# ENEE 663: Systems Theory

## 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.

None

## Topic Prerequisite:

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.

## References:

1. T. Kailath, Linear Systems, Prentice-Hall (1980).
2. W.J. Rugh, Linear System Theory, Prentice-Hall (1993).
3. F.M. Callier and C.A. Desoer, Linear System Theory, Springer (1985).
4. P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd Ed., Academic (1985).

## Core Topics:

1. 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.
2. 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.
3. Stability: definition of Lyapunov and asymptotic stability, conditions for stability of linear time-varying and time-invariant systems.
4. 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.
5. Realization: realizability of input-output maps from the impulse response matrix or the transfer funtions, minimal time-invariant realizations, Markov parameters.
6. Canonical forms: invariant subspaces, the controllable and the unobservable subspace, Kalman canonical form.
7. Feedback: effects of state and output feedback on controllability and observability, eigenvalue assignment by linear state feedback, stabilization.
8. Observers: full-order observers, reduced-order observers, output feedback stabilization.
9. Polynomial fraction descriptions: right and left polynomial fractions, column and row degrees, McMillan degree, minimal realization.

## Optional 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.