LINEAR SYSTEMS
THEORY – EE585
Thu 6:30-9:20pm @ OHE 100
Announcements
Posted on 12/18/01
You can download the solutions to the final exam from here.
Joao
Abstract
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. We develop the continuous and discrete-time cases in parallel. For time-invariant systems both state-space and polynomial methods are studied.
The intended audience for this course includes, but is not restricted to, students in circuits, communications, control, and signal processing.
Recommended readings and homeworks are available on the web.
Office Hours
email: hespanha@usc.edu
campus address: EEB 318
phone: (213) 740-9137
Office
hours: Please email me or phone in
advance to schedule for an appointment.
Preferred times are Tue 2:30-4:00pm, Wed 4:30-6pm
Grader: Selcuk Ateskan
email: ateskan@usc.edu
campus address: EEB 323
phone: (213) 740-6259
Office hours: Fri 2-3pm
Prerequisites
441 Applied Linear Algebra for Engineering
Introduction to linear algebra and matrix theory. Provides a basic understanding of the mathematical concepts that underlie matrix theory and their applications to engineering problems. The material covered includes linear transformations and matrices, determinants, Euclidean spaces, eigenvalues and eigenvectors.
482 Linear Control Systems
Provides the basics of the practical design of linear feedback control systems. The design techniques discussed include lead and lag compensation based on the Bode diagram, root locus design, and Nyquist diagram design. Techniques that are applicable to both discrete and continuous systems are emphasized.
Course's Web Page
The syllabus, homeworks, solutions to homeworks, and all other information relevant to the course will be continuously posted at the course's web page. The URL is
http://www-rcf.usc.edu/~hespanha/EE585/
Assessment format
Homeworks – 30% (the homework with the lowest grade is not considered for the final score)
Mid-term exam – 30%
Final exam – 40%
Homeworks will be posted approximately, weekly on the course's web page and they are due one week after being posted. Homeworks turned in after the due date (but before the solutions are disclosed) will be graded only to 80%. The homework with the lowest grade is not considered for the final score.
Remote students can email their homeworks to denhw@usc.edu or fax them to the following numbers: (213)740-9121, (213)740-8591, or (213)821-3041. See details at http://den.usc.edu/cont/hwsub.html.
ATTENTION: Remote students please do not
forget to fill out your DEN enrollment
form at
http://den.usc.edu/academic/enrollmentform.html
Textbook
The main textbook is:
1. C.-T. Chen. Linear Systems Theory and Design. Oxford Univ. Press, 3rd ed., 1999. (ISBN 0-19-511777-8)
Other recommended textbooks are:
2. P. Antsaklis, A. Michel. Linear Systems. McGraw Hill, 1997.
3. W. Rugh. Linear System Theory, 1996.
The classes will follow Chen's
book [1] closely. Antsaklis' book [2]
will be used for topics not covered by [1] (see syllabus
below). Both [1] and
[2] are available at the bookstore.
Detailed Syllabus
The following is a tentative schedule for the
course. If revisions are needed they
will be posted on the course's web page.
The rightmost column of the schedule contains the recommended reading
for the topics covered on each class. Students
are strongly encouraged to read these materials prior to the class.
|
Class |
Content |
Reference |
|
8/29/01 |
Introduction and course overview Key concepts: system (2.1.0 of [1]), causality (2.1.1 of [1]), state (2.1.1 of [1]), linearity (2.2, pp 7-8 of [1]), time-invariance (2.3 of [1]) Input-output vs. state-space models for causal, linear systems (2.2, pp 8-11 of [1]) Examples: electrical network (Ex. 4.2, p 13 of [2] as in Ex. 2.11 of [1]), spring-mass system (Ex. 2.6 of [1]), etc. |
Secs 2.1-2.2 and 2.5 of [1] Clarification notes here. |
|
9/5/01 |
No class |
|
|
9/12/01 |
Transfer matrix of linear time-invariant (LTI)
systems (2.3 p 12-16 of [1]) Linearization (2.4 of [1]) Example: inverted pendulum (Ex.
2.8 of [1]) Discrete time systems: input-output, transfer matrix, and state-space models (2.6 of [1]) |
Secs 2.3-2.6 of [1] |
|
9/19/01 |
Solutions of homogeneous linear ODEs—Peano-Baker series (4.5 of [1] see also 1.13, pp 55-56 of [2]) Properties of the fundamental and state transition matrices (4.5 of [1], see also 2.3B, pp 140-145 and Th. 3.3 of [2]) Solutions of nonhomogeneous linear ODEs—variations of constants formula (1.13, pp 57-58 of [2]) Solutions of continuous-time state-space models (4.5 of [1]) |
Secs 4.1-4.2 and 4.5 of [1] Secs 1.13 of [2] |
|
9/26/01 |
Elements of linear algebra: linear independence, basis, matrix representation of linear maps (3.2 of [1]), similarity transformations (3.4 of [1]) Elements of matrix algebra: determinant, rank, eigenvalues, matrix polynomials (3.6, p 62 of [1] and 2.1M, p 132 of [2]), Cayley-Hamilton Theorem (3.6, p 63, 65, 68 of [1]) |
Secs 3.1-3.6 of [1] |
|
10/3/01 |
Jordan normal form (3.5 of [1] also p 135 of [2]) State transition matrix for LTI systems—matrix exponential (1.13, p 56 of [2]) Properties of eAt (3.6, pp 68-69 of [1]) |
Secs 3.5 and 3.6 of [1] |
|
10/10/01 |
Computation of eAt: Caley-Hamilton Theorem (2.2J, pp 124-125 of [2]), Laplace transform (3.6, pp 69-70 of [1]), Jordan normal form (2.4B, p 151 of [2]) Solutions of discrete-time, LTI state-space models (4.2.2 of [1]). Input-output stability of continuous-time LTI systems (5.2 of [1]) |
Secs 3.6 and 5.2 of [1] |
|
10/17/01 |
Internal stability of continuous-time LTI systems (5.3 of [1]) Continuous-time Lyapunov Theorem (5.4 of [1]) The discrete-time case (5.2.1, 5.3.1, 5.4.1 of [1]) Equivalent state-space models for linear time-invariant (4.3 of [1]) and time-varying systems (4.6 of [1]) Elementary realization theory for LTI systems—canonical forms (4.4 of [1]) |
Secs 4.3-4.4, 4.6 of [1] |
|
10/24/01 |
Midterm
exam on the material covered up to 10/10/01 class |
|
|
10/31/01 |
Controllability: definitions (3.2A, p 228-229 of [2]), gramians (3.2A, pp 230-233 of [2]), time-invariant case (3.2B, pp 237-241 in particular the. 2.17 of [2]) also in (6.2 of [1]) Observability: definitions (3.3A, p 248,250 of [2]), gramians (3.2A, pp 249,251 of [2]), time-invariant case (3.3B, pp 252-256 in particular the. 3.10 of [2]) also in (6.3 of [1]) |
Secs 6.1-6.3 of [1] |
|
11/7/01 |
Canonical
decomposition (6.4 of [1]) The discrete-time case: controllability, observability, etc. (6.6 of [1]) |
Secs 6.4 and 6.6 of [1] |
|
11/14/01 |
Minimal realizations: definition, connection with controllability/observability, and equivalence of minimal realizations (7.2 of [1]) Balanced realizations (7.4 of [1]) |
Secs 7.1-7.4 [1] |
|
11/21/01 |
State feedback: pole assignment and invariance of zeros (8.2 of [1] also state the multivariable case 8.6 of [1]) |
Sec 8.1-8.2 of [1] |
|
11/28/01 |
State estimation (8.4 of [1] also state the multivariable case 8.7 of [1]) Stabilization through output feedback—separation theorem (8.5 of [1] also state the multivariable case 8.8 of [1]) Robust set-point control—internal model principal (8.3 of [1]) |
Secs 8.3-8.5 of [1] |
|
12/5/01 |
Matrix fractional description of systems: Smith-McMillan form (3.5, pp 298-299 of [2]), multivariable poles and transmission zeros of a transfer matrix (3.5, pp 299-305 of [2]), zero-pole cancellation (3.5, pp 299-305 of [2]) |
Sec 3.5 of [2] |
|
12/12/01 |
Final
exam on the material covered up to 11/28/01 class (7-9PM) |
|
Homeworks
The solutions to the homeworks
can be found here.
|
Number |
Posted on |
Due date |
Exercises |
|
1 |
8/28/01 |
9/12/01 |
2.3, 2.4, 2.5, 2.9 of [1]. In 2.5 assume that the system is linear. |
|
2 |
9/11/01 |
9/21/01 |
2.16, 2.18, 2.19, 2.21 of [1]. Some clarifications/hints: In 2.16, the aerodynamic damping is a torque that opposes angular motion. You only need to write down the differential equations for the angular moment (q ) and the height ( h ). In 2.18, use Example 2.10 as the basis for your answer. In 2.21, assume that the system it at rest for y = q = 0, i.e., that for this configuration the forces exerted by the springs cancel gravity. (see notes on my book) |
|
3 |
9/20/01 |
9/26/01 |
4.17, 4.18, 4.19, 4.20 of [1]. Hints: In 4.17, use property (4.55) of [1]. In 4.18, try to determine a differential equation for the signal v( t ) = det F( t, t0 ). The solutions to 4.20 are wrong, the first column of X(t) should be ecos(t)-1. |
|
4 |
9/27/01 |
10/3/01 (except 4 that is due on 10/10/01) |
Download the homework from here. For problem 3 (3.17 of [1]) you will find the definition
of a generalized eigenvector in page 59 of [1]. Also, there is a typo in the
book: the 1st row/2nd column of the matrix Q
should be lT
2/2. |
|
5 |
10/3/01 |
10/10/01 |
4.1, 4.21, 4.22 of [1] and also compute eAt, eBt, eCt for the matrices given in homework #4 (for each matrix use a distinct method). |
|
6 |
10/11/01 |
5.3, 5.4 due on 10/17/01, remaining due on 10/29/01 |
5.3, 5.4, 5.7, 5.11, 5.14, 5.18 of [1]. Hint for 5.18: start by showing that all eigenvalues of A+m I have real part less then 0 if and only if all eigenvalues of A have real part less than -m. |
|
7 |
10/18/01 |
10/31/01 |
4.8, 4.15 of [1]. Hint for 4.15: expand the transfer function in a series using the Markov parameters and analyze its behavior as s goes to infinity. Note that the difference between the degrees of the numerator and denominator determines if skg(s) converges to 0, infinity, or a constant, for different values of k. |
|
8 |
11/1/01 |
11/7/01 |
4.9, 4.14, 6.1, 6.18 of [1] and also another exercise you can download by clicking here. Leave the questions related to observability for the next homework. Hint for 4.9: start by computing W = [ W1 W2 … Wr ] = C(s I – A)-1 using the technique used to compute Z in page 102. |
|
9 |
11/8/01 |
11/15/01 |
6.8, 6.10, 6.11 of [1] and also another two exercises you can download by clicking here. A hint for problem 6.11 is also given in the link. There is a typo in the book in problem 6.10: the C matrix should be [0 1 1 0 1]. Since the class is pre-recorded, please turn in the homework on my mailbox on EEB308. |
|
10 |
11/13/01 |
11/21/01 |
7.12 of [1] and also 3 other exercises you can download
by clicking here. |
|
11 |
11/18/01 |
12/2/01 |
8.1, 8.2, 8.4 of [1] and also another exercise you can
download by clicking here. |
|
12 |
12/2/01 |
12/9/01 |
|
|
13 |
|
not due |
Download all exercises by clicking here |
Old Announcements
Posted on 8/28/01
There will be no class on September
5th, 2001.
Joao
Posted on 9/11/01
I posted some notes clarifying the
derivation of the formula that relates the input to a linear system with its
forced output (using the impulse response). This is in response to a question
that came up during the lecture. You can find these notes here
or in the table contained the detailed syllabus
of the course below.
Joao
Posted on 9/11/01
Selcuk Ateskan will be the grader for
the course. His contact information is given below.
Joao
Posted on 9/19/01
Due to several requests, the deadline
for the second homework is extended until the end of the week. Please turn in
the homework in my mailbox (in EEB 308). If the room is closed, please ask
someone in rooms EEB322 or EEB300 to put it in my mailbox.
Joao
Posted on 10/1/01
The solutions to the homeworks #1--#3
are now available.
There is a typo in problem 3.17 of [1]
(homework #4): the 1st row/2nd column of the matrix Q should be lT
2/2.
Joao
Posted on 10/17/01
You can download a practice mid-term
from here and the solutions
from here.
Joao
Posted on 10/30/01
The mid-term exams have been graded.
I’ll turn them in during the October 31st class. The average score was 106%
(great !) with a standard deviation of 29%. You can get the solutions to the
exam from here.
Joao
Posted on 11/10/01
There is a typo in the book in problem
6.10: the C matrix should be [0 1 1 0 1].
Joao
Posted on 10/27/01
You can download a practice final from here and the solutions from here. This was the final I used the last
time I taught the course (Fall’99). Question 3 refers to material that will be
taught in the last class and therefore it will not appear in this year’s final
exam. However, our final will cover state estimation (Nov 28th class), which
was not covered in the Fall of 1999.
Joao
Posted on 11/22/01
I will be out of town on the week of
December 2nd through 8th. Please plan accordingly your preparation for the
final exam (on Dec 12th). Please call or email me before going to my office
otherwise you may not find me there.
Joao
Posted on 10/30/01
I will not be able to make it to the
classes on November 14th and December 5th. Because of this I will pre-record
these classes and they will be broadcast on those days without my presence.
You can either go to class on the
scheduled times and see the recording or you can attend the pre-recordings
(studio D). These will be on November 12th (Monday) and 30th (Friday) from
9am-12noon. Please note that December 5th would be our last class.
I encourage you to attend the
pre-recordings if you can make it at those times. Unfortunately, there is
little flexibility in choosing schedules because the studios are heavily
booked.
Joao
Posted on 8/28/01
All students please send me an email to
hespanha@usc.edu
so that I can construct a mailing list for the class to send last minute
announcements. Thanks.
Joao
Posted on 12/3/99
Since the last class is very close to
the final, I will not include in the final the material covered in this class.
The final will therefore cover all the material up to (and including) the
12/2/99 class.
Posted on 10/10/99
As you all know there will be an
in-class mid-term exam on 10/21/99 on the material covered up to and including
the10/7/99 class. During the midterm
you are allowed to consult one letter-size piece of paper with handwritten
notes.
Good Luck!
Posted on 10/10/99
Homework #7 covering the material of
the 10/14/99 class (the one before the mid-term) is only due on 10/28/99, i.e.,
in the first class after the midterm.