ACS6101 –Control Systems Design Assignment 1 ACS6101: Foundations of Control Systems Penalties for Late Submission Late submissions will incur the usual penalties of a 5% reduction in the mark for every working day (or part thereof) that the assignment is late and a mark of zero for submission more than 5 working days late. For more information see http://www.shef.ac.uk/ssid/exams/policies. Unfair Means The assignment should be completed individually. You should not discuss the assignment with other students and should not work together in completing the assignment. The assignment must be wholly your own work. References must be provided to any other work that is used as part of this assignment. Any suspicions of the use of unfair means will be investigated and may lead to penalties. See http://www.shef.ac.uk/ssid/exams/plagiarism for more information. Special Circumstances If you have medical or personal circumstances which cause you to be unable to submit this assignment on time or that may have affected your performance, please complete and submit an appropriate circumstances form along with any documentary evidence of the circumstances. Please see: http://www.sheffield.ac.uk/ssid/forms/circs, for guidance as to which form is appropriate for your circumstances, and what, if any, supporting documentation is required. Help This assignment briefing and the lecture notes provide all the information that is required to complete this assignment. It is not expected that you should need to ask further questions. Remember that you need to decide on what the most appropriate approach to solve the assignment is and also how to present your results. This is part of what you are being assessed on and will assess your knowledge and understanding of the lecture notes. Notes Remember that in an assignment, there is often no standard “staff answer”. You have the freedom to investigate any aspect of the problem that you think is relevant. Credit will be given for those students who display creative and investigative skills in their submitted work. • If you cannot meet the specification of the problem exactly with your solution, submit that part you have succeeded with rather than nothing at all. Credit will be given to those students who have carried out a second iteration of the design process and have succeeded to improve the performance compared with the initial design. • You are reminded that this assignment must be carried out independently and the written submission must be the student’s own work. Your assignments will be submitted through Turn-it in and checked for potential plagiarism – any unfair means identified will be investigated, and penalties may be applied. Assignment Briefing • Your assignment should be written as a worded document, not just a series of mathematical steps. • Within your answer to each question, you should provide an introduction to the solution, provide the solution as the ‘body’ of your answer, and conclude your answer, with respect to the performance specification provided. Each answer should be self-contained, and not refer to a previous or subsequent question. ACS6101 –Control Systems Design Assignment 2 • Make sure that you explain clearly all the steps in your calculations, write down the equations used, explain the notation, and the choices you made during the design process. • The report should be word processed. Use Arial 11 point font throughout. Page margins should be set to ‘normal’ (top-bottom-left-right margins all set at 2.54cm). • Line spacing must be set to “1”. • Figures should be incorporated into the document in appropriate places (not as appendices) and suitably sized for the information contained. • You must include your registration number at the top of every page. Marking Criteria This assignment will be marked in two elements: 1. Technical Content: The technical aspects of this assignment make-up 70% of the final mark for this assignment. The marks awarded for each section of each question are provided in the assignment brief above, and will be used in assessing technical aspects of this assignment (This element will be marked by ACSE staff) 2. General Writing, Presentation & Formatting: Quality of the written English used in this assignment make up 30% of the final mark. (This element will be marked by ELTC staff) 6 ACS6101 –Control Systems Design Assignment Period: Weeks 1 Question 1: Write a script to do the following: a) create the following matrices and vector y: -2 -4.1 2 3 1 3 9 2 -1 -2 1.1 2 3.3 5 8 4 3 y = 2 4.1 2 5 8.1 10 16 8 6 -2 1 3 2.2 1.5 2 3 1 8 4 b) in well presented way and with some attempt to guard against poor user input, ask the user to choose one of the matrices and assign it to a variable A. c) if the determinant of A is nonzero, find the eigenvalues and eigenvectors of A and the solution x to the equation Ax=y, or display an appropriate message otherwise d) clear the workspace and command window at the start of each run and output all results to the command window. Question 2: Differential equations to model the HIRES reaction problem1 are: 1 = −1.711 + 0.432 + 8.323 + 0.007 2 = 1.711 − 8.752 3 = −10.033 + 0.434 + 0.0355 4 = 8.322 + 1.713 − 1.124 5 = −1.7455+ 0.436 + 0.437 6 = −28068 + 0.694 + 1.715 − 0.436 + 0.697 7 = 28068 − 1.817 8 = −28068 + 1.817 a) Create a function called HIRES. − The function should have two input args ordered like this: t, x. − The function should return one output arg (a 8x1 sized vector called xdot) computed within the body of the function using the equations above. (Note: The arg t is unused within the function but is necessary to be in the arg list if HIRES is passed to one of Matlab’s ode solvers). 6 ACS6101 –Control Systems Design Assignment b) Write a script that … − calls the Matlab function ode45, using your HIRES function as an input, to obtain a solution over a time interval 0 to 300 seconds, with initial conditions 1= 1, 2 = 0, 3 = 0, 4 =0, 5 = 0, 6 = 0, 7 = 0, 8 = 0.0057. Make your call of ode45 for the function HIRES in anonymous function style rather than as a string. (See matlab help for guidance). − Use the options arg with your solver to set 'RelTol' to 1e-3 and 'AbsTol' to 1e-6. − Use tic and toc to find the execution time for the ode45 call and save to a suitably named variable. c) Add to your script to repeat the task again, only this time use ode15s. d) Add to your script to … − Plot the eight returned components of xdot from your ode45 call in one set of axes in a subplot, and the eight returned components of xdot from your ode15s call in another subplot, arranging the subplots one above the other in the same figure. − Use a logarithmic x-axis. − Label the axes as Concentration for the y axis (assume units are mol/dm3), Time for the x axis (assume units are seconds), and add a legend placed northeast within each plot to identify the eight reagents. − Set the figure to occupy the left half of full screen when the script is run. − Create a two-line title for each plot, giving the title of the plot in line 1 and the execution speed, and AbsTol, RelTol values on the second line. Make the second line of the title blue colour and fontsize 9. e) Add to your script to produce a second figure and − In the second figure make eight subplots (eight rows, one column) and in each subplot overlay a plot of the first 100 points of data obtained from ode45 and ode15s. − Plot ode15s as a green line and ode45 as black dots. − Give each subplot a title, axes labels and a legend. − Set the figure to occupy the right half of full screen when the script is run. 6 ACS6101 –Control Systems Design Assignment Period: Weeks 2 Question 1: A. What is a mathematical model? List and discuss the advantages and disadvantages of systems modelling. Discuss modelling errors and describe what good modelling practice is. B. Describe the modelling process when using theoretical equations, empirical equations, discuss the need for each methodology, and give a simple example for each modelling technique. C. Discuss the merits of the state space method for the modelling of dynamic systems. Describe a situation where it would be better to use a classical approach to model a system instead of using the state space method. D. What is a signal flow graph, and how it compares against the block diagram representation? Show an example of a signal flow graph for a simple process of your choice. 6 ACS6101 –Control Systems Design Assignment Question 2: A. Derive a linear approximation mathematical model of the pendulum system shown in Figure 1. to describe the relationship between the torque developed on the mass M and the angle theta between the rod and the vertical plane. State clearly your assumptions and your modelling methodology. Figure 1 B. For the mechanical system shown in Figure 2 a. Derive the mathematical model of the system to describe the displacement associated with the two masses M1 and M2. b. Derive the transfer function model of the system, and show the final TF model in matrix format. c. Derive the state space representation of the system, and show the final state space model in matrix format. Assume that a force W(t) is applied on the mass M2 pointing downwards as shown in the diagram. Clearly state other assumptions you may make, initial conditions, linearity etc. ACS6101 –Control Systems Design Assignment 7 Figure 2 C. Answer directly the tasks in the ‘Depth Control of a Torpedo’ simulation problem, in the course’s lab handout (available on MOLE) ‘Practical Simulation Experiments with SIMULINK’ (ACS6101 Week 2 – Systems Modelling and simulation). W(t) ACS6101 –Control Systems Design Assignment 8 Period: Weeks 4 Question 1: Consider the unity feedback control system where the plant transfer function is given by: 5000 () = ( + 0.33)( + 16)( + 120) a. Write the frequency response function of the system. Plot the Bode diagram indicating the stability margins (phase and gain margin). Use the Bode diagram to investigate the stability of the closed-loop system with unity feedback. b. Apply the analytic approach covered in the module to design a PID compensator, in cascade with the plant, to meet the following performance specifications. ▪ Percentage overshoot less than 10% ▪ Settling time less than 2 seconds Explain clearly each step in your design, including the equations used to calculate different quantities as appropriate. c. Use MATLAB to evaluate the performance of your final design in the time and frequency domain. Present these in tabular form – see Table 1, and provide a written conclusion for your design: Quantity Value Steady state error to a unit ramp Rise Time Settling Time Percentage Overshoot Phase Margin Gain Margin Bandwidth Peak Magnitude Resonant frequency Table 1 ACS6101 –Control Systems Design Assignment 9 Question 2: Consider the unity feedback control system where the plant transfer function is given by: 500 () = ( + 0.33)( + 16)( + 120) a. Plot the root locus of the system. Determine analytically the range of K for which the system is stable. Explain clearly the assumption and the calculations and the calculations to get credit for your solution. b. Design a phase lead-lag compensator using the root locus approach to achieve • A velocity error constant Kv=25 • An overshoot of less than 10% • A settling time of less than 2 seconds Explain clearly each step in your design, including the equations used to calculate different quantities as appropriate. c. Use MATLAB to evaluate the performance of your final design in the time and frequency domain. Present these in tabular form – see Table 1, and provide a written conclusion for your design. End of Assignment Questions
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