The University of Texas at Austin
Deptartment of Electrical and Computer Engineering

EE380K: Introduction to System Theory

Fall Semester 2006


Some Basic Information

Instructor: Constantine Caramanis

Email: cmcaram AT ece DOT utexas DOT edu
Phone: (512) 471-9269
Office: ENS 426
Office Hours: M: 3:30 pm - 5:00 pm. Tu: 2:00 pm - 3:30 pm. Th: 11:00 am - 12:00 pm.

TA: Johnson Carroll

Email: sonofjohn AT mail DOT utexas DOT edu
Phone:
Office: ENS 309
Office Hours: Tuesday 11:00 am - 1:00 pm.

Lectures:

Time: Monday and Wednesday, 2:00-3:30 PM,
Location: RLM 5.114

Recitation:

Time: Friday, 2:00-3:00 PM,
Location: RLM 5.114
**Recitations will review the material covered in class (no new material will be presented) and whether you go is entirely up to you.

Course Overview

The concepts from System Theory, including state and dynamics, feedback, stability, robustness, and estimation, are pervasive in many areas of both research and application, well beyond what is traditionally termed "systems and control." System theoretic ideas are playing an increasing role in networks and Internet applications (e.g., congestion control), Communications, Signal Processing, MDPs, Game Theory, Robust Optimization, and also Information Theory (feedback channels, communication constraints, etc.), Hidden Markov Models, and many others.

The purpose of this class will be to develop the mathematical tools as well as the concepts and intuition of System Theory, in particular Linear Dynamical Systems, in order to prepare the students for further work in a wide variety of areas.

Official Course Description

This class gives an introduction to the basic theory of linear dynamical systems in both discrete and continuous time, with applications to circuits, signal processing, communications, and control systems. Applied linear algebra will be a central part of the class, and the first part of the class will be devoted to developing important tools from linear algebra, which will then resurface many times through the rest of the course. Important topics we will cover include:

Course Outline (tentative):

  • Linear algebra review.
  • Least squares, and applications: Solution to overdetermined systems. Least norm solution to underdetermined systems. Robustness and regularization.
  • Symmetric matrices. Eigenvalues and Eigenspaces. Invariant Subspaces.
  • Invariant subsaces. Jordan Canonical Form, and Singular Value Decomposition.
  • Matrix norms. Matrix perturbation. Small gain theorem. Total least squares.
  • Dynamic Systems. State-space models. Linearity and Time Invariance (LTI).
  • Autonomous Linear Dynamical Systems, and solution via Laplace transform and matrix exponential.
  • Solutions of state space models, Transfer functions. Similarity Transformations.
  • Stability and Lyapunov functions. Positive definite functions. Internal Stability. Quadratic Lyapunov functions.
  • System norms. Induced norms.
  • Input-Output stability. Interconnected systems and feedback. Well-posedness, stability, and performance.
  • Affine parameterizations of stabilizing controllers.
  • Review of performance measures in Feedback Control in Single-Input, Single-Output Systems and then extensions to MIMO.
  • Robust Stability in MIMO Systems: Small gain theorem. Stability robustness.
  • Structured singular value. Robust performance measures.
  • Reachability/Controllability and Observability. Robust observability and reachability.
  • Kalman decomposition and minimal realizations. Truncation and Model reduction.

    There will be a big effort to draw interesting examples illustrating the basic concepts from a wide area, in order to give an idea of the applicability and impact ideas from Systems Theory have had, and are currently continuing to have.

    Course Prerequisites

    Officially, the listed co-requisite for this class is a course in Real Analysis I, Math 365C, or the equivalent. This will be quite helpful for what we plan to cover in the class. A good background in Linear Algebra is certainly desirable. I will also assume some basic familiarity with transforms, and ordinary differential equations (at the level of an undergraduate class). Some knowledge of Matlab will also be needed.

    General Note: If you are concerned about the prerequisites or your background, or what the course will cover, please don't hesitate to contact me by e-mail, or come by my office hours.

    Homework and Exams

    In this class there will be weekly homeworks; there will be two mid-term exams in class, and then a final exam, either take-home or in-class. The weighting will be as follows:
  • Homework: 15%
  • Midterm Exams: 35%
  • Final Exam: 45%
  • Class participation: 5%

    Policy on Collaboration: Discussion of homework questions is encouraged. Please be sure to submit your own independent homework solution. This includes any matlab code required for the assignments. Late homework assignments will not be accepted.

    Text and References

    The course will be taught from lecture notes written by Mohammed Dahleh, Munther Dahleh, and George Verghese. These notes will be available for purchase (at only the cost of the materials) from HKN, located at ENS 129. You may also download the notes here. They are all in a zip-file.

    Additional References: Some additional references you will not need but may nonetheless find helpful as a different perspective:

  • Introduction to Dynamic Systems: Theory, Models, and Applications, by David Luenberger. (A classic in the field).
  • Linear System Theory and Design, by Chi-Tsong Chen. (A nice, well-structured and easily understandable textbook).
  • Feedback Control Theory, by Doyle, Francis, and Tannenbaum. (Also readable, and moreover this book is freely available on-line).
  • A Course in Robust Control Theory, by Dullerud and Paganini. (This book is more advanced and covers much material that we will not have a chance to go over. It is a full year treatment of the robustness we cover towards the middle of the class).
  • Introduction to Mathematical Systems Theory: A Behavioral Approach, by Polderman and Willems. (This book introduces the behavioral viewpoint, and while more mathematical, it renders certain elements of the theory quite simple and mathematically elegant).
  • Robust and Optimal Control, by Zhou, Doyle, and Glover. (This book covers all the material we see in this class, and goes well beyond it.)


    Lecture schedule (tentative)

    Lecture No.

    Date

    Homework out

    Homework in

    Assigned Reading

    Exam

    1

    Wed Aug 30

    #0.

    ---

    Chapter 1

    ---

    HOLIDAY

    Mon Sep 4

    ---

    ---

    ---

    ---

    2

    Wed Sep 6

    #1.

    #0.

    Chapters 1,2

    ---

    3

    Mon Sep 11

    ---

    ---

    Chapters 2,3

    ---

    4

    Wed Sep 13

    #2.

    #1.

    Chapter 4

    ---

    5

    Mon Sep 18

    ---

    ---

    Chapter 4,5

    ---

    6

    Wed Sep 20

    #3.

    #2.

    Chapter 5

    ---

    7

    Mon Sep 25

    ---

    ---

    Chapters 6,7

    ---

    8

    Wed Sep 27

    #4.

    #3.

    Chapters 7,8

    ---

    9

    Mon Oct 2

    ---

    ---

    Chapters 10,11

    ---

    10

    Wed Oct 4

    #5.

    #4.

    Chapter 12

    ---

    MIDTERM #1

    Mon Oct 9

    ---

    ---

    ---

    Midterm #1

    11

    Wed Oct 11

    #6.

    #5.

    Chapter 13

    ---

    12

    Mon Oct 16

    ---

    ---

    Chapter 14

    ---

    13

    Wed Oct 18

    #7.

    #6

    Chapter 15

    ---

    14

    Mon Oct 23

    ---

    ---

    Chapter 16

    ---

    15

    Wed Oct 25

    #8.

    #7.

    Chapter 17

    ---

    16

    Mon Oct 30

    ---

    ---

    Chapter 17

    ---

    17

    Wed Nov 1

    #9.

    #8.

    Chapter 18

    ---

    18

    Mon Nov 6

    ---

    ---

    Chapter 19,20

    ---

    19

    Wed Nov 8

    ---

    #9.

    Chapter 19,20

    ---

    20

    Mon Nov 13

    ---

    ---

    Chapter 20

    ---

    21

    Wed Nov 15

    #10.

    ---

    Chapter 21

    ---

    MIDTERM #2

    Mon Nov 20

    ---

    ---

    ---

    Midterm #2

    22

    Wed Nov 22

    ---

    ---

    Chapter 22,23

    ---

    23

    Mon Nov 27

    #11.

    #10.

    Chapter 24

    ---

    24

    Wed Nov 29

    ---

    ---

    Chapter 25

    ---

    25

    Mon Dec 4

    ---

    ---

    ---

    ---

    26

    Wed Dec 6

    ---

    #11.

    ---

    ---

    Final Exam

    Fri Dec 15

    ---

    ---

    ---

    Final

    Homeworks

    Homeworks are due Wednesday at the beginning of class. Early assignments (e.g. Monday in class) are fine, but no late homeworks will be accepted. You are allowed to drop three (3) homeworks. You are free to not hand them in, but if you do, we will drop you three lowest grades.

  • Homework #0: Concepts from linear algebra. Vector spaces, linear operators, linearity, range, nullspace.

  • Homework #1: More linear algebra. Eigenvalues and Eigenspaces. Matrices. Matrices and linear operators. Matrix algebra.

  • Homework #2: More linear algebra. Least squares. Matrix norms and SVD. No bonus on this week's homework... sorry...

  • Homework #3: Singular value decomposition and matrix norms.

  • Homework #4: Linear systems and Jordan canonical form. Asymptotic stability.

  • Homework #5: A lighter problem with some review problems for the test.

  • Homework #6: This is the midterm, given as a problem set. Please do the problems carefully, and thoroughly. The grade on the midterm will be averaged with the grade on this problem set.

  • Homework #7: This problem set focuses on internal stability: asymptotic, marginal (i.s.L), as well as Lyapunov functions, and Lyapunov's indirect method.

  • Homework #8: This is a short little problem set, focusing on system norms and p-stability.

  • Homework #Z: This is a sequence of problems that investigate a frequency-domain approach to time invariance and stability. This is not worth anything, and it is completely optional. The discussion relies on machinery and results that are way beyond the scope of this class. Doing this problem set will not help you in any future tests, or homeworks. But if you are into analysis, perhaps you will find this interesting.

  • Homework #9: This problem set covers material from chapters 16 and 17 in the course notes.

  • Homework #10: This problem set covers material from Chapters 19 and 20, with some additional review from earlier chapters. Note that it is NOT due on Wednesday, but rather Monday after Thanksgiving.

  • Homework #11: This problem set covers material from Chapters 22, 23, 24 and 25. There are 14 problems, but you need to turn in only seven of them (whichever you choose). The remaining 7 problems should serve as a starting point for your review for the final exam.


    Solutions

    (Have been taken off-line).
  • Homework #0 Solutions.

  • Homework #1 Solutions.

  • Homework #2 Solutions.

  • Homework #3 Solutions.

  • Homework #4 Solutions.

  • Homework #5 Solutions.

  • Homework #6 Solutions, a.k.a., Midterm #1 Solutions: These will be handed out in hard copy, and not posted.

  • Homework #7 Solutions.

  • Homework #8 Solutions.

  • Homework #9 Solutions.

  • Homework #10 Solutions.

  • Homework #11 Solutions.

    Announcements

  • Thur., November 16: Problem set 10 has been posted. Note that this is due on Monday after Thanksgiving. It is the second-to-last problem set. There will be (only) one more problem set after this one, due on the last day of class. (I know, I am slipping...).

  • Tue., October 24: In problem set 7, problem 14.2 part (a) asks you to give conditions on when the system is asymptotically stable. Note that the characteristic polynomial here can be any degree N polynomial with real coefficients. This part of the problem is not asking you to compute conditions under which a general polynomial has roots in the left half plane, but rather just to write down the characteristic polynomial, and to state that the condition for asymptotic stability is for the roots of this polynomial to have negative real part.

  • Tue., October 24: In problem set 7, problem 1 part b should read: "For what range of \alpha in [-1,1] can you verify that...".

  • Thu., September 21: In problem set 2, one of the problems with the orthogonal subspaces is missing a converse. This is a good opportunity to introduce the Gramm-Schmidt orthogonalization procedure. This is now included in the currently posted version of the problem set. Thanks to Kalpana for pointing this out.

  • Tue., September 19: The Course Text -- There is no Chapter 9 of the course notes.

  • Wed., September 13: Homework #2 is now posted.

  • Mon., September 11: You may drop 3 homework grades. You are free to not turn them in, but if you do, we will drop your three lowest grades before computing the homework score.

  • Thurs., September 7: The course notes are now posted on this web page. They are packaged into a single zip file, which you can download here. HKN is still selling bound versions of the course notes, so you can still order your copy from them, if you prefer.

  • Wed., September 6: There was a mistake in the hint to problem 5, and also on problem 7, on PS#1 (Thanks Rahul Vaze). The corrected version is posted. Also, there were some typos in the solution set to PS#0. The corrected version is now posted.

  • Wed., September 6: Problem set 1 is now posted. The first recitation will be this Friday, from 2-3 pm in RLM 5.114 (same room as class). Recitations will review some background material, and what was covered during the week. There will not be any new material covered in recitations.

  • Fri., September 1: The course notes are now available from HKN. You can buy them from the HKN office, which is located in ENS 129.

  • Wed., August 30: Problem set 0 is now posted.

  • Tues. August 29: I checked with HKN, and course notes will be available within a few days -- in particular, they are not available today, so an electronic copy of the first chapter is available.

  • Tues. August 29: The course notes will be published by HKN. Please pick them up there.

  • Tues. August 15: Welcome to EE380K. I have posted a tentative lecture, homework, and exam schedule above. If you have any questions about the class, please feel free to e-mail me, or to drop by my office (ENS 426) if you prefer.

    • Questions, Comments, Answers

    If you have questions/comments, I encourage you to e-mail me. I will post questions/comments/answers that might be useful to the entire class here.