Introduction to Model Order Reduction

Ph.D. course, 7 credits, Spring 2014

 

 

In engineering and science it is often desirable to use the simplest possible mathematical model that “does the job”. First-principles modeling or system identification commonly result in unnecessarily high-dimensional mathematical models. Model (order) reduction concerns systematic approximation of such models. There are many advantages of working with models with a low-dimensional state space. For example, low-dimensional models are easier to analyze and much faster to simulate. Model reduction methods have successfully been used to solve large-scale problems in areas such as control engineering, signal processing, image compression, fluid mechanics, and power systems. In this Ph.D. course, we give an introduction to some powerful available reduction techniques, as well as to the required underlying mathematics.

 


 

Instructors: Henrik Sandberg (responsible), Bart Besselink, and Madhu N. Belur (IIT Bombay)

Meeting times: First lecture scheduled for March 31st, 13:15-15:00 (see schedule below).

Location: To be announced. Enroll to receive course updates.

Grading: Grade (pass/fail) is based on turn-in exercises (50%) and written exam (50%).

Prerequisites: Basic linear systems theory (state-space realizations, Laplace transforms etc.), matrix theory, and basic MATLAB-programming.

To enroll, please send an email to Henrik Sandberg


Course goals

After finishing the course, the Ph.D. student will

·         be able to distinguish between difficult and simple model reduction problems;

·         have a thorough understanding of Principle Component Analysis (PCA)/Singular Value Decomposition (SVD);

·         understand the interplay between linear operators on Hilbert spaces, controllability, observability, and model reduction;

·         know the theory behind balanced truncation and Hankel norm approximation;

·         know how convex optimization and Linear Matrix Inequalities (LMIs) can be used to solve model reduction problems;

·         know the basics of the behavioral modeling approach and dissipative systems theory; and

·         be able to reduce systems while preserving dissipativity properties, and reduce linear feedback controllers while taking the overall system performance into account.


Preliminary schedule (subject to change)

(Here some lectures notes from the course in 2010 can be found.)

Program (dates and times subject to change)

Date

Time

Event

Room

Lecturer

Topic

2014-03-31

13:15-15:00

Lecture 1

Q13

HS

Introduction. The model reduction problem. [Slides, Course Program]

2014-04-02

13:15-15:00

Lecture 2

L41

HS

Model truncation, singular perturbation. [Notes]

2014-04-04

10:15-12:00

Exercise 1

Q24

HS

Review of linear systems and Hilbert spaces etc.

2014-04-07

13:15-15:00

Lecture 3

Q24

BB

POD/PCA/SVD-based reduction. [Notes]

2014-04-09

13:15-15:00

Lecture 4

Q24

BB

Gramians and balanced realizations. [Notes]

2014-04-24

13:15-15:00

Exercise 2

Q24

HS/BB

Exercises due April 17

2014-04-25

13:15-15:00

Lecture 5

Q24

BB

Balanced truncation and weighted extensions.

2014-04-28

10:15-12:00

Lecture 6

Q24

BB

Controller reduction and linear matrix inequalities (LMIs).

2014-05-05

10:15-12:00

Lecture 7

Q13

HS

Optimal model order reduction in the Hankel norm.

2014-05-07

15:15-17:00

Lecture 8

Q24

HS

Optimal model order reduction in the H2/H-infinity norms.

2014-05-09

10:15-12:00

Lecture 9

Q13

MB

Behavioral approach, kernel representations

2014-05-12

13:15-15:00

Lecture 10

Q13

MB

Controllability, passivity, dissipativity, storage functions

2014-05-14

10:15-12:00

Lecture 11

Q24

MB

Maximum and minimum storage functions

2014-05-16

10:15-12:00

Exercise 3

Q13

HS/BB

Exercises due May 12

2014-05-19

13:15-15:00

Lecture 12

Q13

MB

The  LQ problem, H-infinity-norm as dissipativity

2014-05-21

10:15-12:00

Lecture 13

L41

MB

Passivity preserving model reduction

2014-05-26

10:15-12:00

Lecture 14

Q13

MB

Dissipativity preserving model reduction

2014-05-28

10:15-12:00

Exercise 4

M24

MB

 

 


Literature

Research papers and parts of the books

·         Antoulas, A.C., "Approximation of Large-Scale Dynamical Systems", Society for Industrial and Applied Mathematics, 2005.

·         Obinata, G. and Anderson, B.D.O., "Model Reduction for Control System Design", Springer-Verlag, London, 2001.

·         Doyle, J.C., Francis, B., and Tannenbaum, A., "Feedback Control Theory", Macmillan Publishing Co., 1990.

·         Green, M., and Limebeer, D.J.N., "Linear Robust Control", Prentice Hall, Englewood Cliffs, 1995.

·         Zhou K., Doyle, J.C., and Glover, K., "Robust and Optimal Control", Prentice Hall, Upper Saddle River, 1996.