LINEAR SYSTEMS I
ECE230A/ME243A — FALL 2008
4 units
Tu Th
1 Course summary
The purpose of this course is to provide the students with the basic tools of modern linear systems theory: stability, controllability, observability, realization theory, state feedback, state estimation, separation theorem, etc. For time-invariant systems both state-space and polynomial methods are studied. The students will also be introduced to the computational tools for linear systems theory available in MATLAB.
The intended audience for this course includes, but is not restricted to, students in circuits, communications, control, signal processing, physics, and mechanical and chemical engineering.
Recommended readings and homework are available on the web.
2
Instrutor
email:
hespanha@ece.ucsb.edu
phone: (805) 893-7042
office: Harold Frank Hall, 5157
Office
hours: Please email
or phone me in advance to schedule for an appointment.
Preferred times are Tue, Thu
3 Prerequisites
3.1.1 ECE 210A Matrix Analysis and Computation
Graduate level-matrix theory with introduction to matrix computations. SVD's, pseudo-inverses, variational characterization of eigenvalues, perturbation theory, direct and iterative methods for matrix computations.
4 Course's Web Page
The syllabus, homework, solutions to homework, and all other information relevant to the course will be continuously posted at the course's web page. The URL is
http://www.ece.ucsb.edu/~hespanha/ece230a/
5 Assessment format
Homework – 30%
Mid-term
exam – 30% (tentatively
on Nov 4, 2008; in class)
Final
exam – 40% (Dec 11,
2008; Phelps 1437 from 12noon-3pm)
6 Textbook
The
course will follow a set of lecture notes that will be made available on web at
http://www.ece.ucsb.edu/~hespanha/published/linearsystems.pdf
Other recommended textbooks are:
[1] P. Antsaklis, A. Michel. Linear Systems. McGraw Hill, 1997.
[2] C.-T. Chen. Linear Systems
Theory and Design. Oxford Univ. Press, 3rd ed., 1999. (ISBN
0-19-511777-8)
[3] W. Rugh. Linear System Theory, 1996.
All
students are strongly encouraged to review linear algebra. Chapter 3 of [2]
provides a brief summary but a review of a Linear Algebra textbook (such as [4]
below) is preferable, especially if one goes through a few exercises.
[4] Gilbert Strang
Linear Algebra and Its Applications, 1988.
7 Study Guide
The following is a tentative schedule for the course.
If revisions are needed they will be posted on the course's web page. Students are strongly encouraged to
read the corresponding chapter of the textbook prior to each class.
|
Class |
Content |
|
|
Lect #1 9/25 |
Introduction and
course overview Systems
Representation System
representation: input-output, block diagrams Continuous vs.
discrete-time Examples |
|
|
Lect #2 9/30 |
Where do state-space
linear systems come from? ·
Local linearization around equilibrium ·
Local linearization around trajectory ·
Feedback linearization |
|
|
Lect #3 10/2 |
Basic system
properties: causality, linearity, time-invariance Forced responses ·
Impulse response ·
Transfer function |
|
|
Lect #4 10/7 |
Impulse response and
transfer function for state-space systems Elementary
realization theory for LTI systems Equivalent state-space representations |
|
|
Let #5 10/9 |
Solution for
state-space linear time-varying (LTV) systems ·
Solution to homogeneous linear systems—Peano-Baker
series ·
State-transition matrix ·
Properties of the state transition matrix ·
Solution to nonhomogeneous linear
systems—variations of constants formula The discrete-time
case |
|
|
Lect #6 10/14 |
Solution for
state-space linear time-invariant (LTI) systems ·
Matrix exponential (definition and properties) ·
Computation of matrix-exponentials using the Laplace
transform ·
The importance of the determinant of A |
|
|
Lect #7 10/16 |
Solution to
state-space linear time-invariant (LTI) systems (cont.) ·
Jordan normal form ·
Computation of matrix-exponentials using the Jordan normal form ·
Poles with multiplicity larger than one (block diagram
interpretations) ·
The discrete-time case |
|
|
Lect #8 10/21 |
Stability Internal stability of continuous-time LTI systems ·
Definitions ·
Eigenvalues condition (block diagram interpretation
of multiplicity) ·
Lyapunov Theorem (LMI) Stability of nonlinear systems from local linearization |
|
|
Lect #9 10/23 |
Input-output stability of LTI systems ·
Definition ·
Time-domain condition ·
Frequency-domain condition |
|
|
Lect #10 10/28 |
Preview of optimal control ·
Linear quadratic regulator problem ·
Algebraic Riccati equation ·
Optimal state-feedback control ·
Stability Preparation for
midterm |
|
|
10/30 class before Halloween! (possibly no class – trip to Yale Workshop) |
No class |
|
|
Lect #11 11/4 Elections! |
Controllability and State Feedback Reachability and controllability subspaces for LTI
systems ·
Definition, Physical example, and block diagrams ·
Reachability and controllability Gramians ·
Controllability matrix ·
Open-loop minimum energy control The discrete-time case |
|
|
Mid-term Exam 11/6 |
Midterm exam on the material
covered up to (and including) lecture #10 (taught on 10/28). The midterm exam is closed book, but during the
exam you are allowed to consult one letter-size piece of paper with
handwritten notes. The exam will take place on PHELPS 1417 from 12noon-2pm. |
Practice midterm 1 & solutions |
|
11/11 |
No class – Veteran’s day |
|
|
Lect #12 11/13 |
Controllable systems ·
Definition ·
Controllability matrix test ·
Popov-Belevitch-Hautus (PBH) test ·
Eigenvector/eigenvalue test ·
Lyapunov test (LME) ·
Feedback stabilization based on the Lyapunov
test |
|
|
Lect #13 11/18 |
Canonical decompositions ·
Invariance with respect to equivalence transformations ·
Controllable canonical form for single-input systems Controllable decomposition |
|
|
Lect #14 11/20 |
Stabilizability |