ENEE 661 - Nonlinear Control Systems (P. S. Krishnaprasad)

ENEE 661 - Nonlinear Control Systems

Spring 2011

Last Update March 16, 2011

Course Information

Special Announcements (in reverse chronological order)

11. Read material from Lecture 5, part (i) (stability of equilibria, basic theorems of Lyapunov and related results) Lecture 5, part (ii) (on instability) corrections to page 2, ALL located in the 2010 part of this website.

10. Homework Set 4 posted (on Lyapunov theory)

9. Homework Set 3 posted

8. First page of Lecture 3(a) corrected (diagonal dominance and contraction).

7. Lecture 3(b) (mean value theorem) posted. Material up to and including this will be part of Mid Term I on February 24.

6. Lecture 2(b) (on Lie brackets in control - examples) posted. Homework assignment 2 posted. A practice exam will be emailed to you.

5. Lecture 3(a), on contraction mapping fixed point theorem, and existence-uniqueness theorem for ODEs posted - for Saturday class (Feb 12)

4. Lecture 2(a), a very basic introduction to matrix Lie groups and Lie algebras, is posted. Problem set 1 is posted. There will be an extra class on Saturday February 12 from 1:00 pm till 4:00 pm, most likely in room 2168, A.V. Williams building.

3. Lecture notes 1(a) is posted. Lecture notes 1(c) on an alternative way to frame a curve is posted. This will be covered in homework sets but no formal lecture will be given. Read pages 6-8 of Lecture 0 for a quick summary of this material. Read pages 9-10 of Lecture 0 for the re-casting of the frame equations back into the original 't' parametrization. Show this (chain rule).

2. Read first few pages of Brockett's 1973 and 1976 papers (see OR below).

1. Look for free download (PDF) of Hans Samelson's book on Lie algebras and the expository volume on Motion, Control, and Geometry (from NAP) under Other Resources (OR) at the end of this page. Read pages 1-5 of Samelson carefully, and my article in the NAP volume for some motivation.

This version of the class website will be under construction and update through the semester. Notes from 2010 are linked here. The new notes from 2011 will be added, starting Saturday, January 22, 2011. Copyrighted material in journals is linked below through journal websites. You can access them when logged into the umd.edu domain.

Weekly Lecture Notes by P. S. Krishnaprasad (from 2011)

Lecture 0 (a survey of some aspects of nonlinear control)

Lecture 1(a) (Introducing nonlinear control systems) Lecture 1(b) (Frenet-Serret framing of curves and control on SE(3)) Lecture 1(c) (natural framing of curves and control on SE(3))

Lecture 2(a) (matrix Lie groups and Lie algebras - very basic introduction) Lecture 2(b) (Lie brackets in control - examples)

Lecture 3(a) (contraction mappings, fixed points, existence and uniqueness of solutions to ODE's) Lecture 3(b) (mean value theorem)


Weekly Lecture Notes by P. S. Krishnaprasad (from 2010)

Lecture 1 (what is nonlinear behavior?)

Lecture 2 (planar systems, Hartman-Grobman theorem, Poincare-Bendixson theorem)

Lecture 3 (index of a plane vector field, elements of bifurcations)

Lecture 4, part (i), pages 1-9 (Banach's fixed point theorem, application to ODE's) Lecture 4, part (i), pages 10-17 (more on the Cauchy-Lipschitz existence and uniqueness theorem)

Lecture 4, part (ii) (some concepts and theorems from elementary analysis)

Lecture 5, part (i) (stability of equilibria, basic theorems of Lyapunov and related results) Lecture 5, part (ii) (on instability) corrections to page 2

Lecture 6 part (i) (stability in time-varying systems) Lecture 6 part (ii) (time-varying linear systems, a converse Lyapunov theorem, Floquet theory of periodic linear systems) Lecture 6 part (iii) (indirect method of Lyapunov) Lecture 6 - additional reading (exponential stability and its converse)

Lecture 7 (input-output stability)

A detour on measure

Lecture 8 (absolute stability, the Lure' problem, positive real lemma)

Lecture 8 (absolute stability, circle criterion, Popov criterion, multipliers)

Lecture 9 (center manifold reduction)

Lecture 10 (feedback linearization)


Homework Assignments (from 2011)

Problem Set 1

Problem Set 2

Problem Set 3

Problem Set 4


Homework Assignments (from 2010)

Problem Set 1

Problem Set 2

Problem Set 3

Problem Set 4

Problem Set 5

Problem Set 6

Problem Set 7

Problem Set 8


Homework solutions are sent by email

For Matlab problems remember to include your code. --------------------------------------------------------------------

Examinations (2011)

Mid Term I

Mid Term II



Other Resources (OR)

My lecture notes on Optimal Control (ENEE 664)

My lecture notes on (Linear) System Theory (ENEE 660)

Website for Feedback Systems: An Introduction for Scientists and Engineers by K. J. Astrom and R. M. Murray On-line readable book

Notes on Lie Algebras by Hans Samelson A classic Roger Howe's 1983 paper (Very basic Lie theory); corrections to Howe's paper (1984)

Paper by Wei and Norman on product of exponentials representation (1964)

Motion, Control and Geometry Four expository articles illustrating nonlinear control in action

Two classic papers by Roger Brockett Lie Algebras and Lie Groups in Control Theory (1973) and Nonlinear Systems and Differential Geometry (1976)

Real Analysis Book by Cinlar and Vanderbei - material useful in Systems courses

About Sophus Lie

About Stefan Banach

About A. M. Lyapunov

About G. D. Birkhoff