**Electrical,
Computer, and Systems Engineering Department**

**ECSE 6460 Multivariable
Control
Fall 2004**

**Instructor:** Professor John
Wen, CII 8213, x8744
wen@cat.rpi.edu

**Office Hour:** T/F

**Class Hours:** M/Th
12:00pm-1:20pm

**Class Room:** JEC 4304

**Course Homepage: **http://www.cat.rpi.edu/~wen/ECSE646F04

**Text: **

K. Zhou, J.C. Doyle, Essentials of Robust Control, Prentice-Hall, 1998.

**Prerequisites:**

ECSE-4440 Control System Engineering and ECSE--6400 System Analysis Techniques required.

**Course objectives:**

This course covers the tools and methods for the analysis and control of linear multivariable feedback systems. The emphasis is on contemporary control system design, connection between frequency domain and state space methods, convex analysis and design, and systematic consideration of model uncertainty and closed loop performance.

**Grade Composition:**

Midterm 30%

Project 25%

Homework,
project, and exams may require the use of MATLAB, Simulink, Control System
Toolbox, and LMI Toolbox. Collaboration
on homework and project is encouraged, but collaboration (including discussion)
on take home exams is **absolutely forbidden**.

The project may be done individually or by a team of two students. In the latter case, a clear statement of individual contribution should be included. An inverted pendulum experiment will be set up for the course. The project may involve this experiment or other control design problems problem from books or journal articles. The system should be of order equal to or higher than four and has multiple inputs and outputs. The control problem formulation should at least include performance specification, stabilization, and robustness with respect to model uncertainty. Additional objectives to consider include: disturbance rejection, output trajectory tracking, robust performance, transient response, performance, time and frequency domain control and state constraints. The design project should be written up and handed in by Monday 12/9/02.

**Suggested Format of
the written report:**

1. Introduction (source of problem, relevance to application, past approaches and results)

2. Problem Statement (modeling assumption, control objective, performance specification, etc.)

3. Design Approach (model simplification, controller design, controller simplification, iteration to achieve spec)

4. Design Result (simulation and experimental results, discussion of controller performance and limitation of approach)

5. Conclusion

Appendix: Computer Simulation Code

**Exam Dates: **

Both midterm and final are take home exams. Midterm exam will be handed out on Thursday, 10/14, and collected on Thursday, 10/21. Final exam will be handed out on Thursday, 12/2, and collected on Thursday, 12/9.

**Additional References**
(on reserve in Library)

·
G.E. Dulerud and F. Paganini, *A
Course in Robust Control Theory: A Convex Approach,* Springer-Verlag, 2000

· J.C. Doyle, B.A. Francis, A.R.Tannenbaum, Feedback Control Theory, Macmillan, 1992.

· K. Zhou, J.C. Doyle, and K. Glover, Robust and Optimal Control, Prentice-Hall, 1996.

· J. Ackermann, Robust Control: systems with uncertain parameters, Springer-Verlag, 1994.

· J.M. Maciejowski, Multivariable Feedback Design, Addison-Wesley, 1989.

· B.A. Francis, A course in H-infinity control theory, Springer--Verlag, 1987.

· F.M. Callier and C.A. Desoer, Multivariable Feedback Systems, Springer--Verlag, 1982.

· C.T. Chen, Linear Systems Theory and Design, Holt, Rinehart and Winston, 1984.

· M. Vidyasagar, Control system synthesis: a factorization approach, MIT Press, 1985.

· W.A. Wolovich, Linear multivariable systems, Springer--Verlag, 1974.

· T. Kailath, Linear systems, Prentice--Hall, 1980.

· H.H. Rosenbrock, State space and multivariable theory, Wiley, 1970.

· V. Kucera, Discrete Linear Control, Wiley, 1979.

· Kwakernaak and Sivan, Linear Optimal Control, Wiley, 1979.

· J. Lunze, Robust Multivariable Feedback Control, Prentice Hall, 1988.

· M. Morari and E. Zafiriou, Robust Process Control, Prentice Hall, 1989.

· S. Boyd and C. Barratt, Linear controller design, Prentice Hall, 1992.

· S. Boyd, L. El Ghaoui, E. Feron, V. Balarkishnan, Linear Matrix Inequality, SIAM Press, 1994.

**Course
Outline** (Total number of classes: 28)

· Introduction (1 lecture)

· Finite dimensional spaces (2 lectures. Ch. 1)

· Linear system theory (4 lectures, Ch. 2)

· Linear system analysis (3 lectures, Ch. 3)

· Realization, reduction, and identification (3 lectures, Ch. 4)

n Midterm

· Stabilizing controllers (3 lectures, Ch. 5)

·
*H _{2} *optimal
control (2 lectures, Ch. 6)

·
*H** _{¥}*
optimal control (2 lectures, Ch. 7)

· Stability analysis of uncertain systems (3 lectures, Ch. 8)

· Feedback control of uncertain systems (2 lectures, Ch. 9)

n In-class project discussion (3 sessions)

n Final