16.31 Feedback Control - Fall 2006

Contact Information

Picture of Steven Hall  

Prof. Steven R. Hall

Room: 33-313
Phone: 253-0869
e-mail: Steven_Hall@mit.edu

Subject Information


The purpose of this subject is to teach you the fundamentals of "modern control," i.e., the use of state-space methods for control analysis and design. The subject includes both practical and theoretical aspects of the topic. At the end of the term, you should be able to design controllers using state-space methods. You will also be able to evaluate whether the controllers you design are "robust," that is, if they are likely to work well in practice, even given the inevitable errors in modeling of the system to be controlled.

The topics covered in this class are:

Intended audience

16.31 is intended primarily for graduate students, but is accessible to well-qualified undergraduates. It should be taken by any student in the SM or PhD program who intends to major in control, since it is the prerequisite to more advanced topics in control.


The necessary prerequisite for this class is a classical control course that includes analysis (root locus, Nyquist, Bode), as well as some controller synthesis. 16.060 is of course acceptable. If you are unsure about a subject taken elsewhere, please see me. .


Most of the readings are from the notes by Prof. Jon How, with some slight revisions by me. There is no required text. The text books

may be helpful, especially for the review of classical control, and are available on reserve in the library. For best results, you should do the reading before the lecture.


The homework is for your benefit, to help you understand the material. You are expected to do and understand the problems in the assignments. The homeworks will be collected and graded informally, on a check plus/minus basis. Homeworks may not be turned in after the problem solutions are posted, generally the day the problem sets are due. Generally, problem sets will go out on Friday, and will be due the following Friday, although there may be some variations to this pattern.

Much of the homework involves computer work, using Matlab. Homework solutions may sometimes be given in terms of Matlab scripts.


There will be two quizzes (~25%) and a final (~40%). Problem sets will count for a smaller share of the grade (~10%).

Academic Honesty

You may collaborate with others in the class on homework, but only to the extent needed to understand the problem statement, and to decide on a solution method. However, the work that you turn in must be your own. Of course, you may consult outside reference material, as long as the reference is adequately cited. However, you may under no circumstances use material from prior terms of 16.31. Collaboration of any sort is not permitted on exams. Violations of this policy are serious and will be treated as such.

Office Hours

I do not keep specific office hours for help. Instead, I have an open-door policy, as follows: If you have a question that I can help with, you should stop by my office or call me. If I can see you right away, I will; if I cannot, I will set up a time to speak with you, usually within 24 hours.

If you need help, it is your responsibility to contact me. If I am not in my office, call me on the phone (253-0869) and leave a message with my secretary or on my voice mail; or send me an e-mail (Steven_Hall@mit.edu), or leave a note on my door.

I encourage you to use e-mail as a way to ask questions as well. Of course, if you prefer to see me in person, you should do so. However, I find that I can usually respond quite quickly to e-mail, and many students find it a very convenient method of communication. Further, I can respond to the class as a whole with answers of interest to everyone.


Homework 1 Issued: 9/8/2006 Due: 9/20/2006
Homework 2 Issued: 9/22/2006 Due: 9/29/2006
Homework 3 Issued: 10/2/2006 Due: 10/11/2006 (but please try to do by 10/6/2006)
Homework 4 Issued: 10/16/2006 Due: 10/23/2006
Homework 5 Issued: 10/23/2006 Due: 10/30/2006
Homework 6 Issued: 11/1/2006 Due: 11/8/2006
Homework 7 Issued: 11/20/2006 Due: 11/27/2006
Homework 8 Issued: 12/4/2006 Due: 12/8/2006


Date Lecture Topic Reading/Notes Comments
9/6 L01 Introduction
Lecture 1
9/8 L02 Root Locus Analysis
Lecture 2
9/11 L03 Root Locus Analysis
     "     "
9/13 L04 Root Locus Design      "     "  
9/15 L05 Bode Analysis and Design Bode Analysis
Stability in the Frequency Domain
Bode Synthesis
9/18 L06 State Space 1 PDF  
9/20 L07 State Space 2 PDF  
9/22 L08 State Space 3 PDF  
9/27 L09 State Space 4 PDF  
9/29 L10 State Space 5 PDF  
10/2 L11 State Space 6 PDF  
10/4 L12 State Space 7    
10/6 L13 Feedback 1 PDF  
10/9   VACATION    
10/11 QUIZ 1
10/13 L14 Feedback 2    
10/16 L15 Feedback 3 PDF  
10/18 L16 Feedback 4 (Dynamic Programming)


10/20 L17 Feedback 5 (Lagrange Multiplier Techniques) PDF  
10/23 L18 Feedback 6 (Solution of the Riccati Equation) PDF  
10/25 L19 Feedback 7 (Symmetric Root Locus) PDF  
10/27 L20 Estimators 1 PDF  
10/30 L21 Random Processes PDF  
11/1 L22 Response to White Noise ""  
11/3 L23 No Class    
11/6 L24 Kalman Filter PDF  
11/8 L25      
11/13 L26 The Linear Quadratic Gaussian (LQG) Problem pdf  
11/15   QUIZ 2    
11/17 L27 Quiz Review, Design of Feedback Controllers PDF PDF  
11/20 L28 Robustness 3    
11/22 L29 Modeling Uncertainties pdf  
11/27 L30 Robustness of SISO Systems pdf  
11/29 L31 MIMO Robustness pdf  
12/1 L32 The infinity norm pdf  
12/4 L33 H-infinity Control pdf  
12/6 L34 Structured Uncertainties pdf  
12/8 L35 Review    
12/11 L36 Review    
12/13 L37 Recitation    


My e-mail is Steven_Hall@mit.edu.

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