ECE 688 - Nonlinear Systems
Instructor:
Professor
Christopher
Nielsen
Office:
EIT
4106.
Office
hours:
By
appointment.
Contact:
cnielsen@uwaterloo.ca
;
telephone
extension
32241.
Website:
http://learn.uwaterloo.ca/
Location and time
Tuesdays, 8:30 am – 11:20 am in room EIT 3141
Course Description
Why should one study nonlinear systems? Virtually all physical systems are nonlinear in nature. Sometimes it is possible to describe the operation of a physical system by a linear model. This is the case, for example, if the mode of operation of the physical system does not deviate too much from the “nominal” set of operating conditions. But in analyzing the behaviour of any physical system, one often encounters situations where the linearized model is inadequate or inaccurate. That is the time that the material covered in this course may prove useful.
In this course we cover classical and modern approaches to the analysis of finite-dimensional, deterministic, nonlinear systems modeled by ordinary differential equations with an emphasis on stability, robustness and the effect of interconnecting dynamical systems. The material in this course may appeal to engineers interested in a rigorous treatment of nonlinear systems and can find applications in all branches of engineering.
Recommended background
Multivariable Control Systems (ECE 682) and undergraduate knowledge of signals and systems (ECE 207), calculus (ECE 205) and linear algebra (MATH 215).
Required text
There is no required text for this course. Instructor will write notes on the black board. An excellent optional textbook is
- Nonlinear Systems, 3rd edition, H.K. Khalil.
Additional references
- Nonlinear Systems Analysis, 2nd edition, M. Vidyasagar.
- Nonlinear Control Systems II, A. Isidori.
- L2-Gain and Passivity Techniques in Nonlinear Control, A. van der Schaft.
- Nonlinear Systems: Analysis, stability and control, S. Sastry.
Evaluation
50%
Final
exam
:
open
book.
40%
Assignments
:
Four
assignments
posted
over
the
course
of
the
term.
10%
Course
project:
report.
Tentative Topics List
-
Introduction
to
nonlinear
nonlinear
models
and
phenomena
- Examples.
-
Mathematical
preliminaries
- Functions, Norms, topology of ℝⁿ, continuity on ℝⁿ, differentiability on ℝⁿ.
-
The
vocabulary
of
dynamical
systems
- Phase and integral curves, phase portraits, state transition function, phase flows, vector fields, existence and uniqueness of solutions, equilibria, closed orbits, invariant sets and limit sets.
-
Lyapunov
stability
- Autonomous systems, invariance principle, sign definite functions, domain of attraction, linearization, converse theorems, stability and small perturbations.
-
Input-output
stability
- “ℒ spaces” and their extensions, input-output maps, small gain theorem, linear systems with nonlinear feedback.
-
Input-to-state
stability
- Cascade connected systems, feedback connected systems, small gain theorem for ISS systems.
-
Dissipative
systems
- Definitions, relationship with Lyapunov stability, classes of dissipative systems, control affine systems with quadratic supply rates, linear systems, absolute stability problem.
-
Introduction
to
output
regulation
- Centre-manifold theory, tracking for nonlinear control systems (local theory), single-input single-output control affine systems with relative degree.
Required inclusions
- Academic integrity: In order to maintain a culture of academic integrity, members of the University of Waterloo community are expected to promote honesty, trust, fairness, respect and responsibility.
- Grievance: A student who believes that a decision affecting some aspect of his/her university life has been unfair or unreasonable may have grounds for initiating a grievance. Read Policy 70, Student Petitions and Grievances, Section 4. When in doubt please be certain to contact the department’s administrative assistant who will provide further assistance.
- Discipline: A student is expected to know what constitutes academic integrity to avoid committing an academic offence, and to take responsibility for his/her actions. A student who is unsure whether an action constitutes an offence, or who needs help in learning how to avoid offences (e.g., plagiarism, cheating) or about “rules” for group work/collaboration should seek guidance from the course instructor, academic advisor, or the undergraduate Associate Dean. For information on categories of offences and types of penalties, students should refer to Policy 71, Student Discipline. For typical penalties check Guidelines for the Assessment of Penalties.
- Appeals: A decision made or penalty imposed under Policy 70 (Student Petitions and Grievances) (other than a petition) or Policy 71 (Student Discipline) may be appealed if there is a ground. A student who believes he/she has a ground for an appeal should refer to Policy 72 (Student Appeals).
- Note for students with disabilities: The AccessAbility Services, located in Needles Hall, Room 1132, collaborates with all academic departments to arrange appropriate accommodations for students with disabilities without compromising the academic integrity of the curriculum. If you require academic accommodations to lessen the impact of your disability, please register with the AccessAbility Services at the beginning of each academic term.