AMME3500 Systems Dynamics and Control Design Project 01 New Due: 11:59pm, Sunday 12th of April (end of Week 7) Weight: 20% of your total mark. 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. We suggest you design and test your controllers using simple linearised models, but then also simulate on the true nonlinear coupled dynamics to verify performance. 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: mv˙ + 1 2 AρcDv 2 = u (1) Here 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 (Wikipedia has an interesting list). For your car, look up, measure, or estimate A and cD. 1. Design a controller that will precisely achieve any desired speed even if there are constant dis- turbances so that the true system is mv˙ + 1 2 AρcDv 2 = u+ d (2) where d is the disturbance. 2. Examine the dynamics of changing from one target speed to another, e.g. 40, 60, 80 km/h. 3. Analyse your controller’s response to a disturbance force corresponding to a sudden transition from flat ground to a very steep uphill slope of 20% grade. Note that the grade of a slope is not the angle of its inclination, but rather the tangent of the angle of inclination times 100. 4. Examine the effect of uncertainty in mass (e.g. due to the number of passengers). To begin the work of this part, you should (1) be familiar with Sec 4.1 of textbook and the lecture material (Lecture 2) on linearisation (2) know how to build Simulink blocks for dynamical systems. You may want to investigate the “Signal Generator” or “Repeating Sequence Stair” blocks in Simulink for some of the reference inputs. 1 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 differential equations (you might like to verify this, but it is not part of the assignment). Note the coupling to longitudinal dynamics through v(t). x˙ = v cos(ψ + β) y˙ = v sin(ψ + β) ψ˙ = v lr sin(β) In addition, we have the following algebraic equation between δf and the CoM rotation angle β: tan(β) = lr lf + lr tan(δf ). 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), and it is the lateral position y that we want to control. If we linearise the dynamics about constant speed motion v(t) ≈ v0 > 0 with small angles, i.e. φ ≈ 0, β ≈ 0, δf ≈ 0, show that we get • a second-order differential 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 G(s) = As+B s2 2 Calculate the values of A and B for your car (note that A and B will depend on v0). 1. Design a controller that will smoothly and accurately transition from lane to lane (meaning: y should change from one position to another). 2. Simulate the closed-loop system response for lane-change manoeuvres at a variety of speeds, e.g. those you considered for the cruise control: 40, 60, 80 km/h. 3. Apply an additional 80 degree turn to your control input δf , for both the original (nonlinear) system dynamics and the linearised dynamics. Examine whether there would be significant difference, and discuss the limits of its applicability in real-world circumstances. 4. Discuss the effect and physical meaning of the system zero (zero of transfer function) when the vehicle is reversing (meaning: v0 < 0). 3 Report Format You must submit a professional-quality report as a machine-readable pdf (i.e. not scanned images) through Canvas. You can use whatever format for your report, but your report must be at most 10 pages and must consist of the following sections: 1. Introduction 2. System modelling and Simulation Setup 3. Longitudinal Controller 4. Lateral Controller 5. Discussion and Conclusions We recommend you use the template for IEEE Transactions Articles, which is a widely used standard in engineering research articles. The template, in Word or Latex, can be found at IEEE Templates. 3.1 Further remarks Unlike your problem sets, 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 guidelines introduced in lecture 1. 3.2 Marking Schedule The mark breakdown is indicated below. The marks should serve as a guideline for how much space to allocate to each section. Introduction (5%) 3 • Clear explanation of the motivation of study • Precise and comprehensive introduction to project scope • Organization of report System modelling and Simulation Setup: (25%) • Model and linearisation of cruise control (8%) • Model, linearisation, and transfer function of lane changing (12%) • Simulink block for numerical simulations (5%) Longitudinal Controller: (25%) • Control aims and specs (5%) • Controller structure and Choice of gains (5%) • Numerical validation on linearised and nonlinear models (15%: tracking 3%, speed changing 3%, uphill slope 5%, uncertainty 4%) Lateral Controller: (30%) • Control aims and specs (5%) • Controller structure and Choice of gains (5%) • Numerical validation on linearised and nonlinear models (15%: lane to lane transition 5%, closed- loop system response 5%, 80 degree turn 5%) • Discussion on effect of system zero (5%) Conclusions (5%) • Summary of the project and results • Highlight the most significant discoveries/understandings • Discussion on possible improvements and future directions Presentation and clarity (10%) • Pointed and critical analysis of model (3%), fluent and logical arguments in the controller design (3%), thoroughness of simulation discussions (4%). 4
