Linear Systems Theory

ENSC 801-3

 

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.

Textbook

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.

Topics

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 Webpage

ENSC 801. Please check this page for further information concerning recent offerings of the course.