Teaching > ECE1647
ECE1647 - Introduction to Nonlinear Control Systems
Instructor
M. Maggiore
Office: GB344
Email address: maggiore (at)
ece.utoronto.ca
Lectures
| Tue 12PM-1:30PM | BA4164 |
| Thu 12PM-1:30PM | BA4164 |
Lectures begin on Thursday, January 10
Course Description
This course is a mathematical introduction to nonlinear control theory, a subject with roots in dynamical systems theory, mechanics, and differential geometry. The focus of this course is on the dynamical systems perspective. The material covered in this course finds application in fields as diverse as orbital mechanics and aerospace engineering, circuit theory, power systems, robotics, and mathematical biology, to name a few. The course is organized in four chapters:
- Vector Fields and Dynamical Systems
- Foundations of Dynamical Systems Theory
- Foundations of Stability Theory
- Introduction to Nonlinear Stabilization
Prerequisite: There is no formal prerequisite for this course,
but the student will benefit from having taken a course on state space
linear control theory such as ECE410 or ECE557 in this
institution.
Learning Objectives
This course presents the foundations of nonlinear control theory from a dynamical systems perspective. The course is addressed to graduate students who are mathematically inclined, but are not necessarily experts in this area. The objective of this course is to provide a language and mathematical tools to begin research in control theory. The course develops four main threads.
- Vector Fields and Dynamical Systems: We will introduce ordinary differential equations (ODEs) and their geometric counterparts, vector fields. After presenting the basic theory of existence and uniqueness of solutions of ODEs, we will show that ODEs generate a so-called local phase flow. This will lead us to defining dynamical systems in some generality. The main outcome of this chapter is the mathematical equivalence of three objects: vector fields, ODEs, and finite-dimensional dynamical systems.
- Foundations of Dynamical Systems Theory: We will introduce the foundational results of dynamical systems theory. We will begin with the notion of invariant set. We will present the Nagumo theorem, giving necessary and sufficient conditions for invariance. We will then discuss limit sets and the Birkhoff theorem, characterizing the asymptotic behaviour of dynamical systems in terms of limit sets. We will present the Poincare'-Bendixson theorem characterizing limit sets of second-order systems. Next, we will talk about linearization of nonlinear ODEs at an equilibrium. In this context, the stable manifold theorem characterizes which solutions exponentially converge to or diverge from the equilibrium. Finally, we will discuss the linearization about closed orbits (the so-called variational equation), presenting necessary and sufficient conditions for exponential stability in terms of the characteristic multipliers of the variational equation.
- Foundations of Stability Theory: This chapter presents the theory of stability of equilibria pioneered by Lyapunov. We will prove the main stability theorems and the Krasovskii-LaSalle invariance principle. We will talk about domains of attraction of asymptotically stable equilibria and a way to estimate them using Lyapunov functions. We will discuss the special case of LTI systems, which will allow us to link the notion of exponential stability of an equilibrium with the stability of the linearization. We will discuss the converse Lyapunov theorems of Massera and Kurzweil.
- Introduction to Nonlinear Stabilization: We will discuss the stabilization of equilibria of nonlinear control-affine systems. We will introduce passive systems and the Byrnes-Isidori-Willems necessary and sufficient conditions for passivity-based stabilization of equilibria. We will discuss the special case of Hamiltonian control systems, gaining insightful mechanical interpretations of various notions. Time permitting, we will present control-Lyapunov functions and the Artstein-Sontag theorem. Finally, we will discuss topological obstructions to equilibrium stabilization and the Brockett necessary conditions.
Composition of Final Mark
| Homework Assignments | 30% |
| Midterm Exam | 30% |
| Final Exam | 40% |
Textbook
M. Maggiore, Foundations of Nonlinear Control Theory - Available for download to registered students on the institutional website.
Additional Reference Texts
- Hassan Khalil, Nonlinear Systems, Third Edition, Prentice Hall, 2002.
- M.W. Hirsch, S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974.
- V.I. Arnold, Ordinary Differential Equations, MIT Press, 1973.
- P. Hartman, Ordinary Differential Equations, Second edition, SIAM, 2002.
- J.K. Hale, Ordinary Differential Equations, Robert E. Krieger Publishing Company, 1980.
Midterm Exam
| Day and Time | Location |
|---|---|
| Tuesday, March 19, 12-2PM | BA4164 |
Homework Assignments
There are four or five homework assignments posted on the institutional website and taken from the textbook. You must work independently on the assignment without discussing it with other students.