John S. Baras

ENEE 660 - System Theory

Fall 2008 Information

MW........12:30pm - 1:45pm

Location: EGR 2112

Catalog Description:

Prerequisite: ENEE460 or equivalent; MATH463 or equivalent; or permission of instructor. Also offered as MAPL460. Credit will be granted for only one of the following: ENEE660, ENEE663, or MAPL640. Formerly ENEE 663. General systems models. State variables and state spaces. Differential dynamical systems. Discrete time systems. Linearity and its implications. Controllability and observability. State space structure and representation. Realization theory and algorithmic solutions. Parameterizations of linear systems; canonical forms. Basic results from stability theory. Stabilizability. Fine structure of linear multivariable systems; minimal indices and polynomial matrices. Inverse nyquist array. Geometric methods in design. Interplay between frequency domain and state space design methods. Interactive computer-aided design methods.

Course Goals

This is a basic course on linear system theory at the graduate level. Linear system theory is important as a cornerstone of control theory and is also useful in areas such as signal processing. The main goal of the course is to familiarize the student with the concepts, tools, and techniques commonly used in linear system theory. Thus, students will not only learn the fundamental results and constructions of the theory, but will also learn to synthesize these results on their own and to produce similar results using the tools learned. The course is taught at a high level of mathematical rigor, and students are expected to understand results as well as their derivations.

Course and Topic Prerequisites

Course prerequisites: ENEE 460 or equivalent, MATH463 or equivalent, or permission of instructor.

Topic prerequisites: Vector spaces. Linear systems of algebraic equations and matrix algebra. Elementary differential equations. Routh and Nyquist criteria for stability of linear systems. Laplace transforms and transfer functions of simple scalar systems.

Core and Optional Topics

Core Topics:

  • Vector spaces and linear operators: changes of coordinates, the Fredholm Alternative, theorems from linear algebra such as the Cayley-Hamiton Theorem and Sylvester's Inequality.
  • State equation representation: the concept of state, state equations, existence and uniqueness of solutions, linearization of nonlinear state equations, solution of linear state equations, transition matrices.
  • Stability: definition of Lyapunov and asymptotic stability, conditions for stability of linear time-varying and time-invariant systems.
  • Controllability and observability: definitions and theorems for linear time-varying and time-invariant systems, use of adjoint operators in proving main theorems and for deriving minimum-energy controls.
  • Realization: realizability of input-output maps from the impulse response matrix or the transfer funtions, minimal time-invariant realizations, Markov parameters.
  • Canonical forms: invariant subspaces, the controllable and the unobservable subspace, Kalman canonical form.
  • Feedback: effects of state and output feedback on controllability and observability, eigenvalue assignment by linear state feedback, stabilization.
  • Observers: full-order observers, reduced-order observers, output feedback stabilization.
  • Polynomial fraction descriptions: right and left polynomial fractions, column and row degrees, McMillan degree, minimal realization.

Other Possible Topics:

  • Input-out stability of linear systems.
  • Controller and observer forms for multivariable systems.
  • Controllability and observability indices.
  • Algorithms for eigenvalue assignment for multivariable systems.
  • Poles and transmission zeros of polynomial fraction descriptions.
  • State feedback design using polynomial fractions.


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