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

EE380K: Introduction to System Theory

Fall Semester 2010



Some Basic Information

Instructor: Constantine Caramanis

Email: caramanis AT mail DOT utexas DOT edu
Phone: (512) 471-9269
Office: ENS 426
Office Hours: M: 2:00 pm - 3:00 pm. W: 2:00 pm - 3:00 pm, and by appointment

Lectures:

Time: Monday and Wednesday, 12:30-2:00 PM,
Location: ENS 127


Course Overview

This class is at once an introduction to Systems Theory, and a class in Advanced Linear Algebra.

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.

Linear Algebra is foundational in many topics important to CommNetS and beyond. Thus, in addition to introducing linear algebra tools important for our study of concepts from System Theory, we will also introduce many topics, methods, concepts and tools that are more generally important.

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.
  • Structured singular value. Robust performance measures.
  • Dynamic Systems. State-space models. Linearity and Time Invariance (LTI).
  • Autonomous Linear Dynamical Systems, Solutions of state space models.
  • Stability and Lyapunov functions. Positive definite functions and the cone of positive definite matrices.
  • Reachability/Controllability and Observability. Robust observability and reachability.
  • An introduction to optimal control. Bellman recursion and the principle of optimality.
  • LQR and Riccati equations.

    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 strong background in Linear Algebra is very desirable. Some basic 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 roughly weekly homeworks; there will be two mid-term exams in class, and then a final exam. 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 not be taught directly from any lecture notes or text book. The lecture notes written by Mohammed Dahleh, Munther Dahleh, and George Verghese may be helpful, and these are freely available, courtesy of the authors: they can be downloaded here. They are all in a zip-file.

    Additional References: For references on systems theory, see:

  • 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.)

    For review of linear algebra, I suggest:

  • Linear Algebra Done Right, by Sheldon Axler,
  • Matrix Analysis, by Horn and Johnson,
  • Topics in Matrix Analysis, by Horn and Johnson.



    Lecture schedule (tentative)

    Lecture No.

    Date

    Homework

    Solutions

    Assigned Reading

    Exam

    1

    Wed Aug 25

    problem set 0

    solution set 0

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    2

    Mon Aug 30

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    3

    Wed Sep 1

    problem set 1

    solution set 1

    ---

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    HOLIDAY

    Mon Sep 6

    ---

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    4

    Wed Sep 8

    ---

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    5

    Mon Sep 13

    problem set 2

    solution set 2

    ---

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    6

    Wed Sep 15

    ---

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    7

    Mon Sep 20

    ---

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    8

    Wed Sep 22

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    9

    Mon Sept 27

    problem set 3

    solution set 3

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    10

    Wed Sept 29

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    11

    Mon Oct 4

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    12

    Wed Oct 6

    problem set 4

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    MIDTERM #1

    Mon Oct 11

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    midterm 1

    13

    Wed Oct 13

    problem set 5

    solution set 5

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    14

    Mon Oct 18

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    15

    Wed Oct 20

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    16

    Mon Oct 25

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    17

    Wed Oct 27

    problem set 6

    solution set 6

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    18

    Mon Nov 1

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    19

    Wed Nov 3

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    20

    Mon Nov 8

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    21

    Wed Nov 10

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    22

    Mon Nov 15

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    MIDTERM #2

    Wed Nov 17

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    Midterm #2

    23

    Mon Nov 22

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    24

    Wed Nov 24

    problem set 7

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    25

    Mon Nov 29

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    26

    Wed Dec 1

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    Final Exam

    TBA

    ---

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    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. Orthogonality. Rank.
  • Homework #2: Matrix norms. Projections. More linear algebra.
  • Homework #3: SVD. matrix perturbation. More linear algebra.
  • Homework #4: Some review for the test.
  • Homework #5: Some practice with dynamical systems, and notions of stability.
  • Homework #6: Observability and Reachability, and some review notes on linear state feedback and linear observers.
  • Homework #7: Optimal control.

    Solutions

  • solution set 0
  • solution set 1
  • solution set 2
  • solution set 3
  • solution set 5
  • solution set 6

    Announcements

  • Wed. August 25: 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.