6.243J | Fall 2003 | Graduate

Dynamics of Nonlinear Systems


Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Course Description

This course studies state-of-the-art methods for modeling, analysis, and design of nonlinear dynamical systems with applications in control. Topics include:

  • Nonlinear Behavior
  • Mathematical Language for Modeling Nonlinear Behavior
    • Discrete Time State Space Equations
    • Differential Equations on Manifolds
    • Input/Output Models
    • Finite State Automata and Hybrid Systems
  • Linearization
    • Linearization Around a Trajectory
    • Singular Perturbations
    • Harmonic Balance
    • Model Reduction
    • Feedback Linearization
  • System Invariants
    • Storage Functions and Lyapunov Functions
    • Implicitly Defined Storage Functions
    • Search for Lyapunov Functions
  • Local Behavior of Differential Equations
    • Local Stability
    • Center Manifold Theorems
    • Bifurcations
  • Controllability of Nonlinear Differential Equations
    • Frobenius Theorem
    • Existence of Feedback Linearization
    • Local Controllability of Nonlinear Systems
  • Nonlinear Feedback Design Techniques
    • Control Lyapunov Functions
    • Feedback Linearization: Backstepping, Dynamic Inversion, etc.
    • Adaptive Control
    • Invariant Probability Density Functions
    • Optimal Control and Dynamic Programming

Prerequisite: 6.241 or an equivalent course.

Information Resources and Literature

This year, there will be no required textbook. All necessary information will be supplied in the lecture notes.

The books Nonlinear Systems by Hassan K. Khalil, published by Prentice Hall, and the more advanced Nonlinear Systems: Analysis, Stability, and Control by Shankar Sastry, published by Springer, can both serve as basic references on Nonlinear Systems Theory, frequently covering the topics skipped in the lectures.


Prof. Alexandre Megretski

Class Schedule


Homework assignments are usually given on Wednesdays. Homework papers are to be submitted during the lecture hours on the following Wednesday. The homework will be corrected, graded, and returned as soon as possible. Solutions to the homework will be distributed when the corrected homework is returned.

Team work on home assignments is strictly encouraged, as far as generating ideas and arriving at the best possible solution is concerned. However, you have to write your own solution texts (and your own code, when needed).


MATLAB®, the “language of technical computing’’, will be used in some assignments. We will need Simulink®, Control Systems, and LMI Control Toolboxes. You may wish to consult its online help for general information and for specific commands for simulating and analysing systems.


There will be two take-home quizzes, to be completed and returned within 24 hours, but no final exam. The quizzes will cover the theory of 6.243J (divided as equally as possible). The questions will be based on the ideas used in the problem set solutions made available at least a week before the test. No homework will be given on the last Wednesdays before the quizzes. No cooperation is allowed on take-home quizzes.


The letter grade will be determined at the end of the semester from a numerical grade N, obtained from the formula


where H is the average homework grade, and Q1, Q2 are quiz grades (H, Q1, Q2 are numbers between 0 and 100). From the distribution of N for the entire class, boundaries will be chosen to define letter grades. For students near the boundaries, other factors may be taken into account to determine the letter grade, such as effort, classroom activity, etc.

Learning Resource Types
Problem Sets with Solutions
Exams with Solutions
Lecture Notes