6.245 | Spring 2004 | Graduate

Multivariable Control Systems


Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Course Description

This course studies state-of-the-art computer-aided design methodologies for multivariable linear time invariant feedback control systems. Topics include:

  • Mathematical Language for Describing Design Objectives
  • Why some Design Objectives cannot be Satisfied
  • Ill-posedness of Optimization Problems and How to Avoid It
  • How to Predict Qualities of Optimal Controller
  • Major Optimization Engines of Multivariable Design:
    • Linear-quadratic

    • H-infinity

    • Mu Synthesis

    • Model Order Reduction

    • Linear Programming

    • LMI Optimization


  • Case Studies of Practical LTI Feedback Design (Mostly as Homework Assignments)

Prerequisites for this course are:

  • mastery of classical feedback concepts for single-input single output (SISO) systems (6.302 or equivalent)
  • an appropriate first year graduate control course (either 6.241 or 16.30)

Information Resources and Literature

This year, there will be no required textbook. All necessary information will be supplied in the lecture notes.

The book Essentials of Robust Control by Kemin Zhou and John C. Doyle, published by Prentice Hall is a detailed and reasonably modern reference to all or almost all technicalities related to the class material. Anyone who wants to go through the fine details of the famous algorithms for H2 and H-infinity optimization, optimal model order reduction, structured singular values computation etc. will be well advised to buy this book. It may also be a good idea to get hold of a copy of the MATLAB® manual for the Mu-Analysis and Synthesis Toolbox: this is a well-written “cookbook” style reference for those primarily interested in applications of H2 and H-Infinity optimization.


Prof. Alexandre Megretski


Homework assignments are usually posted on the Web on Wednesdays evenings. Homework papers are to be submitted during the next Wednesday. All electronic files used to complete the homework or to visualize the solutions are to be made accessible. The homework will be corrected, graded, and returned as soon as possible. Solutions to the homework will be posted on the Web 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 write your own comments and your own code.


MATLAB®, the “language of technical computing,” will be used extensively. We will need Simulink® and the Control Systems, LMI Control, and Mu-Analysis and Synthesis Toolboxes. You may wish to consult its online help for general information and for specific commands for simulating and analysing systems.


There will be two quizzes, given during lecture hours (dates to be announced soon after beginning of the semester), but no final exam. The quizzes will cover the theory of 6.245 (divided as equally as possible). The questions will be based on the ideas used in the problem set solutions made available at least a week before the test.


The letter grade will be determined at the end of the semester from a numerical grade N, obtained from the formula


where H is the average homework grade, and Q1, Q2 are quiz grades (H, Q1, Q2 are numbers between 0 and 100). From the distribution of N for the entire class, boundaries will be chosen to define letter grades. For students near the boundaries, other factors may be taken into account to determine the letter grade, such as effort, classroom activity, etc.