AMME3500 Systems Dynamics and Control Design Project 1 New Due: 10:00am, Friday 23th of April (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. T1.1 Select an equilibrium, and linearize the nonlinear dynamics (1) around the equilibrium to get a new linear model. Design a controller for the linear model that will precisely achieve any desired speed (reference). Demonstrate the effectiveness of your design by numerical experiments. T1.2 Apply your controller established in Step 1 to the original nonlinear model (1), and validate if the vehicle can still achieve any desired speed by simulations. Discuss the similarities and differences of the responses between the linear model and nonlinear model under the same controller. T1.3 Examine the closed-loop dynamics of (1) with the designed controller when the reference speed changes from one target speed to another, e.g. 40, 60, 80 km/h. T1.4 Examine the closed-loop dynamics of (1) with the designed controller when there is uncertainty in mass (e.g. due to the number of passengers). T1.5 In the presence of constant disturbance, the true system becomes mv˙ + 1 2 AρcDv 2 = u+ d (2) 1 where d is the disturbance. Test the effects of the disturbance d under different magnitudes of distur- bances, and discuss how the feedback gains in the controller affect the system response characteristics such as steady-state error. T1.6 Suppose the vehicle encounters a sudden transition from flat ground to a very steep uphill slope of 20% grade.1 Establish the corresponding equation of motion of the vehicle by extending the equation (1) to the case with the slope accounted for. Analyse your controller’s response 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. 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) 1Note that the grade of a slope is not the angle of its inclination, but rather the tangent of the angle of inclination times 100. 2 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 Calculate the values of A and B for your car (note that A and B will depend on v0). T2.1 Design a controller for the linear model that will smoothly and accurately transition from lane to lane (meaning: y should be able to change from one position to another). Explain the reasoning for the choice of gains. T2.2 Simulate the closed-loop system response of the linear model for lane-change manoeuvre at a variety of speeds, e.g. those you considered for the cruise control: 40, 60, 80 km/h. T2.3 Demonstrate the system responses for the linear model with v0 > 0 and v0 < 0, respectively, by numerical examples. Based on these numerical examples, discuss the effect and physical meaning of the system zero (zero of transfer function) when the vehicle is reversing (meaning: v0 < 0). You are also encouraged to test your controller for the original nonlinear plant, but this is not required in the report. 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: 1. Introduction 2. Longitudinal Controller 3. Lateral Controller 4. Discussion and Conclusions 3 The full report must be no more than 8 pages, including the cover page and appendix if you decide to include them. 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 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 The mark breakdown is indicated below. The marks should serve as a guideline for how much space to allocate to each section. Introduction (5%) • Clear explanation of the motivation of study • Precise and comprehensive introduction to project scope • Organization of report Longitudinal Controller: (50%) • T1.1 Linearization and controller design (10%) • T1.2 Nonlinear plant validation (10%) • T1.3 Speed changes (5%) • T1.4 Uncertainty in mass (5%) • T1.5 Effects of disturbance (10%) • T1.6 Uphill slope. (10%) Lateral Controller: (30%) • Derive second-order differential equation by linearization and establish the transfer function (5%) • T2.1 Controller design and Choice of gains (10%) • T2.2 Close-loop responses (5%) • T2.3 Effect of zero (10%) 4 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%). 5
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