SYLLABUS

EENG442/E&AS902/AMTH342 – LINEAR SYSTEMS – Fall 2006

**Instructor: **A. S. Morse, 212 Malone
Lab., X24295, mailto:morse@sysc.eng.yale.edu

**Teaching Assistant: **Jia Fang, 5202B Malone
Lab, X26555, mailto:jia.fang@yale.edu

**Brief Summary: **This is an introductory course
about finite-dimensional, continuous and discrete-time linear dynamical
systems. The course is intended for students interested in the systems, information,
and computer sciences including robotics, control theory, signal and image
processing, computer vision. The course is open to all students. Undergraduates
interested in taking the course should first contact the course instructor.

**Organization: **The course meets on
Mondays and Wednesdays from 1:00pm to 2:30pm in Malone 217. There are
approximately eleven weekly homework assignments. There is a mid-term and a
final exam. Review sessions will be scheduled from time to time.

**Course Text: ***Linear
System Theory*, W. J. Rugh, Prentice Hall, 1996

**Class Notes: ***Lecture Notes on Linear Algebra, Linear Differential
Equations, and Linear Systems; *A. S. Morse.

**Contents:**

** Week 1: ** Matrix algebra: fields and
polynomial rings, Gauss
elimination.

** Week 2:**
Linear algebra: subspaces and linear transformations .

** Week 3: **Basic concepts from
analysis - normed spaces, continuity, convergence

** Week 4: **Linear differential
equations: linearization, state-transition matrix, variation of constants
formula, periodic systems.

** Week 5: **Matrix similarity, matrix
exponentials, characteristic polynomial, Cayley Hamilton
Theorem, Jordan
normal form.

** Week 6: **Inner product spaces,** **orthogonal projections, normal matrices,
symmetric and orthogonal matrices.

** Week 7: **Continuous and
discrete-time linear systems; sampled systems; the
concept of a realization.

** Week 8: **Controllability: reachable
states, control reduction, controllable decompositions.

**Week 9:** Observability:
unobservable states, observability reduction; minimal
systems.

**Week 10:** Transfer matrix realizations; via partial
fractions; via control canonical form; minimal realizations; isomorphic
systems.

**Week 11:** Uniform and exponential stability; stability of continuous and discrete
time-invariant systems,

Routh-Hurwitz test , Lyapunov stability , perturbed systems.

**Week 12: **Feedback control:
state-feedback, spectrum assignment, observers, observer-based control systems**
**

** **