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Megretski, Alexandre, 6.243J Dynamics of Nonlinear Systems, Fall 2003. (Massachusetts Institute of Technology: MIT OpenCourseWare),  (Accessed 07 Jul, 2010). License: Creative Commons BY-NC-SA

Dynamics of Nonlinear Systems

Fall 2003

A CV-22 Osprey, which operates in both rotorcraft and fixed-wing configurations. The complex transition between the two states presents unique control challenges. (Image is taken from U.S. Air Force Web site.)

Course Highlights

This course site features a complete set of lecture notes.

Course Description

This course provides an introduction to nonlinear deterministic dynamical systems. Topics covered include: nonlinear ordinary differential equations; planar autonomous systems; fundamental theory: Picard iteration, contraction mapping theorem, and Bellman-Gronwall lemma; stability of equilibria by Lyapunov's first and second methods; feedback linearization; and application to nonlinear circuits and control systems.

Technical Requirements

MATLAB® software is required to run the .m files found on this course site.

*Some translations represent previous versions of courses.


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

Two sessions / week
1.5 hours / session


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.


1 Input/Output and State-Space Models
2 Differential Equations as System Models
3 Continuous Dependence on Parameters
4 Analysis Based on Continuity
5 Lyapunov Functions and Storage Functions
6 Storage Functions and Stability Analysis
7 Finding Lyapunov Functions
8 Local Behavior at Eqilibria
9 Local Behavior Near Trajectories
10 Singular Perturbations and Averaging
11 Volume Evolution and System Analysis
12 Local Controllability
13 Feedback Linearization   Tell A Friend