## Prof. Steven R. HallRoom: 33-313 |

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:

- Introduction to the state-space approach of analysis and control synthesis.
- State-space representation of dynamic systems; controllability and observability.
- State-space realizations of transfer functions and canonical forms.
- Design of state-space controllers, including pole-placement and optimal control.
- Introduction to the Kalman filter.
- Limitations on performance of control systems.
- Introduction to the robustness of multivariable systems.

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

- Pierre R. Belanger,
*Control Engineering: A Modern Approach*. - John Van de Vegte,
*Feedback Control Systems*.

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%).

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.

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

Solution

Homework 2 Issued: 9/22/2006 Due: 9/29/2006

Solution

Homework
3 Issued: 10/2/2006 Due: 10/11/2006 (but please try to do by 10/6/2006)

Solution

Homework 4 Issued: 10/16/2006 Due: 10/23/2006

Solution

Homework 5 Issued: 10/23/2006 Due: 10/30/2006

Solution

Homework 6 Issued: 11/1/2006 Due: 11/8/2006

Solution

Homework 7 Issued: 11/20/2006 Due: 11/27/2006

Solution

Homework 8 Issued: 12/4/2006 Due: 12/8/2006

Solution

Schedule

Date | Lecture | Topic | Reading/Notes | Comments | |

9/6 | L01 | Introduction | |||

9/8 | L02 | Root Locus Analysis | |||

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 | |||

9/20 | L07 | State Space 2 | |||

9/22 | L08 | State Space 3 | |||

9/25 | VACATION | ||||

9/27 | L09 | State Space 4 | |||

9/29 | L10 | State Space 5 | |||

10/2 | L11 | State Space 6 | |||

10/4 | L12 | State Space 7 | |||

10/6 | L13 | Feedback 1 | |||

10/9 | VACATION | ||||

10/11 | QUIZ 1 | ||||

10/13 | L14 | Feedback 2 | |||

10/16 | L15 | Feedback 3 | |||

10/18 | L16 | Feedback 4 (Dynamic Programming) | |||

10/20 | L17 | Feedback 5 (Lagrange Multiplier Techniques) | |||

10/23 | L18 | Feedback 6 (Solution of the Riccati Equation) | |||

10/25 | L19 | Feedback 7 (Symmetric Root Locus) | |||

10/27 | L20 | Estimators 1 | |||

10/30 | L21 | Random Processes | |||

11/1 | L22 | Response to White Noise | "" | ||

11/3 | L23 | No Class | |||

11/6 | L24 | Kalman Filter | |||

11/8 | L25 | ||||

11/10 | VETERAN'S DAY HOLIDAY | ||||

11/13 | L26 | The Linear Quadratic Gaussian (LQG) Problem | |||

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 | |||

11/24 | THANKSGIVING DAY HOLIDAY | ||||

11/27 | L30 | Robustness of SISO Systems | |||

11/29 | L31 | MIMO Robustness | |||

12/1 | L32 | The infinity norm | |||

12/4 | L33 | H-infinity Control | |||

12/6 | L34 | Structured Uncertainties | |||

12/8 | L35 | Review | |||

12/11 | L36 | Review | |||

12/13 | L37 | Recitation | |||

FINAL EXAM |

My e-mail is *Steven_Hall@mit.edu*.