ENSC 801 (03-3)
ENSC 801 (03-3)
Course Description
Introduction to the analysis of finite and infinite dimensional linear systems. Linear vector spaces, linear operators, normed linear spaces and inner product spaces. Fundamentals of matrix algebra, induced norm and metric measures, functions of a matrix, orthogonal decomposition and projection, duality, and pseudoinverses. Analytical representation of linear systems and state-space formulation. Examples of these concepts will be given from the solution of linear equations and the determination of system response, controllability and observability, canonical forms and realizations, and linear and nonlinear optimization.
Course Objectives
The objective of this course is provide an introduction to algebraic and functional analytic concepts in systems modeling and optimization. It emphasizes geometric insight into the structure of deterministic, continuous and discrete, linear system models, and optimization of linear and nonlinear models. The course is intended for graduate students in Engineering Science.
Course Instructor
William A. Gruver
ASB 9809
Tel: 604-291-4339
Fax: 604-291-4951
Internet: gruver@cs.sfu.ca
www.ensc.sfu.ca/research/idea/personnel/gruver.html
Office hours by appointment (via e-mail)
Zhenwang Yao
Office Hours: Tue, Thu 4:30-5:20 PM
Room ASB 8805
zyao@sfu.ca
Lecture Time and Place
Monday 5:30-6:50 PMTextbook
C. Nelson Dorny, A Vector Space Approach to Models and Optimization, JohnWiley & Sons, reprinted 1983 and 1986. (The text, which is unavailable from the publisher, may be purchased as Custom Courseware at the SFU Book Store)
An on-line version being developed by the author can be accessed at http://www.seas.upenn.edu/~dorny/VectorSp/vector_space.html
Grading Policy
Midterm exam - 40%
Final exam - 50%
Homework assignments - 10%
Homework and laboratory assignments will be available for download from the course web page at least one week prior to the due date. Completed assignments should be submitted by 5:30 PM on the due date. If it will be necessary to submit your assignment late, send an e-mail to Dr. Gruver prior to the deadline. Include the reason for your delay and the date when your assignment will be submitted. Late submissions will not accepted after the assignments have been returned to the class.
Failure to take the mid term exam or the final exam during the assigned class period will result in a mark of zero being recorded unless you have contacted Dr. Gruver before the exam. Make-up exams will be given only under exceptional circumstances.
Course web page:
E-mail: Check your e-mail daily for messages concerning class meetings, exams, homework, lab assignments.
Lectures: You are responsible for all business conducted during the scheduled class period, including announcements that may be given.
Homework: The homework assignments are intended to reinforce the major concepts and to serve as a measure of how well you understand the course. The homework, however, is not necessarily representative of questions that will be asked on the exams.
Exams: It is very important that you understand the concepts covered in this course. I will try to design the exams so that comprehension of the material is emphasized, not merely memorization. You may bring to the exams one 8.5"x11" page with handwritten notes on either side. You must show all work performed on an exam to receive full credit. You may use a pencil instead of pen to write the exam. If I cannot follow or read your work, you may receive no credit.
Course Outline
Vector space and linear mappings: basic definitions and coordinate representation. Examples from differential operator inversion and linear boundary value problems.
Decomposition: invariant subspace, minimal polynomials, generalized eigenvectors, functions of matrices. Examples from modal transformations, controllability and observability, computation of transition matrices and impulse response.
Hilbert space: norm, inner product, convergence, orthogonalization. Examples from Fourier expansion, least square minimization, linear estimation and minimum energy control.
Dual space: bounded linear functional, adjoint transformations, orthogonal decomposition and projection, pseduo inverses. Examples from least squareapproximation and minimum norm control.
Differentials: linearization in Hilbert space, Fréchet and Gateaux derivatives, gradient representation. Examples from nonlinear programming, calculus of variations, and optimal control.
Successive approximation: contraction mapping, gradient descent. Examples from nonlinear equations, unconstrained optimization, and nonlinear programming.
Course Syllabus
| Date | Topic |
Assignment |
Homework |
| Sep 8,10 | Linear vector spaces and linear transformations | Chap. 1.1-1.5, 2.1-2.5 | |
| Sep 15,17 | Differential operators, Green's function | Chap. 3.1-3.4 | HW 1 |
| Sep 22,24 | Generalized eigenvectors, Jordon form | Chap. 4.1-4.3 | |
| Sep 29,Oct 1 | Functions of matrices, impulse response | Chap. 4.4-4.5 | HW 2 |
| Oct. 6,8 | State representation, canonical forms | Chap. 4.6 | HW 3 |
| Oct. 15 | Controllability and observability | Chap. 5.1-5.2 | |
| Oct. 20,22 | Tutorial, midterm exam | Chap. 5.3-5.4 | HW 4 |
| Oct. 27,29 | Hilbert space, orthogonal projection | Chap. 6.1-6.3 | |
| Nov. 3,5 | Adjoint transformations | Chap. 6.4-6.6 | HW 5 |
| Nov. 10,12 | Least square/minimum-norm pseudo-inverses | Chap. 5.1-6.6 | |
| Nov. 17,19 | Frechet and Gateaux differentials, optimality |
Chap. 7.1-7.2 |
HW 6 |
| Nov. 24,26 | Contraction mapping, gradient descent | Chap. 8.1-8.2 | HW 7 |
| Dec 10 | Final exam (AQ 3153) 6:00-9:00PM |
Revised: December 9, 2003, 2003