabled Lab session is dedicated to helping you in this assignment.
2 Lateral Control (Lane Changing)
For this section we look at lateral (side-to-side) motion of the vehicle, in particular for automatic lane changes.
A schematic of the vehicle with relevant quantities is shown below. See textbook Chapter 3, Example 3.10 and Chapter 6, Example 6.12 “Vehicle steering” for a more detailed analysis. For this question, you should assume v > 0 is constant, and the control input is δf, the steering wheel angle.
The motion of the centre of mass (CoM) position (x, y) is described by the following diferential equations (you might like to verify this, but it is not part of the assignment). Note the coupling to longitudinal dynamics through v(t).
In addition, we have the following algebraic equation between δf and the CoM rotation angle β:
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 diferential equation can be ignored), and it is the lateral position y that we want to control.
Task 1 (Linearization): Linearise the dynamics about constant speed motion v(t) ≈ v0 > 0 with small angles, i.e. φ ≈ 0, β ≈ 0, δf ≈ 0. Show the whole process of linearisation and further show that we get
● a second-order diferential equation describing how y(t) depends on δf(t); and thus
● a transfer function from steering-wheel angle δf to lateral position y that has the form.
Calculate the values of A and B for your car (note that A and B will depend on v0 ).
Task 2 (Controller Design): For the second-order diferential equation describing how y(t) depends on δf(t) established in Task 1, design a controller for δf(t) so that y(t) should be able to change from one position to another, i.e., y(t) should be able to track a reference ry = 2, 3, 4 meters from zero. Explain why this means the controller will steer the vehicle for smooth and accurate transition from lane to lane.
Task 3 (Validations): Simulate and plot the closed-loop system response of the linear model for lane-change manoeuvre at speeds of 10, 50, 100 km/h, respectively. Plot the trajectories of y(t) when its reference ry = 4 meters. Explain the performance of the controller in terms of achieving its goal in smooth and accurate lane change.
Assumed Knowledge: Lecture 4 and Lecture 5 cover how to design controller for second-order linear systems and small-angle linearization. Lab 2 in Week 5 will help you practice building Simulink blocks for second-order systems.
Supports: You may ask questions about the project in Week 3 - 5 timetabled Lab sessions; you may also ask questions around the project description in our Ed forum. Week 6 timetabled Lab session is dedicated to helping you in this assignment.
3 Report Format
You must submit a professional-quality report as a machine-readable pdf (i.e. not scanned images) through Canvas. By professional-quality report, it means 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 double-column IEEE Conference Articles. The template, in Word or Latex, can be found at IEEE Templates. Your report must consist of the following sections and subsections:
1. Introduction
2. Longitudinal Controller
2.1 Linearization
2.2 Controller Design
2.3 Validations
3. Lateral Controller
3.1 Linearization
3.2 Controller Design
3.3 Validations
4. Discussion and Conclusions
The subsections 2.1, 2.2, 2.3, and 3.1, 3.2, 3.3 must fully address the required tasks in above project description.
The full report must be no more than 8 pages including EVERYTHING, e.g., the cover page and ap- pendix. Your marks will depend not only on technical correctness, but also the way you motivate your design choices, and the way you analyse and present the results.
The report must be entirely your own work, except where clearly indicated otherwise. Any references to external material (papers, books, or websites) 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 students website:
https://sydney.edu.au/students/academic-integrity.html.
Further information for on research integrity and ethics for postgraduate research students and students undertaking research-focussed coursework such as Honours and capstone research projects can be also be found on the current students website: https://sydney.edu.au/students/research-integrity-ethics.html.
4 Marking Criterion and Procedure
4.1 Mark Breakdown and Criterion
The mark breakdown is indicated below. The marks should serve as a 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: Longitudinal Controller (45%): Thorough investigations, 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 Controller (45%): Thorough investigations, clear explanation of the working, and com- plete 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 on possible improvements and future directions
4.2 Marking Procedure
You report will be assigned to a random marker from our teaching staf, and the marking will follow strictly the above criterion.
4.3 Feedback
You may receive two types of feedback:
(1) A detailed mark breakdown of your total mark under Canvas rubrics: the score for each of the above items listed above. Therefore, you will be able to see how well you have been doing in all parts of the report.
(2) Additional comments and/or suggestions from the marker.