6.242 general information

6.242, Fall 2004: General Information

Course Description

The course offers an introduction to a variety of methods which can be used in deriving simple approximate models for complex systems. In addition to the classical methods for linear time invariant systems (moments matching, proper orthogonal decomposition, balanced truncation and Hankel optimal model reduction), the topics of approximation quality, algorithm complexity, convex optimization, and the interplay between model reduction and system identification, will be covered.

System models (CT LTI, DT LTI, plus other)
- transfer matrices and impulse responses
- state space models
- moments
- nonlinear models
Sources of finite order system models:
- first principles models with examples
- approximations of distributed models
- hidden Markov models
- optimal feedback controllers
System order and other complexity measures
- Minimal state space realizations
- Order of a transfer matrix
- Order as rank of a Hankel operator
- Approximation vs. model reduction
Quality of approximating one system by another
- parameter matching
- presetrvation of stability
- preservation of passivity
- error system gains
- small gain theorems
System identification
- models with incomplete information
- measurement noise
- sensitivity to measurement errors
- noise fitting
Numerical Linear Algebra
- complexity of a numerical algorithm
- sparsity and bounded rank matrices
- solving linear equations
- Schur decomposition
Least squares approximation
- orthogonal decompositions
- Taylor series expansions
- Fourier series expansions
Moments matching via projection methods
- Krylov subspaces
- Arnoldi method
Proper orthogonal decomposition
- snapshots
- singular value decomposition
Balanced truncation
- controllability and observability Gramm matrices
- balancing
- error bounds
- Hankel optimal model reduction
Model reduction via convex optimization
- convex optimization engines
- convexification of model reduction objectives

Familiarity with undergraduate linear system concepts (6.003) and basic linear algebra (matrices, eigenvalues) is a prerequisite. A graduate system class (6.241 or 16.31), as well as better mastery of linear algebra, will be useful.

Information resources and literature

There will be no textbook. All necessary information will be supplied in lecture handouts.

Staff

Alexandre Megretski (Lecturer), Associate Professor of Electrical Engineering:

Room 32-D730, Ext. 3-9828, email ameg@mit.edu

Jacob White, Professor of Electrical Engineering:

Room 36-817, Ext. 3-2543, email white@mit.edu

Luca Daniel, Assistant Professor of Electrical Engineering:

Room 36-849, Ext. 3-2631, email dluca@mit.edu

Karen Willcox, Assistant Professor of Aeronautics and Astronautics:

Room 37-447, Ext. 3-3503, email kwillcox@mit.edu

Fifa Monserrate (Course Secretary):

Room 32-D730, Ext. 3-2184, email fifa@mit.edu

Web page

The URL for the 6.242 home page is

http://web.mit.edu/6.242/www/index.html

The home page will contain homework assignments, important announcements, 6.242 questions and answers collection, errata, and other frequently updated information. We will try to put most of other necessary texts on the web as well. However, you are advised to minimize printing of the class handouts from the web, and use the hard copies provided at the lecture instead.

Class schedule

Lectures occur on Monday and Wednesday, 2.30-4 pm, Room 36-112.

Homework

As a rule, homework assignments will be posted on the web weekly on Wednesday nights. The homework papers are to be returned within 1 week (either submit them during class time or put under the door of 32-D730). Problem sets will be corrected, graded, and returned as soon as possible. Solutions will be distributed when the corrected homework is returned.

Team work on home assignments is strictly encouraged, as far as generating ideas and arriving at the best possible solution is concerned. However, you have to prepare your own write-up, and write your own code, if necessary.

The homework grades will be graded on a 0-100 points scale. After an assignment is graded, one set of make-up problems per problem set will be offered to those who want to improve a particular problem set grade.

MATLAB

MATLAB will be used extensively. We will need SIMULINK and the Control Systems, LMI Control, and Mu-Analysis and Synthesis Toolboxes (available on Athena computers). In addition, some custom-made software will be available from the class locker.

Examinations

There will be no final exam. Two in-class tests, graded on a 0-100 scale basis, will be given on November 1 and December 6. They will be graded in the same way as problem sets. A (substantial) make-up assignment will be offered to those who want to improve an in-class test grade.

Grade

A final numerical grade S for the class will be calculated according to formula S=0.5H+0.2T1+0.2T2+0.1P, where H is the average homework grade, T1 and T2 are the in-class test grades, and P is a participation grade (on the scale 0-100, equals 100 if you are never late with a homework and take part in class discussions). A numerical grade of 80 and above means an "A", 75-79 corresponds to an "A-", 70-74 to a "B+", 65-69 to a "B", etc.