ENEE 663: Systems Theory
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
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.
- T. Kailath, Linear Systems, Prentice-Hall (1980).
- W.J. Rugh, Linear System Theory, Prentice-Hall (1993).
- F.M. Callier and C.A. Desoer, Linear System Theory, Springer (1985).
- P. Lancaster and M. Tismenetsky, The Theory of Matrices,
2nd Ed., Academic (1985).
- 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
- Stability: definition of Lyapunov and asymptotic stability,
conditions for stability of linear time-varying and time-invariant
- 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
- 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
- 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.
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.