This is a core graduate course, meaning that it is given every year, it is broad and fundamental, and it is intended for a wider audience than just students in the Systems Control Group. For example, students studying robotics, computer animation, energy systems, electromagnetics, photonic systems, chemical processes, and aerospace systems are encouraged to check out the course. After all, nonlinear systems crop up in many subjects.
The subject of the course is dynamical systems modeled by nonlinear differential equations. You will learn the mathematical methods for analyzing the behaviour of these systems; the latter part of the course is an introduction to control design for nonlinear systems. Most students with a standard engineering background will find the course challenging because they are not used to formal mathematical treatments; the concepts are defined rigorously and the results are stated as theorems and proved rigorously. The other feature of the course is that there is a lab.
Nonlinear systems is a fascinating subject because new behaviour can appear that is not possible for linear systems. For example, limit cycles or multiple isolated equilibria.
The course was designed by, and the notes were written by, Professor Manfredi Maggiore. The notes (about 120 pages) are self-contained. You can buy them in room SFB540.
Outline of the proof of the Picard-Lindelof theorem
A good undergraduate course on linear systems and a liking of mathematics. A course on state-space theory would be advantageous but is not strictly necessary. Our state-space theory course is ECE557 Systems Control. It's fine, in fact a good thing, to take ECE557 at the same time as you're taking ECE1647.
Here are course notes you may need to get up to speed:
The most popular textbook for this subject is Nonlinear Systems, 3rd ed., by H. Khalil. Other very nice books are Nonlinear Systems Analysis, 2nd ed., by Vidyasagar, and Differential Equations, Dynamical Systems, and Linear Algebra, by Hirsch and Smale. A more elementary, easy to read text for Chapter 2 of the course is Differential Equations with Applications and Historical Notes, by Simmons.
Approximately every week, homework will be assigned for the next week. You must work on the homework and hand it in. You may work in groups, but what you hand in must be your own write-up. I will read your solutions, but they won't count toward your course grade. I'll present some solutions in class, or I may ask a student to give her/his solution. Repeat: To receive a course grade, each student taking the course for credit must hand in homework; I will read it to see how you're doing, but it's not counted towards your grade. | problems | | | Problems 1 and 2 due 9 am Friday Sept | solutions | | Problems 3,4,5,14,17,18,19,26 due 9 am Friday Sept 30 | solutions | | Problems 6,7,9,12 due 9 am Friday October 7 | solutions | | Problems 8,10,11,13,23 due 9 am Friday October 14 | solutions | | Problems 24,29,30,31,4.1 from course notes, due 9 am Friday October 28 | solutions | | Problems 25,27,32,33 due 9 am Friday November 4 | solutions | | Problems 34 - 38 due 9 am Friday November 11 | solutions | | Problem 3.8 in the course notes (feel free to simplify); due 9 am Friday November 18 | solutions | | Problems 39 - 42 due 9 am Friday November 25 | solutions | | Practice Problems | | | | |
October 20, 9 am - 11 am. The test is "limited open book." You may bring the following items only: The typed course notes (dated 2009); notes you yourself took in class. You will be provided with a booklet in which to write your solutions. Some students don't like them. If you are one of those students, you may bring your own paper.
The test covers only Chapter 2. There are four problems: 1. There are three small parts, the first requiring a formal proof in logic notation. 2. Linear and nonlinear functions; derivatives. 3. Motion in the plane; reference Problem 19. 4. The Picard-Lindelof theorem.
test marks results: out of 36 (9 marks per problem), average 25, max 32, min 11
Dec 9, 9 am - noon, BA4164. The exam is "limited open book." You may bring the following items only: The typed course notes (dated 2009); notes you yourself took in class; my posted homework solutions. You will be provided with a booklet in which to write your solutions.
The exam covers the following three subjects:
1. General things about nonlinear differential equations, such as vector fields, flows, orbits, etc. 2. Stability of equilibria via Lyapunov's theorems and LaSalle's theorem.
3. The theory of periodic orbits, including Poincare-Bendixson theory and the use of Nagumo's theorem and indices. The Picard-Lindelof theorem and Brockett's theorem are not on. Exam solution
You will be given a letter grade, A+, A, A-, etc. based on your performance in the lab (20%) and on the midterm test (30%) and the exam (50%). B- is the lowest passing grade. You will receive numerical, percent grades for the test, exam, and lab. These will be totalled, giving a percent grade for the course. That percent grade will be converted subjectively (not by a formula) to a letter grade for the course. (Course ECE557, for example, is different. It has both undergraduates and graduates. The undergraduates receive a percent grade, the graduates a letter grade. Normally this is done by assigning a percent grade to everyone and converting via a formula to a letter grade for graduates.)
The lab is to design a nonlinear controller to swing up and balance the pendulum shown below.
We will all go to the lab at 10 am, December 1 (the second hour of a Thursday morning class). At that time Professor Maggiore will introduce you to the experiment and the software. Here are the details:
1. Each student must have done the lab preparation before the Dec 1 session; I will check that each student has done so. 2. You will work in groups of 3 or 4. Email me your group names and the time slot you want. 3. No lab report is required. Instead, you must demonstrate to me that your experiment works. 4. You will receive full marks (20% of the course grade) if your experiment works.
Groups and schedule:
Group 1: Alireza, Fahimeh, Morteza, Firouz; Dec 14, 10 am - 1 pm Group 2: Sara, Shahir, Mohammad; Dec 15, 3 pm - 6 pm Group 3: Yao-Hong, Vickie, Yatao; Dec 9, 1 pm - 4 pm Group 4: Edoardo, Anjani, Jimmy; Dec 16, 10 am - 1 pm Group 5: Helen, Mark, Sana, Chris; Dec 12, 1 pm - 4 pm
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