CDS 140b, Spring 2014

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Differential Equations and Dynamical Systems

Instructors

  • Richard Murray (CDS/BE), murray@cds.caltech.edu
  • Doug MacMartin (CDS), macmardg@cds.caltech.edu
  • Lectures: MWF, 1-2 pm, 314 ANB
  • Office hours: TBD

Teaching Assistant

  • Katie Broersma (CDS)
  • Contact: cds140-tas@cds.caltech.edu
  • Office hours: Wed, 3:30-4:30 pm, 101 STH

Course Description

CDS 140b is a continuation of CDS 140a. A large part of the course will focus on tools from nonlinear dynamics, such as perturbation theory and averaging, advanced stability analysis, the existence of periodic orbits, bifurcation theory, chaos, etc. In addition, guest lecturers will give an introduction to current research topics in dynamical systems theory. There will be 8 homeworks throughout the term but no exams.

Lecture Schedule

Week Date Topic Suggested Reading/Lecture Notes Homework
0 31 Mar
RMM
Course overview
1 2 Apr
4 Apr*
DGM
Lagrangian and Hamiltonian systems I
  • Definition and properties
  • Newtonian Systems
HW 1
Due: 10 Apr (Thu)
2 9 Apr*
11 Apr
DGM
Lagrangian and Hamiltonian systems II
  • Hamiltonian systems with 2 degrees of freedom
  • Mechanical systems
HW 2
Due: 17 Apr (Thu)
3 18 Apr
21 Apr
RMM
Advanced stability theory
  • Non-autonomous (time-varying) systems
  • Input-to-state stability, input-output stability, small gain theorem
  • Khalil 4.5, 4.6-4.7 (skim), 5.1-5.4
HW 3

Due 1 May (Thu)

4 28 Apr (12-1 pm)
30 Apr
RMM
Stability of perturbed systems
  • Vanishing perturbations
  • Non-vanishing perturbations
  • Slowiy varying systems
  • Khalil, 9.1, 9.2, 9.6
HW 4

Due 8 May (Thu)

5 5 May
7 May
RMM
Perturbation theory
  • Periodic perturbations
  • Averaging
  • Singular Perturbations
  • Khalil, 10.3-10.4, 11.1-11.3, 11.5
HW 5

Due 15 May (Thu)

6 12 May
14 May
RMM
Interconnected systems
  • Absolute stability
  • Describing functions
  • Khalil, 7.1-7.2
HW 6

Due 22 May (Thu)

7 19 May
21 May
DGM
Nonlinear control I
  • Overview of techniques
  • Controllability and Lie brackets
  • Gain scheduling
  • Feedback linearization
  • Khalil 12.2,12.5 (gain scheduling), 13.1-13.3 (feedback linearization)
  • Isidori (chapter 2) or Nijmeijer and van der Schaft, 3.1 (for more on controllability)
  • Lecture notes
HW 7
Due: 29 May (Thu)
8 23 May*
30 May
DGM
Nonlinear control II
  • Backstepping
  • Sliding mode control
HW 8
Due: 6 June (Thu)

References:

Course Textbooks

  • H. Khalil, Nonlinear Systems, Prentice Hall; 3rd edition, 2001. ISBN: 978-0130673893
  • S. Strogatz, Nonlinear Dynamics And Chaos, Westview Press, 1994. ISBN: 978-0738204536
  • F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer; 2ed Edition, 1996. ISBN: 978-3540609346

Additional Sources:

  • L. Perko, Differential Equations and Dynamical Systems (3rd), Springer, 2001. ISBN: 978-0387951164
  • S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer; 2nd edition, 2003. ISBN: 978-0387001777

Policies

Collaboration Policy

Homeworks are to be done and handed in individually. To improve the learning process, students are encouraged to discuss the problems with, provide guidance to and get help from other students, the TAs and instructors. However, to make sure each student understands the concepts, solutions must be written independently and should reflect your understanding of the subject matter at the time of writing. Copying solutions, using solutions from previous years, having someone else type or dictate any part of the solution manual or using publicly available solutions (from the Internet) are not allowed.

Grading Policy

The final grades will be evaluated based on homework assignments, with the lowest score dropped in computing the final grade.